TPTP Problem File: SLH0928^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Equivalence_Relation_Enumeration/0007_Equivalence_Relation_Enumeration/prob_00435_016513__12143610_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1525 ( 788 unt; 253 typ; 0 def)
% Number of atoms : 2898 (2037 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 11493 ( 348 ~; 78 |; 202 &;9790 @)
% ( 0 <=>;1075 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 5 avg)
% Number of types : 45 ( 44 usr)
% Number of type conns : 607 ( 607 >; 0 *; 0 +; 0 <<)
% Number of symbols : 212 ( 209 usr; 14 con; 0-3 aty)
% Number of variables : 3489 ( 143 ^;3213 !; 133 ?;3489 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:14:52.736
%------------------------------------------------------------------------------
% Could-be-implicit typings (44)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
set_Pr4333006031979791559at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mtf__b_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_Itf__b_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_Itf__b_J_J,type,
list_list_b: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__b_J,type,
list_b: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
% Explicit typings (209)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Oequiv__rels,type,
equiva8721718519204927301v_rels: nat > list_s1210847774152347623at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Nat__Onat,type,
equiva2048684438135499664of_nat: list_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
equiva1173177585473067681at_nat: list_s1210847774152347623at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001tf__b,type,
equiva2867628904822520639l_of_b: list_b > set_Pr1261947904930325089at_nat ).
thf(sy_c_Euclidean__Division_Odivmod__nat,type,
euclidean_divmod_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Int__Oint,type,
euclid4774559944035922753ze_int: int > nat ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Nat__Onat,type,
euclid4777050414544973029ze_nat: nat > nat ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Int__Oint,type,
euclid3395696857347342551nt_int: int > int ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Nat__Onat,type,
euclid3398187327856392827nt_nat: nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
append985823374593552924at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
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append8980083764895095685at_nat: list_P6846009644206846448at_nat > list_P6846009644206846448at_nat > list_P6846009644206846448at_nat ).
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append5999596214939841794at_nat: list_P5464809261938338413at_nat > list_P5464809261938338413at_nat > list_P5464809261938338413at_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mtf__b_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__b_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
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thf(sy_c_List_Oappend_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_Oappend_001tf__b,type,
append_b: list_b > list_b > list_b ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
count_6440129622255701469at_nat: list_s1210847774152347623at_nat > set_Pr1261947904930325089at_nat > nat ).
thf(sy_c_List_Ocount__list_001tf__b,type,
count_list_b: list_b > b > nat ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
drop_s5749455908503499943at_nat: nat > list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Odrop_001tf__b,type,
drop_b: nat > list_b > list_b ).
thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oenumerate_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
enumer3886722665900167622at_nat: nat > list_s1210847774152347623at_nat > list_P5181996255455634710at_nat ).
thf(sy_c_List_Oenumerate_001tf__b,type,
enumerate_b: nat > list_b > list_P2922825790777833268_nat_b ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Ofilter_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
filter4791820933415917918at_nat: ( set_Pr1261947904930325089at_nat > $o ) > list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Ofilter_001tf__b,type,
filter_b: ( b > $o ) > list_b > list_b ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
gen_le5092146751752969972at_nat: nat > list_s1210847774152347623at_nat > nat ).
thf(sy_c_List_Ogen__length_001tf__b,type,
gen_length_b: nat > list_b > nat ).
thf(sy_c_List_Olenlex_001t__Nat__Onat,type,
lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olenlex_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
lenlex1357538814655152620at_nat: set_Pr4329608150637261639at_nat > set_Pr4333006031979791559at_nat ).
thf(sy_c_List_Olenlex_001tf__b,type,
lenlex_b: set_Product_prod_b_b > set_Pr7665282455119567943list_b ).
thf(sy_c_List_Olex_001t__Nat__Onat,type,
lex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olex_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
lex_se2245640040323279819at_nat: set_Pr4329608150637261639at_nat > set_Pr4333006031979791559at_nat ).
thf(sy_c_List_Olex_001tf__b,type,
lex_b: set_Product_prod_b_b > set_Pr7665282455119567943list_b ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
cons_s6881495754146722583at_nat: set_Pr1261947904930325089at_nat > list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Olist_OCons_001tf__b,type,
cons_b: b > list_b > list_b ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
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produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc5463602008962177208at_nat: nat > set_Pr1261947904930325089at_nat > produc3313772616054891654at_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__b,type,
product_Pair_nat_b: nat > b > product_prod_nat_b ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
produc3149970401713881818at_nat: set_Pr1261947904930325089at_nat > nat > produc366245978424229472at_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001tf__b,type,
produc5313103311483691637_nat_b: set_Pr1261947904930325089at_nat > b > produc3549329967060017029_nat_b ).
thf(sy_c_Product__Type_OPair_001tf__b_001t__Nat__Onat,type,
product_Pair_b_nat: b > nat > product_prod_b_nat ).
thf(sy_c_Product__Type_OPair_001tf__b_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc1957037382231495865at_nat: b > set_Pr1261947904930325089at_nat > produc1620927101341568713at_nat ).
thf(sy_c_Product__Type_OPair_001tf__b_001tf__b,type,
product_Pair_b_b: b > b > product_prod_b_b ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Stirling_OStirling,type,
stirling: nat > nat > nat ).
thf(sy_c_Stirling_Ostirling,type,
stirling2: nat > nat > nat ).
thf(sy_c_Stirling_Ostirling__row,type,
stirling_row: nat > list_nat ).
thf(sy_c_Stirling_Ostirling__row__aux_001t__Int__Oint,type,
stirling_row_aux_int: int > int > list_int > list_int ).
thf(sy_c_Stirling_Ostirling__row__aux_001t__Nat__Onat,type,
stirling_row_aux_nat: nat > nat > list_nat > list_nat ).
thf(sy_c_Stirling_Ozip__with__prev_001t__Int__Oint_001t__Int__Oint,type,
zip_wi8271306424544069320nt_int: ( int > int > int ) > int > list_int > list_int ).
thf(sy_c_Stirling_Ozip__with__prev_001t__Nat__Onat_001t__Nat__Onat,type,
zip_wi7274443183152065040at_nat: ( nat > nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
prefixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Oprefixes_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
prefix5371233514468679790at_nat: list_s1210847774152347623at_nat > list_l3822697302700470509at_nat ).
thf(sy_c_Sublist_Oprefixes_001tf__b,type,
prefixes_b: list_b > list_list_b ).
thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
sublists_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
suffixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
suffix8250022383976823727at_nat: list_s1210847774152347623at_nat > list_l3822697302700470509at_nat ).
thf(sy_c_Sublist_Osuffixes_001tf__b,type,
suffixes_b: list_b > list_list_b ).
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__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
member4080735728053443344at_nat: produc424102278133772007at_nat > set_Pr4333006031979791559at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_Itf__b_J_J,type,
member2990321877988238992list_b: produc3963297410138542439list_b > set_Pr7665282455119567943list_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
member7862447936710763792od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_xs,type,
xs: list_b ).
% Relevant facts (1266)
thf(fact_0_assms,axiom,
( n
= ( size_size_list_b @ xs ) ) ).
% assms
thf(fact_1_kernel__of__eq__len,axiom,
! [X: list_s1210847774152347623at_nat,Y: list_s1210847774152347623at_nat] :
( ( ( equiva1173177585473067681at_nat @ X )
= ( equiva1173177585473067681at_nat @ Y ) )
=> ( ( size_s8736152011456118867at_nat @ X )
= ( size_s8736152011456118867at_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_2_kernel__of__eq__len,axiom,
! [X: list_s1210847774152347623at_nat,Y: list_nat] :
( ( ( equiva1173177585473067681at_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_s8736152011456118867at_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_3_kernel__of__eq__len,axiom,
! [X: list_nat,Y: list_s1210847774152347623at_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva1173177585473067681at_nat @ Y ) )
=> ( ( size_size_list_nat @ X )
= ( size_s8736152011456118867at_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_4_kernel__of__eq__len,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_5_kernel__of__eq__len,axiom,
! [X: list_s1210847774152347623at_nat,Y: list_b] :
( ( ( equiva1173177585473067681at_nat @ X )
= ( equiva2867628904822520639l_of_b @ Y ) )
=> ( ( size_s8736152011456118867at_nat @ X )
= ( size_size_list_b @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_6_kernel__of__eq__len,axiom,
! [X: list_nat,Y: list_b] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2867628904822520639l_of_b @ Y ) )
=> ( ( size_size_list_nat @ X )
= ( size_size_list_b @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_7_kernel__of__eq__len,axiom,
! [X: list_b,Y: list_s1210847774152347623at_nat] :
( ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva1173177585473067681at_nat @ Y ) )
=> ( ( size_size_list_b @ X )
= ( size_s8736152011456118867at_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_8_kernel__of__eq__len,axiom,
! [X: list_b,Y: list_nat] :
( ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_size_list_b @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_9_kernel__of__eq__len,axiom,
! [X: list_b,Y: list_b] :
( ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2867628904822520639l_of_b @ Y ) )
=> ( ( size_size_list_b @ X )
= ( size_size_list_b @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_10_filter__filter,axiom,
! [P: nat > $o,Q: nat > $o,Xs: list_nat] :
( ( filter_nat @ P @ ( filter_nat @ Q @ Xs ) )
= ( filter_nat
@ ^ [X2: nat] :
( ( Q @ X2 )
& ( P @ X2 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_11_filter__filter,axiom,
! [P: b > $o,Q: b > $o,Xs: list_b] :
( ( filter_b @ P @ ( filter_b @ Q @ Xs ) )
= ( filter_b
@ ^ [X2: b] :
( ( Q @ X2 )
& ( P @ X2 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_12_filter__filter,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,Q: set_Pr1261947904930325089at_nat > $o,Xs: list_s1210847774152347623at_nat] :
( ( filter4791820933415917918at_nat @ P @ ( filter4791820933415917918at_nat @ Q @ Xs ) )
= ( filter4791820933415917918at_nat
@ ^ [X2: set_Pr1261947904930325089at_nat] :
( ( Q @ X2 )
& ( P @ X2 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_13_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_14_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_s1210847774152347623at_nat] :
( ( size_s8736152011456118867at_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_15_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_b] :
( ( size_size_list_b @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_16_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_17_neq__if__length__neq,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs )
!= ( size_s8736152011456118867at_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_18_neq__if__length__neq,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( size_size_list_b @ Xs )
!= ( size_size_list_b @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_19_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_20_size__neq__size__imp__neq,axiom,
! [X: list_s1210847774152347623at_nat,Y: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ X )
!= ( size_s8736152011456118867at_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_21_size__neq__size__imp__neq,axiom,
! [X: list_b,Y: list_b] :
( ( ( size_size_list_b @ X )
!= ( size_size_list_b @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_22_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_23_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_24_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_25_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_26_count__list__expand,axiom,
( count_6440129622255701469at_nat
= ( ^ [Xs3: list_s1210847774152347623at_nat,X2: set_Pr1261947904930325089at_nat] :
( size_s8736152011456118867at_nat
@ ( filter4791820933415917918at_nat
@ ( ^ [Y2: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( Y2 = Z )
@ X2 )
@ Xs3 ) ) ) ) ).
% count_list_expand
thf(fact_27_count__list__expand,axiom,
( count_list_b
= ( ^ [Xs3: list_b,X2: b] :
( size_size_list_b
@ ( filter_b
@ ( ^ [Y2: b,Z: b] : ( Y2 = Z )
@ X2 )
@ Xs3 ) ) ) ) ).
% count_list_expand
thf(fact_28_count__list__expand,axiom,
( count_list_nat
= ( ^ [Xs3: list_nat,X2: nat] :
( size_size_list_nat
@ ( filter_nat
@ ( ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ X2 )
@ Xs3 ) ) ) ) ).
% count_list_expand
thf(fact_29_euclidean__size__1,axiom,
( ( euclid4777050414544973029ze_nat @ one_one_nat )
= one_one_nat ) ).
% euclidean_size_1
thf(fact_30_euclidean__size__1,axiom,
( ( euclid4774559944035922753ze_int @ one_one_int )
= one_one_nat ) ).
% euclidean_size_1
thf(fact_31_sum__length__filter__compl,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,Xs: list_s1210847774152347623at_nat] :
( ( plus_plus_nat @ ( size_s8736152011456118867at_nat @ ( filter4791820933415917918at_nat @ P @ Xs ) )
@ ( size_s8736152011456118867at_nat
@ ( filter4791820933415917918at_nat
@ ^ [X2: set_Pr1261947904930325089at_nat] :
~ ( P @ X2 )
@ Xs ) ) )
= ( size_s8736152011456118867at_nat @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_32_sum__length__filter__compl,axiom,
! [P: b > $o,Xs: list_b] :
( ( plus_plus_nat @ ( size_size_list_b @ ( filter_b @ P @ Xs ) )
@ ( size_size_list_b
@ ( filter_b
@ ^ [X2: b] :
~ ( P @ X2 )
@ Xs ) ) )
= ( size_size_list_b @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_33_sum__length__filter__compl,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( plus_plus_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) )
@ ( size_size_list_nat
@ ( filter_nat
@ ^ [X2: nat] :
~ ( P @ X2 )
@ Xs ) ) )
= ( size_size_list_nat @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_34_replicate__length__filter,axiom,
! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat] :
( ( replic7735362847476918817at_nat
@ ( size_s8736152011456118867at_nat
@ ( filter4791820933415917918at_nat
@ ( ^ [Y2: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( Y2 = Z )
@ X )
@ Xs ) )
@ X )
= ( filter4791820933415917918at_nat
@ ( ^ [Y2: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( Y2 = Z )
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_35_replicate__length__filter,axiom,
! [X: b,Xs: list_b] :
( ( replicate_b
@ ( size_size_list_b
@ ( filter_b
@ ( ^ [Y2: b,Z: b] : ( Y2 = Z )
@ X )
@ Xs ) )
@ X )
= ( filter_b
@ ( ^ [Y2: b,Z: b] : ( Y2 = Z )
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_36_replicate__length__filter,axiom,
! [X: nat,Xs: list_nat] :
( ( replicate_nat
@ ( size_size_list_nat
@ ( filter_nat
@ ( ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ X )
@ Xs ) )
@ X )
= ( filter_nat
@ ( ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_37_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_38_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_39_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_40_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_41_length__replicate,axiom,
! [N: nat,X: set_Pr1261947904930325089at_nat] :
( ( size_s8736152011456118867at_nat @ ( replic7735362847476918817at_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_42_length__replicate,axiom,
! [N: nat,X: b] :
( ( size_size_list_b @ ( replicate_b @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_43_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_44_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_45_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_46_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_47_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_48_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_49_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_50_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_51_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_52_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_53_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_54_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_55_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_56_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_57_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_58_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_59_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_60_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_61_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_62_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_63_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_64_gen__length__def,axiom,
( gen_le5092146751752969972at_nat
= ( ^ [N2: nat,Xs3: list_s1210847774152347623at_nat] : ( plus_plus_nat @ N2 @ ( size_s8736152011456118867at_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_65_gen__length__def,axiom,
( gen_length_b
= ( ^ [N2: nat,Xs3: list_b] : ( plus_plus_nat @ N2 @ ( size_size_list_b @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_66_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N2: nat,Xs3: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_67_length__splice,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( size_s8736152011456118867at_nat @ ( splice8498341623201783354at_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ ( size_s8736152011456118867at_nat @ Ys ) ) ) ).
% length_splice
thf(fact_68_length__splice,axiom,
! [Xs: list_b,Ys: list_b] :
( ( size_size_list_b @ ( splice_b @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_b @ Xs ) @ ( size_size_list_b @ Ys ) ) ) ).
% length_splice
thf(fact_69_length__splice,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( splice_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_splice
thf(fact_70_count__list_Osimps_I2_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( X = Y )
=> ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
= ( plus_plus_nat @ ( count_list_nat @ Xs @ Y ) @ one_one_nat ) ) )
& ( ( X != Y )
=> ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
= ( count_list_nat @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_71_count__list_Osimps_I2_J,axiom,
! [X: b,Y: b,Xs: list_b] :
( ( ( X = Y )
=> ( ( count_list_b @ ( cons_b @ X @ Xs ) @ Y )
= ( plus_plus_nat @ ( count_list_b @ Xs @ Y ) @ one_one_nat ) ) )
& ( ( X != Y )
=> ( ( count_list_b @ ( cons_b @ X @ Xs ) @ Y )
= ( count_list_b @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_72_count__list_Osimps_I2_J,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat] :
( ( ( X = Y )
=> ( ( count_6440129622255701469at_nat @ ( cons_s6881495754146722583at_nat @ X @ Xs ) @ Y )
= ( plus_plus_nat @ ( count_6440129622255701469at_nat @ Xs @ Y ) @ one_one_nat ) ) )
& ( ( X != Y )
=> ( ( count_6440129622255701469at_nat @ ( cons_s6881495754146722583at_nat @ X @ Xs ) @ Y )
= ( count_6440129622255701469at_nat @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_73_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_74_length__prefixes,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( size_s3876073854528961369at_nat @ ( prefix5371233514468679790at_nat @ Xs ) )
= ( plus_plus_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_75_length__prefixes,axiom,
! [Xs: list_b] :
( ( size_s420531428170919671list_b @ ( prefixes_b @ Xs ) )
= ( plus_plus_nat @ ( size_size_list_b @ Xs ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_76_length__prefixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( prefixes_nat @ Xs ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_77_count__list__append,axiom,
! [Xs: list_nat,Ys: list_nat,X: nat] :
( ( count_list_nat @ ( append_nat @ Xs @ Ys ) @ X )
= ( plus_plus_nat @ ( count_list_nat @ Xs @ X ) @ ( count_list_nat @ Ys @ X ) ) ) ).
% count_list_append
thf(fact_78_count__list__append,axiom,
! [Xs: list_b,Ys: list_b,X: b] :
( ( count_list_b @ ( append_b @ Xs @ Ys ) @ X )
= ( plus_plus_nat @ ( count_list_b @ Xs @ X ) @ ( count_list_b @ Ys @ X ) ) ) ).
% count_list_append
thf(fact_79_count__list__append,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,X: set_Pr1261947904930325089at_nat] :
( ( count_6440129622255701469at_nat @ ( append4192317425040545660at_nat @ Xs @ Ys ) @ X )
= ( plus_plus_nat @ ( count_6440129622255701469at_nat @ Xs @ X ) @ ( count_6440129622255701469at_nat @ Ys @ X ) ) ) ).
% count_list_append
thf(fact_80_length__append,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( size_s8736152011456118867at_nat @ ( append4192317425040545660at_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ ( size_s8736152011456118867at_nat @ Ys ) ) ) ).
% length_append
thf(fact_81_length__append,axiom,
! [Xs: list_b,Ys: list_b] :
( ( size_size_list_b @ ( append_b @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_b @ Xs ) @ ( size_size_list_b @ Ys ) ) ) ).
% length_append
thf(fact_82_length__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_83_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_84_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_85_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_86_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_87_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_88_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_89_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_90_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_91_append__eq__append__conv,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,Us: list_s1210847774152347623at_nat,Vs: list_s1210847774152347623at_nat] :
( ( ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_s8736152011456118867at_nat @ Ys ) )
| ( ( size_s8736152011456118867at_nat @ Us )
= ( size_s8736152011456118867at_nat @ Vs ) ) )
=> ( ( ( append4192317425040545660at_nat @ Xs @ Us )
= ( append4192317425040545660at_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_92_append__eq__append__conv,axiom,
! [Xs: list_b,Ys: list_b,Us: list_b,Vs: list_b] :
( ( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
| ( ( size_size_list_b @ Us )
= ( size_size_list_b @ Vs ) ) )
=> ( ( ( append_b @ Xs @ Us )
= ( append_b @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_93_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_94_filter__append,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( filter4791820933415917918at_nat @ P @ ( append4192317425040545660at_nat @ Xs @ Ys ) )
= ( append4192317425040545660at_nat @ ( filter4791820933415917918at_nat @ P @ Xs ) @ ( filter4791820933415917918at_nat @ P @ Ys ) ) ) ).
% filter_append
thf(fact_95_filter__append,axiom,
! [P: nat > $o,Xs: list_nat,Ys: list_nat] :
( ( filter_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( filter_nat @ P @ Xs ) @ ( filter_nat @ P @ Ys ) ) ) ).
% filter_append
thf(fact_96_filter__append,axiom,
! [P: b > $o,Xs: list_b,Ys: list_b] :
( ( filter_b @ P @ ( append_b @ Xs @ Ys ) )
= ( append_b @ ( filter_b @ P @ Xs ) @ ( filter_b @ P @ Ys ) ) ) ).
% filter_append
thf(fact_97_euclidean__size__of__nat,axiom,
! [N: nat] :
( ( euclid4774559944035922753ze_int @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% euclidean_size_of_nat
thf(fact_98_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_99_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_100_splice_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( splice_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( cons_nat @ X @ ( splice_nat @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_101_append__Cons,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_102_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_103_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_104_replicate__app__Cons__same,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( append_nat @ ( replicate_nat @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_105_filter_Osimps_I2_J,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat] :
( ( ( P @ X )
=> ( ( filter4791820933415917918at_nat @ P @ ( cons_s6881495754146722583at_nat @ X @ Xs ) )
= ( cons_s6881495754146722583at_nat @ X @ ( filter4791820933415917918at_nat @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter4791820933415917918at_nat @ P @ ( cons_s6881495754146722583at_nat @ X @ Xs ) )
= ( filter4791820933415917918at_nat @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_106_filter_Osimps_I2_J,axiom,
! [P: b > $o,X: b,Xs: list_b] :
( ( ( P @ X )
=> ( ( filter_b @ P @ ( cons_b @ X @ Xs ) )
= ( cons_b @ X @ ( filter_b @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter_b @ P @ ( cons_b @ X @ Xs ) )
= ( filter_b @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_107_filter_Osimps_I2_J,axiom,
! [P: nat > $o,X: nat,Xs: list_nat] :
( ( ( P @ X )
=> ( ( filter_nat @ P @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( filter_nat @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter_nat @ P @ ( cons_nat @ X @ Xs ) )
= ( filter_nat @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_108_enumerate__append__eq,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( enumer3886722665900167622at_nat @ N @ ( append4192317425040545660at_nat @ Xs @ Ys ) )
= ( append2704238365670982059at_nat @ ( enumer3886722665900167622at_nat @ N @ Xs ) @ ( enumer3886722665900167622at_nat @ ( plus_plus_nat @ N @ ( size_s8736152011456118867at_nat @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_109_enumerate__append__eq,axiom,
! [N: nat,Xs: list_b,Ys: list_b] :
( ( enumerate_b @ N @ ( append_b @ Xs @ Ys ) )
= ( append1694031010730255049_nat_b @ ( enumerate_b @ N @ Xs ) @ ( enumerate_b @ ( plus_plus_nat @ N @ ( size_size_list_b @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_110_enumerate__append__eq,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( enumerate_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_111_nths__append,axiom,
! [L: list_s1210847774152347623at_nat,L2: list_s1210847774152347623at_nat,A2: set_nat] :
( ( nths_s223767802152895845at_nat @ ( append4192317425040545660at_nat @ L @ L2 ) @ A2 )
= ( append4192317425040545660at_nat @ ( nths_s223767802152895845at_nat @ L @ A2 )
@ ( nths_s223767802152895845at_nat @ L2
@ ( collect_nat
@ ^ [J2: nat] : ( member_nat @ ( plus_plus_nat @ J2 @ ( size_s8736152011456118867at_nat @ L ) ) @ A2 ) ) ) ) ) ).
% nths_append
thf(fact_112_nths__append,axiom,
! [L: list_b,L2: list_b,A2: set_nat] :
( ( nths_b @ ( append_b @ L @ L2 ) @ A2 )
= ( append_b @ ( nths_b @ L @ A2 )
@ ( nths_b @ L2
@ ( collect_nat
@ ^ [J2: nat] : ( member_nat @ ( plus_plus_nat @ J2 @ ( size_size_list_b @ L ) ) @ A2 ) ) ) ) ) ).
% nths_append
thf(fact_113_nths__append,axiom,
! [L: list_nat,L2: list_nat,A2: set_nat] :
( ( nths_nat @ ( append_nat @ L @ L2 ) @ A2 )
= ( append_nat @ ( nths_nat @ L @ A2 )
@ ( nths_nat @ L2
@ ( collect_nat
@ ^ [J2: nat] : ( member_nat @ ( plus_plus_nat @ J2 @ ( size_size_list_nat @ L ) ) @ A2 ) ) ) ) ) ).
% nths_append
thf(fact_114_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_115_division__segment__of__nat,axiom,
! [N: nat] :
( ( euclid3395696857347342551nt_int @ ( semiri1314217659103216013at_int @ N ) )
= one_one_int ) ).
% division_segment_of_nat
thf(fact_116_division__segment__of__nat,axiom,
! [N: nat] :
( ( euclid3398187327856392827nt_nat @ ( semiri1316708129612266289at_nat @ N ) )
= one_one_nat ) ).
% division_segment_of_nat
thf(fact_117_list__update__length,axiom,
! [Xs: list_s1210847774152347623at_nat,X: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( list_u8444657558853818831at_nat @ ( append4192317425040545660at_nat @ Xs @ ( cons_s6881495754146722583at_nat @ X @ Ys ) ) @ ( size_s8736152011456118867at_nat @ Xs ) @ Y )
= ( append4192317425040545660at_nat @ Xs @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_118_list__update__length,axiom,
! [Xs: list_b,X: b,Ys: list_b,Y: b] :
( ( list_update_b @ ( append_b @ Xs @ ( cons_b @ X @ Ys ) ) @ ( size_size_list_b @ Xs ) @ Y )
= ( append_b @ Xs @ ( cons_b @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_119_list__update__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y )
= ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_120_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_121_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_122_nth__append__length__plus,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,N: nat] :
( ( nth_se5679702624988424552at_nat @ ( append4192317425040545660at_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ N ) )
= ( nth_se5679702624988424552at_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_123_nth__append__length__plus,axiom,
! [Xs: list_b,Ys: list_b,N: nat] :
( ( nth_b @ ( append_b @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_b @ Xs ) @ N ) )
= ( nth_b @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_124_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_125_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_126_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_127_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_128_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_129_length__list__update,axiom,
! [Xs: list_s1210847774152347623at_nat,I: nat,X: set_Pr1261947904930325089at_nat] :
( ( size_s8736152011456118867at_nat @ ( list_u8444657558853818831at_nat @ Xs @ I @ X ) )
= ( size_s8736152011456118867at_nat @ Xs ) ) ).
% length_list_update
thf(fact_130_length__list__update,axiom,
! [Xs: list_b,I: nat,X: b] :
( ( size_size_list_b @ ( list_update_b @ Xs @ I @ X ) )
= ( size_size_list_b @ Xs ) ) ).
% length_list_update
thf(fact_131_length__list__update,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_132_division__segment__1,axiom,
( ( euclid3395696857347342551nt_int @ one_one_int )
= one_one_int ) ).
% division_segment_1
thf(fact_133_division__segment__1,axiom,
( ( euclid3398187327856392827nt_nat @ one_one_nat )
= one_one_nat ) ).
% division_segment_1
thf(fact_134_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,X: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_135_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_136_length__enumerate,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat] :
( ( size_s8291778405851463082at_nat @ ( enumer3886722665900167622at_nat @ N @ Xs ) )
= ( size_s8736152011456118867at_nat @ Xs ) ) ).
% length_enumerate
thf(fact_137_length__enumerate,axiom,
! [N: nat,Xs: list_b] :
( ( size_s314938103728740808_nat_b @ ( enumerate_b @ N @ Xs ) )
= ( size_size_list_b @ Xs ) ) ).
% length_enumerate
thf(fact_138_length__enumerate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_enumerate
thf(fact_139_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_140_nth__append__length,axiom,
! [Xs: list_s1210847774152347623at_nat,X: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat] :
( ( nth_se5679702624988424552at_nat @ ( append4192317425040545660at_nat @ Xs @ ( cons_s6881495754146722583at_nat @ X @ Ys ) ) @ ( size_s8736152011456118867at_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_141_nth__append__length,axiom,
! [Xs: list_b,X: b,Ys: list_b] :
( ( nth_b @ ( append_b @ Xs @ ( cons_b @ X @ Ys ) ) @ ( size_size_list_b @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_142_nth__append__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_143_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_144_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_145_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_146_nat__int__comparison_I1_J,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_147_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_148_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_149_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_150_list__update__code_I3_J,axiom,
! [X: nat,Xs: list_nat,I: nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_nat @ X @ ( list_update_nat @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_151_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_152_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_153_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_154_division__segment__eq__iff,axiom,
! [A: nat,B: nat] :
( ( ( euclid3398187327856392827nt_nat @ A )
= ( euclid3398187327856392827nt_nat @ B ) )
=> ( ( ( euclid4777050414544973029ze_nat @ A )
= ( euclid4777050414544973029ze_nat @ B ) )
=> ( A = B ) ) ) ).
% division_segment_eq_iff
thf(fact_155_Suc__length__conv,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat] :
( ( ( suc @ N )
= ( size_s8736152011456118867at_nat @ Xs ) )
= ( ? [Y3: set_Pr1261947904930325089at_nat,Ys2: list_s1210847774152347623at_nat] :
( ( Xs
= ( cons_s6881495754146722583at_nat @ Y3 @ Ys2 ) )
& ( ( size_s8736152011456118867at_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_156_Suc__length__conv,axiom,
! [N: nat,Xs: list_b] :
( ( ( suc @ N )
= ( size_size_list_b @ Xs ) )
= ( ? [Y3: b,Ys2: list_b] :
( ( Xs
= ( cons_b @ Y3 @ Ys2 ) )
& ( ( size_size_list_b @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_157_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y3: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_158_length__Suc__conv,axiom,
! [Xs: list_s1210847774152347623at_nat,N: nat] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: set_Pr1261947904930325089at_nat,Ys2: list_s1210847774152347623at_nat] :
( ( Xs
= ( cons_s6881495754146722583at_nat @ Y3 @ Ys2 ) )
& ( ( size_s8736152011456118867at_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_159_length__Suc__conv,axiom,
! [Xs: list_b,N: nat] :
( ( ( size_size_list_b @ Xs )
= ( suc @ N ) )
= ( ? [Y3: b,Ys2: list_b] :
( ( Xs
= ( cons_b @ Y3 @ Ys2 ) )
& ( ( size_size_list_b @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_160_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_161_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_162_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_163_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_164_replicate__Suc,axiom,
! [N: nat,X: nat] :
( ( replicate_nat @ ( suc @ N ) @ X )
= ( cons_nat @ X @ ( replicate_nat @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_165_gen__length__code_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_166_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_167_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_168_length__Cons,axiom,
! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat] :
( ( size_s8736152011456118867at_nat @ ( cons_s6881495754146722583at_nat @ X @ Xs ) )
= ( suc @ ( size_s8736152011456118867at_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_169_length__Cons,axiom,
! [X: b,Xs: list_b] :
( ( size_size_list_b @ ( cons_b @ X @ Xs ) )
= ( suc @ ( size_size_list_b @ Xs ) ) ) ).
% length_Cons
thf(fact_170_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_171_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_172_length__suffixes,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( size_s3876073854528961369at_nat @ ( suffix8250022383976823727at_nat @ Xs ) )
= ( suc @ ( size_s8736152011456118867at_nat @ Xs ) ) ) ).
% length_suffixes
thf(fact_173_length__suffixes,axiom,
! [Xs: list_b] :
( ( size_s420531428170919671list_b @ ( suffixes_b @ Xs ) )
= ( suc @ ( size_size_list_b @ Xs ) ) ) ).
% length_suffixes
thf(fact_174_length__suffixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_suffixes
thf(fact_175_list_Osize__gen_I2_J,axiom,
! [X: nat > nat,X21: nat,X22: list_nat] :
( ( size_list_nat @ X @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( X @ X21 ) @ ( size_list_nat @ X @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size_gen(2)
thf(fact_176_filter__eq__nths,axiom,
( filter4791820933415917918at_nat
= ( ^ [P2: set_Pr1261947904930325089at_nat > $o,Xs3: list_s1210847774152347623at_nat] :
( nths_s223767802152895845at_nat @ Xs3
@ ( collect_nat
@ ^ [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8736152011456118867at_nat @ Xs3 ) )
& ( P2 @ ( nth_se5679702624988424552at_nat @ Xs3 @ I2 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_177_filter__eq__nths,axiom,
( filter_b
= ( ^ [P2: b > $o,Xs3: list_b] :
( nths_b @ Xs3
@ ( collect_nat
@ ^ [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs3 ) )
& ( P2 @ ( nth_b @ Xs3 @ I2 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_178_filter__eq__nths,axiom,
( filter_nat
= ( ^ [P2: nat > $o,Xs3: list_nat] :
( nths_nat @ Xs3
@ ( collect_nat
@ ^ [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs3 ) )
& ( P2 @ ( nth_nat @ Xs3 @ I2 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_179_list_Osize_I4_J,axiom,
! [X21: set_Pr1261947904930325089at_nat,X22: list_s1210847774152347623at_nat] :
( ( size_s8736152011456118867at_nat @ ( cons_s6881495754146722583at_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_s8736152011456118867at_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_180_list_Osize_I4_J,axiom,
! [X21: b,X22: list_b] :
( ( size_size_list_b @ ( cons_b @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_b @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_181_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_182_length__Suc__conv__rev,axiom,
! [Xs: list_s1210847774152347623at_nat,N: nat] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: set_Pr1261947904930325089at_nat,Ys2: list_s1210847774152347623at_nat] :
( ( Xs
= ( append4192317425040545660at_nat @ Ys2 @ ( cons_s6881495754146722583at_nat @ Y3 @ nil_se357566008730718055at_nat ) ) )
& ( ( size_s8736152011456118867at_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_183_length__Suc__conv__rev,axiom,
! [Xs: list_b,N: nat] :
( ( ( size_size_list_b @ Xs )
= ( suc @ N ) )
= ( ? [Y3: b,Ys2: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ Y3 @ nil_b ) ) )
& ( ( size_size_list_b @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_184_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_185_division__segment__euclidean__size,axiom,
! [A: int] :
( ( times_times_int @ ( euclid3395696857347342551nt_int @ A ) @ ( semiri1314217659103216013at_int @ ( euclid4774559944035922753ze_int @ A ) ) )
= A ) ).
% division_segment_euclidean_size
thf(fact_186_division__segment__euclidean__size,axiom,
! [A: nat] :
( ( times_times_nat @ ( euclid3398187327856392827nt_nat @ A ) @ ( semiri1316708129612266289at_nat @ ( euclid4777050414544973029ze_nat @ A ) ) )
= A ) ).
% division_segment_euclidean_size
thf(fact_187_nth__list__update__eq,axiom,
! [I: nat,Xs: list_s1210847774152347623at_nat,X: set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( nth_se5679702624988424552at_nat @ ( list_u8444657558853818831at_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_188_nth__list__update__eq,axiom,
! [I: nat,Xs: list_b,X: b] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( ( nth_b @ ( list_update_b @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_189_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_190_enumerate__simps_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( enumerate_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_191_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_192_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_193_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_194_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_195_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_196_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_197_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_198_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_199_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_200_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_201_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_202_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_203_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_204_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_205_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_206_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_207_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_208_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_209_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_210_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_211_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_212_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_213_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_214_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_215_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_216_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_217_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_218_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_219_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_220_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_221_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_222_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_223_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_224_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_225_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_226_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_227_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_228_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_229_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_230_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_231_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_232_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_233_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_234_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_235_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_236_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_237_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_238_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_239_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_240_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_241_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_242_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_243_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_244_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_245_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_246_list__update__nonempty,axiom,
! [Xs: list_nat,K: nat,X: nat] :
( ( ( list_update_nat @ Xs @ K @ X )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% list_update_nonempty
thf(fact_247_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_nat @ nil_nat @ A2 )
= nil_nat ) ).
% nths_nil
thf(fact_248_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_nat @ N @ nil_nat )
= nil_Pr5478986624290739719at_nat ) ).
% enumerate_simps(1)
thf(fact_249_splice__Nil2,axiom,
! [Xs: list_nat] :
( ( splice_nat @ Xs @ nil_nat )
= Xs ) ).
% splice_Nil2
thf(fact_250_split__Nil__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( splice_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% split_Nil_iff
thf(fact_251_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_252_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_253_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_254_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_255_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_256_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_257_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_258_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_259_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_260_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_261_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_262_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_263_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_264_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_265_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_266_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_267_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_268_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_269_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_270_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_271_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_272_length__0__conv,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_se357566008730718055at_nat ) ) ).
% length_0_conv
thf(fact_273_length__0__conv,axiom,
! [Xs: list_b] :
( ( ( size_size_list_b @ Xs )
= zero_zero_nat )
= ( Xs = nil_b ) ) ).
% length_0_conv
thf(fact_274_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_275_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_276_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_277_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_278_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_279_replicate__empty,axiom,
! [N: nat,X: nat] :
( ( ( replicate_nat @ N @ X )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_280_empty__replicate,axiom,
! [N: nat,X: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_281_nth__replicate,axiom,
! [I: nat,N: nat,X: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_282_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_283_length__greater__0__conv,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s8736152011456118867at_nat @ Xs ) )
= ( Xs != nil_se357566008730718055at_nat ) ) ).
% length_greater_0_conv
thf(fact_284_length__greater__0__conv,axiom,
! [Xs: list_b] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ Xs ) )
= ( Xs != nil_b ) ) ).
% length_greater_0_conv
thf(fact_285_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_286_nths__singleton,axiom,
! [A2: set_nat,X: nat] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_nat @ ( cons_nat @ X @ nil_nat ) @ A2 )
= ( cons_nat @ X @ nil_nat ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_nat @ ( cons_nat @ X @ nil_nat ) @ A2 )
= nil_nat ) ) ) ).
% nths_singleton
thf(fact_287_prefixes__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_288_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_289_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_290_suffixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( suffixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z3: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs2 ) )
& ( Xs
= ( suffixes_nat @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_291_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_292_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_293_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_294_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_295_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_296_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_297_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_298_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_299_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_300_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_301_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_302_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_303_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_304_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_305_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_306_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_307_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_308_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_309_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_310_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_311_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_312_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_313_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_314_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_315_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_316_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_317_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_318_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_319_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_320_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_321_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_322_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_323_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_324_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_325_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_326_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_327_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_328_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_329_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_330_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_331_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_332_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_333_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_334_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_335_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_336_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_337_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_338_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_339_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_340_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_341_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_342_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_343_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_344_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_345_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_346_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_347_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_348_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_349_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_350_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_351_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_352_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_353_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_354_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_355_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_356_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_357_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_358_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_359_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_360_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_361_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_362_list_Osize_I3_J,axiom,
( ( size_s8736152011456118867at_nat @ nil_se357566008730718055at_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_363_list_Osize_I3_J,axiom,
( ( size_size_list_b @ nil_b )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_364_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_365_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_366_suffixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_367_replicate__0,axiom,
! [X: nat] :
( ( replicate_nat @ zero_zero_nat @ X )
= nil_nat ) ).
% replicate_0
thf(fact_368_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_369_count__list_Osimps_I1_J,axiom,
! [Y: nat] :
( ( count_list_nat @ nil_nat @ Y )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_370_count__list_Osimps_I1_J,axiom,
! [Y: b] :
( ( count_list_b @ nil_b @ Y )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_371_count__list_Osimps_I1_J,axiom,
! [Y: set_Pr1261947904930325089at_nat] :
( ( count_6440129622255701469at_nat @ nil_se357566008730718055at_nat @ Y )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_372_list_Osize__gen_I1_J,axiom,
! [X: nat > nat] :
( ( size_list_nat @ X @ nil_nat )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_373_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_374_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_375_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_376_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_377_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_378_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_379_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_380_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_381_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_382_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_383_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_384_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_385_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_386_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_387_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_388_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_389_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_390_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_391_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_392_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_393_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_394_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_395_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_396_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_397_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_398_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_399_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_400_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_401_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_402_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_403_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_404_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_405_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_406_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_407_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_408_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_409_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_410_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_411_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_412_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_413_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_414_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_415_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_416_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_417_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_418_length__induct,axiom,
! [P: list_s1210847774152347623at_nat > $o,Xs: list_s1210847774152347623at_nat] :
( ! [Xs2: list_s1210847774152347623at_nat] :
( ! [Ys3: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ ( size_s8736152011456118867at_nat @ Ys3 ) @ ( size_s8736152011456118867at_nat @ Xs2 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_419_length__induct,axiom,
! [P: list_b > $o,Xs: list_b] :
( ! [Xs2: list_b] :
( ! [Ys3: list_b] :
( ( ord_less_nat @ ( size_size_list_b @ Ys3 ) @ ( size_size_list_b @ Xs2 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_420_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_421_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_422_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_423_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_424_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs2 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_425_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X3: nat] :
( X
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y4: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_426_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y3: nat,Ys2: list_nat] :
( Xs
= ( cons_nat @ Y3 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_427_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y4: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y4 @ Ys4 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_428_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_429_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_430_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_431_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_432_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_433_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_434_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_435_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_436_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_437_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_438_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_439_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_440_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_441_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_442_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_443_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_444_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_445_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_446_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_447_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_448_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_449_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_450_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_451_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_452_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_453_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_454_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_455_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P @ X3 @ Y4 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_456_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_457_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_458_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_459_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_460_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_461_enum__rgfs_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% enum_rgfs.cases
thf(fact_462_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_463_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_464_filter_Osimps_I1_J,axiom,
! [P: set_Pr1261947904930325089at_nat > $o] :
( ( filter4791820933415917918at_nat @ P @ nil_se357566008730718055at_nat )
= nil_se357566008730718055at_nat ) ).
% filter.simps(1)
thf(fact_465_filter_Osimps_I1_J,axiom,
! [P: b > $o] :
( ( filter_b @ P @ nil_b )
= nil_b ) ).
% filter.simps(1)
thf(fact_466_filter_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( filter_nat @ P @ nil_nat )
= nil_nat ) ).
% filter.simps(1)
thf(fact_467_list__update__code_I1_J,axiom,
! [I: nat,Y: nat] :
( ( list_update_nat @ nil_nat @ I @ Y )
= nil_nat ) ).
% list_update_code(1)
thf(fact_468_list__update_Osimps_I1_J,axiom,
! [I: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_469_splice_Osimps_I1_J,axiom,
! [Ys: list_nat] :
( ( splice_nat @ nil_nat @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_470_prefixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( prefixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z3: nat,Zs2: list_nat] :
( ( Ys
= ( append_nat @ Zs2 @ ( cons_nat @ Z3 @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_471_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_472_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_473_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_s1210847774152347623at_nat,N: nat] :
( ( ord_less_nat @ M @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( nth_Pr8633021227891005631at_nat @ ( enumer3886722665900167622at_nat @ N @ Xs ) @ M )
= ( produc5463602008962177208at_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_se5679702624988424552at_nat @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_474_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_b,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_b @ Xs ) )
=> ( ( nth_Pr2476257085692543965_nat_b @ ( enumerate_b @ N @ Xs ) @ M )
= ( product_Pair_nat_b @ ( plus_plus_nat @ N @ M ) @ ( nth_b @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_475_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_nat,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_476_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_477_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_478_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_479_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_480_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_481_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_482_list__induct4,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b,Ws: list_b,P: list_b > list_b > list_b > list_b > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_b @ Ws ) )
=> ( ( P @ nil_b @ nil_b @ nil_b @ nil_b )
=> ( ! [X3: b,Xs2: list_b,Y4: b,Ys4: list_b,Z4: b,Zs3: list_b,W2: b,Ws2: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( ( size_size_list_b @ Ys4 )
= ( size_size_list_b @ Zs3 ) )
=> ( ( ( size_size_list_b @ Zs3 )
= ( size_size_list_b @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) @ ( cons_b @ Z4 @ Zs3 ) @ ( cons_b @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_483_list__induct4,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b,Ws: list_nat,P: list_b > list_b > list_b > list_nat > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_b @ nil_b @ nil_b @ nil_nat )
=> ( ! [X3: b,Xs2: list_b,Y4: b,Ys4: list_b,Z4: b,Zs3: list_b,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( ( size_size_list_b @ Ys4 )
= ( size_size_list_b @ Zs3 ) )
=> ( ( ( size_size_list_b @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) @ ( cons_b @ Z4 @ Zs3 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_484_list__induct4,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_nat,Ws: list_b,P: list_b > list_b > list_nat > list_b > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_b @ Ws ) )
=> ( ( P @ nil_b @ nil_b @ nil_nat @ nil_b )
=> ( ! [X3: b,Xs2: list_b,Y4: b,Ys4: list_b,Z4: nat,Zs3: list_nat,W2: b,Ws2: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( ( size_size_list_b @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_b @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs3 ) @ ( cons_b @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_485_list__induct4,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_nat,Ws: list_nat,P: list_b > list_b > list_nat > list_nat > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_b @ nil_b @ nil_nat @ nil_nat )
=> ( ! [X3: b,Xs2: list_b,Y4: b,Ys4: list_b,Z4: nat,Zs3: list_nat,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( ( size_size_list_b @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs3 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_486_list__induct4,axiom,
! [Xs: list_b,Ys: list_nat,Zs: list_b,Ws: list_b,P: list_b > list_nat > list_b > list_b > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_b @ Ws ) )
=> ( ( P @ nil_b @ nil_nat @ nil_b @ nil_b )
=> ( ! [X3: b,Xs2: list_b,Y4: nat,Ys4: list_nat,Z4: b,Zs3: list_b,W2: b,Ws2: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_b @ Zs3 ) )
=> ( ( ( size_size_list_b @ Zs3 )
= ( size_size_list_b @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_b @ Z4 @ Zs3 ) @ ( cons_b @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_487_list__induct4,axiom,
! [Xs: list_b,Ys: list_nat,Zs: list_b,Ws: list_nat,P: list_b > list_nat > list_b > list_nat > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_b @ nil_nat @ nil_b @ nil_nat )
=> ( ! [X3: b,Xs2: list_b,Y4: nat,Ys4: list_nat,Z4: b,Zs3: list_b,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_b @ Zs3 ) )
=> ( ( ( size_size_list_b @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_b @ Z4 @ Zs3 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_488_list__induct4,axiom,
! [Xs: list_b,Ys: list_nat,Zs: list_nat,Ws: list_b,P: list_b > list_nat > list_nat > list_b > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_b @ Ws ) )
=> ( ( P @ nil_b @ nil_nat @ nil_nat @ nil_b )
=> ( ! [X3: b,Xs2: list_b,Y4: nat,Ys4: list_nat,Z4: nat,Zs3: list_nat,W2: b,Ws2: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_b @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs3 ) @ ( cons_b @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_489_list__induct4,axiom,
! [Xs: list_b,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_b > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_b @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: b,Xs2: list_b,Y4: nat,Ys4: list_nat,Z4: nat,Zs3: list_nat,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs3 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_490_list__induct4,axiom,
! [Xs: list_nat,Ys: list_b,Zs: list_b,Ws: list_b,P: list_nat > list_b > list_b > list_b > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_b @ Ws ) )
=> ( ( P @ nil_nat @ nil_b @ nil_b @ nil_b )
=> ( ! [X3: nat,Xs2: list_nat,Y4: b,Ys4: list_b,Z4: b,Zs3: list_b,W2: b,Ws2: list_b] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( ( size_size_list_b @ Ys4 )
= ( size_size_list_b @ Zs3 ) )
=> ( ( ( size_size_list_b @ Zs3 )
= ( size_size_list_b @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) @ ( cons_b @ Z4 @ Zs3 ) @ ( cons_b @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_491_list__induct4,axiom,
! [Xs: list_nat,Ys: list_b,Zs: list_b,Ws: list_nat,P: list_nat > list_b > list_b > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_b @ nil_b @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y4: b,Ys4: list_b,Z4: b,Zs3: list_b,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( ( size_size_list_b @ Ys4 )
= ( size_size_list_b @ Zs3 ) )
=> ( ( ( size_size_list_b @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) @ ( cons_b @ Z4 @ Zs3 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_492_list__induct3,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b,P: list_b > list_b > list_b > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( P @ nil_b @ nil_b @ nil_b )
=> ( ! [X3: b,Xs2: list_b,Y4: b,Ys4: list_b,Z4: b,Zs3: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( ( size_size_list_b @ Ys4 )
= ( size_size_list_b @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) @ ( cons_b @ Z4 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_493_list__induct3,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_nat,P: list_b > list_b > list_nat > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_b @ nil_b @ nil_nat )
=> ( ! [X3: b,Xs2: list_b,Y4: b,Ys4: list_b,Z4: nat,Zs3: list_nat] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( ( size_size_list_b @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_494_list__induct3,axiom,
! [Xs: list_b,Ys: list_nat,Zs: list_b,P: list_b > list_nat > list_b > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( P @ nil_b @ nil_nat @ nil_b )
=> ( ! [X3: b,Xs2: list_b,Y4: nat,Ys4: list_nat,Z4: b,Zs3: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_b @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_b @ Z4 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_495_list__induct3,axiom,
! [Xs: list_b,Ys: list_nat,Zs: list_nat,P: list_b > list_nat > list_nat > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_b @ nil_nat @ nil_nat )
=> ( ! [X3: b,Xs2: list_b,Y4: nat,Ys4: list_nat,Z4: nat,Zs3: list_nat] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_496_list__induct3,axiom,
! [Xs: list_nat,Ys: list_b,Zs: list_b,P: list_nat > list_b > list_b > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( P @ nil_nat @ nil_b @ nil_b )
=> ( ! [X3: nat,Xs2: list_nat,Y4: b,Ys4: list_b,Z4: b,Zs3: list_b] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( ( size_size_list_b @ Ys4 )
= ( size_size_list_b @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) @ ( cons_b @ Z4 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_497_list__induct3,axiom,
! [Xs: list_nat,Ys: list_b,Zs: list_nat,P: list_nat > list_b > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_b @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y4: b,Ys4: list_b,Z4: nat,Zs3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( ( size_size_list_b @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_498_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_b,P: list_nat > list_nat > list_b > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_b )
=> ( ! [X3: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat,Z4: b,Zs3: list_b] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_b @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_b @ Z4 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_499_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat,Z4: nat,Zs3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_500_list__induct3,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_b,Zs: list_b,P: list_s1210847774152347623at_nat > list_b > list_b > $o] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( P @ nil_se357566008730718055at_nat @ nil_b @ nil_b )
=> ( ! [X3: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,Y4: b,Ys4: list_b,Z4: b,Zs3: list_b] :
( ( ( size_s8736152011456118867at_nat @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( ( size_size_list_b @ Ys4 )
= ( size_size_list_b @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 )
=> ( P @ ( cons_s6881495754146722583at_nat @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) @ ( cons_b @ Z4 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_501_list__induct3,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_b,Zs: list_nat,P: list_s1210847774152347623at_nat > list_b > list_nat > $o] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_se357566008730718055at_nat @ nil_b @ nil_nat )
=> ( ! [X3: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,Y4: b,Ys4: list_b,Z4: nat,Zs3: list_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( ( size_size_list_b @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs3 )
=> ( P @ ( cons_s6881495754146722583at_nat @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_502_list__induct2,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,P: list_s1210847774152347623at_nat > list_s1210847774152347623at_nat > $o] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_s8736152011456118867at_nat @ Ys ) )
=> ( ( P @ nil_se357566008730718055at_nat @ nil_se357566008730718055at_nat )
=> ( ! [X3: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,Y4: set_Pr1261947904930325089at_nat,Ys4: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs2 )
= ( size_s8736152011456118867at_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_s6881495754146722583at_nat @ X3 @ Xs2 ) @ ( cons_s6881495754146722583at_nat @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_503_list__induct2,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_b,P: list_s1210847774152347623at_nat > list_b > $o] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( P @ nil_se357566008730718055at_nat @ nil_b )
=> ( ! [X3: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,Y4: b,Ys4: list_b] :
( ( ( size_s8736152011456118867at_nat @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_s6881495754146722583at_nat @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_504_list__induct2,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_nat,P: list_s1210847774152347623at_nat > list_nat > $o] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_se357566008730718055at_nat @ nil_nat )
=> ( ! [X3: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,Y4: nat,Ys4: list_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_s6881495754146722583at_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_505_list__induct2,axiom,
! [Xs: list_b,Ys: list_s1210847774152347623at_nat,P: list_b > list_s1210847774152347623at_nat > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_s8736152011456118867at_nat @ Ys ) )
=> ( ( P @ nil_b @ nil_se357566008730718055at_nat )
=> ( ! [X3: b,Xs2: list_b,Y4: set_Pr1261947904930325089at_nat,Ys4: list_s1210847774152347623at_nat] :
( ( ( size_size_list_b @ Xs2 )
= ( size_s8736152011456118867at_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_s6881495754146722583at_nat @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_506_list__induct2,axiom,
! [Xs: list_b,Ys: list_b,P: list_b > list_b > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( P @ nil_b @ nil_b )
=> ( ! [X3: b,Xs2: list_b,Y4: b,Ys4: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_507_list__induct2,axiom,
! [Xs: list_b,Ys: list_nat,P: list_b > list_nat > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_b @ nil_nat )
=> ( ! [X3: b,Xs2: list_b,Y4: nat,Ys4: list_nat] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_b @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_508_list__induct2,axiom,
! [Xs: list_nat,Ys: list_s1210847774152347623at_nat,P: list_nat > list_s1210847774152347623at_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s8736152011456118867at_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_se357566008730718055at_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y4: set_Pr1261947904930325089at_nat,Ys4: list_s1210847774152347623at_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s8736152011456118867at_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_s6881495754146722583at_nat @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_509_list__induct2,axiom,
! [Xs: list_nat,Ys: list_b,P: list_nat > list_b > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( P @ nil_nat @ nil_b )
=> ( ! [X3: nat,Xs2: list_nat,Y4: b,Ys4: list_b] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_b @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_510_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_511_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_512_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys4: list_nat,Y4: nat] :
( Xs
!= ( append_nat @ Ys4 @ ( cons_nat @ Y4 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_513_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs ) )
| ? [Ys5: list_nat] :
( ( ( cons_nat @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_nat @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_514_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys5: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys5 ) )
& ( ( append_nat @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_515_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_516_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_s1210847774152347623at_nat,Z: list_s1210847774152347623at_nat] : ( Y2 = Z ) )
= ( ^ [Xs3: list_s1210847774152347623at_nat,Ys2: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs3 )
= ( size_s8736152011456118867at_nat @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8736152011456118867at_nat @ Xs3 ) )
=> ( ( nth_se5679702624988424552at_nat @ Xs3 @ I2 )
= ( nth_se5679702624988424552at_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_517_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_b,Z: list_b] : ( Y2 = Z ) )
= ( ^ [Xs3: list_b,Ys2: list_b] :
( ( ( size_size_list_b @ Xs3 )
= ( size_size_list_b @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs3 ) )
=> ( ( nth_b @ Xs3 @ I2 )
= ( nth_b @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_518_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_nat,Z: list_nat] : ( Y2 = Z ) )
= ( ^ [Xs3: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I2 )
= ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_519_Skolem__list__nth,axiom,
! [K: nat,P: nat > set_Pr1261947904930325089at_nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X4: set_Pr1261947904930325089at_nat] : ( P @ I2 @ X4 ) ) )
= ( ? [Xs3: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_se5679702624988424552at_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_520_Skolem__list__nth,axiom,
! [K: nat,P: nat > b > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X4: b] : ( P @ I2 @ X4 ) ) )
= ( ? [Xs3: list_b] :
( ( ( size_size_list_b @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_b @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_521_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X4: nat] : ( P @ I2 @ X4 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_522_nth__equalityI,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_s8736152011456118867at_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( nth_se5679702624988424552at_nat @ Xs @ I3 )
= ( nth_se5679702624988424552at_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_523_nth__equalityI,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
=> ( ( nth_b @ Xs @ I3 )
= ( nth_b @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_524_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_525_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_526_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_527_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_528_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_529_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_530_nths__Cons,axiom,
! [X: nat,L: list_nat,A2: set_nat] :
( ( nths_nat @ ( cons_nat @ X @ L ) @ A2 )
= ( append_nat @ ( if_list_nat @ ( member_nat @ zero_zero_nat @ A2 ) @ ( cons_nat @ X @ nil_nat ) @ nil_nat )
@ ( nths_nat @ L
@ ( collect_nat
@ ^ [J2: nat] : ( member_nat @ ( suc @ J2 ) @ A2 ) ) ) ) ) ).
% nths_Cons
thf(fact_531_nths__all,axiom,
! [Xs: list_s1210847774152347623at_nat,I4: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( member_nat @ I3 @ I4 ) )
=> ( ( nths_s223767802152895845at_nat @ Xs @ I4 )
= Xs ) ) ).
% nths_all
thf(fact_532_nths__all,axiom,
! [Xs: list_b,I4: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
=> ( member_nat @ I3 @ I4 ) )
=> ( ( nths_b @ Xs @ I4 )
= Xs ) ) ).
% nths_all
thf(fact_533_nths__all,axiom,
! [Xs: list_nat,I4: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ I3 @ I4 ) )
=> ( ( nths_nat @ Xs @ I4 )
= Xs ) ) ).
% nths_all
thf(fact_534_filter__replicate,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,X: set_Pr1261947904930325089at_nat,N: nat] :
( ( ( P @ X )
=> ( ( filter4791820933415917918at_nat @ P @ ( replic7735362847476918817at_nat @ N @ X ) )
= ( replic7735362847476918817at_nat @ N @ X ) ) )
& ( ~ ( P @ X )
=> ( ( filter4791820933415917918at_nat @ P @ ( replic7735362847476918817at_nat @ N @ X ) )
= nil_se357566008730718055at_nat ) ) ) ).
% filter_replicate
thf(fact_535_filter__replicate,axiom,
! [P: b > $o,X: b,N: nat] :
( ( ( P @ X )
=> ( ( filter_b @ P @ ( replicate_b @ N @ X ) )
= ( replicate_b @ N @ X ) ) )
& ( ~ ( P @ X )
=> ( ( filter_b @ P @ ( replicate_b @ N @ X ) )
= nil_b ) ) ) ).
% filter_replicate
thf(fact_536_filter__replicate,axiom,
! [P: nat > $o,X: nat,N: nat] :
( ( ( P @ X )
=> ( ( filter_nat @ P @ ( replicate_nat @ N @ X ) )
= ( replicate_nat @ N @ X ) ) )
& ( ~ ( P @ X )
=> ( ( filter_nat @ P @ ( replicate_nat @ N @ X ) )
= nil_nat ) ) ) ).
% filter_replicate
thf(fact_537_list__update__code_I2_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_538_splice_Oelims,axiom,
! [X: list_nat,Xa: list_nat,Y: list_nat] :
( ( ( splice_nat @ X @ Xa )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != Xa ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X3 @ Xs2 ) )
=> ( Y
!= ( cons_nat @ X3 @ ( splice_nat @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_539_length__append__singleton,axiom,
! [Xs: list_s1210847774152347623at_nat,X: set_Pr1261947904930325089at_nat] :
( ( size_s8736152011456118867at_nat @ ( append4192317425040545660at_nat @ Xs @ ( cons_s6881495754146722583at_nat @ X @ nil_se357566008730718055at_nat ) ) )
= ( suc @ ( size_s8736152011456118867at_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_540_length__append__singleton,axiom,
! [Xs: list_b,X: b] :
( ( size_size_list_b @ ( append_b @ Xs @ ( cons_b @ X @ nil_b ) ) )
= ( suc @ ( size_size_list_b @ Xs ) ) ) ).
% length_append_singleton
thf(fact_541_length__append__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_542_length__code,axiom,
( size_s8736152011456118867at_nat
= ( gen_le5092146751752969972at_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_543_length__code,axiom,
( size_size_list_b
= ( gen_length_b @ zero_zero_nat ) ) ).
% length_code
thf(fact_544_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_545_same__length__different,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( Xs != Ys )
=> ( ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_s8736152011456118867at_nat @ Ys ) )
=> ? [Pre: list_s1210847774152347623at_nat,X3: set_Pr1261947904930325089at_nat,Xs4: list_s1210847774152347623at_nat,Y4: set_Pr1261947904930325089at_nat,Ys6: list_s1210847774152347623at_nat] :
( ( X3 != Y4 )
& ( Xs
= ( append4192317425040545660at_nat @ Pre @ ( append4192317425040545660at_nat @ ( cons_s6881495754146722583at_nat @ X3 @ nil_se357566008730718055at_nat ) @ Xs4 ) ) )
& ( Ys
= ( append4192317425040545660at_nat @ Pre @ ( append4192317425040545660at_nat @ ( cons_s6881495754146722583at_nat @ Y4 @ nil_se357566008730718055at_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_546_same__length__different,axiom,
! [Xs: list_b,Ys: list_b] :
( ( Xs != Ys )
=> ( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ? [Pre: list_b,X3: b,Xs4: list_b,Y4: b,Ys6: list_b] :
( ( X3 != Y4 )
& ( Xs
= ( append_b @ Pre @ ( append_b @ ( cons_b @ X3 @ nil_b ) @ Xs4 ) ) )
& ( Ys
= ( append_b @ Pre @ ( append_b @ ( cons_b @ Y4 @ nil_b ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_547_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X3: nat,Xs4: list_nat,Y4: nat,Ys6: list_nat] :
( ( X3 != Y4 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs4 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y4 @ nil_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_548_list__induct__2__rev,axiom,
! [X: list_s1210847774152347623at_nat,Y: list_s1210847774152347623at_nat,P: list_s1210847774152347623at_nat > list_s1210847774152347623at_nat > $o] :
( ( ( size_s8736152011456118867at_nat @ X )
= ( size_s8736152011456118867at_nat @ Y ) )
=> ( ( P @ nil_se357566008730718055at_nat @ nil_se357566008730718055at_nat )
=> ( ! [X3: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,Y4: set_Pr1261947904930325089at_nat,Ys4: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs2 )
= ( size_s8736152011456118867at_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( append4192317425040545660at_nat @ Xs2 @ ( cons_s6881495754146722583at_nat @ X3 @ nil_se357566008730718055at_nat ) ) @ ( append4192317425040545660at_nat @ Ys4 @ ( cons_s6881495754146722583at_nat @ Y4 @ nil_se357566008730718055at_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_549_list__induct__2__rev,axiom,
! [X: list_s1210847774152347623at_nat,Y: list_b,P: list_s1210847774152347623at_nat > list_b > $o] :
( ( ( size_s8736152011456118867at_nat @ X )
= ( size_size_list_b @ Y ) )
=> ( ( P @ nil_se357566008730718055at_nat @ nil_b )
=> ( ! [X3: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,Y4: b,Ys4: list_b] :
( ( ( size_s8736152011456118867at_nat @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( append4192317425040545660at_nat @ Xs2 @ ( cons_s6881495754146722583at_nat @ X3 @ nil_se357566008730718055at_nat ) ) @ ( append_b @ Ys4 @ ( cons_b @ Y4 @ nil_b ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_550_list__induct__2__rev,axiom,
! [X: list_s1210847774152347623at_nat,Y: list_nat,P: list_s1210847774152347623at_nat > list_nat > $o] :
( ( ( size_s8736152011456118867at_nat @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( P @ nil_se357566008730718055at_nat @ nil_nat )
=> ( ! [X3: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,Y4: nat,Ys4: list_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( append4192317425040545660at_nat @ Xs2 @ ( cons_s6881495754146722583at_nat @ X3 @ nil_se357566008730718055at_nat ) ) @ ( append_nat @ Ys4 @ ( cons_nat @ Y4 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_551_list__induct__2__rev,axiom,
! [X: list_b,Y: list_s1210847774152347623at_nat,P: list_b > list_s1210847774152347623at_nat > $o] :
( ( ( size_size_list_b @ X )
= ( size_s8736152011456118867at_nat @ Y ) )
=> ( ( P @ nil_b @ nil_se357566008730718055at_nat )
=> ( ! [X3: b,Xs2: list_b,Y4: set_Pr1261947904930325089at_nat,Ys4: list_s1210847774152347623at_nat] :
( ( ( size_size_list_b @ Xs2 )
= ( size_s8736152011456118867at_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( append_b @ Xs2 @ ( cons_b @ X3 @ nil_b ) ) @ ( append4192317425040545660at_nat @ Ys4 @ ( cons_s6881495754146722583at_nat @ Y4 @ nil_se357566008730718055at_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_552_list__induct__2__rev,axiom,
! [X: list_b,Y: list_b,P: list_b > list_b > $o] :
( ( ( size_size_list_b @ X )
= ( size_size_list_b @ Y ) )
=> ( ( P @ nil_b @ nil_b )
=> ( ! [X3: b,Xs2: list_b,Y4: b,Ys4: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( append_b @ Xs2 @ ( cons_b @ X3 @ nil_b ) ) @ ( append_b @ Ys4 @ ( cons_b @ Y4 @ nil_b ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_553_list__induct__2__rev,axiom,
! [X: list_b,Y: list_nat,P: list_b > list_nat > $o] :
( ( ( size_size_list_b @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( P @ nil_b @ nil_nat )
=> ( ! [X3: b,Xs2: list_b,Y4: nat,Ys4: list_nat] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( append_b @ Xs2 @ ( cons_b @ X3 @ nil_b ) ) @ ( append_nat @ Ys4 @ ( cons_nat @ Y4 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_554_list__induct__2__rev,axiom,
! [X: list_nat,Y: list_s1210847774152347623at_nat,P: list_nat > list_s1210847774152347623at_nat > $o] :
( ( ( size_size_list_nat @ X )
= ( size_s8736152011456118867at_nat @ Y ) )
=> ( ( P @ nil_nat @ nil_se357566008730718055at_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y4: set_Pr1261947904930325089at_nat,Ys4: list_s1210847774152347623at_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s8736152011456118867at_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append4192317425040545660at_nat @ Ys4 @ ( cons_s6881495754146722583at_nat @ Y4 @ nil_se357566008730718055at_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_555_list__induct__2__rev,axiom,
! [X: list_nat,Y: list_b,P: list_nat > list_b > $o] :
( ( ( size_size_list_nat @ X )
= ( size_size_list_b @ Y ) )
=> ( ( P @ nil_nat @ nil_b )
=> ( ! [X3: nat,Xs2: list_nat,Y4: b,Ys4: list_b] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_b @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_b @ Ys4 @ ( cons_b @ Y4 @ nil_b ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_556_list__induct__2__rev,axiom,
! [X: list_nat,Y: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_nat @ Ys4 @ ( cons_nat @ Y4 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_557_list__update__append1,axiom,
! [I: nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,X: set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( list_u8444657558853818831at_nat @ ( append4192317425040545660at_nat @ Xs @ Ys ) @ I @ X )
= ( append4192317425040545660at_nat @ ( list_u8444657558853818831at_nat @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_558_list__update__append1,axiom,
! [I: nat,Xs: list_b,Ys: list_b,X: b] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( ( list_update_b @ ( append_b @ Xs @ Ys ) @ I @ X )
= ( append_b @ ( list_update_b @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_559_list__update__append1,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I @ X )
= ( append_nat @ ( list_update_nat @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_560_replicate__append__same,axiom,
! [I: nat,X: nat] :
( ( append_nat @ ( replicate_nat @ I @ X ) @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ ( replicate_nat @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_561_list__update__same__conv,axiom,
! [I: nat,Xs: list_s1210847774152347623at_nat,X: set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( ( list_u8444657558853818831at_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_se5679702624988424552at_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_562_list__update__same__conv,axiom,
! [I: nat,Xs: list_b,X: b] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( ( ( list_update_b @ Xs @ I @ X )
= Xs )
= ( ( nth_b @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_563_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_564_nth__list__update,axiom,
! [I: nat,Xs: list_s1210847774152347623at_nat,J: nat,X: set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_se5679702624988424552at_nat @ ( list_u8444657558853818831at_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_se5679702624988424552at_nat @ ( list_u8444657558853818831at_nat @ Xs @ I @ X ) @ J )
= ( nth_se5679702624988424552at_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_565_nth__list__update,axiom,
! [I: nat,Xs: list_b,J: nat,X: b] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_b @ ( list_update_b @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_b @ ( list_update_b @ Xs @ I @ X ) @ J )
= ( nth_b @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_566_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_567_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_568_kernel__of__eq,axiom,
! [X: list_b,Y: list_b] :
( ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2867628904822520639l_of_b @ Y ) )
= ( ( ( size_size_list_b @ X )
= ( size_size_list_b @ Y ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_b @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_b @ X @ I2 )
= ( nth_b @ X @ J2 ) )
= ( ( nth_b @ Y @ I2 )
= ( nth_b @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_569_kernel__of__eq,axiom,
! [X: list_b,Y: list_nat] :
( ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
= ( ( ( size_size_list_b @ X )
= ( size_size_list_nat @ Y ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_b @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_b @ X @ I2 )
= ( nth_b @ X @ J2 ) )
= ( ( nth_nat @ Y @ I2 )
= ( nth_nat @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_570_kernel__of__eq,axiom,
! [X: list_b,Y: list_s1210847774152347623at_nat] :
( ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva1173177585473067681at_nat @ Y ) )
= ( ( ( size_size_list_b @ X )
= ( size_s8736152011456118867at_nat @ Y ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_b @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_b @ X @ I2 )
= ( nth_b @ X @ J2 ) )
= ( ( nth_se5679702624988424552at_nat @ Y @ I2 )
= ( nth_se5679702624988424552at_nat @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_571_kernel__of__eq,axiom,
! [X: list_nat,Y: list_b] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2867628904822520639l_of_b @ Y ) )
= ( ( ( size_size_list_nat @ X )
= ( size_size_list_b @ Y ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J2 ) )
= ( ( nth_b @ Y @ I2 )
= ( nth_b @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_572_kernel__of__eq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
= ( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J2 ) )
= ( ( nth_nat @ Y @ I2 )
= ( nth_nat @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_573_kernel__of__eq,axiom,
! [X: list_nat,Y: list_s1210847774152347623at_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva1173177585473067681at_nat @ Y ) )
= ( ( ( size_size_list_nat @ X )
= ( size_s8736152011456118867at_nat @ Y ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J2 ) )
= ( ( nth_se5679702624988424552at_nat @ Y @ I2 )
= ( nth_se5679702624988424552at_nat @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_574_kernel__of__eq,axiom,
! [X: list_s1210847774152347623at_nat,Y: list_b] :
( ( ( equiva1173177585473067681at_nat @ X )
= ( equiva2867628904822520639l_of_b @ Y ) )
= ( ( ( size_s8736152011456118867at_nat @ X )
= ( size_size_list_b @ Y ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s8736152011456118867at_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_se5679702624988424552at_nat @ X @ I2 )
= ( nth_se5679702624988424552at_nat @ X @ J2 ) )
= ( ( nth_b @ Y @ I2 )
= ( nth_b @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_575_kernel__of__eq,axiom,
! [X: list_s1210847774152347623at_nat,Y: list_nat] :
( ( ( equiva1173177585473067681at_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
= ( ( ( size_s8736152011456118867at_nat @ X )
= ( size_size_list_nat @ Y ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s8736152011456118867at_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_se5679702624988424552at_nat @ X @ I2 )
= ( nth_se5679702624988424552at_nat @ X @ J2 ) )
= ( ( nth_nat @ Y @ I2 )
= ( nth_nat @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_576_kernel__of__eq,axiom,
! [X: list_s1210847774152347623at_nat,Y: list_s1210847774152347623at_nat] :
( ( ( equiva1173177585473067681at_nat @ X )
= ( equiva1173177585473067681at_nat @ Y ) )
= ( ( ( size_s8736152011456118867at_nat @ X )
= ( size_s8736152011456118867at_nat @ Y ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s8736152011456118867at_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_se5679702624988424552at_nat @ X @ I2 )
= ( nth_se5679702624988424552at_nat @ X @ J2 ) )
= ( ( nth_se5679702624988424552at_nat @ Y @ I2 )
= ( nth_se5679702624988424552at_nat @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_577_euclidean__size__greater__0__iff,axiom,
! [B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( euclid4777050414544973029ze_nat @ B ) )
= ( B != zero_zero_nat ) ) ).
% euclidean_size_greater_0_iff
thf(fact_578_euclidean__size__greater__0__iff,axiom,
! [B: int] :
( ( ord_less_nat @ zero_zero_nat @ ( euclid4774559944035922753ze_int @ B ) )
= ( B != zero_zero_int ) ) ).
% euclidean_size_greater_0_iff
thf(fact_579_size__0,axiom,
( ( euclid4777050414544973029ze_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% size_0
thf(fact_580_size__0,axiom,
( ( euclid4774559944035922753ze_int @ zero_zero_int )
= zero_zero_nat ) ).
% size_0
thf(fact_581_euclidean__size__eq__0__iff,axiom,
! [B: nat] :
( ( ( euclid4777050414544973029ze_nat @ B )
= zero_zero_nat )
= ( B = zero_zero_nat ) ) ).
% euclidean_size_eq_0_iff
thf(fact_582_euclidean__size__eq__0__iff,axiom,
! [B: int] :
( ( ( euclid4774559944035922753ze_int @ B )
= zero_zero_nat )
= ( B = zero_zero_int ) ) ).
% euclidean_size_eq_0_iff
thf(fact_583_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_584_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_585_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_586_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_587_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_588_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_589_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_590_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_591_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_592_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_593_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_594_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_595_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_596_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_597_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_598_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_599_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_600_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_601_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_602_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_603_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_604_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_605_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_606_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_607_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_608_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_609_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_610_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_611_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_612_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_613_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_614_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_615_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W3: int,Z3: int] :
? [N2: nat] :
( Z3
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_616_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_617_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_618_division__segment__nat__def,axiom,
( euclid3398187327856392827nt_nat
= ( ^ [N2: nat] : one_one_nat ) ) ).
% division_segment_nat_def
thf(fact_619_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_620_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_621_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_622_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_623_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_624_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_625_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_626_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_627_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_628_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_629_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_630_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_631_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_632_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_633_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_634_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_635_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_636_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_637_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_638_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_639_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_640_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_641_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_642_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_643_euclidean__size__mult,axiom,
! [A: nat,B: nat] :
( ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A @ B ) )
= ( times_times_nat @ ( euclid4777050414544973029ze_nat @ A ) @ ( euclid4777050414544973029ze_nat @ B ) ) ) ).
% euclidean_size_mult
thf(fact_644_euclidean__size__mult,axiom,
! [A: int,B: int] :
( ( euclid4774559944035922753ze_int @ ( times_times_int @ A @ B ) )
= ( times_times_nat @ ( euclid4774559944035922753ze_int @ A ) @ ( euclid4774559944035922753ze_int @ B ) ) ) ).
% euclidean_size_mult
thf(fact_645_division__segment__not__0,axiom,
! [A: int] :
( ( euclid3395696857347342551nt_int @ A )
!= zero_zero_int ) ).
% division_segment_not_0
thf(fact_646_division__segment__not__0,axiom,
! [A: nat] :
( ( euclid3398187327856392827nt_nat @ A )
!= zero_zero_nat ) ).
% division_segment_not_0
thf(fact_647_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_648_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_649_lambda__one,axiom,
( ( ^ [X2: nat] : X2 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_650_lambda__one,axiom,
( ( ^ [X2: int] : X2 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_651_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_652_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_653_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_654_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_655_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_656_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_657_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_658_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_659_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_660_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_661_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_662_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_663_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_664_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_665_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_666_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_667_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_668_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_669_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_670_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_671_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_672_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_673_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_674_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_675_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_676_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_677_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_678_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_679_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_680_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_681_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_682_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_683_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_684_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_685_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_686_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_687_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_688_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_689_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_690_division__segment__mult,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( euclid3395696857347342551nt_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( euclid3395696857347342551nt_int @ A ) @ ( euclid3395696857347342551nt_int @ B ) ) ) ) ) ).
% division_segment_mult
thf(fact_691_division__segment__mult,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( euclid3398187327856392827nt_nat @ ( times_times_nat @ A @ B ) )
= ( times_times_nat @ ( euclid3398187327856392827nt_nat @ A ) @ ( euclid3398187327856392827nt_nat @ B ) ) ) ) ) ).
% division_segment_mult
thf(fact_692_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_693_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_694_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_695_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_696_divides__aux__eq,axiom,
! [Q3: nat,R: nat] :
( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q3 @ R ) )
= ( R = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_697_divides__aux__eq,axiom,
! [Q3: int,R: int] :
( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q3 @ R ) )
= ( R = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_698_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_699_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N2: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I2: nat] : ( plus_plus_nat @ I2 @ one_one_nat )
@ N2
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_700_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] :
( semiri8420488043553186161ux_int
@ ^ [I2: int] : ( plus_plus_int @ I2 @ one_one_int )
@ N2
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_701_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_702_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_703_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc4471711990508489141at_nat] :
~ ! [F2: nat > nat > nat,A4: nat,B4: nat,Acc: nat] :
( X
!= ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A4 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_704_successively_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P3: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ nil_nat ) )
=> ( ! [P3: nat > nat > $o,X3: nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X3 @ nil_nat ) ) )
=> ~ ! [P3: nat > nat > $o,X3: nat,Y4: nat,Xs2: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X3 @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_705_sorted__wrt_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P3: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ nil_nat ) )
=> ~ ! [P3: nat > nat > $o,X3: nat,Ys4: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X3 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_706_splice_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
=> ~ ! [X3: nat,Xs2: list_nat,Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_707_shuffles_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
=> ( ! [Xs2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
=> ~ ! [X3: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_708_longest__common__prefix_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [X3: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) )
=> ( ! [Uv: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Uv ) )
=> ~ ! [Uu: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Uu @ nil_nat ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_709_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_710_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_711_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_712_add__scale__eq__noteq,axiom,
! [R: int,A: int,B: int,C: int,D: int] :
( ( R != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_713_length__product,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( size_s1588839187904960729at_nat @ ( produc8019046267696426629at_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ ( size_s8736152011456118867at_nat @ Ys ) ) ) ).
% length_product
thf(fact_714_length__product,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_b] :
( ( size_s8683744545080183927_nat_b @ ( produc1711209128353775907_nat_b @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ ( size_size_list_b @ Ys ) ) ) ).
% length_product
thf(fact_715_length__product,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_nat] :
( ( size_s732419757747899012at_nat @ ( produc3722856675314693292at_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_716_length__product,axiom,
! [Xs: list_b,Ys: list_s1210847774152347623at_nat] :
( ( size_s2552963303476550075at_nat @ ( produc7578515235956355943at_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_b @ Xs ) @ ( size_s8736152011456118867at_nat @ Ys ) ) ) ).
% length_product
thf(fact_717_length__product,axiom,
! [Xs: list_b,Ys: list_b] :
( ( size_s3321063670277389913od_b_b @ ( product_b_b @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_b @ Xs ) @ ( size_size_list_b @ Ys ) ) ) ).
% length_product
thf(fact_718_length__product,axiom,
! [Xs: list_b,Ys: list_nat] :
( ( size_s6581216242392499234_b_nat @ ( product_b_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_b @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_719_length__product,axiom,
! [Xs: list_nat,Ys: list_s1210847774152347623at_nat] :
( ( size_s8291778405851463082at_nat @ ( produc6036488282562988682at_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s8736152011456118867at_nat @ Ys ) ) ) ).
% length_product
thf(fact_720_length__product,axiom,
! [Xs: list_nat,Ys: list_b] :
( ( size_s314938103728740808_nat_b @ ( product_nat_b @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_b @ Ys ) ) ) ).
% length_product
thf(fact_721_length__product,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_722_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_723_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_724_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_725_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_726_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_727_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_728_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A5: nat,B5: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_729_prod_Oinject,axiom,
! [X1: nat,X23: nat,Y1: nat,Y23: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X23 )
= ( product_Pair_nat_nat @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_730_Pair__inject,axiom,
! [A: nat,B: nat,A5: nat,B5: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_731_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
( ! [A4: nat,B4: nat] : ( P @ ( product_Pair_nat_nat @ A4 @ B4 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_732_surj__pair,axiom,
! [P4: product_prod_nat_nat] :
? [X3: nat,Y4: nat] :
( P4
= ( product_Pair_nat_nat @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_733_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A4: nat,B4: nat] :
( Y
!= ( product_Pair_nat_nat @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_734_mult__less__iff1,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_735_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_736_Succ__def,axiom,
( bNF_Gr6352880689984616693cc_nat
= ( ^ [Kl: set_list_nat,Kl2: list_nat] :
( collect_nat
@ ^ [K3: nat] : ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K3 @ nil_nat ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_737_SuccD,axiom,
! [K: nat,Kl3: set_list_nat,Kl4: list_nat] :
( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl3 @ Kl4 ) )
=> ( member_list_nat @ ( append_nat @ Kl4 @ ( cons_nat @ K @ nil_nat ) ) @ Kl3 ) ) ).
% SuccD
thf(fact_738_SuccI,axiom,
! [Kl4: list_nat,K: nat,Kl3: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl4 @ ( cons_nat @ K @ nil_nat ) ) @ Kl3 )
=> ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl3 @ Kl4 ) ) ) ).
% SuccI
thf(fact_739_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_740_empty__Shift,axiom,
! [Kl3: set_list_nat,K: nat] :
( ( member_list_nat @ nil_nat @ Kl3 )
=> ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl3 @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl3 @ K ) ) ) ) ).
% empty_Shift
thf(fact_741_Succ__Shift,axiom,
! [Kl3: set_list_nat,K: nat,Kl4: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl3 @ K ) @ Kl4 )
= ( bNF_Gr6352880689984616693cc_nat @ Kl3 @ ( cons_nat @ K @ Kl4 ) ) ) ).
% Succ_Shift
thf(fact_742_ShiftD,axiom,
! [Kl4: list_nat,Kl3: set_list_nat,K: nat] :
( ( member_list_nat @ Kl4 @ ( bNF_Gr1872714664788909425ft_nat @ Kl3 @ K ) )
=> ( member_list_nat @ ( cons_nat @ K @ Kl4 ) @ Kl3 ) ) ).
% ShiftD
thf(fact_743_Shift__def,axiom,
( bNF_Gr1872714664788909425ft_nat
= ( ^ [Kl: set_list_nat,K3: nat] :
( collect_list_nat
@ ^ [Kl2: list_nat] : ( member_list_nat @ ( cons_nat @ K3 @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_744_Cons__lenlex__iff,axiom,
! [M: set_Pr1261947904930325089at_nat,Ms: list_s1210847774152347623at_nat,N: set_Pr1261947904930325089at_nat,Ns: list_s1210847774152347623at_nat,R: set_Pr4329608150637261639at_nat] :
( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ M @ Ms ) @ ( cons_s6881495754146722583at_nat @ N @ Ns ) ) @ ( lenlex1357538814655152620at_nat @ R ) )
= ( ( ord_less_nat @ ( size_s8736152011456118867at_nat @ Ms ) @ ( size_s8736152011456118867at_nat @ Ns ) )
| ( ( ( size_s8736152011456118867at_nat @ Ms )
= ( size_s8736152011456118867at_nat @ Ns ) )
& ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ M @ N ) @ R ) )
| ( ( M = N )
& ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Ms @ Ns ) @ ( lenlex1357538814655152620at_nat @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_745_Cons__lenlex__iff,axiom,
! [M: b,Ms: list_b,N: b,Ns: list_b,R: set_Product_prod_b_b] :
( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( cons_b @ M @ Ms ) @ ( cons_b @ N @ Ns ) ) @ ( lenlex_b @ R ) )
= ( ( ord_less_nat @ ( size_size_list_b @ Ms ) @ ( size_size_list_b @ Ns ) )
| ( ( ( size_size_list_b @ Ms )
= ( size_size_list_b @ Ns ) )
& ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ M @ N ) @ R ) )
| ( ( M = N )
& ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Ms @ Ns ) @ ( lenlex_b @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_746_Cons__lenlex__iff,axiom,
! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
| ( ( ( size_size_list_nat @ Ms )
= ( size_size_list_nat @ Ns ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ R ) )
| ( ( M = N )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_747_Cons__in__lex,axiom,
! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R: set_Pr4329608150637261639at_nat] :
( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ X @ Xs ) @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) ) @ ( lex_se2245640040323279819at_nat @ R ) )
= ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
& ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_s8736152011456118867at_nat @ Ys ) ) )
| ( ( X = Y )
& ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs @ Ys ) @ ( lex_se2245640040323279819at_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_748_Cons__in__lex,axiom,
! [X: b,Xs: list_b,Y: b,Ys: list_b,R: set_Product_prod_b_b] :
( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( cons_b @ X @ Xs ) @ ( cons_b @ Y @ Ys ) ) @ ( lex_b @ R ) )
= ( ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ R )
& ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) ) )
| ( ( X = Y )
& ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( lex_b @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_749_Cons__in__lex,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_750_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_751_listrel1__iff__update,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,R: set_Pr4329608150637261639at_nat] :
( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs @ Ys ) @ ( listre8180072825634266816at_nat @ R ) )
= ( ? [Y3: set_Pr1261947904930325089at_nat,N2: nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ N2 ) @ Y3 ) @ R )
& ( ord_less_nat @ N2 @ ( size_s8736152011456118867at_nat @ Xs ) )
& ( Ys
= ( list_u8444657558853818831at_nat @ Xs @ N2 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_752_listrel1__iff__update,axiom,
! [Xs: list_b,Ys: list_b,R: set_Product_prod_b_b] :
( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( listrel1_b @ R ) )
= ( ? [Y3: b,N2: nat] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Xs @ N2 ) @ Y3 ) @ R )
& ( ord_less_nat @ N2 @ ( size_size_list_b @ Xs ) )
& ( Ys
= ( list_update_b @ Xs @ N2 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_753_listrel1__iff__update,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
= ( ? [Y3: nat,N2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y3 ) @ R )
& ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
& ( Ys
= ( list_update_nat @ Xs @ N2 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_754_Nil__lenlex__iff1,axiom,
! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R ) )
= ( Ns != nil_nat ) ) ).
% Nil_lenlex_iff1
thf(fact_755_Cons__listrel1__Cons,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_756_listrel1I2,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,X: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ X @ Ys ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I2
thf(fact_757_not__Nil__listrel1,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R ) ) ).
% not_Nil_listrel1
thf(fact_758_not__listrel1__Nil,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R ) ) ).
% not_listrel1_Nil
thf(fact_759_listrel1__eq__len,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,R: set_Pr4329608150637261639at_nat] :
( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs @ Ys ) @ ( listre8180072825634266816at_nat @ R ) )
=> ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_s8736152011456118867at_nat @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_760_listrel1__eq__len,axiom,
! [Xs: list_b,Ys: list_b,R: set_Product_prod_b_b] :
( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( listrel1_b @ R ) )
=> ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_761_listrel1__eq__len,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_762_Nil__notin__lex,axiom,
! [Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) @ ( lex_nat @ R ) ) ).
% Nil_notin_lex
thf(fact_763_Nil2__notin__lex,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R ) ) ).
% Nil2_notin_lex
thf(fact_764_lenlex__irreflexive,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_765_Nil__lenlex__iff2,axiom,
! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_766_listrel1I1,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I1
thf(fact_767_Cons__listrel1E1,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ Ys ) @ ( listrel1_nat @ R ) )
=> ( ! [Y4: nat] :
( ( Ys
= ( cons_nat @ Y4 @ Xs ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R ) )
=> ~ ! [Zs3: list_nat] :
( ( Ys
= ( cons_nat @ X @ Zs3 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs3 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_768_Cons__listrel1E2,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R ) )
=> ( ! [X3: nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R ) )
=> ~ ! [Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Zs3 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs3 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_769_lex__append__left__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_770_lex__append__leftD,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_771_lex__append__rightI,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,R: set_Pr4329608150637261639at_nat,Vs: list_s1210847774152347623at_nat,Us: list_s1210847774152347623at_nat] :
( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs @ Ys ) @ ( lex_se2245640040323279819at_nat @ R ) )
=> ( ( ( size_s8736152011456118867at_nat @ Vs )
= ( size_s8736152011456118867at_nat @ Us ) )
=> ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( append4192317425040545660at_nat @ Xs @ Us ) @ ( append4192317425040545660at_nat @ Ys @ Vs ) ) @ ( lex_se2245640040323279819at_nat @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_772_lex__append__rightI,axiom,
! [Xs: list_b,Ys: list_b,R: set_Product_prod_b_b,Vs: list_b,Us: list_b] :
( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( lex_b @ R ) )
=> ( ( ( size_size_list_b @ Vs )
= ( size_size_list_b @ Us ) )
=> ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( append_b @ Xs @ Us ) @ ( append_b @ Ys @ Vs ) ) @ ( lex_b @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_773_lex__append__rightI,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,Vs: list_nat,Us: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Us ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_774_listrel1I,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Us: list_nat,Vs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( ( Xs
= ( append_nat @ Us @ ( cons_nat @ X @ Vs ) ) )
=> ( ( Ys
= ( append_nat @ Us @ ( cons_nat @ Y @ Vs ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% listrel1I
thf(fact_775_listrel1E,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
=> ~ ! [X3: nat,Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R )
=> ! [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ ( cons_nat @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_nat @ Us2 @ ( cons_nat @ Y4 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_776_lenlex__append1,axiom,
! [Us: list_s1210847774152347623at_nat,Xs: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat,Vs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Us @ Xs ) @ ( lenlex1357538814655152620at_nat @ R2 ) )
=> ( ( ( size_s8736152011456118867at_nat @ Vs )
= ( size_s8736152011456118867at_nat @ Ys ) )
=> ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( append4192317425040545660at_nat @ Us @ Vs ) @ ( append4192317425040545660at_nat @ Xs @ Ys ) ) @ ( lenlex1357538814655152620at_nat @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_777_lenlex__append1,axiom,
! [Us: list_b,Xs: list_b,R2: set_Product_prod_b_b,Vs: list_b,Ys: list_b] :
( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Us @ Xs ) @ ( lenlex_b @ R2 ) )
=> ( ( ( size_size_list_b @ Vs )
= ( size_size_list_b @ Ys ) )
=> ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( append_b @ Us @ Vs ) @ ( append_b @ Xs @ Ys ) ) @ ( lenlex_b @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_778_lenlex__append1,axiom,
! [Us: list_nat,Xs: list_nat,R2: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Xs ) @ ( lenlex_nat @ R2 ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Ys ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us @ Vs ) @ ( append_nat @ Xs @ Ys ) ) @ ( lenlex_nat @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_779_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_780_stirling_Ocases,axiom,
! [X: product_prod_nat_nat] :
( ( X
!= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) )
=> ( ! [K2: nat] :
( X
!= ( product_Pair_nat_nat @ zero_zero_nat @ ( suc @ K2 ) ) )
=> ( ! [N3: nat] :
( X
!= ( product_Pair_nat_nat @ ( suc @ N3 ) @ zero_zero_nat ) )
=> ~ ! [N3: nat,K2: nat] :
( X
!= ( product_Pair_nat_nat @ ( suc @ N3 ) @ ( suc @ K2 ) ) ) ) ) ) ).
% stirling.cases
thf(fact_781_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( take_s4815058814712505167at_nat @ ( suc @ I ) @ Xs )
= ( append4192317425040545660at_nat @ ( take_s4815058814712505167at_nat @ I @ Xs ) @ ( cons_s6881495754146722583at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ I ) @ nil_se357566008730718055at_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_782_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_b] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( ( take_b @ ( suc @ I ) @ Xs )
= ( append_b @ ( take_b @ I @ Xs ) @ ( cons_b @ ( nth_b @ Xs @ I ) @ nil_b ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_783_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_784_take__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_785_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs3: list_nat] : nil_nat ) ) ).
% take0
thf(fact_786_take__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= nil_nat )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil
thf(fact_787_take__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( take_nat @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil2
thf(fact_788_nth__take,axiom,
! [I: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ I ) ) ) ).
% nth_take
thf(fact_789_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_790_take__0,axiom,
! [Xs: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs )
= nil_nat ) ).
% take_0
thf(fact_791_lex__take__index,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,R: set_Pr4329608150637261639at_nat] :
( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs @ Ys ) @ ( lex_se2245640040323279819at_nat @ R ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_s8736152011456118867at_nat @ Ys ) )
=> ( ( ( take_s4815058814712505167at_nat @ I3 @ Xs )
= ( take_s4815058814712505167at_nat @ I3 @ Ys ) )
=> ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ I3 ) @ ( nth_se5679702624988424552at_nat @ Ys @ I3 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_792_lex__take__index,axiom,
! [Xs: list_b,Ys: list_b,R: set_Product_prod_b_b] :
( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( lex_b @ R ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
=> ( ( ( take_b @ I3 @ Xs )
= ( take_b @ I3 @ Ys ) )
=> ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Xs @ I3 ) @ ( nth_b @ Ys @ I3 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_793_lex__take__index,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( ( ( take_nat @ I3 @ Xs )
= ( take_nat @ I3 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_794_stirling__row__aux__correct,axiom,
( stirling_row_aux_int
= ( ^ [N2: int,Y3: int,Xs3: list_int] :
( append_int
@ ( zip_wi8271306424544069320nt_int
@ ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( times_times_int @ N2 @ B3 ) )
@ Y3
@ Xs3 )
@ ( cons_int @ one_one_int @ nil_int ) ) ) ) ).
% stirling_row_aux_correct
thf(fact_795_stirling__row__aux__correct,axiom,
( stirling_row_aux_nat
= ( ^ [N2: nat,Y3: nat,Xs3: list_nat] :
( append_nat
@ ( zip_wi7274443183152065040at_nat
@ ^ [A3: nat,B3: nat] : ( plus_plus_nat @ A3 @ ( times_times_nat @ N2 @ B3 ) )
@ Y3
@ Xs3 )
@ ( cons_nat @ one_one_nat @ nil_nat ) ) ) ) ).
% stirling_row_aux_correct
thf(fact_796_stirling__row__aux_Osimps_I1_J,axiom,
! [N: int,Y: int] :
( ( stirling_row_aux_int @ N @ Y @ nil_int )
= ( cons_int @ one_one_int @ nil_int ) ) ).
% stirling_row_aux.simps(1)
thf(fact_797_stirling__row__aux_Osimps_I1_J,axiom,
! [N: nat,Y: nat] :
( ( stirling_row_aux_nat @ N @ Y @ nil_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_aux.simps(1)
thf(fact_798_stirling__row__aux_Osimps_I2_J,axiom,
! [N: int,Y: int,X: int,Xs: list_int] :
( ( stirling_row_aux_int @ N @ Y @ ( cons_int @ X @ Xs ) )
= ( cons_int @ ( plus_plus_int @ Y @ ( times_times_int @ N @ X ) ) @ ( stirling_row_aux_int @ N @ X @ Xs ) ) ) ).
% stirling_row_aux.simps(2)
thf(fact_799_stirling__row__aux_Osimps_I2_J,axiom,
! [N: nat,Y: nat,X: nat,Xs: list_nat] :
( ( stirling_row_aux_nat @ N @ Y @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ ( plus_plus_nat @ Y @ ( times_times_nat @ N @ X ) ) @ ( stirling_row_aux_nat @ N @ X @ Xs ) ) ) ).
% stirling_row_aux.simps(2)
thf(fact_800_Stirling_Oelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( ( stirling @ X @ Xa )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( ( Xa = zero_zero_nat )
=> ( Y != one_one_nat ) ) )
=> ( ( ( X = zero_zero_nat )
=> ( ? [K2: nat] :
( Xa
= ( suc @ K2 ) )
=> ( Y != zero_zero_nat ) ) )
=> ( ( ? [N3: nat] :
( X
= ( suc @ N3 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y != zero_zero_nat ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ! [K2: nat] :
( ( Xa
= ( suc @ K2 ) )
=> ( Y
!= ( plus_plus_nat @ ( times_times_nat @ ( suc @ K2 ) @ ( stirling @ N3 @ ( suc @ K2 ) ) ) @ ( stirling @ N3 @ K2 ) ) ) ) ) ) ) ) ) ).
% Stirling.elims
thf(fact_801_Stirling__same,axiom,
! [N: nat] :
( ( stirling @ N @ N )
= one_one_nat ) ).
% Stirling_same
thf(fact_802_Stirling__less,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( stirling @ N @ K )
= zero_zero_nat ) ) ).
% Stirling_less
thf(fact_803_Stirling__1,axiom,
! [N: nat] :
( ( stirling @ ( suc @ N ) @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% Stirling_1
thf(fact_804_Stirling_Osimps_I2_J,axiom,
! [K: nat] :
( ( stirling @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% Stirling.simps(2)
thf(fact_805_Stirling_Osimps_I3_J,axiom,
! [N: nat] :
( ( stirling @ ( suc @ N ) @ zero_zero_nat )
= zero_zero_nat ) ).
% Stirling.simps(3)
thf(fact_806_Stirling_Osimps_I1_J,axiom,
( ( stirling @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% Stirling.simps(1)
thf(fact_807_Stirling_Osimps_I4_J,axiom,
! [N: nat,K: nat] :
( ( stirling @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( suc @ K ) @ ( stirling @ N @ ( suc @ K ) ) ) @ ( stirling @ N @ K ) ) ) ).
% Stirling.simps(4)
thf(fact_808_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_s1210847774152347623at_nat,A: set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( list_u8444657558853818831at_nat @ Xs @ I @ A )
= ( append4192317425040545660at_nat @ ( take_s4815058814712505167at_nat @ I @ Xs ) @ ( cons_s6881495754146722583at_nat @ A @ ( drop_s5749455908503499943at_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_809_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_b,A: b] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( ( list_update_b @ Xs @ I @ A )
= ( append_b @ ( take_b @ I @ Xs ) @ ( cons_b @ A @ ( drop_b @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_810_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_nat,A: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I @ A )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_811_id__take__nth__drop,axiom,
! [I: nat,Xs: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( Xs
= ( append4192317425040545660at_nat @ ( take_s4815058814712505167at_nat @ I @ Xs ) @ ( cons_s6881495754146722583at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ I ) @ ( drop_s5749455908503499943at_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_812_id__take__nth__drop,axiom,
! [I: nat,Xs: list_b] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( Xs
= ( append_b @ ( take_b @ I @ Xs ) @ ( cons_b @ ( nth_b @ Xs @ I ) @ ( drop_b @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_813_id__take__nth__drop,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_814_stirling_Oelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( ( stirling2 @ X @ Xa )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( ( Xa = zero_zero_nat )
=> ( Y != one_one_nat ) ) )
=> ( ( ( X = zero_zero_nat )
=> ( ? [K2: nat] :
( Xa
= ( suc @ K2 ) )
=> ( Y != zero_zero_nat ) ) )
=> ( ( ? [N3: nat] :
( X
= ( suc @ N3 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y != zero_zero_nat ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ! [K2: nat] :
( ( Xa
= ( suc @ K2 ) )
=> ( Y
!= ( plus_plus_nat @ ( times_times_nat @ N3 @ ( stirling2 @ N3 @ ( suc @ K2 ) ) ) @ ( stirling2 @ N3 @ K2 ) ) ) ) ) ) ) ) ) ).
% stirling.elims
thf(fact_815_stirling__same,axiom,
! [N: nat] :
( ( stirling2 @ N @ N )
= one_one_nat ) ).
% stirling_same
thf(fact_816_drop__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_817_stirling__less,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( stirling2 @ N @ K )
= zero_zero_nat ) ) ).
% stirling_less
thf(fact_818_stirling__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( stirling2 @ N @ zero_zero_nat )
= zero_zero_nat ) ) ).
% stirling_0
thf(fact_819_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_820_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ( ( nth_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_821_stirling_Osimps_I3_J,axiom,
! [N: nat] :
( ( stirling2 @ ( suc @ N ) @ zero_zero_nat )
= zero_zero_nat ) ).
% stirling.simps(3)
thf(fact_822_stirling_Osimps_I2_J,axiom,
! [K: nat] :
( ( stirling2 @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% stirling.simps(2)
thf(fact_823_stirling_Osimps_I1_J,axiom,
( ( stirling2 @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% stirling.simps(1)
thf(fact_824_append__eq__conv__conj,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,Zs: list_s1210847774152347623at_nat] :
( ( ( append4192317425040545660at_nat @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_s4815058814712505167at_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ Zs ) )
& ( Ys
= ( drop_s5749455908503499943at_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_825_append__eq__conv__conj,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b] :
( ( ( append_b @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_b @ ( size_size_list_b @ Xs ) @ Zs ) )
& ( Ys
= ( drop_b @ ( size_size_list_b @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_826_append__eq__conv__conj,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) )
& ( Ys
= ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_827_stirling_Osimps_I4_J,axiom,
! [N: nat,K: nat] :
( ( stirling2 @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( stirling2 @ N @ ( suc @ K ) ) ) @ ( stirling2 @ N @ K ) ) ) ).
% stirling.simps(4)
thf(fact_828_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( cons_s6881495754146722583at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ I ) @ ( drop_s5749455908503499943at_nat @ ( suc @ I ) @ Xs ) )
= ( drop_s5749455908503499943at_nat @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_829_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_b] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( ( cons_b @ ( nth_b @ Xs @ I ) @ ( drop_b @ ( suc @ I ) @ Xs ) )
= ( drop_b @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_830_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) )
= ( drop_nat @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_831_stirling__code,axiom,
( stirling2
= ( ^ [N2: nat,K3: nat] : ( if_nat @ ( K3 = zero_zero_nat ) @ ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat @ zero_zero_nat ) @ ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( K3 = N2 ) @ one_one_nat @ ( nth_nat @ ( stirling_row @ N2 ) @ K3 ) ) ) ) ) ) ).
% stirling_code
thf(fact_832_take__hd__drop,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ N @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( append4192317425040545660at_nat @ ( take_s4815058814712505167at_nat @ N @ Xs ) @ ( cons_s6881495754146722583at_nat @ ( hd_set5761236171258712524at_nat @ ( drop_s5749455908503499943at_nat @ N @ Xs ) ) @ nil_se357566008730718055at_nat ) )
= ( take_s4815058814712505167at_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_833_take__hd__drop,axiom,
! [N: nat,Xs: list_b] :
( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ( append_b @ ( take_b @ N @ Xs ) @ ( cons_b @ ( hd_b @ ( drop_b @ N @ Xs ) ) @ nil_b ) )
= ( take_b @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_834_take__hd__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( append_nat @ ( take_nat @ N @ Xs ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs ) ) @ nil_nat ) )
= ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_835_nth__zip,axiom,
! [I: nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Ys ) )
=> ( ( nth_Pr1239524528086697198at_nat @ ( zip_se5600341670672612855at_nat @ Xs @ Ys ) @ I )
= ( produc2922128104949294807at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ I ) @ ( nth_se5679702624988424552at_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_836_nth__zip,axiom,
! [I: nat,Xs: list_s1210847774152347623at_nat,Ys: list_b] :
( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_b @ Ys ) )
=> ( ( nth_Pr1155505181600352140_nat_b @ ( zip_se1037981971522095509_nat_b @ Xs @ Ys ) @ I )
= ( produc5313103311483691637_nat_b @ ( nth_se5679702624988424552at_nat @ Xs @ I ) @ ( nth_b @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_837_nth__zip,axiom,
! [I: nat,Xs: list_s1210847774152347623at_nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr5685494590260343449at_nat @ ( zip_se6935005982215590842at_nat @ Xs @ Ys ) @ I )
= ( produc3149970401713881818at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_838_nth__zip,axiom,
! [I: nat,Xs: list_b,Ys: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Ys ) )
=> ( ( nth_Pr8450474352736679632at_nat @ ( zip_b_6905288079124675545at_nat @ Xs @ Ys ) @ I )
= ( produc1957037382231495865at_nat @ ( nth_b @ Xs @ I ) @ ( nth_se5679702624988424552at_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_839_nth__zip,axiom,
! [I: nat,Xs: list_b,Ys: list_b] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_b @ Ys ) )
=> ( ( nth_Product_prod_b_b @ ( zip_b_b @ Xs @ Ys ) @ I )
= ( product_Pair_b_b @ ( nth_b @ Xs @ I ) @ ( nth_b @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_840_nth__zip,axiom,
! [I: nat,Xs: list_b,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr473537946622032695_b_nat @ ( zip_b_nat @ Xs @ Ys ) @ I )
= ( product_Pair_b_nat @ ( nth_b @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_841_nth__zip,axiom,
! [I: nat,Xs: list_nat,Ys: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s8736152011456118867at_nat @ Ys ) )
=> ( ( nth_Pr8633021227891005631at_nat @ ( zip_na25265552609110424at_nat @ Xs @ Ys ) @ I )
= ( produc5463602008962177208at_nat @ ( nth_nat @ Xs @ I ) @ ( nth_se5679702624988424552at_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_842_nth__zip,axiom,
! [I: nat,Xs: list_nat,Ys: list_b] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_b @ Ys ) )
=> ( ( nth_Pr2476257085692543965_nat_b @ ( zip_nat_b @ Xs @ Ys ) @ I )
= ( product_Pair_nat_b @ ( nth_nat @ Xs @ I ) @ ( nth_b @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_843_nth__zip,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_844_hd__prefixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( prefixes_nat @ Xs ) )
= nil_nat ) ).
% hd_prefixes
thf(fact_845_hd__suffixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( suffixes_nat @ Xs ) )
= nil_nat ) ).
% hd_suffixes
thf(fact_846_length__stirling__row,axiom,
! [N: nat] :
( ( size_size_list_nat @ ( stirling_row @ N ) )
= ( suc @ N ) ) ).
% length_stirling_row
thf(fact_847_hd__append2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_append2
thf(fact_848_zip__eq__Nil__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( zip_nat_nat @ Xs @ Ys )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs = nil_nat )
| ( Ys = nil_nat ) ) ) ).
% zip_eq_Nil_iff
thf(fact_849_Nil__eq__zip__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( zip_nat_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
| ( Ys = nil_nat ) ) ) ).
% Nil_eq_zip_iff
thf(fact_850_zip__Cons__Cons,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( zip_nat_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_851_zip__append,axiom,
! [Xs: list_s1210847774152347623at_nat,Us: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,Vs: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_s8736152011456118867at_nat @ Us ) )
=> ( ( zip_se5600341670672612855at_nat @ ( append4192317425040545660at_nat @ Xs @ Ys ) @ ( append4192317425040545660at_nat @ Us @ Vs ) )
= ( append5999596214939841794at_nat @ ( zip_se5600341670672612855at_nat @ Xs @ Us ) @ ( zip_se5600341670672612855at_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_852_zip__append,axiom,
! [Xs: list_s1210847774152347623at_nat,Us: list_b,Ys: list_s1210847774152347623at_nat,Vs: list_b] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_size_list_b @ Us ) )
=> ( ( zip_se1037981971522095509_nat_b @ ( append4192317425040545660at_nat @ Xs @ Ys ) @ ( append_b @ Us @ Vs ) )
= ( append5461451288015094176_nat_b @ ( zip_se1037981971522095509_nat_b @ Xs @ Us ) @ ( zip_se1037981971522095509_nat_b @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_853_zip__append,axiom,
! [Xs: list_s1210847774152347623at_nat,Us: list_nat,Ys: list_s1210847774152347623at_nat,Vs: list_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_se6935005982215590842at_nat @ ( append4192317425040545660at_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append8980083764895095685at_nat @ ( zip_se6935005982215590842at_nat @ Xs @ Us ) @ ( zip_se6935005982215590842at_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_854_zip__append,axiom,
! [Xs: list_b,Us: list_s1210847774152347623at_nat,Ys: list_b,Vs: list_s1210847774152347623at_nat] :
( ( ( size_size_list_b @ Xs )
= ( size_s8736152011456118867at_nat @ Us ) )
=> ( ( zip_b_6905288079124675545at_nat @ ( append_b @ Xs @ Ys ) @ ( append4192317425040545660at_nat @ Us @ Vs ) )
= ( append3533048422296645860at_nat @ ( zip_b_6905288079124675545at_nat @ Xs @ Us ) @ ( zip_b_6905288079124675545at_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_855_zip__append,axiom,
! [Xs: list_b,Us: list_b,Ys: list_b,Vs: list_b] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Us ) )
=> ( ( zip_b_b @ ( append_b @ Xs @ Ys ) @ ( append_b @ Us @ Vs ) )
= ( append2547753241377386114od_b_b @ ( zip_b_b @ Xs @ Us ) @ ( zip_b_b @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_856_zip__append,axiom,
! [Xs: list_b,Us: list_nat,Ys: list_b,Vs: list_nat] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_b_nat @ ( append_b @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append8914683908514519587_b_nat @ ( zip_b_nat @ Xs @ Us ) @ ( zip_b_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_857_zip__append,axiom,
! [Xs: list_nat,Us: list_s1210847774152347623at_nat,Ys: list_nat,Vs: list_s1210847774152347623at_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s8736152011456118867at_nat @ Us ) )
=> ( ( zip_na25265552609110424at_nat @ ( append_nat @ Xs @ Ys ) @ ( append4192317425040545660at_nat @ Us @ Vs ) )
= ( append2704238365670982059at_nat @ ( zip_na25265552609110424at_nat @ Xs @ Us ) @ ( zip_na25265552609110424at_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_858_zip__append,axiom,
! [Xs: list_nat,Us: list_b,Ys: list_nat,Vs: list_b] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_b @ Us ) )
=> ( ( zip_nat_b @ ( append_nat @ Xs @ Ys ) @ ( append_b @ Us @ Vs ) )
= ( append1694031010730255049_nat_b @ ( zip_nat_b @ Xs @ Us ) @ ( zip_nat_b @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_859_zip__append,axiom,
! [Xs: list_nat,Us: list_nat,Ys: list_nat,Vs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_860_stirling__row__nonempty,axiom,
! [N: nat] :
( ( stirling_row @ N )
!= nil_nat ) ).
% stirling_row_nonempty
thf(fact_861_hd__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_nat )
=> ( ( hd_Pro3460610213475200108at_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= ( product_Pair_nat_nat @ ( hd_nat @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_862_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_863_zip__update,axiom,
! [Xs: list_nat,I: nat,X: nat,Ys: list_nat,Y: nat] :
( ( zip_nat_nat @ ( list_update_nat @ Xs @ I @ X ) @ ( list_update_nat @ Ys @ I @ Y ) )
= ( list_u6180841689913720943at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I @ ( product_Pair_nat_nat @ X @ Y ) ) ) ).
% zip_update
thf(fact_864_longest__common__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ps: list_nat,Xs4: list_nat,Ys6: list_nat] :
( ( Xs
= ( append_nat @ Ps @ Xs4 ) )
& ( Ys
= ( append_nat @ Ps @ Ys6 ) )
& ( ( Xs4 = nil_nat )
| ( Ys6 = nil_nat )
| ( ( hd_nat @ Xs4 )
!= ( hd_nat @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_865_hd__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Ys ) ) )
& ( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Xs ) ) ) ) ).
% hd_append
thf(fact_866_zip__eq__ConsE,axiom,
! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
( ( ( zip_nat_nat @ Xs @ Ys )
= ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
=> ~ ! [X3: nat,Xs4: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs4 ) )
=> ! [Y4: nat,Ys6: list_nat] :
( ( Ys
= ( cons_nat @ Y4 @ Ys6 ) )
=> ( ( Xy
= ( product_Pair_nat_nat @ X3 @ Y4 ) )
=> ( Xys
!= ( zip_nat_nat @ Xs4 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_867_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_868_stirling__row__code_I2_J,axiom,
! [N: nat] :
( ( stirling_row @ ( suc @ N ) )
= ( stirling_row_aux_nat @ N @ zero_zero_nat @ ( stirling_row @ N ) ) ) ).
% stirling_row_code(2)
thf(fact_869_stirling__row__code_I1_J,axiom,
( ( stirling_row @ zero_zero_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_code(1)
thf(fact_870_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ N @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( hd_set5761236171258712524at_nat @ ( drop_s5749455908503499943at_nat @ N @ Xs ) )
= ( nth_se5679702624988424552at_nat @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_871_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_b] :
( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ( hd_b @ ( drop_b @ N @ Xs ) )
= ( nth_b @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_872_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( hd_nat @ ( drop_nat @ N @ Xs ) )
= ( nth_nat @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_873_zip__Cons,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat] :
( ( zip_nat_nat @ Xs @ ( cons_nat @ Y @ Ys ) )
= ( case_l2371295980111794673at_nat @ nil_Pr5478986624290739719at_nat
@ ^ [Z3: nat,Zs2: list_nat] : ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ Z3 @ Y ) @ ( zip_nat_nat @ Zs2 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_874_zip__Cons1,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( zip_nat_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( case_l2371295980111794673at_nat @ nil_Pr5478986624290739719at_nat
@ ^ [Y3: nat,Ys2: list_nat] : ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( zip_nat_nat @ Xs @ Ys2 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_875_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_876_rgf__limit_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs2 ) ) ) ).
% rgf_limit.cases
thf(fact_877_remdups__adj__singleton__iff,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ ( remdup5594536096481411741at_nat @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_se357566008730718055at_nat )
& ( Xs
= ( replic7735362847476918817at_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ ( hd_set5761236171258712524at_nat @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_878_remdups__adj__singleton__iff,axiom,
! [Xs: list_b] :
( ( ( size_size_list_b @ ( remdups_adj_b @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_b )
& ( Xs
= ( replicate_b @ ( size_size_list_b @ Xs ) @ ( hd_b @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_879_remdups__adj__singleton__iff,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ ( hd_nat @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_880_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_881_div__bounded,axiom,
! [B: int,R: int,Q3: int] :
( ( B != zero_zero_int )
=> ( ( ( euclid3395696857347342551nt_int @ R )
= ( euclid3395696857347342551nt_int @ B ) )
=> ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ R ) @ ( euclid4774559944035922753ze_int @ B ) )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ Q3 @ B ) @ R ) @ B )
= Q3 ) ) ) ) ).
% div_bounded
thf(fact_882_div__bounded,axiom,
! [B: nat,R: nat,Q3: nat] :
( ( B != zero_zero_nat )
=> ( ( ( euclid3398187327856392827nt_nat @ R )
= ( euclid3398187327856392827nt_nat @ B ) )
=> ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ R ) @ ( euclid4777050414544973029ze_nat @ B ) )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ Q3 @ B ) @ R ) @ B )
= Q3 ) ) ) ) ).
% div_bounded
thf(fact_883_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_884_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_885_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_886_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_887_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_888_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_889_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_890_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_891_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_892_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_893_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_894_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_895_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_896_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_897_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_898_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_899_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_900_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_901_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_902_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_903_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_904_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_905_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_906_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_907_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_908_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_909_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_910_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_911_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_912_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_913_remdups__adj__Nil__iff,axiom,
! [Xs: list_nat] :
( ( ( remdups_adj_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% remdups_adj_Nil_iff
thf(fact_914_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_915_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_916_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_917_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_918_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_919_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_920_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_921_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_922_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_923_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_924_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_925_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_926_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_927_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_928_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_929_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_930_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_931_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_932_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_933_length__drop,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat] :
( ( size_s8736152011456118867at_nat @ ( drop_s5749455908503499943at_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ N ) ) ).
% length_drop
thf(fact_934_length__drop,axiom,
! [N: nat,Xs: list_b] :
( ( size_size_list_b @ ( drop_b @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_b @ Xs ) @ N ) ) ).
% length_drop
thf(fact_935_length__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% length_drop
thf(fact_936_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_937_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_938_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_939_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_940_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_941_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_942_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_943_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_944_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_945_take__append,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( take_s4815058814712505167at_nat @ N @ ( append4192317425040545660at_nat @ Xs @ Ys ) )
= ( append4192317425040545660at_nat @ ( take_s4815058814712505167at_nat @ N @ Xs ) @ ( take_s4815058814712505167at_nat @ ( minus_minus_nat @ N @ ( size_s8736152011456118867at_nat @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_946_take__append,axiom,
! [N: nat,Xs: list_b,Ys: list_b] :
( ( take_b @ N @ ( append_b @ Xs @ Ys ) )
= ( append_b @ ( take_b @ N @ Xs ) @ ( take_b @ ( minus_minus_nat @ N @ ( size_size_list_b @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_947_take__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_948_drop__append,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( drop_s5749455908503499943at_nat @ N @ ( append4192317425040545660at_nat @ Xs @ Ys ) )
= ( append4192317425040545660at_nat @ ( drop_s5749455908503499943at_nat @ N @ Xs ) @ ( drop_s5749455908503499943at_nat @ ( minus_minus_nat @ N @ ( size_s8736152011456118867at_nat @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_949_drop__append,axiom,
! [N: nat,Xs: list_b,Ys: list_b] :
( ( drop_b @ N @ ( append_b @ Xs @ Ys ) )
= ( append_b @ ( drop_b @ N @ Xs ) @ ( drop_b @ ( minus_minus_nat @ N @ ( size_size_list_b @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_950_drop__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_951_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_952_remdups__adj_Osimps_I3_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( X = Y )
=> ( ( remdups_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( cons_nat @ X @ ( remdups_adj_nat @ ( cons_nat @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_953_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_nat @ nil_nat )
= nil_nat ) ).
% remdups_adj.simps(1)
thf(fact_954_div__if,axiom,
( divide_divide_nat
= ( ^ [M3: nat,N2: nat] :
( if_nat
@ ( ( ord_less_nat @ M3 @ N2 )
| ( N2 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_955_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_956_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_957_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_958_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_959_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_960_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_961_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_962_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_963_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_964_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_965_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_966_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_967_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_968_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_969_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_970_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_971_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_972_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_973_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_974_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_975_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_976_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_977_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_978_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_979_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_980_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_981_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_982_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_983_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_984_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_985_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_986_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_987_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_988_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_989_zdiv__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_990_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_991_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_992_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_993_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_994_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_995_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_996_div__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q3 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) ).
% div_mult2_eq
thf(fact_997_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_998_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_999_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1000_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_1001_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_1002_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1003_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1004_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_1005_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% div_mult2_eq'
thf(fact_1006_div__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% div_mult2_eq'
thf(fact_1007_remdups__adj_Oelims,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( remdups_adj_nat @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != nil_nat ) )
=> ( ! [X3: nat] :
( ( X
= ( cons_nat @ X3 @ nil_nat ) )
=> ( Y
!= ( cons_nat @ X3 @ nil_nat ) ) )
=> ~ ! [X3: nat,Y4: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X3 @ ( cons_nat @ Y4 @ Xs2 ) ) )
=> ~ ( ( ( X3 = Y4 )
=> ( Y
= ( remdups_adj_nat @ ( cons_nat @ X3 @ Xs2 ) ) ) )
& ( ( X3 != Y4 )
=> ( Y
= ( cons_nat @ X3 @ ( remdups_adj_nat @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_1008_remdups__adj_Osimps_I2_J,axiom,
! [X: nat] :
( ( remdups_adj_nat @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ nil_nat ) ) ).
% remdups_adj.simps(2)
thf(fact_1009_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1010_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1011_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1012_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1013_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1014_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1015_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1016_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1017_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1018_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1019_list_Odisc__eq__case_I1_J,axiom,
! [List: list_nat] :
( ( List = nil_nat )
= ( case_list_o_nat @ $true
@ ^ [Uu2: nat,Uv2: list_nat] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_1020_list_Odisc__eq__case_I2_J,axiom,
! [List: list_nat] :
( ( List != nil_nat )
= ( case_list_o_nat @ $false
@ ^ [Uu2: nat,Uv2: list_nat] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_1021_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1022_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1023_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1024_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_1025_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1026_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_1027_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_1028_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_1029_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_1030_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_1031_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1032_remdups__adj__append__two,axiom,
! [Xs: list_nat,X: nat,Y: nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ ( cons_nat @ Y @ nil_nat ) ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) @ ( if_list_nat @ ( X = Y ) @ nil_nat @ ( cons_nat @ Y @ nil_nat ) ) ) ) ).
% remdups_adj_append_two
thf(fact_1033_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1034_div__less__iff__less__mult,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1035_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1036_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1037_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1038_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1039_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1040_of__nat__euclidean__size,axiom,
! [A: int] :
( ( semiri1314217659103216013at_int @ ( euclid4774559944035922753ze_int @ A ) )
= ( divide_divide_int @ A @ ( euclid3395696857347342551nt_int @ A ) ) ) ).
% of_nat_euclidean_size
thf(fact_1041_of__nat__euclidean__size,axiom,
! [A: nat] :
( ( semiri1316708129612266289at_nat @ ( euclid4777050414544973029ze_nat @ A ) )
= ( divide_divide_nat @ A @ ( euclid3398187327856392827nt_nat @ A ) ) ) ).
% of_nat_euclidean_size
thf(fact_1042_remdups__adj__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( remdups_adj_nat @ ( cons_nat @ X @ Xs ) )
= ( case_l2340614614379431832at_nat @ ( cons_nat @ X @ nil_nat )
@ ^ [Y3: nat,Xs3: list_nat] : ( if_list_nat @ ( X = Y3 ) @ ( cons_nat @ Y3 @ Xs3 ) @ ( cons_nat @ X @ ( cons_nat @ Y3 @ Xs3 ) ) )
@ ( remdups_adj_nat @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_1043_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I2: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_1044_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1045_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1046_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1047_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1048_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1049_remdups__adj__adjacent,axiom,
! [I: nat,Xs: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_s8736152011456118867at_nat @ ( remdup5594536096481411741at_nat @ Xs ) ) )
=> ( ( nth_se5679702624988424552at_nat @ ( remdup5594536096481411741at_nat @ Xs ) @ I )
!= ( nth_se5679702624988424552at_nat @ ( remdup5594536096481411741at_nat @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_1050_remdups__adj__adjacent,axiom,
! [I: nat,Xs: list_b] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_b @ ( remdups_adj_b @ Xs ) ) )
=> ( ( nth_b @ ( remdups_adj_b @ Xs ) @ I )
!= ( nth_b @ ( remdups_adj_b @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_1051_remdups__adj__adjacent,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) )
=> ( ( nth_nat @ ( remdups_adj_nat @ Xs ) @ I )
!= ( nth_nat @ ( remdups_adj_nat @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_1052_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1053_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1054_nth__append,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( ( ord_less_nat @ N @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( nth_se5679702624988424552at_nat @ ( append4192317425040545660at_nat @ Xs @ Ys ) @ N )
= ( nth_se5679702624988424552at_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( nth_se5679702624988424552at_nat @ ( append4192317425040545660at_nat @ Xs @ Ys ) @ N )
= ( nth_se5679702624988424552at_nat @ Ys @ ( minus_minus_nat @ N @ ( size_s8736152011456118867at_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1055_nth__append,axiom,
! [N: nat,Xs: list_b,Ys: list_b] :
( ( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ( nth_b @ ( append_b @ Xs @ Ys ) @ N )
= ( nth_b @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ( nth_b @ ( append_b @ Xs @ Ys ) @ N )
= ( nth_b @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_b @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1056_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1057_remdups__adj__replicate,axiom,
! [N: nat,X: nat] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X ) )
= ( cons_nat @ X @ nil_nat ) ) ) ) ).
% remdups_adj_replicate
thf(fact_1058_remdups__adj__singleton,axiom,
! [Xs: list_s1210847774152347623at_nat,X: set_Pr1261947904930325089at_nat] :
( ( ( remdup5594536096481411741at_nat @ Xs )
= ( cons_s6881495754146722583at_nat @ X @ nil_se357566008730718055at_nat ) )
=> ( Xs
= ( replic7735362847476918817at_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_1059_remdups__adj__singleton,axiom,
! [Xs: list_b,X: b] :
( ( ( remdups_adj_b @ Xs )
= ( cons_b @ X @ nil_b ) )
=> ( Xs
= ( replicate_b @ ( size_size_list_b @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_1060_remdups__adj__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( ( remdups_adj_nat @ Xs )
= ( cons_nat @ X @ nil_nat ) )
=> ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_1061_drop__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_1062_list__update__append,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,X: set_Pr1261947904930325089at_nat] :
( ( ( ord_less_nat @ N @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( list_u8444657558853818831at_nat @ ( append4192317425040545660at_nat @ Xs @ Ys ) @ N @ X )
= ( append4192317425040545660at_nat @ ( list_u8444657558853818831at_nat @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s8736152011456118867at_nat @ Xs ) )
=> ( ( list_u8444657558853818831at_nat @ ( append4192317425040545660at_nat @ Xs @ Ys ) @ N @ X )
= ( append4192317425040545660at_nat @ Xs @ ( list_u8444657558853818831at_nat @ Ys @ ( minus_minus_nat @ N @ ( size_s8736152011456118867at_nat @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_1063_list__update__append,axiom,
! [N: nat,Xs: list_b,Ys: list_b,X: b] :
( ( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ( list_update_b @ ( append_b @ Xs @ Ys ) @ N @ X )
= ( append_b @ ( list_update_b @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ( list_update_b @ ( append_b @ Xs @ Ys ) @ N @ X )
= ( append_b @ Xs @ ( list_update_b @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_b @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_1064_list__update__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
= ( append_nat @ ( list_update_nat @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
= ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_1065_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( euclid3395696857347342551nt_int @ A )
= ( euclid3395696857347342551nt_int @ B ) )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ A ) @ ( euclid4774559944035922753ze_int @ B ) )
| ( B = zero_zero_int ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_1066_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( euclid3398187327856392827nt_nat @ A )
= ( euclid3398187327856392827nt_nat @ B ) )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ A ) @ ( euclid4777050414544973029ze_nat @ B ) )
| ( B = zero_zero_nat ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_1067_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1068_take__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_1069_Cons__replicate__eq,axiom,
! [X: nat,Xs: list_nat,N: nat,Y: nat] :
( ( ( cons_nat @ X @ Xs )
= ( replicate_nat @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_1070_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1071_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M4: nat,N3: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_1072_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1073_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1074_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1075_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1076_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_1077_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_1078_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_1079_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_1080_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_1081_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_1082_plusinfinity,axiom,
! [D: int,P5: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P5 @ X3 )
= ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [X_1: int] : ( P5 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_1083_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1084_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q4 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1085_pinf_I1_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q4: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q4 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1086_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q4 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1087_pinf_I2_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q4: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q4 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1088_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1089_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1090_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1091_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1092_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1093_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1094_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1095_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1096_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q4 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1097_minf_I1_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q4: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q4 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1098_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q4 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1099_minf_I2_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q4: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q4 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1100_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1101_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1102_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1103_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1104_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_1105_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_1106_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_1107_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_1108_inf__period_I1_J,axiom,
! [P: int > $o,D3: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
& ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1109_inf__period_I2_J,axiom,
! [P: int > $o,D3: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
| ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1110_product__nth,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ ( size_s8736152011456118867at_nat @ Ys ) ) )
=> ( ( nth_Pr1239524528086697198at_nat @ ( produc8019046267696426629at_nat @ Xs @ Ys ) @ N )
= ( produc2922128104949294807at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s8736152011456118867at_nat @ Ys ) ) ) @ ( nth_se5679702624988424552at_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_s8736152011456118867at_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1111_product__nth,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat,Ys: list_b] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ ( size_size_list_b @ Ys ) ) )
=> ( ( nth_Pr1155505181600352140_nat_b @ ( produc1711209128353775907_nat_b @ Xs @ Ys ) @ N )
= ( produc5313103311483691637_nat_b @ ( nth_se5679702624988424552at_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_b @ Ys ) ) ) @ ( nth_b @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_b @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1112_product__nth,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr5685494590260343449at_nat @ ( produc3722856675314693292at_nat @ Xs @ Ys ) @ N )
= ( produc3149970401713881818at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1113_product__nth,axiom,
! [N: nat,Xs: list_b,Ys: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_b @ Xs ) @ ( size_s8736152011456118867at_nat @ Ys ) ) )
=> ( ( nth_Pr8450474352736679632at_nat @ ( produc7578515235956355943at_nat @ Xs @ Ys ) @ N )
= ( produc1957037382231495865at_nat @ ( nth_b @ Xs @ ( divide_divide_nat @ N @ ( size_s8736152011456118867at_nat @ Ys ) ) ) @ ( nth_se5679702624988424552at_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_s8736152011456118867at_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1114_product__nth,axiom,
! [N: nat,Xs: list_b,Ys: list_b] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_b @ Xs ) @ ( size_size_list_b @ Ys ) ) )
=> ( ( nth_Product_prod_b_b @ ( product_b_b @ Xs @ Ys ) @ N )
= ( product_Pair_b_b @ ( nth_b @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_b @ Ys ) ) ) @ ( nth_b @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_b @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1115_product__nth,axiom,
! [N: nat,Xs: list_b,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_b @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr473537946622032695_b_nat @ ( product_b_nat @ Xs @ Ys ) @ N )
= ( product_Pair_b_nat @ ( nth_b @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1116_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s8736152011456118867at_nat @ Ys ) ) )
=> ( ( nth_Pr8633021227891005631at_nat @ ( produc6036488282562988682at_nat @ Xs @ Ys ) @ N )
= ( produc5463602008962177208at_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s8736152011456118867at_nat @ Ys ) ) ) @ ( nth_se5679702624988424552at_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_s8736152011456118867at_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1117_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys: list_b] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_b @ Ys ) ) )
=> ( ( nth_Pr2476257085692543965_nat_b @ ( product_nat_b @ Xs @ Ys ) @ N )
= ( product_Pair_nat_b @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_b @ Ys ) ) ) @ ( nth_b @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_b @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1118_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys ) @ N )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1119_mod__mod__trivial,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_1120_mod__mod__trivial,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_1121_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_1122_mod__self,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ A )
= zero_zero_int ) ).
% mod_self
thf(fact_1123_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_1124_mod__by__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ zero_zero_int )
= A ) ).
% mod_by_0
thf(fact_1125_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_1126_mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mod_0
thf(fact_1127_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_1128_bits__mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_1129_mod__add__self2,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self2
thf(fact_1130_mod__add__self2,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self2
thf(fact_1131_mod__add__self1,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self1
thf(fact_1132_mod__add__self1,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self1
thf(fact_1133_minus__mod__self2,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_mod_self2
thf(fact_1134_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_1135_mod__mult__self2__is__0,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_1136_mod__mult__self2__is__0,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_1137_mod__mult__self1__is__0,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_1138_mod__mult__self1__is__0,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_1139_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_1140_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_1141_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_1142_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_1143_mod__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self4
thf(fact_1144_mod__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self4
thf(fact_1145_mod__mult__self3,axiom,
! [C: nat,B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self3
thf(fact_1146_mod__mult__self3,axiom,
! [C: int,B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self3
thf(fact_1147_mod__mult__self2,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self2
thf(fact_1148_mod__mult__self2,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self2
thf(fact_1149_mod__mult__self1,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self1
thf(fact_1150_mod__mult__self1,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self1
thf(fact_1151_mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_div_trivial
thf(fact_1152_mod__div__trivial,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_div_trivial
thf(fact_1153_bits__mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% bits_mod_div_trivial
thf(fact_1154_bits__mod__div__trivial,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
= zero_zero_int ) ).
% bits_mod_div_trivial
thf(fact_1155_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_1156_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1157_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1158_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1159_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1160_mod__eqE,axiom,
! [A: int,C: int,B: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ B @ C ) )
=> ~ ! [D4: int] :
( B
!= ( plus_plus_int @ A @ ( times_times_int @ C @ D4 ) ) ) ) ).
% mod_eqE
thf(fact_1161_mod__eq__self__iff__div__eq__0,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= A )
= ( ( divide_divide_nat @ A @ B )
= zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1162_mod__eq__self__iff__div__eq__0,axiom,
! [A: int,B: int] :
( ( ( modulo_modulo_int @ A @ B )
= A )
= ( ( divide_divide_int @ A @ B )
= zero_zero_int ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1163_div__add1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_1164_div__add1__eq,axiom,
! [A: int,B: int,C: int] :
( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_1165_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_1166_mod__induct,axiom,
! [P: nat > $o,N: nat,P4: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P4 )
=> ( ( ord_less_nat @ M @ P4 )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P4 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_1167_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_1168_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
=> ( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
=> ( P @ M4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1169_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_1170_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mod
thf(fact_1171_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mod
thf(fact_1172_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_1173_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_1174_mod__add__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_1175_mod__add__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_1176_mod__add__cong,axiom,
! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A5 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B5 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_1177_mod__add__cong,axiom,
! [A: int,C: int,A5: int,B: int,B5: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_1178_mod__add__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_1179_mod__add__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_1180_mod__add__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_1181_mod__add__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_1182_mod__mult__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_1183_mod__mult__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_1184_mod__mult__cong,axiom,
! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A5 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B5 @ C ) )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_1185_mod__mult__cong,axiom,
! [A: int,C: int,A5: int,B: int,B5: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_1186_mod__mult__mult2,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_1187_mod__mult__mult2,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_1188_mult__mod__right,axiom,
! [C: nat,A: nat,B: nat] :
( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
= ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_1189_mult__mod__right,axiom,
! [C: int,A: int,B: int] :
( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
= ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_1190_mod__mult__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_1191_mod__mult__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_1192_mod__mult__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_1193_mod__mult__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_1194_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N2 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).
% mod_if
thf(fact_1195_mod__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_eq
thf(fact_1196_mod__diff__cong,axiom,
! [A: int,C: int,A5: int,B: int,B5: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_1197_mod__diff__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_left_eq
thf(fact_1198_mod__diff__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_right_eq
thf(fact_1199_div__mult1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_1200_div__mult1__eq,axiom,
! [A: int,B: int,C: int] :
( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_1201_mult__div__mod__eq,axiom,
! [B: nat,A: nat] :
( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_1202_mult__div__mod__eq,axiom,
! [B: int,A: int] :
( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_1203_mod__mult__div__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_1204_mod__mult__div__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_1205_mod__div__mult__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_1206_mod__div__mult__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_1207_div__mult__mod__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_1208_div__mult__mod__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_1209_mod__div__decomp,axiom,
! [A: nat,B: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_1210_mod__div__decomp,axiom,
! [A: int,B: int] :
( A
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_1211_cancel__div__mod__rules_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
= ( plus_plus_nat @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1212_cancel__div__mod__rules_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
= ( plus_plus_int @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1213_cancel__div__mod__rules_I2_J,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
= ( plus_plus_nat @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1214_cancel__div__mod__rules_I2_J,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
= ( plus_plus_int @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1215_minus__div__mult__eq__mod,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
= ( modulo_modulo_nat @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_1216_minus__div__mult__eq__mod,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_1217_minus__mod__eq__div__mult,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_1218_minus__mod__eq__div__mult,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
= ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_1219_minus__mod__eq__mult__div,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
= ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_1220_minus__mod__eq__mult__div,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
= ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_1221_minus__mult__div__eq__mod,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
= ( modulo_modulo_nat @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_1222_minus__mult__div__eq__mod,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_1223_mod__size__less,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ ( modulo_modulo_nat @ A @ B ) ) @ ( euclid4777050414544973029ze_nat @ B ) ) ) ).
% mod_size_less
thf(fact_1224_mod__size__less,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ord_less_nat @ ( euclid4774559944035922753ze_int @ ( modulo_modulo_int @ A @ B ) ) @ ( euclid4774559944035922753ze_int @ B ) ) ) ).
% mod_size_less
thf(fact_1225_div__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( suc @ ( divide_divide_nat @ M @ N ) ) ) )
& ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
!= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% div_Suc
thf(fact_1226_div__less__mono,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A2 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B2 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1227_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q3 ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_1228_div__mod__decomp,axiom,
! [A2: nat,N: nat] :
( A2
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% div_mod_decomp
thf(fact_1229_modulo__nat__def,axiom,
( modulo_modulo_nat
= ( ^ [M3: nat,N2: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).
% modulo_nat_def
thf(fact_1230_mod__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_1231_mod__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_1232_split__mod,axiom,
! [Q: nat > $o,M: nat,N: nat] :
( ( Q @ ( modulo_modulo_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( Q @ M ) )
& ( ( N != zero_zero_nat )
=> ! [I2: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J2 ) ) )
=> ( Q @ J2 ) ) ) ) ) ).
% split_mod
thf(fact_1233_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_1234_Euclidean__Division_Odivmod__nat__def,axiom,
( euclidean_divmod_nat
= ( ^ [M3: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N2 ) @ ( modulo_modulo_nat @ M3 @ N2 ) ) ) ) ).
% Euclidean_Division.divmod_nat_def
thf(fact_1235_euclidean__relationI,axiom,
! [B: int,Q3: int,R: int,A: int] :
( ( ( B = zero_zero_int )
=> ( ( Q3 = zero_zero_int )
& ( R = A ) ) )
=> ( ( ( B != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( R = zero_zero_int )
& ( A
= ( times_times_int @ Q3 @ B ) ) ) ) )
=> ( ( ( B != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ B @ A )
=> ( ( ( euclid3395696857347342551nt_int @ R )
= ( euclid3395696857347342551nt_int @ B ) )
& ( ord_less_nat @ ( euclid4774559944035922753ze_int @ R ) @ ( euclid4774559944035922753ze_int @ B ) )
& ( A
= ( plus_plus_int @ ( times_times_int @ Q3 @ B ) @ R ) ) ) ) )
=> ( ( product_Pair_int_int @ ( divide_divide_int @ A @ B ) @ ( modulo_modulo_int @ A @ B ) )
= ( product_Pair_int_int @ Q3 @ R ) ) ) ) ) ).
% euclidean_relationI
thf(fact_1236_euclidean__relationI,axiom,
! [B: nat,Q3: nat,R: nat,A: nat] :
( ( ( B = zero_zero_nat )
=> ( ( Q3 = zero_zero_nat )
& ( R = A ) ) )
=> ( ( ( B != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( R = zero_zero_nat )
& ( A
= ( times_times_nat @ Q3 @ B ) ) ) ) )
=> ( ( ( B != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ B @ A )
=> ( ( ( euclid3398187327856392827nt_nat @ R )
= ( euclid3398187327856392827nt_nat @ B ) )
& ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ R ) @ ( euclid4777050414544973029ze_nat @ B ) )
& ( A
= ( plus_plus_nat @ ( times_times_nat @ Q3 @ B ) @ R ) ) ) ) )
=> ( ( product_Pair_nat_nat @ ( divide_divide_nat @ A @ B ) @ ( modulo_modulo_nat @ A @ B ) )
= ( product_Pair_nat_nat @ Q3 @ R ) ) ) ) ) ).
% euclidean_relationI
thf(fact_1237_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1238_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_1239_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_1240_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_1241_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_1242_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_1243_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_1244_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_1245_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_1246_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_1247_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_1248_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_1249_div__dvd__div,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_1250_div__dvd__div,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_1251_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1252_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1253_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_1254_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_1255_int__ops_I9_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(9)
thf(fact_1256_zmod__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zmod_int
thf(fact_1257_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_1258_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1259_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1260_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X3: nat,Y4: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y4 ) )
= D4 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
= D4 ) ) ) ).
% bezout1_nat
thf(fact_1261_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1262_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_1263_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1264_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1265_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( size_s8736152011456118867at_nat
@ ( filter4791820933415917918at_nat
@ ( ^ [Y2: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( Y2 = Z )
@ ( equiva2867628904822520639l_of_b @ xs ) )
@ ( equiva8721718519204927301v_rels @ ( size_size_list_b @ xs ) ) ) )
= one_one_nat ) ).
%------------------------------------------------------------------------------