%------------------------------------------------------------------------------
% 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    : Frequency_Moments/0087_Frequency_Moment_k/prob_00119_004374__19991406_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1632 ( 835 unt; 502 typ;   0 def)
%            Number of atoms       : 2475 (2231 equ;   0 cnn)
%            Maximal formula atoms :   15 (   2 avg)
%            Number of connectives : 8319 ( 436   ~;  55   |; 194   &;7103   @)
%                                         (   0 <=>; 531  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   4 avg)
%            Number of types       :  110 ( 109 usr)
%            Number of type conns  :  851 ( 851   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  396 ( 393 usr;  57 con; 0-4 aty)
%            Number of variables   : 2820 ( 209   ^;2540   !;  71   ?;2820   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:19:05.997
%------------------------------------------------------------------------------
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thf(sy_c_Nat__Bijection_Oset__decode,type,
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thf(sy_c_Nat__Bijection_Oset__encode,type,
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thf(sy_c_Nat__Bijection_Otriangle,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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thf(sy_c_Permutations_Oinverse__permutation__of__list_001t__Nat__Onat,type,
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thf(sy_c_Permutations_Opermutation__of__list_001t__Nat__Onat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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thf(sy_c_Probability__Mass__Function_Obernoulli__pmf,type,
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thf(sy_c_Probability__Mass__Function_Obind__pmf_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Probability__Mass__Function_Obind__pmf_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_Probability__Mass__Function_Obind__pmf_001tf__a_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Probability__Mass__Function_Oreturn__pmf_001tf__a,type,
    probab7188105566731915033_pmf_a: a > probab3364570286911266904_pmf_a ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Float__Ofloat_Mt__Float__Ofloat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Float__Ofloat_J_J,type,
    produc4856752708852917839_float: ( nat > float > float ) > produc4016807649911610908_float > produc5314510131211054487_float ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Interval__Ointerval_It__Float__Ofloat_J_Mt__Interval__Ointerval_It__Float__Ofloat_J_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Interval__Ointerval_It__Float__Ofloat_J_J_J,type,
    produc4861995570058492479_float: ( nat > interval_float > interval_float ) > produc4476928629730791788_float > produc6074858485957672967_float ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc9191340486664053748_a_nat: ( nat > product_prod_a_nat > product_prod_a_nat ) > produc4186718826624989395_a_nat > produc2997556308993345468_a_nat ).

thf(sy_c_Product__Type_OPair_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
    produc8111569692950616493list_a: ( a > a > $o ) > list_a > produc5032551385658279741list_a ).

thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
    product_Pair_o_o: $o > $o > product_prod_o_o ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
    product_Pair_o_nat: $o > nat > product_prod_o_nat ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: int > int > product_prod_int_int ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,type,
    product_Pair_int_nat: int > nat > product_prod_int_nat ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Real__Oreal,type,
    produc801115645435158769t_real: int > real > produc679980390762269497t_real ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001tf__a,type,
    product_Pair_int_a: int > a > product_prod_int_a ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    produc8424349340415155968at_nat: list_P6011104703257516679at_nat > nat > produc4008378413191047942at_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
    product_Pair_nat_o: nat > $o > product_prod_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Float__Ofloat,type,
    produc518625033508411951_float: nat > float > produc5883183851519154423_float ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
    product_Pair_nat_int: nat > int > product_prod_nat_int ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Interval__Ointerval_It__Float__Ofloat_J,type,
    produc6067866782486577919_float: nat > interval_float > produc1161000763195272519_float ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Float__Ofloat_J_J,type,
    produc7773819858125983828_float: nat > list_P454701268746800253_float > produc8277557972349479330_float ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Interval__Ointerval_It__Float__Ofloat_J_J_J,type,
    produc2655563877181872036_float: nat > list_P1609389641493977421_float > produc6499262172985382258_float ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc6109913384486294878at_nat: nat > list_P6011104703257516679at_nat > produc8472197452120411308at_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc7526178182849817355_a_nat: nat > list_P4901192995000098612_a_nat > produc5672971924406059225_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Float__Ofloat_J,type,
    produc3893871907814880334_float: nat > produc5883183851519154423_float > produc4016807649911610908_float ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Interval__Ointerval_It__Float__Ofloat_J_J,type,
    produc889832209282122526_float: nat > produc1161000763195272519_float > produc4476928629730791788_float ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Float__Ofloat_J_J,type,
    produc6596454488040397317_float: nat > produc4016807649911610908_float > produc9095508395758718541_float ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc6385450045882626063at_nat: nat > produc7248412053542808358at_nat > produc8642769642335960151at_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Float__Ofloat_J_J_J,type,
    produc2626873021911329572_float: nat > produc9095508395758718541_float > produc985389371198253426_float ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    produc6096211462017127045_a_nat: nat > produc6774132644148096814_a_nat > produc4186718826624989395_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    produc4490577844368043572al_nat: nat > produc3741383161447143261al_nat > produc9155354469731034754al_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    produc148073511828866022_a_nat: nat > product_prod_a_nat > produc6774132644148096814_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Real__Oreal,type,
    produc7837566107596912789t_real: nat > real > produc7716430852924023517t_real ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4207506657711014383et_nat: nat > set_nat > produc2400336064389900727et_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__a,type,
    product_Pair_nat_a: nat > a > product_prod_nat_a ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_001_Eo,type,
    produc8940506901291673601_nat_o: produc6774132644148096814_a_nat > $o > produc3153433774902879761_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    produc2837828401664308135at_nat: produc6774132644148096814_a_nat > nat > produc54233759696573485at_nat ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Int__Oint,type,
    produc3179012173361985393al_int: real > int > produc8786904178792722361al_int ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
    produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
    produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    produc641871753055645167at_nat: set_nat > nat > produc7491599851749785783at_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).

thf(sy_c_Product__Type_OPair_001tf__a_001t__Int__Oint,type,
    product_Pair_a_int: a > int > product_prod_a_int ).

thf(sy_c_Product__Type_OPair_001tf__a_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    produc6674818935846295348_a_nat: a > list_P3592885314253461005_a_nat > produc3864505136547489594_a_nat ).

thf(sy_c_Product__Type_OPair_001tf__a_001t__Nat__Onat,type,
    product_Pair_a_nat: a > nat > product_prod_a_nat ).

thf(sy_c_Product__Type_OPair_001tf__a_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    produc3776435479397687751_a_nat: a > produc6774132644148096814_a_nat > produc3791183549783780439_a_nat ).

thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
    product_Pair_a_a: a > a > product_prod_a_a ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
    divide1155267253282662278s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).

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_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: real > real > real ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: $o > int ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: $o > nat ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
    zero_n3304061248610475627l_real: $o > real ).

thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
    collect_list_a: ( list_a > $o ) > set_list_a ).

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001tf__a,type,
    collect_a: ( a > $o ) > set_a ).

thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
    is_empty_nat: set_nat > $o ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: nat > set_nat ).

thf(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

thf(sy_c_Sublist_Oprefixes_001tf__a,type,
    prefixes_a: list_a > list_list_a ).

thf(sy_c_Sublist_Osublists_001tf__a,type,
    sublists_a: list_a > list_list_a ).

thf(sy_c_Sublist_Osuffixes_001tf__a,type,
    suffixes_a: list_a > list_list_a ).

thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: real > real ).

thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: real > real ).

thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    accp_list_list_a: ( list_list_a > list_list_a > $o ) > list_list_a > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    accp_P9051044500959620704at_nat: ( produc8642769642335960151at_nat > produc8642769642335960151at_nat > $o ) > produc8642769642335960151at_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    accp_P5245135987034508768_a_nat: ( produc3791183549783780439_a_nat > produc3791183549783780439_a_nat > $o ) > produc3791183549783780439_a_nat > $o ).

thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
    member_list_a: list_a > set_list_a > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_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_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Float__Ofloat_J,type,
    member2086872401261188768_float: produc5883183851519154423_float > set_Pr7206334407337572951_float > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Interval__Ointerval_It__Float__Ofloat_J_J,type,
    member3149793176805732464_float: produc1161000763195272519_float > set_Pr6933375829817990695_float > $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__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    member3063082432339977431_a_nat: produc6774132644148096814_a_nat > set_Pr412391540666252558_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_v_as,type,
    as: list_a ).

thf(sy_v_x____,type,
    x: a ).

thf(sy_v_xs____,type,
    xs: list_a ).

% Relevant facts (1116)
thf(fact_0_assms,axiom,
    as != nil_a ).

% assms
thf(fact_1_snoc_I1_J,axiom,
    xs != nil_a ).

% snoc(1)
thf(fact_2_snoc_I2_J,axiom,
    ( ( fold_a7750079645758252821_a_nat
      @ ^ [X: a,S: probab4139393509369344520_a_nat] : ( probab5175300157245341303_a_nat @ S @ ( freque5687656950882125236te_2_a @ X ) )
      @ xs
      @ ( probab6296365750099732169_a_nat @ ( produc148073511828866022_a_nat @ zero_zero_nat @ ( product_Pair_a_nat @ zero_zero_a @ zero_zero_nat ) ) ) )
    = ( probab1029661319033422245_a_nat @ ( probab1830274953030043784et_nat @ ( set_ord_lessThan_nat @ ( size_size_list_a @ xs ) ) )
      @ ^ [K: nat] : ( probab6296365750099732169_a_nat @ ( produc148073511828866022_a_nat @ ( size_size_list_a @ xs ) @ ( freque6565899915408260854etch_a @ xs @ K ) ) ) ) ) ).

% snoc(2)
thf(fact_3_append1__eq__conv,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X2: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
      ( ( ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) )
        = ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_4_append1__eq__conv,axiom,
    ! [Xs: list_P2851791750731487283_nat_a,X2: product_prod_nat_a,Ys: list_P2851791750731487283_nat_a,Y: product_prod_nat_a] :
      ( ( ( append1694031006427026248_nat_a @ Xs @ ( cons_P8443330267410185325_nat_a @ X2 @ nil_Pr1417316670369895453_nat_a ) )
        = ( append1694031006427026248_nat_a @ Ys @ ( cons_P8443330267410185325_nat_a @ Y @ nil_Pr1417316670369895453_nat_a ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_5_append1__eq__conv,axiom,
    ! [Xs: list_P3592885314253461005_a_nat,X2: product_prod_a_nat,Ys: list_P3592885314253461005_a_nat,Y: product_prod_a_nat] :
      ( ( ( append7679239579558125090_a_nat @ Xs @ ( cons_P5205166803686508359_a_nat @ X2 @ nil_Pr7402525243500994295_a_nat ) )
        = ( append7679239579558125090_a_nat @ Ys @ ( cons_P5205166803686508359_a_nat @ Y @ nil_Pr7402525243500994295_a_nat ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_6_append1__eq__conv,axiom,
    ! [Xs: list_interval_float,X2: interval_float,Ys: list_interval_float,Y: interval_float] :
      ( ( ( append3774529519228663611_float @ Xs @ ( cons_interval_float @ X2 @ nil_interval_float ) )
        = ( append3774529519228663611_float @ Ys @ ( cons_interval_float @ Y @ nil_interval_float ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_7_append1__eq__conv,axiom,
    ! [Xs: list_float,X2: float,Ys: list_float,Y: float] :
      ( ( ( append_float @ Xs @ ( cons_float @ X2 @ nil_float ) )
        = ( append_float @ Ys @ ( cons_float @ Y @ nil_float ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_8_append1__eq__conv,axiom,
    ! [Xs: list_real,X2: real,Ys: list_real,Y: real] :
      ( ( ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) )
        = ( append_real @ Ys @ ( cons_real @ Y @ nil_real ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_9_append1__eq__conv,axiom,
    ! [Xs: list_list_a,X2: list_a,Ys: list_list_a,Y: list_a] :
      ( ( ( append_list_a @ Xs @ ( cons_list_a @ X2 @ nil_list_a ) )
        = ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_10_append1__eq__conv,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat,Y: nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
        = ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_11_append1__eq__conv,axiom,
    ! [Xs: list_a,X2: a,Ys: list_a,Y: a] :
      ( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
        = ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_12_length__0__conv,axiom,
    ! [Xs: list_P3592885314253461005_a_nat] :
      ( ( ( size_s984997627204368545_a_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr7402525243500994295_a_nat ) ) ).

% length_0_conv
thf(fact_13_length__0__conv,axiom,
    ! [Xs: list_real] :
      ( ( ( size_size_list_real @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_real ) ) ).

% length_0_conv
thf(fact_14_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_15_length__0__conv,axiom,
    ! [Xs: list_list_a] :
      ( ( ( size_s349497388124573686list_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_list_a ) ) ).

% length_0_conv
thf(fact_16_length__0__conv,axiom,
    ! [Xs: list_P2851791750731487283_nat_a] :
      ( ( ( size_s243904063682394823_nat_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr1417316670369895453_nat_a ) ) ).

% length_0_conv
thf(fact_17_length__0__conv,axiom,
    ! [Xs: list_P1396940483166286381od_a_a] :
      ( ( ( size_s3885678630836030617od_a_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Product_prod_a_a ) ) ).

% length_0_conv
thf(fact_18_length__0__conv,axiom,
    ! [Xs: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_a ) ) ).

% length_0_conv
thf(fact_19_append__eq__append__conv,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Us: list_P6011104703257516679at_nat,Vs: list_P6011104703257516679at_nat] :
      ( ( ( ( size_s5460976970255530739at_nat @ Xs )
          = ( size_s5460976970255530739at_nat @ Ys ) )
        | ( ( size_s5460976970255530739at_nat @ Us )
          = ( size_s5460976970255530739at_nat @ Vs ) ) )
     => ( ( ( append985823374593552924at_nat @ Xs @ Us )
          = ( append985823374593552924at_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_20_append__eq__append__conv,axiom,
    ! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Us: list_P3592885314253461005_a_nat,Vs: list_P3592885314253461005_a_nat] :
      ( ( ( ( size_s984997627204368545_a_nat @ Xs )
          = ( size_s984997627204368545_a_nat @ Ys ) )
        | ( ( size_s984997627204368545_a_nat @ Us )
          = ( size_s984997627204368545_a_nat @ Vs ) ) )
     => ( ( ( append7679239579558125090_a_nat @ Xs @ Us )
          = ( append7679239579558125090_a_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_21_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_22_append__eq__append__conv,axiom,
    ! [Xs: list_list_a,Ys: list_list_a,Us: list_list_a,Vs: list_list_a] :
      ( ( ( ( size_s349497388124573686list_a @ Xs )
          = ( size_s349497388124573686list_a @ Ys ) )
        | ( ( size_s349497388124573686list_a @ Us )
          = ( size_s349497388124573686list_a @ Vs ) ) )
     => ( ( ( append_list_a @ Xs @ Us )
          = ( append_list_a @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_23_append__eq__append__conv,axiom,
    ! [Xs: list_P2851791750731487283_nat_a,Ys: list_P2851791750731487283_nat_a,Us: list_P2851791750731487283_nat_a,Vs: list_P2851791750731487283_nat_a] :
      ( ( ( ( size_s243904063682394823_nat_a @ Xs )
          = ( size_s243904063682394823_nat_a @ Ys ) )
        | ( ( size_s243904063682394823_nat_a @ Us )
          = ( size_s243904063682394823_nat_a @ Vs ) ) )
     => ( ( ( append1694031006427026248_nat_a @ Xs @ Us )
          = ( append1694031006427026248_nat_a @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_24_append__eq__append__conv,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,Us: list_P1396940483166286381od_a_a,Vs: list_P1396940483166286381od_a_a] :
      ( ( ( ( size_s3885678630836030617od_a_a @ Xs )
          = ( size_s3885678630836030617od_a_a @ Ys ) )
        | ( ( size_s3885678630836030617od_a_a @ Us )
          = ( size_s3885678630836030617od_a_a @ Vs ) ) )
     => ( ( ( append5335208819046833346od_a_a @ Xs @ Us )
          = ( append5335208819046833346od_a_a @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_25_append__eq__append__conv,axiom,
    ! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
      ( ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_a @ Ys ) )
        | ( ( size_size_list_a @ Us )
          = ( size_size_list_a @ Vs ) ) )
     => ( ( ( append_a @ Xs @ Us )
          = ( append_a @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_26_append_Oright__neutral,axiom,
    ! [A: list_P6011104703257516679at_nat] :
      ( ( append985823374593552924at_nat @ A @ nil_Pr5478986624290739719at_nat )
      = A ) ).

% append.right_neutral
thf(fact_27_append_Oright__neutral,axiom,
    ! [A: list_P2851791750731487283_nat_a] :
      ( ( append1694031006427026248_nat_a @ A @ nil_Pr1417316670369895453_nat_a )
      = A ) ).

% append.right_neutral
thf(fact_28_append_Oright__neutral,axiom,
    ! [A: list_P3592885314253461005_a_nat] :
      ( ( append7679239579558125090_a_nat @ A @ nil_Pr7402525243500994295_a_nat )
      = A ) ).

% append.right_neutral
thf(fact_29_append_Oright__neutral,axiom,
    ! [A: list_real] :
      ( ( append_real @ A @ nil_real )
      = A ) ).

% append.right_neutral
thf(fact_30_append_Oright__neutral,axiom,
    ! [A: list_list_a] :
      ( ( append_list_a @ A @ nil_list_a )
      = A ) ).

% append.right_neutral
thf(fact_31_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_32_append_Oright__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ A @ nil_a )
      = A ) ).

% append.right_neutral
thf(fact_33_append__Nil2,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( append985823374593552924at_nat @ Xs @ nil_Pr5478986624290739719at_nat )
      = Xs ) ).

% append_Nil2
thf(fact_34_append__Nil2,axiom,
    ! [Xs: list_P2851791750731487283_nat_a] :
      ( ( append1694031006427026248_nat_a @ Xs @ nil_Pr1417316670369895453_nat_a )
      = Xs ) ).

% append_Nil2
thf(fact_35_append__Nil2,axiom,
    ! [Xs: list_P3592885314253461005_a_nat] :
      ( ( append7679239579558125090_a_nat @ Xs @ nil_Pr7402525243500994295_a_nat )
      = Xs ) ).

% append_Nil2
thf(fact_36_append__Nil2,axiom,
    ! [Xs: list_real] :
      ( ( append_real @ Xs @ nil_real )
      = Xs ) ).

% append_Nil2
thf(fact_37_append__Nil2,axiom,
    ! [Xs: list_list_a] :
      ( ( append_list_a @ Xs @ nil_list_a )
      = Xs ) ).

% append_Nil2
thf(fact_38_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_39_append__Nil2,axiom,
    ! [Xs: list_a] :
      ( ( append_a @ Xs @ nil_a )
      = Xs ) ).

% append_Nil2
thf(fact_40_append__self__conv,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( ( append985823374593552924at_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Pr5478986624290739719at_nat ) ) ).

% append_self_conv
thf(fact_41_append__self__conv,axiom,
    ! [Xs: list_P2851791750731487283_nat_a,Ys: list_P2851791750731487283_nat_a] :
      ( ( ( append1694031006427026248_nat_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Pr1417316670369895453_nat_a ) ) ).

% append_self_conv
thf(fact_42_append__self__conv,axiom,
    ! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( ( append7679239579558125090_a_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Pr7402525243500994295_a_nat ) ) ).

% append_self_conv
thf(fact_43_append__self__conv,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( ( append_real @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_real ) ) ).

% append_self_conv
thf(fact_44_append__self__conv,axiom,
    ! [Xs: list_list_a,Ys: list_list_a] :
      ( ( ( append_list_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_list_a ) ) ).

% append_self_conv
thf(fact_45_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_46_append__self__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_a ) ) ).

% append_self_conv
thf(fact_47_self__append__conv,axiom,
    ! [Y: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( Y
        = ( append985823374593552924at_nat @ Y @ Ys ) )
      = ( Ys = nil_Pr5478986624290739719at_nat ) ) ).

% self_append_conv
thf(fact_48_self__append__conv,axiom,
    ! [Y: list_P2851791750731487283_nat_a,Ys: list_P2851791750731487283_nat_a] :
      ( ( Y
        = ( append1694031006427026248_nat_a @ Y @ Ys ) )
      = ( Ys = nil_Pr1417316670369895453_nat_a ) ) ).

% self_append_conv
thf(fact_49_self__append__conv,axiom,
    ! [Y: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( Y
        = ( append7679239579558125090_a_nat @ Y @ Ys ) )
      = ( Ys = nil_Pr7402525243500994295_a_nat ) ) ).

% self_append_conv
thf(fact_50_self__append__conv,axiom,
    ! [Y: list_real,Ys: list_real] :
      ( ( Y
        = ( append_real @ Y @ Ys ) )
      = ( Ys = nil_real ) ) ).

% self_append_conv
thf(fact_51_self__append__conv,axiom,
    ! [Y: list_list_a,Ys: list_list_a] :
      ( ( Y
        = ( append_list_a @ Y @ Ys ) )
      = ( Ys = nil_list_a ) ) ).

% self_append_conv
thf(fact_52_self__append__conv,axiom,
    ! [Y: list_nat,Ys: list_nat] :
      ( ( Y
        = ( append_nat @ Y @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_53_self__append__conv,axiom,
    ! [Y: list_a,Ys: list_a] :
      ( ( Y
        = ( append_a @ Y @ Ys ) )
      = ( Ys = nil_a ) ) ).

% self_append_conv
thf(fact_54_append__self__conv2,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( ( append985823374593552924at_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Pr5478986624290739719at_nat ) ) ).

% append_self_conv2
thf(fact_55_append__self__conv2,axiom,
    ! [Xs: list_P2851791750731487283_nat_a,Ys: list_P2851791750731487283_nat_a] :
      ( ( ( append1694031006427026248_nat_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Pr1417316670369895453_nat_a ) ) ).

% append_self_conv2
thf(fact_56_append__self__conv2,axiom,
    ! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( ( append7679239579558125090_a_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Pr7402525243500994295_a_nat ) ) ).

% append_self_conv2
thf(fact_57_append__self__conv2,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( ( append_real @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_real ) ) ).

% append_self_conv2
thf(fact_58_append__self__conv2,axiom,
    ! [Xs: list_list_a,Ys: list_list_a] :
      ( ( ( append_list_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_list_a ) ) ).

% append_self_conv2
thf(fact_59_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_60_append__self__conv2,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_a ) ) ).

% append_self_conv2
thf(fact_61_self__append__conv2,axiom,
    ! [Y: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( Y
        = ( append985823374593552924at_nat @ Xs @ Y ) )
      = ( Xs = nil_Pr5478986624290739719at_nat ) ) ).

% self_append_conv2
thf(fact_62_self__append__conv2,axiom,
    ! [Y: list_P2851791750731487283_nat_a,Xs: list_P2851791750731487283_nat_a] :
      ( ( Y
        = ( append1694031006427026248_nat_a @ Xs @ Y ) )
      = ( Xs = nil_Pr1417316670369895453_nat_a ) ) ).

% self_append_conv2
thf(fact_63_self__append__conv2,axiom,
    ! [Y: list_P3592885314253461005_a_nat,Xs: list_P3592885314253461005_a_nat] :
      ( ( Y
        = ( append7679239579558125090_a_nat @ Xs @ Y ) )
      = ( Xs = nil_Pr7402525243500994295_a_nat ) ) ).

% self_append_conv2
thf(fact_64_self__append__conv2,axiom,
    ! [Y: list_real,Xs: list_real] :
      ( ( Y
        = ( append_real @ Xs @ Y ) )
      = ( Xs = nil_real ) ) ).

% self_append_conv2
thf(fact_65_self__append__conv2,axiom,
    ! [Y: list_list_a,Xs: list_list_a] :
      ( ( Y
        = ( append_list_a @ Xs @ Y ) )
      = ( Xs = nil_list_a ) ) ).

% self_append_conv2
thf(fact_66_self__append__conv2,axiom,
    ! [Y: list_nat,Xs: list_nat] :
      ( ( Y
        = ( append_nat @ Xs @ Y ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_67_self__append__conv2,axiom,
    ! [Y: list_a,Xs: list_a] :
      ( ( Y
        = ( append_a @ Xs @ Y ) )
      = ( Xs = nil_a ) ) ).

% self_append_conv2
thf(fact_68_Nil__is__append__conv,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( nil_Pr5478986624290739719at_nat
        = ( append985823374593552924at_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr5478986624290739719at_nat )
        & ( Ys = nil_Pr5478986624290739719at_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_69_Nil__is__append__conv,axiom,
    ! [Xs: list_P2851791750731487283_nat_a,Ys: list_P2851791750731487283_nat_a] :
      ( ( nil_Pr1417316670369895453_nat_a
        = ( append1694031006427026248_nat_a @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr1417316670369895453_nat_a )
        & ( Ys = nil_Pr1417316670369895453_nat_a ) ) ) ).

% Nil_is_append_conv
thf(fact_70_Nil__is__append__conv,axiom,
    ! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( nil_Pr7402525243500994295_a_nat
        = ( append7679239579558125090_a_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr7402525243500994295_a_nat )
        & ( Ys = nil_Pr7402525243500994295_a_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_71_Nil__is__append__conv,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( nil_real
        = ( append_real @ Xs @ Ys ) )
      = ( ( Xs = nil_real )
        & ( Ys = nil_real ) ) ) ).

% Nil_is_append_conv
thf(fact_72_Nil__is__append__conv,axiom,
    ! [Xs: list_list_a,Ys: list_list_a] :
      ( ( nil_list_a
        = ( append_list_a @ Xs @ Ys ) )
      = ( ( Xs = nil_list_a )
        & ( Ys = nil_list_a ) ) ) ).

% Nil_is_append_conv
thf(fact_73_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_74_Nil__is__append__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( nil_a
        = ( append_a @ Xs @ Ys ) )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% Nil_is_append_conv
thf(fact_75_append__is__Nil__conv,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( ( append985823374593552924at_nat @ Xs @ Ys )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Xs = nil_Pr5478986624290739719at_nat )
        & ( Ys = nil_Pr5478986624290739719at_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_76_append__is__Nil__conv,axiom,
    ! [Xs: list_P2851791750731487283_nat_a,Ys: list_P2851791750731487283_nat_a] :
      ( ( ( append1694031006427026248_nat_a @ Xs @ Ys )
        = nil_Pr1417316670369895453_nat_a )
      = ( ( Xs = nil_Pr1417316670369895453_nat_a )
        & ( Ys = nil_Pr1417316670369895453_nat_a ) ) ) ).

% append_is_Nil_conv
thf(fact_77_append__is__Nil__conv,axiom,
    ! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( ( append7679239579558125090_a_nat @ Xs @ Ys )
        = nil_Pr7402525243500994295_a_nat )
      = ( ( Xs = nil_Pr7402525243500994295_a_nat )
        & ( Ys = nil_Pr7402525243500994295_a_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_78_append__is__Nil__conv,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( ( append_real @ Xs @ Ys )
        = nil_real )
      = ( ( Xs = nil_real )
        & ( Ys = nil_real ) ) ) ).

% append_is_Nil_conv
thf(fact_79_append__is__Nil__conv,axiom,
    ! [Xs: list_list_a,Ys: list_list_a] :
      ( ( ( append_list_a @ Xs @ Ys )
        = nil_list_a )
      = ( ( Xs = nil_list_a )
        & ( Ys = nil_list_a ) ) ) ).

% append_is_Nil_conv
thf(fact_80_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_81_append__is__Nil__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = nil_a )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% append_is_Nil_conv
thf(fact_82_non__empty_I2_J,axiom,
    ( ( set_ord_lessThan_nat @ ( size_size_list_a @ xs ) )
   != bot_bot_set_nat ) ).

% non_empty(2)
thf(fact_83_list_Oinject,axiom,
    ! [X21: interval_float,X22: list_interval_float,Y21: interval_float,Y22: list_interval_float] :
      ( ( ( cons_interval_float @ X21 @ X22 )
        = ( cons_interval_float @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_84_list_Oinject,axiom,
    ! [X21: float,X22: list_float,Y21: float,Y22: list_float] :
      ( ( ( cons_float @ X21 @ X22 )
        = ( cons_float @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_85_list_Oinject,axiom,
    ! [X21: real,X22: list_real,Y21: real,Y22: list_real] :
      ( ( ( cons_real @ X21 @ X22 )
        = ( cons_real @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_86_list_Oinject,axiom,
    ! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
      ( ( ( cons_list_a @ X21 @ X22 )
        = ( cons_list_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_87_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_88_list_Oinject,axiom,
    ! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
      ( ( ( cons_a @ X21 @ X22 )
        = ( cons_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_89_same__append__eq,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
      ( ( ( append985823374593552924at_nat @ Xs @ Ys )
        = ( append985823374593552924at_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_90_same__append__eq,axiom,
    ! [Xs: list_P2851791750731487283_nat_a,Ys: list_P2851791750731487283_nat_a,Zs: list_P2851791750731487283_nat_a] :
      ( ( ( append1694031006427026248_nat_a @ Xs @ Ys )
        = ( append1694031006427026248_nat_a @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_91_same__append__eq,axiom,
    ! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat] :
      ( ( ( append7679239579558125090_a_nat @ Xs @ Ys )
        = ( append7679239579558125090_a_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_92_same__append__eq,axiom,
    ! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
      ( ( ( append_list_a @ Xs @ Ys )
        = ( append_list_a @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_93_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_94_same__append__eq,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = ( append_a @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_95_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_96_append__same__eq,axiom,
    ! [Ys: list_a,Xs: list_a,Zs: list_a] :
      ( ( ( append_a @ Ys @ Xs )
        = ( append_a @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_97_append__assoc,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
      = ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_98_append_Oassoc,axiom,
    ! [A: list_a,B: list_a,C: list_a] :
      ( ( append_a @ ( append_a @ A @ B ) @ C )
      = ( append_a @ A @ ( append_a @ B @ C ) ) ) ).

% append.assoc
thf(fact_99_not__Cons__self2,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( cons_a @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_100_neq__if__length__neq,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( size_size_list_a @ Xs )
       != ( size_size_list_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_101_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_a] :
      ( ( size_size_list_a @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_102_append__eq__append__conv2,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = ( append_a @ Zs @ Ts ) )
      = ( ? [Us2: list_a] :
            ( ( ( Xs
                = ( append_a @ Zs @ Us2 ) )
              & ( ( append_a @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_a @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_103_append__eq__appendI,axiom,
    ! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
      ( ( ( append_a @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_a @ Xs1 @ Us ) )
       => ( ( append_a @ Xs @ Ys )
          = ( append_a @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_104_list__nonempty__induct,axiom,
    ! [Xs: list_a,P: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
       => ( ! [X3: a,Xs2: list_a] :
              ( ( Xs2 != nil_a )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_a @ X3 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_105_list__induct2_H,axiom,
    ! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
      ( ( P @ nil_a @ nil_a )
     => ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_a )
       => ( ! [Y2: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys2 ) )
         => ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_106_neq__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
      = ( ? [Y3: a,Ys3: list_a] :
            ( Xs
            = ( cons_a @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_107_remdups__adj_Ocases,axiom,
    ! [X2: list_a] :
      ( ( X2 != nil_a )
     => ( ! [X3: a] :
            ( X2
           != ( cons_a @ X3 @ nil_a ) )
       => ~ ! [X3: a,Y2: a,Xs2: list_a] :
              ( X2
             != ( cons_a @ X3 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_108_transpose_Ocases,axiom,
    ! [X2: list_list_a] :
      ( ( X2 != nil_list_a )
     => ( ! [Xss: list_list_a] :
            ( X2
           != ( cons_list_a @ nil_a @ Xss ) )
       => ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
              ( X2
             != ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_109_list_Oexhaust,axiom,
    ! [Y: list_a] :
      ( ( Y != nil_a )
     => ~ ! [X212: a,X222: list_a] :
            ( Y
           != ( cons_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_110_list_OdiscI,axiom,
    ! [List: list_a,X21: a,X22: list_a] :
      ( ( List
        = ( cons_a @ X21 @ X22 ) )
     => ( List != nil_a ) ) ).

% list.discI
thf(fact_111_list_Odistinct_I1_J,axiom,
    ! [X21: a,X22: list_a] :
      ( nil_a
     != ( cons_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_112_Cons__eq__appendI,axiom,
    ! [X2: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
      ( ( ( cons_a @ X2 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_a @ Xs1 @ Zs ) )
       => ( ( cons_a @ X2 @ Xs )
          = ( append_a @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_113_append__Cons,axiom,
    ! [X2: a,Xs: list_a,Ys: list_a] :
      ( ( append_a @ ( cons_a @ X2 @ Xs ) @ Ys )
      = ( cons_a @ X2 @ ( append_a @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_114_eq__Nil__appendI,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_a @ nil_a @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_115_append_Oleft__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ nil_a @ A )
      = A ) ).

% append.left_neutral
thf(fact_116_append__Nil,axiom,
    ! [Ys: list_a] :
      ( ( append_a @ nil_a @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_117_fold__simps_I2_J,axiom,
    ! [F: a > probab4139393509369344520_a_nat > probab4139393509369344520_a_nat,X2: a,Xs: list_a,S2: probab4139393509369344520_a_nat] :
      ( ( fold_a7750079645758252821_a_nat @ F @ ( cons_a @ X2 @ Xs ) @ S2 )
      = ( fold_a7750079645758252821_a_nat @ F @ Xs @ ( F @ X2 @ S2 ) ) ) ).

% fold_simps(2)
thf(fact_118_fold__simps_I1_J,axiom,
    ! [F: a > probab4139393509369344520_a_nat > probab4139393509369344520_a_nat,S2: probab4139393509369344520_a_nat] :
      ( ( fold_a7750079645758252821_a_nat @ F @ nil_a @ S2 )
      = S2 ) ).

% fold_simps(1)
thf(fact_119_list_Osize_I3_J,axiom,
    ( ( size_size_list_a @ nil_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_120_list__induct4,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( ( size_size_list_a @ Zs )
            = ( size_size_list_a @ Ws ) )
         => ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
           => ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
                  ( ( ( size_size_list_a @ Xs2 )
                    = ( size_size_list_a @ Ys2 ) )
                 => ( ( ( size_size_list_a @ Ys2 )
                      = ( size_size_list_a @ Zs2 ) )
                   => ( ( ( size_size_list_a @ Zs2 )
                        = ( size_size_list_a @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_121_list__induct3,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( P @ nil_a @ nil_a @ nil_a )
         => ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a] :
                ( ( ( size_size_list_a @ Xs2 )
                  = ( size_size_list_a @ Ys2 ) )
               => ( ( ( size_size_list_a @ Ys2 )
                    = ( size_size_list_a @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys2 @ Zs2 )
                   => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_122_list__induct2,axiom,
    ! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( P @ nil_a @ nil_a )
       => ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
              ( ( ( size_size_list_a @ Xs2 )
                = ( size_size_list_a @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_123_rev__nonempty__induct,axiom,
    ! [Xs: list_a,P: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
       => ( ! [X3: a,Xs2: list_a] :
              ( ( Xs2 != nil_a )
             => ( ( P @ Xs2 )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_124_append__eq__Cons__conv,axiom,
    ! [Ys: list_a,Zs: list_a,X2: a,Xs: list_a] :
      ( ( ( append_a @ Ys @ Zs )
        = ( cons_a @ X2 @ Xs ) )
      = ( ( ( Ys = nil_a )
          & ( Zs
            = ( cons_a @ X2 @ Xs ) ) )
        | ? [Ys4: list_a] :
            ( ( Ys
              = ( cons_a @ X2 @ Ys4 ) )
            & ( ( append_a @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_125_Cons__eq__append__conv,axiom,
    ! [X2: a,Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( append_a @ Ys @ Zs ) )
      = ( ( ( Ys = nil_a )
          & ( ( cons_a @ X2 @ Xs )
            = Zs ) )
        | ? [Ys4: list_a] :
            ( ( ( cons_a @ X2 @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_a @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_126_rev__exhaust,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ~ ! [Ys2: list_a,Y2: a] :
            ( Xs
           != ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).

% rev_exhaust
thf(fact_127_rev__induct,axiom,
    ! [P: list_a > $o,Xs: list_a] :
      ( ( P @ nil_a )
     => ( ! [X3: a,Xs2: list_a] :
            ( ( P @ Xs2 )
           => ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_128_same__length__different,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( Xs != Ys )
     => ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_a @ Ys ) )
       => ? [Pre: list_a,X3: a,Xs3: list_a,Y2: a,Ys5: list_a] :
            ( ( X3 != Y2 )
            & ( Xs
              = ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs3 ) ) )
            & ( Ys
              = ( append_a @ Pre @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_129_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_130_non__empty_I1_J,axiom,
    ( ( set_ord_lessThan_nat @ ( suc @ ( size_size_list_a @ xs ) ) )
   != bot_bot_set_nat ) ).

% non_empty(1)
thf(fact_131_bind__pmf__const,axiom,
    ! [M: probab4139393509369344520_a_nat,C: probab4139393509369344520_a_nat] :
      ( ( probab5175300157245341303_a_nat @ M
        @ ^ [X: produc6774132644148096814_a_nat] : C )
      = C ) ).

% bind_pmf_const
thf(fact_132_bind__pmf__const,axiom,
    ! [M: probab469873468395307276mf_nat,C: probab4139393509369344520_a_nat] :
      ( ( probab1029661319033422245_a_nat @ M
        @ ^ [X: nat] : C )
      = C ) ).

% bind_pmf_const
thf(fact_133_bind__pmf__const,axiom,
    ! [M: probab1498759712122475378_pmf_o,C: probab469873468395307276mf_nat] :
      ( ( probab4549177196568532785_o_nat @ M
        @ ^ [X: $o] : C )
      = C ) ).

% bind_pmf_const
thf(fact_134_bind__pmf__const,axiom,
    ! [M: probab469873468395307276mf_nat,C: probab469873468395307276mf_nat] :
      ( ( probab5865774939380454169at_nat @ M
        @ ^ [X: nat] : C )
      = C ) ).

% bind_pmf_const
thf(fact_135_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_136_Iio__eq__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
thf(fact_137_return__pmf__inj,axiom,
    ! [X2: produc6774132644148096814_a_nat,Y: produc6774132644148096814_a_nat] :
      ( ( ( probab6296365750099732169_a_nat @ X2 )
        = ( probab6296365750099732169_a_nat @ Y ) )
      = ( X2 = Y ) ) ).

% return_pmf_inj
thf(fact_138_return__pmf__inj,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ( probab4138676067695765237mf_nat @ X2 )
        = ( probab4138676067695765237mf_nat @ Y ) )
      = ( X2 = Y ) ) ).

% return_pmf_inj
thf(fact_139_return__pmf__inj,axiom,
    ! [X2: $o,Y: $o] :
      ( ( ( probab7739007152833094963_pmf_o @ X2 )
        = ( probab7739007152833094963_pmf_o @ Y ) )
      = ( X2 = Y ) ) ).

% return_pmf_inj
thf(fact_140_lessThan__eq__iff,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ( set_ord_lessThan_nat @ X2 )
        = ( set_ord_lessThan_nat @ Y ) )
      = ( X2 = Y ) ) ).

% lessThan_eq_iff
thf(fact_141_old_Oprod_Oinject,axiom,
    ! [A: nat,B: product_prod_a_nat,A2: nat,B2: product_prod_a_nat] :
      ( ( ( produc148073511828866022_a_nat @ A @ B )
        = ( produc148073511828866022_a_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_142_old_Oprod_Oinject,axiom,
    ! [A: nat,B: interval_float,A2: nat,B2: interval_float] :
      ( ( ( produc6067866782486577919_float @ A @ B )
        = ( produc6067866782486577919_float @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_143_old_Oprod_Oinject,axiom,
    ! [A: nat,B: float,A2: nat,B2: float] :
      ( ( ( produc518625033508411951_float @ A @ B )
        = ( produc518625033508411951_float @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_144_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_145_old_Oprod_Oinject,axiom,
    ! [A: a,B: nat,A2: a,B2: nat] :
      ( ( ( product_Pair_a_nat @ A @ B )
        = ( product_Pair_a_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_146_prod_Oinject,axiom,
    ! [X1: nat,X23: product_prod_a_nat,Y1: nat,Y23: product_prod_a_nat] :
      ( ( ( produc148073511828866022_a_nat @ X1 @ X23 )
        = ( produc148073511828866022_a_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_147_prod_Oinject,axiom,
    ! [X1: nat,X23: interval_float,Y1: nat,Y23: interval_float] :
      ( ( ( produc6067866782486577919_float @ X1 @ X23 )
        = ( produc6067866782486577919_float @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_148_prod_Oinject,axiom,
    ! [X1: nat,X23: float,Y1: nat,Y23: float] :
      ( ( ( produc518625033508411951_float @ X1 @ X23 )
        = ( produc518625033508411951_float @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_149_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_150_prod_Oinject,axiom,
    ! [X1: a,X23: nat,Y1: a,Y23: nat] :
      ( ( ( product_Pair_a_nat @ X1 @ X23 )
        = ( product_Pair_a_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_151_mem__simps_I2_J,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% mem_simps(2)
thf(fact_152_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X2: produc2997556308993345468_a_nat] :
      ~ ! [F2: nat > product_prod_a_nat > product_prod_a_nat,A3: nat,B3: nat,Acc: product_prod_a_nat] :
          ( X2
         != ( produc9191340486664053748_a_nat @ F2 @ ( produc6096211462017127045_a_nat @ A3 @ ( produc148073511828866022_a_nat @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_153_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X2: produc6074858485957672967_float] :
      ~ ! [F2: nat > interval_float > interval_float,A3: nat,B3: nat,Acc: interval_float] :
          ( X2
         != ( produc4861995570058492479_float @ F2 @ ( produc889832209282122526_float @ A3 @ ( produc6067866782486577919_float @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_154_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X2: produc5314510131211054487_float] :
      ~ ! [F2: nat > float > float,A3: nat,B3: nat,Acc: float] :
          ( X2
         != ( produc4856752708852917839_float @ F2 @ ( produc3893871907814880334_float @ A3 @ ( produc518625033508411951_float @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_155_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X2: produc4471711990508489141at_nat] :
      ~ ! [F2: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
          ( X2
         != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_156_sorted__wrt_Ocases,axiom,
    ! [X2: produc5032551385658279741list_a] :
      ( ! [P2: a > a > $o] :
          ( X2
         != ( produc8111569692950616493list_a @ P2 @ nil_a ) )
     => ~ ! [P2: a > a > $o,X3: a,Ys2: list_a] :
            ( X2
           != ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X3 @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_157_successively_Ocases,axiom,
    ! [X2: produc5032551385658279741list_a] :
      ( ! [P2: a > a > $o] :
          ( X2
         != ( produc8111569692950616493list_a @ P2 @ nil_a ) )
     => ( ! [P2: a > a > $o,X3: a] :
            ( X2
           != ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X3 @ nil_a ) ) )
       => ~ ! [P2: a > a > $o,X3: a,Y2: a,Xs2: list_a] :
              ( X2
             != ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X3 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_158_splice_Ocases,axiom,
    ! [X2: produc9164743771328383783list_a] :
      ( ! [Ys2: list_a] :
          ( X2
         != ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
     => ~ ! [X3: a,Xs2: list_a,Ys2: list_a] :
            ( X2
           != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_159_shuffles_Ocases,axiom,
    ! [X2: produc9164743771328383783list_a] :
      ( ! [Ys2: list_a] :
          ( X2
         != ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
     => ( ! [Xs2: list_a] :
            ( X2
           != ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
       => ~ ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
              ( X2
             != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_160_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( ( suc @ N )
        = ( size_size_list_a @ Xs ) )
      = ( ? [Y3: a,Ys3: list_a] :
            ( ( Xs
              = ( cons_a @ Y3 @ Ys3 ) )
            & ( ( size_size_list_a @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_161_length__Suc__conv,axiom,
    ! [Xs: list_a,N: nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: a,Ys3: list_a] :
            ( ( Xs
              = ( cons_a @ Y3 @ Ys3 ) )
            & ( ( size_size_list_a @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_162_fk__update__2_Ocases,axiom,
    ! [X2: produc8642769642335960151at_nat] :
      ~ ! [A3: nat,M2: nat,X3: nat,L: nat] :
          ( X2
         != ( produc6385450045882626063at_nat @ A3 @ ( produc487386426758144856at_nat @ M2 @ ( product_Pair_nat_nat @ X3 @ L ) ) ) ) ).

% fk_update_2.cases
thf(fact_163_fk__update__2_Ocases,axiom,
    ! [X2: produc3791183549783780439_a_nat] :
      ~ ! [A3: a,M2: nat,X3: a,L: nat] :
          ( X2
         != ( produc3776435479397687751_a_nat @ A3 @ ( produc148073511828866022_a_nat @ M2 @ ( product_Pair_a_nat @ X3 @ L ) ) ) ) ).

% fk_update_2.cases
thf(fact_164_prod_Oexhaust,axiom,
    ! [Y: produc6774132644148096814_a_nat] :
      ~ ! [X12: nat,X24: product_prod_a_nat] :
          ( Y
         != ( produc148073511828866022_a_nat @ X12 @ X24 ) ) ).

% prod.exhaust
thf(fact_165_prod_Oexhaust,axiom,
    ! [Y: produc1161000763195272519_float] :
      ~ ! [X12: nat,X24: interval_float] :
          ( Y
         != ( produc6067866782486577919_float @ X12 @ X24 ) ) ).

% prod.exhaust
thf(fact_166_prod_Oexhaust,axiom,
    ! [Y: produc5883183851519154423_float] :
      ~ ! [X12: nat,X24: float] :
          ( Y
         != ( produc518625033508411951_float @ X12 @ X24 ) ) ).

% prod.exhaust
thf(fact_167_prod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [X12: nat,X24: nat] :
          ( Y
         != ( product_Pair_nat_nat @ X12 @ X24 ) ) ).

% prod.exhaust
thf(fact_168_prod_Oexhaust,axiom,
    ! [Y: product_prod_a_nat] :
      ~ ! [X12: a,X24: nat] :
          ( Y
         != ( product_Pair_a_nat @ X12 @ X24 ) ) ).

% prod.exhaust
thf(fact_169_surj__pair,axiom,
    ! [P3: produc6774132644148096814_a_nat] :
    ? [X3: nat,Y2: product_prod_a_nat] :
      ( P3
      = ( produc148073511828866022_a_nat @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_170_surj__pair,axiom,
    ! [P3: produc1161000763195272519_float] :
    ? [X3: nat,Y2: interval_float] :
      ( P3
      = ( produc6067866782486577919_float @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_171_surj__pair,axiom,
    ! [P3: produc5883183851519154423_float] :
    ? [X3: nat,Y2: float] :
      ( P3
      = ( produc518625033508411951_float @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_172_surj__pair,axiom,
    ! [P3: product_prod_nat_nat] :
    ? [X3: nat,Y2: nat] :
      ( P3
      = ( product_Pair_nat_nat @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_173_surj__pair,axiom,
    ! [P3: product_prod_a_nat] :
    ? [X3: a,Y2: nat] :
      ( P3
      = ( product_Pair_a_nat @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_174_prod__cases,axiom,
    ! [P: produc6774132644148096814_a_nat > $o,P3: produc6774132644148096814_a_nat] :
      ( ! [A3: nat,B3: product_prod_a_nat] : ( P @ ( produc148073511828866022_a_nat @ A3 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_175_prod__cases,axiom,
    ! [P: produc1161000763195272519_float > $o,P3: produc1161000763195272519_float] :
      ( ! [A3: nat,B3: interval_float] : ( P @ ( produc6067866782486577919_float @ A3 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_176_prod__cases,axiom,
    ! [P: produc5883183851519154423_float > $o,P3: produc5883183851519154423_float] :
      ( ! [A3: nat,B3: float] : ( P @ ( produc518625033508411951_float @ A3 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_177_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_178_prod__cases,axiom,
    ! [P: product_prod_a_nat > $o,P3: product_prod_a_nat] :
      ( ! [A3: a,B3: nat] : ( P @ ( product_Pair_a_nat @ A3 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_179_Pair__inject,axiom,
    ! [A: nat,B: product_prod_a_nat,A2: nat,B2: product_prod_a_nat] :
      ( ( ( produc148073511828866022_a_nat @ A @ B )
        = ( produc148073511828866022_a_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_180_Pair__inject,axiom,
    ! [A: nat,B: interval_float,A2: nat,B2: interval_float] :
      ( ( ( produc6067866782486577919_float @ A @ B )
        = ( produc6067866782486577919_float @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_181_Pair__inject,axiom,
    ! [A: nat,B: float,A2: nat,B2: float] :
      ( ( ( produc518625033508411951_float @ A @ B )
        = ( produc518625033508411951_float @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_182_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_183_Pair__inject,axiom,
    ! [A: a,B: nat,A2: a,B2: nat] :
      ( ( ( product_Pair_a_nat @ A @ B )
        = ( product_Pair_a_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_184_length__Suc__conv__rev,axiom,
    ! [Xs: list_a,N: nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: a,Ys3: list_a] :
            ( ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ Y3 @ nil_a ) ) )
            & ( ( size_size_list_a @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_185_bind__assoc__pmf,axiom,
    ! [A4: probab469873468395307276mf_nat,B4: nat > probab4139393509369344520_a_nat,C2: produc6774132644148096814_a_nat > probab469873468395307276mf_nat] :
      ( ( probab6994650295535379143at_nat @ ( probab1029661319033422245_a_nat @ A4 @ B4 ) @ C2 )
      = ( probab5865774939380454169at_nat @ A4
        @ ^ [X: nat] : ( probab6994650295535379143at_nat @ ( B4 @ X ) @ C2 ) ) ) ).

% bind_assoc_pmf
thf(fact_186_bind__assoc__pmf,axiom,
    ! [A4: probab4139393509369344520_a_nat,B4: produc6774132644148096814_a_nat > probab4139393509369344520_a_nat,C2: produc6774132644148096814_a_nat > probab4139393509369344520_a_nat] :
      ( ( probab5175300157245341303_a_nat @ ( probab5175300157245341303_a_nat @ A4 @ B4 ) @ C2 )
      = ( probab5175300157245341303_a_nat @ A4
        @ ^ [X: produc6774132644148096814_a_nat] : ( probab5175300157245341303_a_nat @ ( B4 @ X ) @ C2 ) ) ) ).

% bind_assoc_pmf
thf(fact_187_bind__assoc__pmf,axiom,
    ! [A4: probab469873468395307276mf_nat,B4: nat > probab4139393509369344520_a_nat,C2: produc6774132644148096814_a_nat > probab4139393509369344520_a_nat] :
      ( ( probab5175300157245341303_a_nat @ ( probab1029661319033422245_a_nat @ A4 @ B4 ) @ C2 )
      = ( probab1029661319033422245_a_nat @ A4
        @ ^ [X: nat] : ( probab5175300157245341303_a_nat @ ( B4 @ X ) @ C2 ) ) ) ).

% bind_assoc_pmf
thf(fact_188_bind__assoc__pmf,axiom,
    ! [A4: probab4139393509369344520_a_nat,B4: produc6774132644148096814_a_nat > probab469873468395307276mf_nat,C2: nat > probab4139393509369344520_a_nat] :
      ( ( probab1029661319033422245_a_nat @ ( probab6994650295535379143at_nat @ A4 @ B4 ) @ C2 )
      = ( probab5175300157245341303_a_nat @ A4
        @ ^ [X: produc6774132644148096814_a_nat] : ( probab1029661319033422245_a_nat @ ( B4 @ X ) @ C2 ) ) ) ).

% bind_assoc_pmf
thf(fact_189_bind__assoc__pmf,axiom,
    ! [A4: probab1498759712122475378_pmf_o,B4: $o > probab469873468395307276mf_nat,C2: nat > probab4139393509369344520_a_nat] :
      ( ( probab1029661319033422245_a_nat @ ( probab4549177196568532785_o_nat @ A4 @ B4 ) @ C2 )
      = ( probab7780334331721979789_a_nat @ A4
        @ ^ [X: $o] : ( probab1029661319033422245_a_nat @ ( B4 @ X ) @ C2 ) ) ) ).

% bind_assoc_pmf
thf(fact_190_bind__assoc__pmf,axiom,
    ! [A4: probab469873468395307276mf_nat,B4: nat > probab469873468395307276mf_nat,C2: nat > probab4139393509369344520_a_nat] :
      ( ( probab1029661319033422245_a_nat @ ( probab5865774939380454169at_nat @ A4 @ B4 ) @ C2 )
      = ( probab1029661319033422245_a_nat @ A4
        @ ^ [X: nat] : ( probab1029661319033422245_a_nat @ ( B4 @ X ) @ C2 ) ) ) ).

% bind_assoc_pmf
thf(fact_191_bind__assoc__pmf,axiom,
    ! [A4: probab1498759712122475378_pmf_o,B4: $o > probab1498759712122475378_pmf_o,C2: $o > probab469873468395307276mf_nat] :
      ( ( probab4549177196568532785_o_nat @ ( probab5433728543877361143mf_o_o @ A4 @ B4 ) @ C2 )
      = ( probab4549177196568532785_o_nat @ A4
        @ ^ [X: $o] : ( probab4549177196568532785_o_nat @ ( B4 @ X ) @ C2 ) ) ) ).

% bind_assoc_pmf
thf(fact_192_bind__assoc__pmf,axiom,
    ! [A4: probab469873468395307276mf_nat,B4: nat > probab1498759712122475378_pmf_o,C2: $o > probab469873468395307276mf_nat] :
      ( ( probab4549177196568532785_o_nat @ ( probab7651848557676718095_nat_o @ A4 @ B4 ) @ C2 )
      = ( probab5865774939380454169at_nat @ A4
        @ ^ [X: nat] : ( probab4549177196568532785_o_nat @ ( B4 @ X ) @ C2 ) ) ) ).

% bind_assoc_pmf
thf(fact_193_bind__assoc__pmf,axiom,
    ! [A4: probab1498759712122475378_pmf_o,B4: $o > probab469873468395307276mf_nat,C2: nat > probab469873468395307276mf_nat] :
      ( ( probab5865774939380454169at_nat @ ( probab4549177196568532785_o_nat @ A4 @ B4 ) @ C2 )
      = ( probab4549177196568532785_o_nat @ A4
        @ ^ [X: $o] : ( probab5865774939380454169at_nat @ ( B4 @ X ) @ C2 ) ) ) ).

% bind_assoc_pmf
thf(fact_194_bind__assoc__pmf,axiom,
    ! [A4: probab469873468395307276mf_nat,B4: nat > probab469873468395307276mf_nat,C2: nat > probab469873468395307276mf_nat] :
      ( ( probab5865774939380454169at_nat @ ( probab5865774939380454169at_nat @ A4 @ B4 ) @ C2 )
      = ( probab5865774939380454169at_nat @ A4
        @ ^ [X: nat] : ( probab5865774939380454169at_nat @ ( B4 @ X ) @ C2 ) ) ) ).

% bind_assoc_pmf
thf(fact_195_bind__commute__pmf,axiom,
    ! [A4: probab4139393509369344520_a_nat,B4: probab4139393509369344520_a_nat,C2: produc6774132644148096814_a_nat > produc6774132644148096814_a_nat > probab4139393509369344520_a_nat] :
      ( ( probab5175300157245341303_a_nat @ A4
        @ ^ [X: produc6774132644148096814_a_nat] : ( probab5175300157245341303_a_nat @ B4 @ ( C2 @ X ) ) )
      = ( probab5175300157245341303_a_nat @ B4
        @ ^ [Y3: produc6774132644148096814_a_nat] :
            ( probab5175300157245341303_a_nat @ A4
            @ ^ [X: produc6774132644148096814_a_nat] : ( C2 @ X @ Y3 ) ) ) ) ).

% bind_commute_pmf
thf(fact_196_bind__commute__pmf,axiom,
    ! [A4: probab4139393509369344520_a_nat,B4: probab469873468395307276mf_nat,C2: produc6774132644148096814_a_nat > nat > probab4139393509369344520_a_nat] :
      ( ( probab5175300157245341303_a_nat @ A4
        @ ^ [X: produc6774132644148096814_a_nat] : ( probab1029661319033422245_a_nat @ B4 @ ( C2 @ X ) ) )
      = ( probab1029661319033422245_a_nat @ B4
        @ ^ [Y3: nat] :
            ( probab5175300157245341303_a_nat @ A4
            @ ^ [X: produc6774132644148096814_a_nat] : ( C2 @ X @ Y3 ) ) ) ) ).

% bind_commute_pmf
thf(fact_197_bind__commute__pmf,axiom,
    ! [A4: probab469873468395307276mf_nat,B4: probab4139393509369344520_a_nat,C2: nat > produc6774132644148096814_a_nat > probab4139393509369344520_a_nat] :
      ( ( probab1029661319033422245_a_nat @ A4
        @ ^ [X: nat] : ( probab5175300157245341303_a_nat @ B4 @ ( C2 @ X ) ) )
      = ( probab5175300157245341303_a_nat @ B4
        @ ^ [Y3: produc6774132644148096814_a_nat] :
            ( probab1029661319033422245_a_nat @ A4
            @ ^ [X: nat] : ( C2 @ X @ Y3 ) ) ) ) ).

% bind_commute_pmf
thf(fact_198_bind__commute__pmf,axiom,
    ! [A4: probab469873468395307276mf_nat,B4: probab469873468395307276mf_nat,C2: nat > nat > probab4139393509369344520_a_nat] :
      ( ( probab1029661319033422245_a_nat @ A4
        @ ^ [X: nat] : ( probab1029661319033422245_a_nat @ B4 @ ( C2 @ X ) ) )
      = ( probab1029661319033422245_a_nat @ B4
        @ ^ [Y3: nat] :
            ( probab1029661319033422245_a_nat @ A4
            @ ^ [X: nat] : ( C2 @ X @ Y3 ) ) ) ) ).

% bind_commute_pmf
thf(fact_199_bind__commute__pmf,axiom,
    ! [A4: probab1498759712122475378_pmf_o,B4: probab1498759712122475378_pmf_o,C2: $o > $o > probab469873468395307276mf_nat] :
      ( ( probab4549177196568532785_o_nat @ A4
        @ ^ [X: $o] : ( probab4549177196568532785_o_nat @ B4 @ ( C2 @ X ) ) )
      = ( probab4549177196568532785_o_nat @ B4
        @ ^ [Y3: $o] :
            ( probab4549177196568532785_o_nat @ A4
            @ ^ [X: $o] : ( C2 @ X @ Y3 ) ) ) ) ).

% bind_commute_pmf
thf(fact_200_bind__commute__pmf,axiom,
    ! [A4: probab1498759712122475378_pmf_o,B4: probab469873468395307276mf_nat,C2: $o > nat > probab469873468395307276mf_nat] :
      ( ( probab4549177196568532785_o_nat @ A4
        @ ^ [X: $o] : ( probab5865774939380454169at_nat @ B4 @ ( C2 @ X ) ) )
      = ( probab5865774939380454169at_nat @ B4
        @ ^ [Y3: nat] :
            ( probab4549177196568532785_o_nat @ A4
            @ ^ [X: $o] : ( C2 @ X @ Y3 ) ) ) ) ).

% bind_commute_pmf
thf(fact_201_bind__commute__pmf,axiom,
    ! [A4: probab469873468395307276mf_nat,B4: probab1498759712122475378_pmf_o,C2: nat > $o > probab469873468395307276mf_nat] :
      ( ( probab5865774939380454169at_nat @ A4
        @ ^ [X: nat] : ( probab4549177196568532785_o_nat @ B4 @ ( C2 @ X ) ) )
      = ( probab4549177196568532785_o_nat @ B4
        @ ^ [Y3: $o] :
            ( probab5865774939380454169at_nat @ A4
            @ ^ [X: nat] : ( C2 @ X @ Y3 ) ) ) ) ).

% bind_commute_pmf
thf(fact_202_bind__commute__pmf,axiom,
    ! [A4: probab469873468395307276mf_nat,B4: probab469873468395307276mf_nat,C2: nat > nat > probab469873468395307276mf_nat] :
      ( ( probab5865774939380454169at_nat @ A4
        @ ^ [X: nat] : ( probab5865774939380454169at_nat @ B4 @ ( C2 @ X ) ) )
      = ( probab5865774939380454169at_nat @ B4
        @ ^ [Y3: nat] :
            ( probab5865774939380454169at_nat @ A4
            @ ^ [X: nat] : ( C2 @ X @ Y3 ) ) ) ) ).

% bind_commute_pmf
thf(fact_203_prod__cases3,axiom,
    ! [Y: produc6774132644148096814_a_nat] :
      ~ ! [A3: nat,B3: a,C3: nat] :
          ( Y
         != ( produc148073511828866022_a_nat @ A3 @ ( product_Pair_a_nat @ B3 @ C3 ) ) ) ).

% prod_cases3
thf(fact_204_prod__induct3,axiom,
    ! [P: produc6774132644148096814_a_nat > $o,X2: produc6774132644148096814_a_nat] :
      ( ! [A3: nat,B3: a,C3: nat] : ( P @ ( produc148073511828866022_a_nat @ A3 @ ( product_Pair_a_nat @ B3 @ C3 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_205_bind__return__pmf,axiom,
    ! [X2: produc6774132644148096814_a_nat,F: produc6774132644148096814_a_nat > probab4139393509369344520_a_nat] :
      ( ( probab5175300157245341303_a_nat @ ( probab6296365750099732169_a_nat @ X2 ) @ F )
      = ( F @ X2 ) ) ).

% bind_return_pmf
thf(fact_206_bind__return__pmf,axiom,
    ! [X2: nat,F: nat > probab4139393509369344520_a_nat] :
      ( ( probab1029661319033422245_a_nat @ ( probab4138676067695765237mf_nat @ X2 ) @ F )
      = ( F @ X2 ) ) ).

% bind_return_pmf
thf(fact_207_bind__return__pmf,axiom,
    ! [X2: nat,F: nat > probab469873468395307276mf_nat] :
      ( ( probab5865774939380454169at_nat @ ( probab4138676067695765237mf_nat @ X2 ) @ F )
      = ( F @ X2 ) ) ).

% bind_return_pmf
thf(fact_208_bind__return__pmf,axiom,
    ! [X2: $o,F: $o > probab469873468395307276mf_nat] :
      ( ( probab4549177196568532785_o_nat @ ( probab7739007152833094963_pmf_o @ X2 ) @ F )
      = ( F @ X2 ) ) ).

% bind_return_pmf
thf(fact_209_bind__return__pmf_H,axiom,
    ! [N2: probab4139393509369344520_a_nat] :
      ( ( probab5175300157245341303_a_nat @ N2 @ probab6296365750099732169_a_nat )
      = N2 ) ).

% bind_return_pmf'
thf(fact_210_bind__return__pmf_H,axiom,
    ! [N2: probab469873468395307276mf_nat] :
      ( ( probab5865774939380454169at_nat @ N2 @ probab4138676067695765237mf_nat )
      = N2 ) ).

% bind_return_pmf'
thf(fact_211_bind__return__pmf_H,axiom,
    ! [N2: probab1498759712122475378_pmf_o] :
      ( ( probab5433728543877361143mf_o_o @ N2 @ probab7739007152833094963_pmf_o )
      = N2 ) ).

% bind_return_pmf'
thf(fact_212_length__append__singleton,axiom,
    ! [Xs: list_a,X2: a] :
      ( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
      = ( suc @ ( size_size_list_a @ Xs ) ) ) ).

% length_append_singleton
thf(fact_213_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_a @ N @ nil_a )
          = ( cons_list_a @ nil_a @ nil_list_a ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_a @ N @ nil_a )
          = nil_list_a ) ) ) ).

% n_lists_Nil
thf(fact_214_length__Cons,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( size_size_list_a @ ( cons_a @ X2 @ Xs ) )
      = ( suc @ ( size_size_list_a @ Xs ) ) ) ).

% length_Cons
thf(fact_215_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_216_nat_Oinject,axiom,
    ! [X23: nat,Y23: nat] :
      ( ( ( suc @ X23 )
        = ( suc @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% nat.inject
thf(fact_217_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_218_Suc__inject,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y ) )
     => ( X2 = Y ) ) ).

% Suc_inject
thf(fact_219_Suc__n__not__n,axiom,
    ! [N: nat] :
      ( ( suc @ N )
     != N ) ).

% Suc_n_not_n
thf(fact_220_size__neq__size__imp__neq,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( ( size_size_list_a @ X2 )
       != ( size_size_list_a @ Y ) )
     => ( X2 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_221_size__neq__size__imp__neq,axiom,
    ! [X2: char,Y: char] :
      ( ( ( size_size_char @ X2 )
       != ( size_size_char @ Y ) )
     => ( X2 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_222_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M2: nat] :
          ( N
          = ( suc @ M2 ) ) ) ).

% not0_implies_Suc
thf(fact_223_Zero__not__Suc,axiom,
    ! [M3: nat] :
      ( zero_zero_nat
     != ( suc @ M3 ) ) ).

% Zero_not_Suc
thf(fact_224_Zero__neq__Suc,axiom,
    ! [M3: nat] :
      ( zero_zero_nat
     != ( suc @ M3 ) ) ).

% Zero_neq_Suc
thf(fact_225_Suc__neq__Zero,axiom,
    ! [M3: nat] :
      ( ( suc @ M3 )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_226_zero__induct,axiom,
    ! [P: nat > $o,K2: nat] :
      ( ( P @ K2 )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_227_diff__induct,axiom,
    ! [P: nat > nat > $o,M3: nat,N: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
       => ( ! [X3: nat,Y2: nat] :
              ( ( P @ X3 @ Y2 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
         => ( P @ M3 @ N ) ) ) ) ).

% diff_induct
thf(fact_228_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_229_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_230_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat
        = ( suc @ X23 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_231_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_232_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_233_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] :
      ( zero_zero_nat
     != ( suc @ X23 ) ) ).

% nat.distinct(1)
thf(fact_234_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_a] :
      ( ( n_lists_a @ zero_zero_nat @ Xs )
      = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% n_lists.simps(1)
thf(fact_235_longest__common__prefix_Ocases,axiom,
    ! [X2: produc9164743771328383783list_a] :
      ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
          ( X2
         != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
     => ( ! [Uv: list_a] :
            ( X2
           != ( produc6837034575241423639list_a @ nil_a @ Uv ) )
       => ~ ! [Uu: list_a] :
              ( X2
             != ( produc6837034575241423639list_a @ Uu @ nil_a ) ) ) ) ).

% longest_common_prefix.cases
thf(fact_236_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_237_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_238_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_239_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_240_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A4 ) )
      = ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_241_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y2: nat] :
          ~ ( member_nat @ Y2 @ A4 )
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_242_equals0D,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_243_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_244_gcd_Ocases,axiom,
    ! [X2: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( X2
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% gcd.cases
thf(fact_245_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% Set.empty_def
thf(fact_246_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C4: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_247_prefixes__snoc,axiom,
    ! [Xs: list_a,X2: a] :
      ( ( prefixes_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
      = ( append_list_a @ ( prefixes_a @ Xs ) @ ( cons_list_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) @ nil_list_a ) ) ) ).

% prefixes_snoc
thf(fact_248_Succ__def,axiom,
    ( bNF_Greatest_Succ_a
    = ( ^ [Kl: set_list_a,Kl2: list_a] :
          ( collect_a
          @ ^ [K: a] : ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl ) ) ) ) ).

% Succ_def
thf(fact_249_zero__reorient,axiom,
    ! [X2: nat] :
      ( ( zero_zero_nat = X2 )
      = ( X2 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_250_zero__reorient,axiom,
    ! [X2: a] :
      ( ( zero_zero_a = X2 )
      = ( X2 = zero_zero_a ) ) ).

% zero_reorient
thf(fact_251_zero__reorient,axiom,
    ! [X2: int] :
      ( ( zero_zero_int = X2 )
      = ( X2 = zero_zero_int ) ) ).

% zero_reorient
thf(fact_252_prefixes_Osimps_I1_J,axiom,
    ( ( prefixes_a @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% prefixes.simps(1)
thf(fact_253_prefixes__eq__snoc,axiom,
    ! [Ys: list_a,Xs: list_list_a,X2: list_a] :
      ( ( ( prefixes_a @ Ys )
        = ( append_list_a @ Xs @ ( cons_list_a @ X2 @ nil_list_a ) ) )
      = ( ( ( ( Ys = nil_a )
            & ( Xs = nil_list_a ) )
          | ? [Z2: a,Zs3: list_a] :
              ( ( Ys
                = ( append_a @ Zs3 @ ( cons_a @ Z2 @ nil_a ) ) )
              & ( Xs
                = ( prefixes_a @ Zs3 ) ) ) )
        & ( X2 = Ys ) ) ) ).

% prefixes_eq_snoc
thf(fact_254_SuccI,axiom,
    ! [Kl3: list_a,K2: a,Kl4: set_list_a] :
      ( ( member_list_a @ ( append_a @ Kl3 @ ( cons_a @ K2 @ nil_a ) ) @ Kl4 )
     => ( member_a @ K2 @ ( bNF_Greatest_Succ_a @ Kl4 @ Kl3 ) ) ) ).

% SuccI
thf(fact_255_SuccD,axiom,
    ! [K2: a,Kl4: set_list_a,Kl3: list_a] :
      ( ( member_a @ K2 @ ( bNF_Greatest_Succ_a @ Kl4 @ Kl3 ) )
     => ( member_list_a @ ( append_a @ Kl3 @ ( cons_a @ K2 @ nil_a ) ) @ Kl4 ) ) ).

% SuccD
thf(fact_256_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C4: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_257_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_258_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_259_empty__Shift,axiom,
    ! [Kl4: set_list_a,K2: a] :
      ( ( member_list_a @ nil_a @ Kl4 )
     => ( ( member_a @ K2 @ ( bNF_Greatest_Succ_a @ Kl4 @ nil_a ) )
       => ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl4 @ K2 ) ) ) ) ).

% empty_Shift
thf(fact_260_Succ__Shift,axiom,
    ! [Kl4: set_list_a,K2: a,Kl3: list_a] :
      ( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl4 @ K2 ) @ Kl3 )
      = ( bNF_Greatest_Succ_a @ Kl4 @ ( cons_a @ K2 @ Kl3 ) ) ) ).

% Succ_Shift
thf(fact_261_Cons__in__lex,axiom,
    ! [X2: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X2 = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_262_Cons__in__lex,axiom,
    ! [X2: a,Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R ) )
      = ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
          & ( ( size_size_list_a @ Xs )
            = ( size_size_list_a @ Ys ) ) )
        | ( ( X2 = Y )
          & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_263_gen__fib_Ocases,axiom,
    ! [X2: produc7248412053542808358at_nat] :
      ( ! [A3: nat,B3: nat] :
          ( X2
         != ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ zero_zero_nat ) ) )
     => ( ! [A3: nat,B3: nat] :
            ( X2
           != ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ ( suc @ zero_zero_nat ) ) ) )
       => ~ ! [A3: nat,B3: nat,N3: nat] :
              ( X2
             != ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ ( suc @ ( suc @ N3 ) ) ) ) ) ) ) ).

% gen_fib.cases
thf(fact_264_Nil2__notin__lex,axiom,
    ! [Xs: list_a,R: set_Product_prod_a_a] :
      ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( lex_a @ R ) ) ).

% Nil2_notin_lex
thf(fact_265_Nil__notin__lex,axiom,
    ! [Ys: list_a,R: set_Product_prod_a_a] :
      ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ys ) @ ( lex_a @ R ) ) ).

% Nil_notin_lex
thf(fact_266_lex__append__leftI,axiom,
    ! [Ys: list_a,Zs: list_a,R: set_Product_prod_a_a,Xs: list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R ) )
     => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R ) ) ) ).

% lex_append_leftI
thf(fact_267_ShiftD,axiom,
    ! [Kl3: list_a,Kl4: set_list_a,K2: a] :
      ( ( member_list_a @ Kl3 @ ( bNF_Greatest_Shift_a @ Kl4 @ K2 ) )
     => ( member_list_a @ ( cons_a @ K2 @ Kl3 ) @ Kl4 ) ) ).

% ShiftD
thf(fact_268_lex__append__leftD,axiom,
    ! [R: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ! [X3: a] :
          ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R )
     => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R ) )
       => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_269_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_270_lex__append__left__iff,axiom,
    ! [R: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ! [X3: a] :
          ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R )
     => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R ) )
        = ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_271_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_272_lex__append__rightI,axiom,
    ! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a,Vs: list_a,Us: list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) )
     => ( ( ( size_size_list_a @ Vs )
          = ( size_size_list_a @ Us ) )
       => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us ) @ ( append_a @ Ys @ Vs ) ) @ ( lex_a @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_273_Shift__def,axiom,
    ( bNF_Greatest_Shift_a
    = ( ^ [Kl: set_list_a,K: a] :
          ( collect_list_a
          @ ^ [Kl2: list_a] : ( member_list_a @ ( cons_a @ K @ Kl2 ) @ Kl ) ) ) ) ).

% Shift_def
thf(fact_274_bot__empty__eq2,axiom,
    ( bot_bo7769610567818710501_nat_o
    = ( ^ [X: nat,Y3: product_prod_a_nat] : ( member3063082432339977431_a_nat @ ( produc148073511828866022_a_nat @ X @ Y3 ) @ bot_bo5446707223867026810_a_nat ) ) ) ).

% bot_empty_eq2
thf(fact_275_bot__empty__eq2,axiom,
    ( bot_bo2715172778979421708loat_o
    = ( ^ [X: nat,Y3: interval_float] : ( member3149793176805732464_float @ ( produc6067866782486577919_float @ X @ Y3 ) @ bot_bo8806220978814136467_float ) ) ) ).

% bot_empty_eq2
thf(fact_276_bot__empty__eq2,axiom,
    ( bot_bot_nat_float_o
    = ( ^ [X: nat,Y3: float] : ( member2086872401261188768_float @ ( produc518625033508411951_float @ X @ Y3 ) @ bot_bo4936707370879519427_float ) ) ) ).

% bot_empty_eq2
thf(fact_277_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_278_bot__empty__eq2,axiom,
    ( bot_bot_a_nat_o
    = ( ^ [X: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y3 ) @ bot_bo9049108969261143879_a_nat ) ) ) ).

% bot_empty_eq2
thf(fact_279_pred__equals__eq2,axiom,
    ! [R2: set_Pr412391540666252558_a_nat,S3: set_Pr412391540666252558_a_nat] :
      ( ( ( ^ [X: nat,Y3: product_prod_a_nat] : ( member3063082432339977431_a_nat @ ( produc148073511828866022_a_nat @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: nat,Y3: product_prod_a_nat] : ( member3063082432339977431_a_nat @ ( produc148073511828866022_a_nat @ X @ Y3 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_280_pred__equals__eq2,axiom,
    ! [R2: set_Pr6933375829817990695_float,S3: set_Pr6933375829817990695_float] :
      ( ( ( ^ [X: nat,Y3: interval_float] : ( member3149793176805732464_float @ ( produc6067866782486577919_float @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: nat,Y3: interval_float] : ( member3149793176805732464_float @ ( produc6067866782486577919_float @ X @ Y3 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_281_pred__equals__eq2,axiom,
    ! [R2: set_Pr7206334407337572951_float,S3: set_Pr7206334407337572951_float] :
      ( ( ( ^ [X: nat,Y3: float] : ( member2086872401261188768_float @ ( produc518625033508411951_float @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: nat,Y3: float] : ( member2086872401261188768_float @ ( produc518625033508411951_float @ X @ Y3 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_282_pred__equals__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_283_pred__equals__eq2,axiom,
    ! [R2: set_Pr4934435412358123699_a_nat,S3: set_Pr4934435412358123699_a_nat] :
      ( ( ( ^ [X: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y3 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_284_fib_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ( ( X2
         != ( suc @ zero_zero_nat ) )
       => ~ ! [N3: nat] :
              ( X2
             != ( suc @ ( suc @ N3 ) ) ) ) ) ).

% fib.cases
thf(fact_285_encode__bounded__nat_Ocases,axiom,
    ! [X2: product_prod_nat_nat] :
      ( ! [L: nat,N3: nat] :
          ( X2
         != ( product_Pair_nat_nat @ ( suc @ L ) @ N3 ) )
     => ~ ! [Uu: nat] :
            ( X2
           != ( product_Pair_nat_nat @ zero_zero_nat @ Uu ) ) ) ).

% encode_bounded_nat.cases
thf(fact_286_lb__ln__horner_Ocases,axiom,
    ! [X2: produc9095508395758718541_float] :
      ( ! [Prec: nat,I: nat,X3: float] :
          ( X2
         != ( produc6596454488040397317_float @ Prec @ ( produc3893871907814880334_float @ zero_zero_nat @ ( produc518625033508411951_float @ I @ X3 ) ) ) )
     => ~ ! [Prec: nat,N3: nat,I: nat,X3: float] :
            ( X2
           != ( produc6596454488040397317_float @ Prec @ ( produc3893871907814880334_float @ ( suc @ N3 ) @ ( produc518625033508411951_float @ I @ X3 ) ) ) ) ) ).

% lb_ln_horner.cases
thf(fact_287_subdivide__interval_Ocases,axiom,
    ! [X2: produc1161000763195272519_float] :
      ( ! [I2: interval_float] :
          ( X2
         != ( produc6067866782486577919_float @ zero_zero_nat @ I2 ) )
     => ~ ! [N3: nat,I2: interval_float] :
            ( X2
           != ( produc6067866782486577919_float @ ( suc @ N3 ) @ I2 ) ) ) ).

% subdivide_interval.cases
thf(fact_288_lb__exp__horner_Ocases,axiom,
    ! [X2: produc985389371198253426_float] :
      ( ! [Prec: nat,I: nat,K3: nat,X3: float] :
          ( X2
         != ( produc2626873021911329572_float @ Prec @ ( produc6596454488040397317_float @ zero_zero_nat @ ( produc3893871907814880334_float @ I @ ( produc518625033508411951_float @ K3 @ X3 ) ) ) ) )
     => ~ ! [Prec: nat,N3: nat,I: nat,K3: nat,X3: float] :
            ( X2
           != ( produc2626873021911329572_float @ Prec @ ( produc6596454488040397317_float @ ( suc @ N3 ) @ ( produc3893871907814880334_float @ I @ ( produc518625033508411951_float @ K3 @ X3 ) ) ) ) ) ) ).

% lb_exp_horner.cases
thf(fact_289_sqrt__iteration_Ocases,axiom,
    ! [X2: produc4016807649911610908_float] :
      ( ! [Prec: nat,X3: float] :
          ( X2
         != ( produc3893871907814880334_float @ Prec @ ( produc518625033508411951_float @ zero_zero_nat @ X3 ) ) )
     => ~ ! [Prec: nat,M2: nat,X3: float] :
            ( X2
           != ( produc3893871907814880334_float @ Prec @ ( produc518625033508411951_float @ ( suc @ M2 ) @ X3 ) ) ) ) ).

% sqrt_iteration.cases
thf(fact_290_encode__unary__nat_Ocases,axiom,
    ! [X2: nat] :
      ( ! [L: nat] :
          ( X2
         != ( suc @ L ) )
     => ( X2 = zero_zero_nat ) ) ).

% encode_unary_nat.cases
thf(fact_291_power__up_Ocases,axiom,
    ! [X2: produc9155354469731034754al_nat] :
      ( ! [P4: nat,X3: real] :
          ( X2
         != ( produc4490577844368043572al_nat @ P4 @ ( produc3181502643871035669al_nat @ X3 @ zero_zero_nat ) ) )
     => ~ ! [P4: nat,X3: real,N3: nat] :
            ( X2
           != ( produc4490577844368043572al_nat @ P4 @ ( produc3181502643871035669al_nat @ X3 @ ( suc @ N3 ) ) ) ) ) ).

% power_up.cases
thf(fact_292_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N3: nat] :
            ( ~ ( P @ N3 )
            & ( P @ ( suc @ N3 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_293_list__decode_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ~ ! [N3: nat] :
            ( X2
           != ( suc @ N3 ) ) ) ).

% list_decode.cases
thf(fact_294_suffixes__eq__snoc,axiom,
    ! [Ys: list_a,Xs: list_list_a,X2: list_a] :
      ( ( ( suffixes_a @ Ys )
        = ( append_list_a @ Xs @ ( cons_list_a @ X2 @ nil_list_a ) ) )
      = ( ( ( ( Ys = nil_a )
            & ( Xs = nil_list_a ) )
          | ? [Z2: a,Zs3: list_a] :
              ( ( Ys
                = ( cons_a @ Z2 @ Zs3 ) )
              & ( Xs
                = ( suffixes_a @ Zs3 ) ) ) )
        & ( X2 = Ys ) ) ) ).

% suffixes_eq_snoc
thf(fact_295_length__suffixes,axiom,
    ! [Xs: list_a] :
      ( ( size_s349497388124573686list_a @ ( suffixes_a @ Xs ) )
      = ( suc @ ( size_size_list_a @ Xs ) ) ) ).

% length_suffixes
thf(fact_296_suffixes_Osimps_I1_J,axiom,
    ( ( suffixes_a @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% suffixes.simps(1)
thf(fact_297_suffixes_Osimps_I2_J,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( suffixes_a @ ( cons_a @ X2 @ Xs ) )
      = ( append_list_a @ ( suffixes_a @ Xs ) @ ( cons_list_a @ ( cons_a @ X2 @ Xs ) @ nil_list_a ) ) ) ).

% suffixes.simps(2)
thf(fact_298_bot__prod__def,axiom,
    ( bot_bo3047382831089536473et_nat
    = ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) ) ).

% bot_prod_def
thf(fact_299_bot__prod__def,axiom,
    ( bot_bo8329445699581754659at_nat
    = ( produc641871753055645167at_nat @ bot_bot_set_nat @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_300_bot__prod__def,axiom,
    ( bot_bo3238181912221869603et_nat
    = ( produc4207506657711014383et_nat @ bot_bot_nat @ bot_bot_set_nat ) ) ).

% bot_prod_def
thf(fact_301_bot__prod__def,axiom,
    ( bot_bo2769642828321324397at_nat
    = ( product_Pair_nat_nat @ bot_bot_nat @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_302_zero__prod__def,axiom,
    ( zero_z4826781987016425390_float
    = ( produc518625033508411951_float @ zero_zero_nat @ zero_zero_float ) ) ).

% zero_prod_def
thf(fact_303_zero__prod__def,axiom,
    ( zero_z3979849011205770936at_nat
    = ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).

% zero_prod_def
thf(fact_304_zero__prod__def,axiom,
    ( zero_z2373507747348899244_nat_a
    = ( product_Pair_nat_a @ zero_zero_nat @ zero_zero_a ) ) ).

% zero_prod_def
thf(fact_305_zero__prod__def,axiom,
    ( zero_z9025370028551350036at_int
    = ( product_Pair_nat_int @ zero_zero_nat @ zero_zero_int ) ) ).

% zero_prod_def
thf(fact_306_zero__prod__def,axiom,
    ( zero_z8358716320479998086_a_nat
    = ( product_Pair_a_nat @ zero_zero_a @ zero_zero_nat ) ) ).

% zero_prod_def
thf(fact_307_zero__prod__def,axiom,
    ( zero_z6294860398188059486od_a_a
    = ( product_Pair_a_a @ zero_zero_a @ zero_zero_a ) ) ).

% zero_prod_def
thf(fact_308_zero__prod__def,axiom,
    ( zero_z4180865300970801378_a_int
    = ( product_Pair_a_int @ zero_zero_a @ zero_zero_int ) ) ).

% zero_prod_def
thf(fact_309_zero__prod__def,axiom,
    ( zero_z4979202723106825492nt_nat
    = ( product_Pair_int_nat @ zero_zero_int @ zero_zero_nat ) ) ).

% zero_prod_def
thf(fact_310_zero__prod__def,axiom,
    ( zero_z128362779678453328_int_a
    = ( product_Pair_int_a @ zero_zero_int @ zero_zero_a ) ) ).

% zero_prod_def
thf(fact_311_zero__prod__def,axiom,
    ( zero_z801351703597628784nt_int
    = ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) ).

% zero_prod_def
thf(fact_312_nths__Cons,axiom,
    ! [X2: a,L2: list_a,A4: set_nat] :
      ( ( nths_a @ ( cons_a @ X2 @ L2 ) @ A4 )
      = ( append_a @ ( if_list_a @ ( member_nat @ zero_zero_nat @ A4 ) @ ( cons_a @ X2 @ nil_a ) @ nil_a )
        @ ( nths_a @ L2
          @ ( collect_nat
            @ ^ [J: nat] : ( member_nat @ ( suc @ J ) @ A4 ) ) ) ) ) ).

% nths_Cons
thf(fact_313_sublists_Osimps_I1_J,axiom,
    ( ( sublists_a @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% sublists.simps(1)
thf(fact_314_suffixes__snoc,axiom,
    ! [Xs: list_a,X2: a] :
      ( ( suffixes_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
      = ( cons_list_a @ nil_a
        @ ( map_list_a_list_a
          @ ^ [Ys3: list_a] : ( append_a @ Ys3 @ ( cons_a @ X2 @ nil_a ) )
          @ ( suffixes_a @ Xs ) ) ) ) ).

% suffixes_snoc
thf(fact_315_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_a @ nil_list_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% product_lists.simps(1)
thf(fact_316_map__is__Nil__conv,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( ( map_a_a @ F @ Xs )
        = nil_a )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_317_Nil__is__map__conv,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( nil_a
        = ( map_a_a @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_318_list_Omap__disc__iff,axiom,
    ! [F: a > a,A: list_a] :
      ( ( ( map_a_a @ F @ A )
        = nil_a )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_319_length__map,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( size_size_list_a @ ( map_a_a @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_320_map__append,axiom,
    ! [F: a > a,Xs: list_a,Ys: list_a] :
      ( ( map_a_a @ F @ ( append_a @ Xs @ Ys ) )
      = ( append_a @ ( map_a_a @ F @ Xs ) @ ( map_a_a @ F @ Ys ) ) ) ).

% map_append
thf(fact_321_nths__nil,axiom,
    ! [A4: set_nat] :
      ( ( nths_a @ nil_a @ A4 )
      = nil_a ) ).

% nths_nil
thf(fact_322_nths__empty,axiom,
    ! [Xs: list_a] :
      ( ( nths_a @ Xs @ bot_bot_set_nat )
      = nil_a ) ).

% nths_empty
thf(fact_323_nths__singleton,axiom,
    ! [A4: set_nat,X2: a] :
      ( ( ( member_nat @ zero_zero_nat @ A4 )
       => ( ( nths_a @ ( cons_a @ X2 @ nil_a ) @ A4 )
          = ( cons_a @ X2 @ nil_a ) ) )
      & ( ~ ( member_nat @ zero_zero_nat @ A4 )
       => ( ( nths_a @ ( cons_a @ X2 @ nil_a ) @ A4 )
          = nil_a ) ) ) ).

% nths_singleton
thf(fact_324_map__eq__Cons__conv,axiom,
    ! [F: a > a,Xs: list_a,Y: a,Ys: list_a] :
      ( ( ( map_a_a @ F @ Xs )
        = ( cons_a @ Y @ Ys ) )
      = ( ? [Z2: a,Zs3: list_a] :
            ( ( Xs
              = ( cons_a @ Z2 @ Zs3 ) )
            & ( ( F @ Z2 )
              = Y )
            & ( ( map_a_a @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_325_Cons__eq__map__conv,axiom,
    ! [X2: a,Xs: list_a,F: a > a,Ys: list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( map_a_a @ F @ Ys ) )
      = ( ? [Z2: a,Zs3: list_a] :
            ( ( Ys
              = ( cons_a @ Z2 @ Zs3 ) )
            & ( X2
              = ( F @ Z2 ) )
            & ( Xs
              = ( map_a_a @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_326_map__eq__Cons__D,axiom,
    ! [F: a > a,Xs: list_a,Y: a,Ys: list_a] :
      ( ( ( map_a_a @ F @ Xs )
        = ( cons_a @ Y @ Ys ) )
     => ? [Z: a,Zs2: list_a] :
          ( ( Xs
            = ( cons_a @ Z @ Zs2 ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_a_a @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_327_Cons__eq__map__D,axiom,
    ! [X2: a,Xs: list_a,F: a > a,Ys: list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( map_a_a @ F @ Ys ) )
     => ? [Z: a,Zs2: list_a] :
          ( ( Ys
            = ( cons_a @ Z @ Zs2 ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_a_a @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_328_list_Osimps_I9_J,axiom,
    ! [F: a > a,X21: a,X22: list_a] :
      ( ( map_a_a @ F @ ( cons_a @ X21 @ X22 ) )
      = ( cons_a @ ( F @ X21 ) @ ( map_a_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_329_list_Osimps_I8_J,axiom,
    ! [F: a > a] :
      ( ( map_a_a @ F @ nil_a )
      = nil_a ) ).

% list.simps(8)
thf(fact_330_map__eq__append__conv,axiom,
    ! [F: a > a,Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( ( map_a_a @ F @ Xs )
        = ( append_a @ Ys @ Zs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_a @ F @ Us2 ) )
            & ( Zs
              = ( map_a_a @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_331_append__eq__map__conv,axiom,
    ! [Ys: list_a,Zs: list_a,F: a > a,Xs: list_a] :
      ( ( ( append_a @ Ys @ Zs )
        = ( map_a_a @ F @ Xs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_a @ F @ Us2 ) )
            & ( Zs
              = ( map_a_a @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_332_sublists_Osimps_I2_J,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( sublists_a @ ( cons_a @ X2 @ Xs ) )
      = ( append_list_a @ ( sublists_a @ Xs ) @ ( map_list_a_list_a @ ( cons_a @ X2 ) @ ( prefixes_a @ Xs ) ) ) ) ).

% sublists.simps(2)
thf(fact_333_prefixes_Osimps_I2_J,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( prefixes_a @ ( cons_a @ X2 @ Xs ) )
      = ( cons_list_a @ nil_a @ ( map_list_a_list_a @ ( cons_a @ X2 ) @ ( prefixes_a @ Xs ) ) ) ) ).

% prefixes.simps(2)
thf(fact_334_subseqs_Osimps_I2_J,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( subseqs_a @ ( cons_a @ X2 @ Xs ) )
      = ( append_list_a @ ( map_list_a_list_a @ ( cons_a @ X2 ) @ ( subseqs_a @ Xs ) ) @ ( subseqs_a @ Xs ) ) ) ).

% subseqs.simps(2)
thf(fact_335_enumerate__simps_I2_J,axiom,
    ! [N: nat,X2: a,Xs: list_a] :
      ( ( enumerate_a @ N @ ( cons_a @ X2 @ Xs ) )
      = ( cons_P8443330267410185325_nat_a @ ( product_Pair_nat_a @ N @ X2 ) @ ( enumerate_a @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_336_enumerate__simps_I2_J,axiom,
    ! [N: nat,X2: product_prod_a_nat,Xs: list_P3592885314253461005_a_nat] :
      ( ( enumer3239626523597437528_a_nat @ N @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs ) )
      = ( cons_P729672029421400548_a_nat @ ( produc148073511828866022_a_nat @ N @ X2 ) @ ( enumer3239626523597437528_a_nat @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_337_enumerate__simps_I2_J,axiom,
    ! [N: nat,X2: interval_float,Xs: list_interval_float] :
      ( ( enumer4582120449759364209_float @ N @ ( cons_interval_float @ X2 @ Xs ) )
      = ( cons_P6013094588511510781_float @ ( produc6067866782486577919_float @ N @ X2 ) @ ( enumer4582120449759364209_float @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_338_enumerate__simps_I2_J,axiom,
    ! [N: nat,X2: float,Xs: list_float] :
      ( ( enumerate_float @ N @ ( cons_float @ X2 @ Xs ) )
      = ( cons_P5322595555988802477_float @ ( produc518625033508411951_float @ N @ X2 ) @ ( enumerate_float @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_339_enumerate__simps_I2_J,axiom,
    ! [N: nat,X2: nat,Xs: list_nat] :
      ( ( enumerate_nat @ N @ ( cons_nat @ X2 @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X2 ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_340_pair__return__pmf,axiom,
    ! [X2: a,Y: nat] :
      ( ( probab7145198258937415252_a_nat @ ( probab7188105566731915033_pmf_a @ X2 ) @ ( probab4138676067695765237mf_nat @ Y ) )
      = ( probab289271046735565986_a_nat @ ( product_Pair_a_nat @ X2 @ Y ) ) ) ).

% pair_return_pmf
thf(fact_341_pair__return__pmf,axiom,
    ! [X2: nat,Y: float] :
      ( ( probab982631873860687634_float @ ( probab4138676067695765237mf_nat @ X2 ) @ ( probab3649709003403758315_float @ Y ) )
      = ( probab5153617962208536722_float @ ( produc518625033508411951_float @ X2 @ Y ) ) ) ).

% pair_return_pmf
thf(fact_342_pair__return__pmf,axiom,
    ! [X2: nat,Y: nat] :
      ( ( probab2751493682063440412at_nat @ ( probab4138676067695765237mf_nat @ X2 ) @ ( probab4138676067695765237mf_nat @ Y ) )
      = ( probab2446207027195173788at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) ) ) ).

% pair_return_pmf
thf(fact_343_pair__return__pmf,axiom,
    ! [X2: nat,Y: $o] :
      ( ( probab6450283233425377868_nat_o @ ( probab4138676067695765237mf_nat @ X2 ) @ ( probab7739007152833094963_pmf_o @ Y ) )
      = ( probab2459633660216095970_nat_o @ ( product_Pair_nat_o @ X2 @ Y ) ) ) ).

% pair_return_pmf
thf(fact_344_pair__return__pmf,axiom,
    ! [X2: $o,Y: nat] :
      ( ( probab3347611872317192558_o_nat @ ( probab7739007152833094963_pmf_o @ X2 ) @ ( probab4138676067695765237mf_nat @ Y ) )
      = ( probab8174471173015378748_o_nat @ ( product_Pair_o_nat @ X2 @ Y ) ) ) ).

% pair_return_pmf
thf(fact_345_pair__return__pmf,axiom,
    ! [X2: $o,Y: $o] :
      ( ( probab4104135168549319290mf_o_o @ ( probab7739007152833094963_pmf_o @ X2 ) @ ( probab7739007152833094963_pmf_o @ Y ) )
      = ( probab7555217079304768834od_o_o @ ( product_Pair_o_o @ X2 @ Y ) ) ) ).

% pair_return_pmf
thf(fact_346_pair__return__pmf,axiom,
    ! [X2: nat,Y: interval_float] :
      ( ( probab7611888048673320034_float @ ( probab4138676067695765237mf_nat @ X2 ) @ ( probab7441574048391196091_float @ Y ) )
      = ( probab1129944425973497186_float @ ( produc6067866782486577919_float @ X2 @ Y ) ) ) ).

% pair_return_pmf
thf(fact_347_pair__return__pmf,axiom,
    ! [X2: nat,Y: product_prod_a_nat] :
      ( ( probab199821339450058953_a_nat @ ( probab4138676067695765237mf_nat @ X2 ) @ ( probab289271046735565986_a_nat @ Y ) )
      = ( probab6296365750099732169_a_nat @ ( produc148073511828866022_a_nat @ X2 @ Y ) ) ) ).

% pair_return_pmf
thf(fact_348_pair__return__pmf,axiom,
    ! [X2: produc6774132644148096814_a_nat,Y: nat] :
      ( ( probab6900048016774989572at_nat @ ( probab6296365750099732169_a_nat @ X2 ) @ ( probab4138676067695765237mf_nat @ Y ) )
      = ( probab4986987171760901842at_nat @ ( produc2837828401664308135at_nat @ X2 @ Y ) ) ) ).

% pair_return_pmf
thf(fact_349_pair__return__pmf,axiom,
    ! [X2: produc6774132644148096814_a_nat,Y: $o] :
      ( ( probab4903275611151479652_nat_o @ ( probab6296365750099732169_a_nat @ X2 ) @ ( probab7739007152833094963_pmf_o @ Y ) )
      = ( probab8751481573885896748_nat_o @ ( produc8940506901291673601_nat_o @ X2 @ Y ) ) ) ).

% pair_return_pmf
thf(fact_350_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
      ( ( ( concat_a @ Xss2 )
        = ( append_a @ Ys @ Zs ) )
      = ( ( ( Xss2 = nil_list_a )
         => ( ( Ys = nil_a )
            & ( Zs = nil_a ) ) )
        & ( ( Xss2 != nil_list_a )
         => ? [Xss1: list_list_a,Xs4: list_a,Xs5: list_a,Xss22: list_list_a] :
              ( ( Xss2
                = ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs4 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys
                = ( append_a @ ( concat_a @ Xss1 ) @ Xs4 ) )
              & ( Zs
                = ( append_a @ Xs5 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_351_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_a,X2: a,Ys: list_a,Y: a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) @ ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) ) @ ( listrel1_a @ R ) )
      = ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
          & ( X2 = Y ) )
        | ( ( Xs = Ys )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_352_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
          & ( X2 = Y ) )
        | ( ( Xs = Ys )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_353_pair__pmf__def,axiom,
    ( probab7611888048673320034_float
    = ( ^ [A5: probab469873468395307276mf_nat,B5: probab3791811577382853330_float] :
          ( probab6486310893334782526_float @ A5
          @ ^ [X: nat] :
              ( probab8402744624613384056_float @ B5
              @ ^ [Y3: interval_float] : ( probab1129944425973497186_float @ ( produc6067866782486577919_float @ X @ Y3 ) ) ) ) ) ) ).

% pair_pmf_def
thf(fact_354_pair__pmf__def,axiom,
    ( probab982631873860687634_float
    = ( ^ [A5: probab469873468395307276mf_nat,B5: probab6629792655741328898_float] :
          ( probab4248178704490373998_float @ A5
          @ ^ [X: nat] :
              ( probab7594616382697444600_float @ B5
              @ ^ [Y3: float] : ( probab5153617962208536722_float @ ( produc518625033508411951_float @ X @ Y3 ) ) ) ) ) ) ).

% pair_pmf_def
thf(fact_355_pair__pmf__def,axiom,
    ( probab2751493682063440412at_nat
    = ( ^ [A5: probab469873468395307276mf_nat,B5: probab469873468395307276mf_nat] :
          ( probab2287559031411225208at_nat @ A5
          @ ^ [X: nat] :
              ( probab2287559031411225208at_nat @ B5
              @ ^ [Y3: nat] : ( probab2446207027195173788at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) ) ) ) ) ) ).

% pair_pmf_def
thf(fact_356_pair__pmf__def,axiom,
    ( probab7145198258937415252_a_nat
    = ( ^ [A5: probab3364570286911266904_pmf_a,B5: probab469873468395307276mf_nat] :
          ( probab2526244042035113540_a_nat @ A5
          @ ^ [X: a] :
              ( probab2528856301151534278_a_nat @ B5
              @ ^ [Y3: nat] : ( probab289271046735565986_a_nat @ ( product_Pair_a_nat @ X @ Y3 ) ) ) ) ) ) ).

% pair_pmf_def
thf(fact_357_pair__pmf__def,axiom,
    ( probab199821339450058953_a_nat
    = ( ^ [A5: probab469873468395307276mf_nat,B5: probab2020455115029699641_a_nat] :
          ( probab1029661319033422245_a_nat @ A5
          @ ^ [X: nat] :
              ( probab904420511876414392_a_nat @ B5
              @ ^ [Y3: product_prod_a_nat] : ( probab6296365750099732169_a_nat @ ( produc148073511828866022_a_nat @ X @ Y3 ) ) ) ) ) ) ).

% pair_pmf_def
thf(fact_358_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_a @ N @ nil_a )
      = nil_Pr1417316670369895453_nat_a ) ).

% enumerate_simps(1)
thf(fact_359_length__enumerate,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( size_s243904063682394823_nat_a @ ( enumerate_a @ N @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_enumerate
thf(fact_360_concat__append,axiom,
    ! [Xs: list_list_a,Ys: list_list_a] :
      ( ( concat_a @ ( append_list_a @ Xs @ Ys ) )
      = ( append_a @ ( concat_a @ Xs ) @ ( concat_a @ Ys ) ) ) ).

% concat_append
thf(fact_361_Cons__listrel1__Cons,axiom,
    ! [X2: a,Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R ) )
      = ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X2 = Y )
          & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_362_Cons__listrel1__Cons,axiom,
    ! [X2: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X2 = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_363_concat_Osimps_I2_J,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( concat_a @ ( cons_list_a @ X2 @ Xs ) )
      = ( append_a @ X2 @ ( concat_a @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_364_concat_Osimps_I1_J,axiom,
    ( ( concat_a @ nil_list_a )
    = nil_a ) ).

% concat.simps(1)
thf(fact_365_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_a,Xss2: list_list_a] :
      ( ( product_lists_a @ ( cons_list_a @ Xs @ Xss2 ) )
      = ( concat_list_a
        @ ( map_a_list_list_a
          @ ^ [X: a] : ( map_list_a_list_a @ ( cons_a @ X ) @ ( product_lists_a @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_366_listrel1I2,axiom,
    ! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a,X2: a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
     => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ X2 @ Ys ) ) @ ( listrel1_a @ R ) ) ) ).

% listrel1I2
thf(fact_367_not__listrel1__Nil,axiom,
    ! [Xs: list_a,R: set_Product_prod_a_a] :
      ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel1_a @ R ) ) ).

% not_listrel1_Nil
thf(fact_368_not__Nil__listrel1,axiom,
    ! [Xs: list_a,R: set_Product_prod_a_a] :
      ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel1_a @ R ) ) ).

% not_Nil_listrel1
thf(fact_369_listrel1__eq__len,axiom,
    ! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_370_append__listrel1I,axiom,
    ! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a,Us: list_a,Vs: list_a] :
      ( ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Vs ) @ ( listrel1_a @ R ) ) ) )
     => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us ) @ ( append_a @ Ys @ Vs ) ) @ ( listrel1_a @ R ) ) ) ).

% append_listrel1I
thf(fact_371_listrel1I1,axiom,
    ! [X2: a,Y: a,R: set_Product_prod_a_a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
     => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Xs ) ) @ ( listrel1_a @ R ) ) ) ).

% listrel1I1
thf(fact_372_listrel1I1,axiom,
    ! [X2: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I1
thf(fact_373_Cons__listrel1E1,axiom,
    ! [X2: a,Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs ) @ Ys ) @ ( listrel1_a @ R ) )
     => ( ! [Y2: a] :
            ( ( Ys
              = ( cons_a @ Y2 @ Xs ) )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ R ) )
       => ~ ! [Zs2: list_a] :
              ( ( Ys
                = ( cons_a @ X2 @ Zs2 ) )
             => ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs2 ) @ ( listrel1_a @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_374_Cons__listrel1E1,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ Ys ) @ ( listrel1_nat @ R ) )
     => ( ! [Y2: nat] :
            ( ( Ys
              = ( cons_nat @ Y2 @ Xs ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Ys
                = ( cons_nat @ X2 @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_375_Cons__listrel1E2,axiom,
    ! [Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R ) )
     => ( ! [X3: a] :
            ( ( Xs
              = ( cons_a @ X3 @ Ys ) )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R ) )
       => ~ ! [Zs2: list_a] :
              ( ( Xs
                = ( cons_a @ Y @ Zs2 ) )
             => ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Zs2 @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_376_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 ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Xs
                = ( cons_nat @ Y @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_377_listrel1E,axiom,
    ! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
     => ~ ! [X3: a,Y2: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R )
           => ! [Us3: list_a,Vs3: list_a] :
                ( ( Xs
                  = ( append_a @ Us3 @ ( cons_a @ X3 @ Vs3 ) ) )
               => ( Ys
                 != ( append_a @ Us3 @ ( cons_a @ Y2 @ Vs3 ) ) ) ) ) ) ).

% listrel1E
thf(fact_378_listrel1E,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
     => ~ ! [X3: nat,Y2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R )
           => ! [Us3: list_nat,Vs3: list_nat] :
                ( ( Xs
                  = ( append_nat @ Us3 @ ( cons_nat @ X3 @ Vs3 ) ) )
               => ( Ys
                 != ( append_nat @ Us3 @ ( cons_nat @ Y2 @ Vs3 ) ) ) ) ) ) ).

% listrel1E
thf(fact_379_listrel1I,axiom,
    ! [X2: a,Y: a,R: set_Product_prod_a_a,Xs: list_a,Us: list_a,Vs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
     => ( ( Xs
          = ( append_a @ Us @ ( cons_a @ X2 @ Vs ) ) )
       => ( ( Ys
            = ( append_a @ Us @ ( cons_a @ Y @ Vs ) ) )
         => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).

% listrel1I
thf(fact_380_listrel1I,axiom,
    ! [X2: 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 @ X2 @ Y ) @ R )
     => ( ( Xs
          = ( append_nat @ Us @ ( cons_nat @ X2 @ Vs ) ) )
       => ( ( Ys
            = ( append_nat @ Us @ ( cons_nat @ Y @ Vs ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% listrel1I
thf(fact_381_concat__eq__appendD,axiom,
    ! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
      ( ( ( concat_a @ Xss2 )
        = ( append_a @ Ys @ Zs ) )
     => ( ( Xss2 != nil_list_a )
       => ? [Xss12: list_list_a,Xs2: list_a,Xs3: list_a,Xss23: list_list_a] :
            ( ( Xss2
              = ( append_list_a @ Xss12 @ ( cons_list_a @ ( append_a @ Xs2 @ Xs3 ) @ Xss23 ) ) )
            & ( Ys
              = ( append_a @ ( concat_a @ Xss12 ) @ Xs2 ) )
            & ( Zs
              = ( append_a @ Xs3 @ ( concat_a @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_382_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_a @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% subseqs.simps(1)
thf(fact_383_n__lists_Osimps_I2_J,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( n_lists_a @ ( suc @ N ) @ Xs )
      = ( concat_list_a
        @ ( map_li5729356230488778442list_a
          @ ^ [Ys3: list_a] :
              ( map_a_list_a
              @ ^ [Y3: a] : ( cons_a @ Y3 @ Ys3 )
              @ Xs )
          @ ( n_lists_a @ N @ Xs ) ) ) ) ).

% n_lists.simps(2)
thf(fact_384_inverse__permutation__of__list_Ocases,axiom,
    ! [X2: produc4008378413191047942at_nat] :
      ( ! [X3: nat] :
          ( X2
         != ( produc8424349340415155968at_nat @ nil_Pr5478986624290739719at_nat @ X3 ) )
     => ~ ! [Y2: nat,X4: nat,Xs2: list_P6011104703257516679at_nat,X3: nat] :
            ( X2
           != ( produc8424349340415155968at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ Y2 @ X4 ) @ Xs2 ) @ X3 ) ) ) ).

% inverse_permutation_of_list.cases
thf(fact_385_delete__aux_Ocases,axiom,
    ! [X2: produc5672971924406059225_a_nat] :
      ( ! [K3: nat] :
          ( X2
         != ( produc7526178182849817355_a_nat @ K3 @ nil_Pr4135541677726875188_a_nat ) )
     => ~ ! [K3: nat,K4: nat,V: product_prod_a_nat,Xs2: list_P4901192995000098612_a_nat] :
            ( X2
           != ( produc7526178182849817355_a_nat @ K3 @ ( cons_P729672029421400548_a_nat @ ( produc148073511828866022_a_nat @ K4 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_386_delete__aux_Ocases,axiom,
    ! [X2: produc6499262172985382258_float] :
      ( ! [K3: nat] :
          ( X2
         != ( produc2655563877181872036_float @ K3 @ nil_Pr8949446520521958733_float ) )
     => ~ ! [K3: nat,K4: nat,V: interval_float,Xs2: list_P1609389641493977421_float] :
            ( X2
           != ( produc2655563877181872036_float @ K3 @ ( cons_P6013094588511510781_float @ ( produc6067866782486577919_float @ K4 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_387_delete__aux_Ocases,axiom,
    ! [X2: produc8277557972349479330_float] :
      ( ! [K3: nat] :
          ( X2
         != ( produc7773819858125983828_float @ K3 @ nil_Pr3827615237818013693_float ) )
     => ~ ! [K3: nat,K4: nat,V: float,Xs2: list_P454701268746800253_float] :
            ( X2
           != ( produc7773819858125983828_float @ K3 @ ( cons_P5322595555988802477_float @ ( produc518625033508411951_float @ K4 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_388_delete__aux_Ocases,axiom,
    ! [X2: produc8472197452120411308at_nat] :
      ( ! [K3: nat] :
          ( X2
         != ( produc6109913384486294878at_nat @ K3 @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [K3: nat,K4: nat,V: nat,Xs2: list_P6011104703257516679at_nat] :
            ( X2
           != ( produc6109913384486294878at_nat @ K3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_389_delete__aux_Ocases,axiom,
    ! [X2: produc3864505136547489594_a_nat] :
      ( ! [K3: a] :
          ( X2
         != ( produc6674818935846295348_a_nat @ K3 @ nil_Pr7402525243500994295_a_nat ) )
     => ~ ! [K3: a,K4: a,V: nat,Xs2: list_P3592885314253461005_a_nat] :
            ( X2
           != ( produc6674818935846295348_a_nat @ K3 @ ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ K4 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_390_maps__simps_I2_J,axiom,
    ! [F: a > list_a] :
      ( ( maps_a_a @ F @ nil_a )
      = nil_a ) ).

% maps_simps(2)
thf(fact_391_maps__simps_I1_J,axiom,
    ! [F: a > list_a,X2: a,Xs: list_a] :
      ( ( maps_a_a @ F @ ( cons_a @ X2 @ Xs ) )
      = ( append_a @ ( F @ X2 ) @ ( maps_a_a @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_392_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: product_prod_a_nat,K2: nat,F: product_prod_a_nat > product_prod_a_nat] :
      ( ( update3995510672940499665at_nat @ V2 @ K2 @ F @ nil_Pr4135541677726875188_a_nat )
      = ( cons_P729672029421400548_a_nat @ ( produc148073511828866022_a_nat @ K2 @ ( F @ V2 ) ) @ nil_Pr4135541677726875188_a_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_393_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: interval_float,K2: nat,F: interval_float > interval_float] :
      ( ( update44210888014591210at_nat @ V2 @ K2 @ F @ nil_Pr8949446520521958733_float )
      = ( cons_P6013094588511510781_float @ ( produc6067866782486577919_float @ K2 @ ( F @ V2 ) ) @ nil_Pr8949446520521958733_float ) ) ).

% update_with_aux.simps(1)
thf(fact_394_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: float,K2: nat,F: float > float] :
      ( ( update8899324884019004698at_nat @ V2 @ K2 @ F @ nil_Pr3827615237818013693_float )
      = ( cons_P5322595555988802477_float @ ( produc518625033508411951_float @ K2 @ ( F @ V2 ) ) @ nil_Pr3827615237818013693_float ) ) ).

% update_with_aux.simps(1)
thf(fact_395_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: nat,K2: nat,F: nat > nat] :
      ( ( update528237659335440164at_nat @ V2 @ K2 @ F @ nil_Pr5478986624290739719at_nat )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( F @ V2 ) ) @ nil_Pr5478986624290739719at_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_396_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: nat,K2: a,F: nat > nat] :
      ( ( update5219137139480407210_nat_a @ V2 @ K2 @ F @ nil_Pr7402525243500994295_a_nat )
      = ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ K2 @ ( F @ V2 ) ) @ nil_Pr7402525243500994295_a_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_397_map__default_Osimps_I1_J,axiom,
    ! [K2: nat,V2: product_prod_a_nat,F: product_prod_a_nat > product_prod_a_nat] :
      ( ( map_de8993022399731873447_a_nat @ K2 @ V2 @ F @ nil_Pr4135541677726875188_a_nat )
      = ( cons_P729672029421400548_a_nat @ ( produc148073511828866022_a_nat @ K2 @ V2 ) @ nil_Pr4135541677726875188_a_nat ) ) ).

% map_default.simps(1)
thf(fact_398_map__default_Osimps_I1_J,axiom,
    ! [K2: nat,V2: interval_float,F: interval_float > interval_float] :
      ( ( map_de4322568951476430656_float @ K2 @ V2 @ F @ nil_Pr8949446520521958733_float )
      = ( cons_P6013094588511510781_float @ ( produc6067866782486577919_float @ K2 @ V2 ) @ nil_Pr8949446520521958733_float ) ) ).

% map_default.simps(1)
thf(fact_399_map__default_Osimps_I1_J,axiom,
    ! [K2: nat,V2: float,F: float > float] :
      ( ( map_de9143837120676091120_float @ K2 @ V2 @ F @ nil_Pr3827615237818013693_float )
      = ( cons_P5322595555988802477_float @ ( produc518625033508411951_float @ K2 @ V2 ) @ nil_Pr3827615237818013693_float ) ) ).

% map_default.simps(1)
thf(fact_400_map__default_Osimps_I1_J,axiom,
    ! [K2: nat,V2: nat,F: nat > nat] :
      ( ( map_default_nat_nat @ K2 @ V2 @ F @ nil_Pr5478986624290739719at_nat )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ nil_Pr5478986624290739719at_nat ) ) ).

% map_default.simps(1)
thf(fact_401_map__default_Osimps_I1_J,axiom,
    ! [K2: a,V2: nat,F: nat > nat] :
      ( ( map_default_a_nat @ K2 @ V2 @ F @ nil_Pr7402525243500994295_a_nat )
      = ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ K2 @ V2 ) @ nil_Pr7402525243500994295_a_nat ) ) ).

% map_default.simps(1)
thf(fact_402_delete__aux_Oelims,axiom,
    ! [X2: nat,Xa: list_P4901192995000098612_a_nat,Y: list_P4901192995000098612_a_nat] :
      ( ( ( delete5594580531140670069_a_nat @ X2 @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr4135541677726875188_a_nat )
         => ( Y != nil_Pr4135541677726875188_a_nat ) )
       => ~ ! [K4: nat,V: product_prod_a_nat,Xs2: list_P4901192995000098612_a_nat] :
              ( ( Xa
                = ( cons_P729672029421400548_a_nat @ ( produc148073511828866022_a_nat @ K4 @ V ) @ Xs2 ) )
             => ~ ( ( ( X2 = K4 )
                   => ( Y = Xs2 ) )
                  & ( ( X2 != K4 )
                   => ( Y
                      = ( cons_P729672029421400548_a_nat @ ( produc148073511828866022_a_nat @ K4 @ V ) @ ( delete5594580531140670069_a_nat @ X2 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_403_delete__aux_Oelims,axiom,
    ! [X2: nat,Xa: list_P1609389641493977421_float,Y: list_P1609389641493977421_float] :
      ( ( ( delete7196171427534157838_float @ X2 @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr8949446520521958733_float )
         => ( Y != nil_Pr8949446520521958733_float ) )
       => ~ ! [K4: nat,V: interval_float,Xs2: list_P1609389641493977421_float] :
              ( ( Xa
                = ( cons_P6013094588511510781_float @ ( produc6067866782486577919_float @ K4 @ V ) @ Xs2 ) )
             => ~ ( ( ( X2 = K4 )
                   => ( Y = Xs2 ) )
                  & ( ( X2 != K4 )
                   => ( Y
                      = ( cons_P6013094588511510781_float @ ( produc6067866782486577919_float @ K4 @ V ) @ ( delete7196171427534157838_float @ X2 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_404_delete__aux_Oelims,axiom,
    ! [X2: nat,Xa: list_P454701268746800253_float,Y: list_P454701268746800253_float] :
      ( ( ( delete_aux_nat_float @ X2 @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr3827615237818013693_float )
         => ( Y != nil_Pr3827615237818013693_float ) )
       => ~ ! [K4: nat,V: float,Xs2: list_P454701268746800253_float] :
              ( ( Xa
                = ( cons_P5322595555988802477_float @ ( produc518625033508411951_float @ K4 @ V ) @ Xs2 ) )
             => ~ ( ( ( X2 = K4 )
                   => ( Y = Xs2 ) )
                  & ( ( X2 != K4 )
                   => ( Y
                      = ( cons_P5322595555988802477_float @ ( produc518625033508411951_float @ K4 @ V ) @ ( delete_aux_nat_float @ X2 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_405_delete__aux_Oelims,axiom,
    ! [X2: nat,Xa: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( delete_aux_nat_nat @ X2 @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
         => ( Y != nil_Pr5478986624290739719at_nat ) )
       => ~ ! [K4: nat,V: nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( Xa
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V ) @ Xs2 ) )
             => ~ ( ( ( X2 = K4 )
                   => ( Y = Xs2 ) )
                  & ( ( X2 != K4 )
                   => ( Y
                      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V ) @ ( delete_aux_nat_nat @ X2 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_406_delete__aux_Oelims,axiom,
    ! [X2: a,Xa: list_P3592885314253461005_a_nat,Y: list_P3592885314253461005_a_nat] :
      ( ( ( delete_aux_a_nat @ X2 @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr7402525243500994295_a_nat )
         => ( Y != nil_Pr7402525243500994295_a_nat ) )
       => ~ ! [K4: a,V: nat,Xs2: list_P3592885314253461005_a_nat] :
              ( ( Xa
                = ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ K4 @ V ) @ Xs2 ) )
             => ~ ( ( ( X2 = K4 )
                   => ( Y = Xs2 ) )
                  & ( ( X2 != K4 )
                   => ( Y
                      = ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ K4 @ V ) @ ( delete_aux_a_nat @ X2 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_407_delete__aux__eq__Nil__conv,axiom,
    ! [K2: nat,Ts: list_P4901192995000098612_a_nat] :
      ( ( ( delete5594580531140670069_a_nat @ K2 @ Ts )
        = nil_Pr4135541677726875188_a_nat )
      = ( ( Ts = nil_Pr4135541677726875188_a_nat )
        | ? [V3: product_prod_a_nat] :
            ( Ts
            = ( cons_P729672029421400548_a_nat @ ( produc148073511828866022_a_nat @ K2 @ V3 ) @ nil_Pr4135541677726875188_a_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_408_delete__aux__eq__Nil__conv,axiom,
    ! [K2: nat,Ts: list_P1609389641493977421_float] :
      ( ( ( delete7196171427534157838_float @ K2 @ Ts )
        = nil_Pr8949446520521958733_float )
      = ( ( Ts = nil_Pr8949446520521958733_float )
        | ? [V3: interval_float] :
            ( Ts
            = ( cons_P6013094588511510781_float @ ( produc6067866782486577919_float @ K2 @ V3 ) @ nil_Pr8949446520521958733_float ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_409_delete__aux__eq__Nil__conv,axiom,
    ! [K2: nat,Ts: list_P454701268746800253_float] :
      ( ( ( delete_aux_nat_float @ K2 @ Ts )
        = nil_Pr3827615237818013693_float )
      = ( ( Ts = nil_Pr3827615237818013693_float )
        | ? [V3: float] :
            ( Ts
            = ( cons_P5322595555988802477_float @ ( produc518625033508411951_float @ K2 @ V3 ) @ nil_Pr3827615237818013693_float ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_410_delete__aux__eq__Nil__conv,axiom,
    ! [K2: nat,Ts: list_P6011104703257516679at_nat] :
      ( ( ( delete_aux_nat_nat @ K2 @ Ts )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Ts = nil_Pr5478986624290739719at_nat )
        | ? [V3: nat] :
            ( Ts
            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V3 ) @ nil_Pr5478986624290739719at_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_411_delete__aux__eq__Nil__conv,axiom,
    ! [K2: a,Ts: list_P3592885314253461005_a_nat] :
      ( ( ( delete_aux_a_nat @ K2 @ Ts )
        = nil_Pr7402525243500994295_a_nat )
      = ( ( Ts = nil_Pr7402525243500994295_a_nat )
        | ? [V3: nat] :
            ( Ts
            = ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ K2 @ V3 ) @ nil_Pr7402525243500994295_a_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_412_length__code,axiom,
    ( size_size_list_a
    = ( gen_length_a @ zero_zero_nat ) ) ).

% length_code
thf(fact_413_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_a @ N @ nil_a )
      = N ) ).

% gen_length_code(1)
thf(fact_414_delete__aux_Osimps_I2_J,axiom,
    ! [K2: nat,K5: nat,V2: product_prod_a_nat,Xs: list_P4901192995000098612_a_nat] :
      ( ( ( K2 = K5 )
       => ( ( delete5594580531140670069_a_nat @ K2 @ ( cons_P729672029421400548_a_nat @ ( produc148073511828866022_a_nat @ K5 @ V2 ) @ Xs ) )
          = Xs ) )
      & ( ( K2 != K5 )
       => ( ( delete5594580531140670069_a_nat @ K2 @ ( cons_P729672029421400548_a_nat @ ( produc148073511828866022_a_nat @ K5 @ V2 ) @ Xs ) )
          = ( cons_P729672029421400548_a_nat @ ( produc148073511828866022_a_nat @ K5 @ V2 ) @ ( delete5594580531140670069_a_nat @ K2 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_415_delete__aux_Osimps_I2_J,axiom,
    ! [K2: nat,K5: nat,V2: interval_float,Xs: list_P1609389641493977421_float] :
      ( ( ( K2 = K5 )
       => ( ( delete7196171427534157838_float @ K2 @ ( cons_P6013094588511510781_float @ ( produc6067866782486577919_float @ K5 @ V2 ) @ Xs ) )
          = Xs ) )
      & ( ( K2 != K5 )
       => ( ( delete7196171427534157838_float @ K2 @ ( cons_P6013094588511510781_float @ ( produc6067866782486577919_float @ K5 @ V2 ) @ Xs ) )
          = ( cons_P6013094588511510781_float @ ( produc6067866782486577919_float @ K5 @ V2 ) @ ( delete7196171427534157838_float @ K2 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_416_delete__aux_Osimps_I2_J,axiom,
    ! [K2: nat,K5: nat,V2: float,Xs: list_P454701268746800253_float] :
      ( ( ( K2 = K5 )
       => ( ( delete_aux_nat_float @ K2 @ ( cons_P5322595555988802477_float @ ( produc518625033508411951_float @ K5 @ V2 ) @ Xs ) )
          = Xs ) )
      & ( ( K2 != K5 )
       => ( ( delete_aux_nat_float @ K2 @ ( cons_P5322595555988802477_float @ ( produc518625033508411951_float @ K5 @ V2 ) @ Xs ) )
          = ( cons_P5322595555988802477_float @ ( produc518625033508411951_float @ K5 @ V2 ) @ ( delete_aux_nat_float @ K2 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_417_delete__aux_Osimps_I2_J,axiom,
    ! [K2: nat,K5: nat,V2: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( K2 = K5 )
       => ( ( delete_aux_nat_nat @ K2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V2 ) @ Xs ) )
          = Xs ) )
      & ( ( K2 != K5 )
       => ( ( delete_aux_nat_nat @ K2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V2 ) @ Xs ) )
          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V2 ) @ ( delete_aux_nat_nat @ K2 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_418_delete__aux_Osimps_I2_J,axiom,
    ! [K2: a,K5: a,V2: nat,Xs: list_P3592885314253461005_a_nat] :
      ( ( ( K2 = K5 )
       => ( ( delete_aux_a_nat @ K2 @ ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ K5 @ V2 ) @ Xs ) )
          = Xs ) )
      & ( ( K2 != K5 )
       => ( ( delete_aux_a_nat @ K2 @ ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ K5 @ V2 ) @ Xs ) )
          = ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ K5 @ V2 ) @ ( delete_aux_a_nat @ K2 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_419_gen__length__code_I2_J,axiom,
    ! [N: nat,X2: a,Xs: list_a] :
      ( ( gen_length_a @ N @ ( cons_a @ X2 @ Xs ) )
      = ( gen_length_a @ ( suc @ N ) @ Xs ) ) ).

% gen_length_code(2)
thf(fact_420_transpose_Oelims,axiom,
    ! [X2: list_list_a,Y: list_list_a] :
      ( ( ( transpose_a @ X2 )
        = Y )
     => ( ( ( X2 = nil_list_a )
         => ( Y != nil_list_a ) )
       => ( ! [Xss: list_list_a] :
              ( ( X2
                = ( cons_list_a @ nil_a @ Xss ) )
             => ( Y
               != ( transpose_a @ Xss ) ) )
         => ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
                ( ( X2
                  = ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) )
               => ( Y
                 != ( cons_list_a
                    @ ( cons_a @ X3
                      @ ( concat_a
                        @ ( map_list_a_list_a
                          @ ( case_list_list_a_a @ nil_a
                            @ ^ [H: a,T: list_a] : ( cons_a @ H @ nil_a ) )
                          @ Xss ) ) )
                    @ ( transpose_a
                      @ ( cons_list_a @ Xs2
                        @ ( concat_list_a
                          @ ( map_li5729356230488778442list_a
                            @ ( case_l8408404631611421914st_a_a @ nil_list_a
                              @ ^ [H: a,T: list_a] : ( cons_list_a @ T @ nil_list_a ) )
                            @ Xss ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_421_transpose_Osimps_I3_J,axiom,
    ! [X2: a,Xs: list_a,Xss2: list_list_a] :
      ( ( transpose_a @ ( cons_list_a @ ( cons_a @ X2 @ Xs ) @ Xss2 ) )
      = ( cons_list_a
        @ ( cons_a @ X2
          @ ( concat_a
            @ ( map_list_a_list_a
              @ ( case_list_list_a_a @ nil_a
                @ ^ [H: a,T: list_a] : ( cons_a @ H @ nil_a ) )
              @ Xss2 ) ) )
        @ ( transpose_a
          @ ( cons_list_a @ Xs
            @ ( concat_list_a
              @ ( map_li5729356230488778442list_a
                @ ( case_l8408404631611421914st_a_a @ nil_list_a
                  @ ^ [H: a,T: list_a] : ( cons_list_a @ T @ nil_list_a ) )
                @ Xss2 ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_422_transpose_Opinduct,axiom,
    ! [A0: list_list_a,P: list_list_a > $o] :
      ( ( accp_list_list_a @ transpose_rel_a @ A0 )
     => ( ( ( accp_list_list_a @ transpose_rel_a @ nil_list_a )
         => ( P @ nil_list_a ) )
       => ( ! [Xss: list_list_a] :
              ( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ nil_a @ Xss ) )
             => ( ( P @ Xss )
               => ( P @ ( cons_list_a @ nil_a @ Xss ) ) ) )
         => ( ! [X3: a,Xs2: list_a,Xss: list_list_a] :
                ( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) )
               => ( ( P
                    @ ( cons_list_a @ Xs2
                      @ ( concat_list_a
                        @ ( map_li5729356230488778442list_a
                          @ ( case_l8408404631611421914st_a_a @ nil_list_a
                            @ ^ [H: a,T: list_a] : ( cons_list_a @ T @ nil_list_a ) )
                          @ Xss ) ) ) )
                 => ( P @ ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_423_eval__permutation__of__list_I2_J,axiom,
    ! [X2: nat,X5: nat,Y: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( X2 = X5 )
     => ( ( permut3281995586759737075st_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ Xs ) @ X2 )
        = Y ) ) ).

% eval_permutation_of_list(2)
thf(fact_424_eval__permutation__of__list_I3_J,axiom,
    ! [X2: nat,X5: nat,Y4: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( X2 != X5 )
     => ( ( permut3281995586759737075st_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ Xs ) @ X2 )
        = ( permut3281995586759737075st_nat @ Xs @ X2 ) ) ) ).

% eval_permutation_of_list(3)
thf(fact_425_bind__simps_I2_J,axiom,
    ! [X2: a,Xs: list_a,F: a > list_a] :
      ( ( bind_a_a @ ( cons_a @ X2 @ Xs ) @ F )
      = ( append_a @ ( F @ X2 ) @ ( bind_a_a @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_426_bind__simps_I1_J,axiom,
    ! [F: a > list_a] :
      ( ( bind_a_a @ nil_a @ F )
      = nil_a ) ).

% bind_simps(1)
thf(fact_427_transpose_Opsimps_I2_J,axiom,
    ! [Xss2: list_list_a] :
      ( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ nil_a @ Xss2 ) )
     => ( ( transpose_a @ ( cons_list_a @ nil_a @ Xss2 ) )
        = ( transpose_a @ Xss2 ) ) ) ).

% transpose.psimps(2)
thf(fact_428_list_Odisc__eq__case_I1_J,axiom,
    ! [List: list_a] :
      ( ( List = nil_a )
      = ( case_list_o_a @ $true
        @ ^ [Uu2: a,Uv2: list_a] : $false
        @ List ) ) ).

% list.disc_eq_case(1)
thf(fact_429_list_Odisc__eq__case_I2_J,axiom,
    ! [List: list_a] :
      ( ( List != nil_a )
      = ( case_list_o_a @ $false
        @ ^ [Uu2: a,Uv2: list_a] : $true
        @ List ) ) ).

% list.disc_eq_case(2)
thf(fact_430_transpose_Osimps_I2_J,axiom,
    ! [Xss2: list_list_a] :
      ( ( transpose_a @ ( cons_list_a @ nil_a @ Xss2 ) )
      = ( transpose_a @ Xss2 ) ) ).

% transpose.simps(2)
thf(fact_431_permutation__of__list__Cons,axiom,
    ! [X2: nat,X5: nat,Y: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( X2 = X5 )
       => ( ( permut3281995586759737075st_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ Xs ) @ X5 )
          = Y ) )
      & ( ( X2 != X5 )
       => ( ( permut3281995586759737075st_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ Xs ) @ X5 )
          = ( permut3281995586759737075st_nat @ Xs @ X5 ) ) ) ) ).

% permutation_of_list_Cons
thf(fact_432_transpose_Opelims,axiom,
    ! [X2: list_list_a,Y: list_list_a] :
      ( ( ( transpose_a @ X2 )
        = Y )
     => ( ( accp_list_list_a @ transpose_rel_a @ X2 )
       => ( ( ( X2 = nil_list_a )
           => ( ( Y = nil_list_a )
             => ~ ( accp_list_list_a @ transpose_rel_a @ nil_list_a ) ) )
         => ( ! [Xss: list_list_a] :
                ( ( X2
                  = ( cons_list_a @ nil_a @ Xss ) )
               => ( ( Y
                    = ( transpose_a @ Xss ) )
                 => ~ ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ nil_a @ Xss ) ) ) )
           => ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
                  ( ( X2
                    = ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) )
                 => ( ( Y
                      = ( cons_list_a
                        @ ( cons_a @ X3
                          @ ( concat_a
                            @ ( map_list_a_list_a
                              @ ( case_list_list_a_a @ nil_a
                                @ ^ [H: a,T: list_a] : ( cons_a @ H @ nil_a ) )
                              @ Xss ) ) )
                        @ ( transpose_a
                          @ ( cons_list_a @ Xs2
                            @ ( concat_list_a
                              @ ( map_li5729356230488778442list_a
                                @ ( case_l8408404631611421914st_a_a @ nil_list_a
                                  @ ^ [H: a,T: list_a] : ( cons_list_a @ T @ nil_list_a ) )
                                @ Xss ) ) ) ) ) )
                   => ~ ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_433_transpose_Opsimps_I3_J,axiom,
    ! [X2: a,Xs: list_a,Xss2: list_list_a] :
      ( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ ( cons_a @ X2 @ Xs ) @ Xss2 ) )
     => ( ( transpose_a @ ( cons_list_a @ ( cons_a @ X2 @ Xs ) @ Xss2 ) )
        = ( cons_list_a
          @ ( cons_a @ X2
            @ ( concat_a
              @ ( map_list_a_list_a
                @ ( case_list_list_a_a @ nil_a
                  @ ^ [H: a,T: list_a] : ( cons_a @ H @ nil_a ) )
                @ Xss2 ) ) )
          @ ( transpose_a
            @ ( cons_list_a @ Xs
              @ ( concat_list_a
                @ ( map_li5729356230488778442list_a
                  @ ( case_l8408404631611421914st_a_a @ nil_list_a
                    @ ^ [H: a,T: list_a] : ( cons_list_a @ T @ nil_list_a ) )
                  @ Xss2 ) ) ) ) ) ) ) ).

% transpose.psimps(3)
thf(fact_434_inverse__permutation__of__list_Oelims,axiom,
    ! [X2: list_P6011104703257516679at_nat,Xa: nat,Y: nat] :
      ( ( ( invers7730505255807555637st_nat @ X2 @ Xa )
        = Y )
     => ( ( ( X2 = nil_Pr5478986624290739719at_nat )
         => ( Y != Xa ) )
       => ~ ! [Y2: nat,X4: nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( X2
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ Y2 @ X4 ) @ Xs2 ) )
             => ~ ( ( ( Xa = X4 )
                   => ( Y = Y2 ) )
                  & ( ( Xa != X4 )
                   => ( Y
                      = ( invers7730505255807555637st_nat @ Xs2 @ Xa ) ) ) ) ) ) ) ).

% inverse_permutation_of_list.elims
thf(fact_435_replicate__pmf_Osimps_I2_J,axiom,
    ! [N: nat,P3: probab3364570286911266904_pmf_a] :
      ( ( probab2734007981130594152_pmf_a @ ( suc @ N ) @ P3 )
      = ( probab8840707091229225981list_a @ P3
        @ ^ [X: a] :
            ( probab8407533088143979511list_a @ ( probab2734007981130594152_pmf_a @ N @ P3 )
            @ ^ [Xs4: list_a] : ( probab4480815210451182111list_a @ ( cons_a @ X @ Xs4 ) ) ) ) ) ).

% replicate_pmf.simps(2)
thf(fact_436_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A5: set_nat] : ( A5 = bot_bot_set_nat ) ) ) ).

% Set.is_empty_def
thf(fact_437_eval__inverse__permutation__of__list_I2_J,axiom,
    ! [X2: nat,X5: nat,Y: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( X2 = X5 )
     => ( ( invers7730505255807555637st_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ Y @ X5 ) @ Xs ) @ X2 )
        = Y ) ) ).

% eval_inverse_permutation_of_list(2)
thf(fact_438_eval__inverse__permutation__of__list_I3_J,axiom,
    ! [X2: nat,X5: nat,Y4: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( X2 != X5 )
     => ( ( invers7730505255807555637st_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ Y4 @ X5 ) @ Xs ) @ X2 )
        = ( invers7730505255807555637st_nat @ Xs @ X2 ) ) ) ).

% eval_inverse_permutation_of_list(3)
thf(fact_439_inverse__permutation__of__list_Osimps_I2_J,axiom,
    ! [X2: nat,X5: nat,Y: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( X2 = X5 )
       => ( ( invers7730505255807555637st_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ Y @ X5 ) @ Xs ) @ X2 )
          = Y ) )
      & ( ( X2 != X5 )
       => ( ( invers7730505255807555637st_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ Y @ X5 ) @ Xs ) @ X2 )
          = ( invers7730505255807555637st_nat @ Xs @ X2 ) ) ) ) ).

% inverse_permutation_of_list.simps(2)
thf(fact_440_replicate__pmf_Osimps_I1_J,axiom,
    ! [Uu3: probab3364570286911266904_pmf_a] :
      ( ( probab2734007981130594152_pmf_a @ zero_zero_nat @ Uu3 )
      = ( probab4480815210451182111list_a @ nil_a ) ) ).

% replicate_pmf.simps(1)
thf(fact_441_replicate__pmf__distrib,axiom,
    ! [M3: nat,N: nat,P3: probab3364570286911266904_pmf_a] :
      ( ( probab2734007981130594152_pmf_a @ ( plus_plus_nat @ M3 @ N ) @ P3 )
      = ( probab8407533088143979511list_a @ ( probab2734007981130594152_pmf_a @ M3 @ P3 )
        @ ^ [Xs4: list_a] :
            ( probab8407533088143979511list_a @ ( probab2734007981130594152_pmf_a @ N @ P3 )
            @ ^ [Ys3: list_a] : ( probab4480815210451182111list_a @ ( append_a @ Xs4 @ Ys3 ) ) ) ) ) ).

% replicate_pmf_distrib
thf(fact_442_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_443_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_444_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_445_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_446_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_447_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_448_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_449_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_450_add__0__left,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0_left
thf(fact_451_add__0__left,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0_left
thf(fact_452_add__0__left,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0_left
thf(fact_453_add__0__right,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add_0_right
thf(fact_454_add__0__right,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add_0_right
thf(fact_455_add__0__right,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add_0_right
thf(fact_456_linordered__ab__group__add__class_Odouble__zero,axiom,
    ! [A: real] :
      ( ( ( plus_plus_real @ A @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% linordered_ab_group_add_class.double_zero
thf(fact_457_linordered__ab__group__add__class_Odouble__zero,axiom,
    ! [A: int] :
      ( ( ( plus_plus_int @ A @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% linordered_ab_group_add_class.double_zero
thf(fact_458_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_459_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_460_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( ( plus_plus_real @ B @ A )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_461_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_462_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_463_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_464_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_465_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_466_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ B @ A ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_467_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_468_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_469_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ A @ B ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_470_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_471_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_472_add__eq__0__iff__both__eq__0,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X2 @ Y )
        = zero_zero_nat )
      = ( ( X2 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_473_zero__eq__add__iff__both__eq__0,axiom,
    ! [X2: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X2 @ Y ) )
      = ( ( X2 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_474_add__Pair,axiom,
    ! [A: nat,B: float,C: nat,D: float] :
      ( ( plus_p4657267053260338478_float @ ( produc518625033508411951_float @ A @ B ) @ ( produc518625033508411951_float @ C @ D ) )
      = ( produc518625033508411951_float @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_float @ B @ D ) ) ) ).

% add_Pair
thf(fact_475_add__Pair,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( plus_p9057090461656269880at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( product_Pair_nat_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_476_add__Pair,axiom,
    ! [A: nat,B: int,C: nat,D: int] :
      ( ( plus_p4879239442147073172at_int @ ( product_Pair_nat_int @ A @ B ) @ ( product_Pair_nat_int @ C @ D ) )
      = ( product_Pair_nat_int @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ).

% add_Pair
thf(fact_477_add__Pair,axiom,
    ! [A: nat,B: real,C: nat,D: real] :
      ( ( plus_p8900843186509212308t_real @ ( produc7837566107596912789t_real @ A @ B ) @ ( produc7837566107596912789t_real @ C @ D ) )
      = ( produc7837566107596912789t_real @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).

% add_Pair
thf(fact_478_add__Pair,axiom,
    ! [A: int,B: nat,C: int,D: nat] :
      ( ( plus_p833072136702548628nt_nat @ ( product_Pair_int_nat @ A @ B ) @ ( product_Pair_int_nat @ C @ D ) )
      = ( product_Pair_int_nat @ ( plus_plus_int @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_479_add__Pair,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( plus_p5878593154048127728nt_int @ ( product_Pair_int_int @ A @ B ) @ ( product_Pair_int_int @ C @ D ) )
      = ( product_Pair_int_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ).

% add_Pair
thf(fact_480_add__Pair,axiom,
    ! [A: int,B: real,C: int,D: real] :
      ( ( plus_p1864392724347458288t_real @ ( produc801115645435158769t_real @ A @ B ) @ ( produc801115645435158769t_real @ C @ D ) )
      = ( produc801115645435158769t_real @ ( plus_plus_int @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).

% add_Pair
thf(fact_481_add__Pair,axiom,
    ! [A: real,B: nat,C: real,D: nat] :
      ( ( plus_p4925795495032332052al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc3181502643871035669al_nat @ C @ D ) )
      = ( produc3181502643871035669al_nat @ ( plus_plus_real @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_482_add__Pair,axiom,
    ! [A: real,B: int,C: real,D: int] :
      ( ( plus_p747944475523135344al_int @ ( produc3179012173361985393al_int @ A @ B ) @ ( produc3179012173361985393al_int @ C @ D ) )
      = ( produc3179012173361985393al_int @ ( plus_plus_real @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ).

% add_Pair
thf(fact_483_add__Pair,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( plus_p1196244663705802608l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( produc4511245868158468465l_real @ C @ D ) )
      = ( produc4511245868158468465l_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).

% add_Pair
thf(fact_484_Nat_Oadd__0__right,axiom,
    ! [M3: nat] :
      ( ( plus_plus_nat @ M3 @ zero_zero_nat )
      = M3 ) ).

% Nat.add_0_right
thf(fact_485_add__is__0,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( plus_plus_nat @ M3 @ N )
        = zero_zero_nat )
      = ( ( M3 = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_486_add__Suc__right,axiom,
    ! [M3: nat,N: nat] :
      ( ( plus_plus_nat @ M3 @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M3 @ N ) ) ) ).

% add_Suc_right
thf(fact_487_length__append,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).

% length_append
thf(fact_488_nat__arith_Oadd2,axiom,
    ! [B4: nat,K2: nat,B: nat,A: nat] :
      ( ( B4
        = ( plus_plus_nat @ K2 @ B ) )
     => ( ( plus_plus_nat @ A @ B4 )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% nat_arith.add2
thf(fact_489_nat__arith_Oadd2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4
        = ( plus_plus_int @ K2 @ B ) )
     => ( ( plus_plus_int @ A @ B4 )
        = ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).

% nat_arith.add2
thf(fact_490_nat__arith_Oadd2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4
        = ( plus_plus_real @ K2 @ B ) )
     => ( ( plus_plus_real @ A @ B4 )
        = ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).

% nat_arith.add2
thf(fact_491_nat__arith_Oadd1,axiom,
    ! [A4: nat,K2: nat,A: nat,B: nat] :
      ( ( A4
        = ( plus_plus_nat @ K2 @ A ) )
     => ( ( plus_plus_nat @ A4 @ B )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% nat_arith.add1
thf(fact_492_nat__arith_Oadd1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4
        = ( plus_plus_int @ K2 @ A ) )
     => ( ( plus_plus_int @ A4 @ B )
        = ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).

% nat_arith.add1
thf(fact_493_nat__arith_Oadd1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4
        = ( plus_plus_real @ K2 @ A ) )
     => ( ( plus_plus_real @ A4 @ B )
        = ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).

% nat_arith.add1
thf(fact_494_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_495_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_496_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_497_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_498_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_499_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_500_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_501_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_502_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.left_commute
thf(fact_503_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A6: nat,B6: nat] : ( plus_plus_nat @ B6 @ A6 ) ) ) ).

% add.commute
thf(fact_504_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A6: int,B6: int] : ( plus_plus_int @ B6 @ A6 ) ) ) ).

% add.commute
thf(fact_505_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A6: real,B6: real] : ( plus_plus_real @ B6 @ A6 ) ) ) ).

% add.commute
thf(fact_506_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_507_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_508_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_509_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_510_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_511_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_512_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.assoc
thf(fact_513_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: nat,J2: nat,K2: nat,L2: nat] :
      ( ( ( I3 = J2 )
        & ( K2 = L2 ) )
     => ( ( plus_plus_nat @ I3 @ K2 )
        = ( plus_plus_nat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_514_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: int,J2: int,K2: int,L2: int] :
      ( ( ( I3 = J2 )
        & ( K2 = L2 ) )
     => ( ( plus_plus_int @ I3 @ K2 )
        = ( plus_plus_int @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_515_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: real,J2: real,K2: real,L2: real] :
      ( ( ( I3 = J2 )
        & ( K2 = L2 ) )
     => ( ( plus_plus_real @ I3 @ K2 )
        = ( plus_plus_real @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_516_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_517_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_518_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_519_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_520_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_521_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_522_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_523_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_524_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_525_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_526_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_527_add__eq__self__zero,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( plus_plus_nat @ M3 @ N )
        = M3 )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_528_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_529_plus__nat_Osimps_I2_J,axiom,
    ! [M3: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M3 ) @ N )
      = ( suc @ ( plus_plus_nat @ M3 @ N ) ) ) ).

% plus_nat.simps(2)
thf(fact_530_nat__arith_Osuc1,axiom,
    ! [A4: nat,K2: nat,A: nat] :
      ( ( A4
        = ( plus_plus_nat @ K2 @ A ) )
     => ( ( suc @ A4 )
        = ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_531_add__Suc__shift,axiom,
    ! [M3: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M3 ) @ N )
      = ( plus_plus_nat @ M3 @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_532_add__is__1,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( plus_plus_nat @ M3 @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M3
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M3 = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_533_one__is__add,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M3 @ N ) )
      = ( ( ( M3
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M3 = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_534_gen__length__def,axiom,
    ( gen_length_a
    = ( ^ [N4: nat,Xs4: list_a] : ( plus_plus_nat @ N4 @ ( size_size_list_a @ Xs4 ) ) ) ) ).

% gen_length_def
thf(fact_535_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_a,Ys: list_a] :
      ( ( enumerate_a @ N @ ( append_a @ Xs @ Ys ) )
      = ( append1694031006427026248_nat_a @ ( enumerate_a @ N @ Xs ) @ ( enumerate_a @ ( plus_plus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_536_nths__append,axiom,
    ! [L2: list_a,L3: list_a,A4: set_nat] :
      ( ( nths_a @ ( append_a @ L2 @ L3 ) @ A4 )
      = ( append_a @ ( nths_a @ L2 @ A4 )
        @ ( nths_a @ L3
          @ ( collect_nat
            @ ^ [J: nat] : ( member_nat @ ( plus_plus_nat @ J @ ( size_size_list_a @ L2 ) ) @ A4 ) ) ) ) ) ).

% nths_append
thf(fact_537_list_Osize_I4_J,axiom,
    ! [X21: a,X22: list_a] :
      ( ( size_size_list_a @ ( cons_a @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_a @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_538_pth__7_I2_J,axiom,
    ! [X2: real] :
      ( ( plus_plus_real @ X2 @ zero_zero_real )
      = X2 ) ).

% pth_7(2)
thf(fact_539_pth__7_I1_J,axiom,
    ! [X2: real] :
      ( ( plus_plus_real @ zero_zero_real @ X2 )
      = X2 ) ).

% pth_7(1)
thf(fact_540_lattice__ab__group__add__class_Odouble__zero,axiom,
    ! [A: real] :
      ( ( ( plus_plus_real @ A @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% lattice_ab_group_add_class.double_zero
thf(fact_541_lattice__ab__group__add__class_Odouble__zero,axiom,
    ! [A: int] :
      ( ( ( plus_plus_int @ A @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% lattice_ab_group_add_class.double_zero
thf(fact_542_sumset__empty_I2_J,axiom,
    ! [A4: set_nat] :
      ( ( plus_plus_set_nat @ bot_bot_set_nat @ A4 )
      = bot_bot_set_nat ) ).

% sumset_empty(2)
thf(fact_543_sumset__empty_I1_J,axiom,
    ! [A4: set_nat] :
      ( ( plus_plus_set_nat @ A4 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% sumset_empty(1)
thf(fact_544_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P @ A3 @ B3 )
          = ( P @ B3 @ A3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ A3 @ B3 )
             => ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_545_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_546_eq__add__iff,axiom,
    ! [X2: real,Y: real] :
      ( ( X2
        = ( plus_plus_real @ X2 @ Y ) )
      = ( Y = zero_zero_real ) ) ).

% eq_add_iff
thf(fact_547_eq__add__iff,axiom,
    ! [X2: int,Y: int] :
      ( ( X2
        = ( plus_plus_int @ X2 @ Y ) )
      = ( Y = zero_zero_int ) ) ).

% eq_add_iff
thf(fact_548_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_549_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_550_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_551_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_552_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_553_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_554_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_555_product_Osimps_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( produc315302346078083732_a_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
      = ( append7569360072597011785_a_nat @ ( map_Pr3171194087735490347_a_nat @ ( produc148073511828866022_a_nat @ X2 ) @ Ys ) @ ( produc315302346078083732_a_nat @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_556_product_Osimps_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_interval_float] :
      ( ( produc1157954343226064557_float @ ( cons_nat @ X2 @ Xs ) @ Ys )
      = ( append5631936165717808098_float @ ( map_in1989472262963647467_float @ ( produc6067866782486577919_float @ X2 ) @ Ys ) @ ( produc1157954343226064557_float @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_557_product_Osimps_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_float] :
      ( ( product_nat_float @ ( cons_nat @ X2 @ Xs ) @ Ys )
      = ( append2749283807726223122_float @ ( map_fl5540703493322096747_float @ ( produc518625033508411951_float @ X2 ) @ Ys ) @ ( product_nat_float @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_558_product_Osimps_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( product_nat_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
      = ( append985823374593552924at_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X2 ) @ Ys ) @ ( product_nat_nat @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_559_product_Osimps_I2_J,axiom,
    ! [X2: a,Xs: list_a,Ys: list_nat] :
      ( ( product_a_nat @ ( cons_a @ X2 @ Xs ) @ Ys )
      = ( append7679239579558125090_a_nat @ ( map_na4679982104245117459_a_nat @ ( product_Pair_a_nat @ X2 ) @ Ys ) @ ( product_a_nat @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_560_length__prefixes,axiom,
    ! [Xs: list_a] :
      ( ( size_s349497388124573686list_a @ ( prefixes_a @ Xs ) )
      = ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).

% length_prefixes
thf(fact_561_list_Osize__gen_I2_J,axiom,
    ! [X2: a > nat,X21: a,X22: list_a] :
      ( ( size_list_a @ X2 @ ( cons_a @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( X2 @ X21 ) @ ( size_list_a @ X2 @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size_gen(2)
thf(fact_562_size__list__append,axiom,
    ! [F: a > nat,Xs: list_a,Ys: list_a] :
      ( ( size_list_a @ F @ ( append_a @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_list_a @ F @ Xs ) @ ( size_list_a @ F @ Ys ) ) ) ).

% size_list_append
thf(fact_563_verit__eq__simplify_I24_J,axiom,
    one_one_real != zero_zero_real ).

% verit_eq_simplify(24)
thf(fact_564_verit__eq__simplify_I24_J,axiom,
    one_one_nat != zero_zero_nat ).

% verit_eq_simplify(24)
thf(fact_565_verit__eq__simplify_I24_J,axiom,
    one_one_int != zero_zero_int ).

% verit_eq_simplify(24)
thf(fact_566_one__reorient,axiom,
    ! [X2: nat] :
      ( ( one_one_nat = X2 )
      = ( X2 = one_one_nat ) ) ).

% one_reorient
thf(fact_567_one__reorient,axiom,
    ! [X2: int] :
      ( ( one_one_int = X2 )
      = ( X2 = one_one_int ) ) ).

% one_reorient
thf(fact_568_one__reorient,axiom,
    ! [X2: real] :
      ( ( one_one_real = X2 )
      = ( X2 = one_one_real ) ) ).

% one_reorient
thf(fact_569_numeral__nat_I7_J,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% numeral_nat(7)
thf(fact_570_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_571_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_572_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_573_product__concat__map,axiom,
    ( produc315302346078083732_a_nat
    = ( ^ [Xs4: list_nat,Ys3: list_P3592885314253461005_a_nat] :
          ( concat6318098542637659887_a_nat
          @ ( map_na481557189381479198_a_nat
            @ ^ [X: nat] : ( map_Pr3171194087735490347_a_nat @ ( produc148073511828866022_a_nat @ X ) @ Ys3 )
            @ Xs4 ) ) ) ) ).

% product_concat_map
thf(fact_574_product__concat__map,axiom,
    ( produc1157954343226064557_float
    = ( ^ [Xs4: list_nat,Ys3: list_interval_float] :
          ( concat1123897152797845128_float
          @ ( map_na7465345748648547383_float
            @ ^ [X: nat] : ( map_in1989472262963647467_float @ ( produc6067866782486577919_float @ X ) @ Ys3 )
            @ Xs4 ) ) ) ) ).

% product_concat_map
thf(fact_575_product__concat__map,axiom,
    ( product_nat_float
    = ( ^ [Xs4: list_nat,Ys3: list_float] :
          ( concat3892468926543407288_float
          @ ( map_na8201320331418081383_float
            @ ^ [X: nat] : ( map_fl5540703493322096747_float @ ( produc518625033508411951_float @ X ) @ Ys3 )
            @ Xs4 ) ) ) ) ).

% product_concat_map
thf(fact_576_product__concat__map,axiom,
    ( product_nat_nat
    = ( ^ [Xs4: list_nat,Ys3: list_nat] :
          ( concat7691415812945658306at_nat
          @ ( map_na4561905831291441265at_nat
            @ ^ [X: nat] : ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X ) @ Ys3 )
            @ Xs4 ) ) ) ) ).

% product_concat_map
thf(fact_577_product__concat__map,axiom,
    ( product_a_nat
    = ( ^ [Xs4: list_a,Ys3: list_nat] :
          ( concat6937981737187208188_a_nat
          @ ( map_a_7240520880688972999_a_nat
            @ ^ [X: a] : ( map_na4679982104245117459_a_nat @ ( product_Pair_a_nat @ X ) @ Ys3 )
            @ Xs4 ) ) ) ) ).

% product_concat_map
thf(fact_578_list_Osize_I1_J,axiom,
    ! [X2: a > nat] :
      ( ( size_list_a @ X2 @ nil_a )
      = zero_zero_nat ) ).

% list.size(1)
thf(fact_579_length__product,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( size_s3885678630836030617od_a_a @ ( product_a_a @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).

% length_product
thf(fact_580_gen__fib_Oelims,axiom,
    ! [X2: nat,Xa: nat,Xb: nat,Y: nat] :
      ( ( ( gen_fib @ X2 @ Xa @ Xb )
        = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != X2 ) )
       => ( ( ( Xb
              = ( suc @ zero_zero_nat ) )
           => ( Y != Xa ) )
         => ~ ! [N3: nat] :
                ( ( Xb
                  = ( suc @ ( suc @ N3 ) ) )
               => ( Y
                 != ( gen_fib @ Xa @ ( plus_plus_nat @ X2 @ Xa ) @ ( suc @ N3 ) ) ) ) ) ) ) ).

% gen_fib.elims
thf(fact_581_arith__simps_I63_J,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% arith_simps(63)
thf(fact_582_arith__simps_I63_J,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% arith_simps(63)
thf(fact_583_arith__simps_I62_J,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% arith_simps(62)
thf(fact_584_arith__simps_I62_J,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% arith_simps(62)
thf(fact_585_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_586_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_587_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_588_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_589_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_590_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_591_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_592_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_593_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_594_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_595_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_596_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_597_mult__cancel2,axiom,
    ! [M3: nat,K2: nat,N: nat] :
      ( ( ( times_times_nat @ M3 @ K2 )
        = ( times_times_nat @ N @ K2 ) )
      = ( ( M3 = N )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_598_mult__cancel1,axiom,
    ! [K2: nat,M3: nat,N: nat] :
      ( ( ( times_times_nat @ K2 @ M3 )
        = ( times_times_nat @ K2 @ N ) )
      = ( ( M3 = N )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_599_mult__0__right,axiom,
    ! [M3: nat] :
      ( ( times_times_nat @ M3 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_600_mult__is__0,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( times_times_nat @ M3 @ N )
        = zero_zero_nat )
      = ( ( M3 = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_601_nat__mult__eq__1__iff,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( times_times_nat @ M3 @ N )
        = one_one_nat )
      = ( ( M3 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_602_nat__1__eq__mult__iff,axiom,
    ! [M3: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M3 @ N ) )
      = ( ( M3 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_603_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_604_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_605_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ( times_times_real @ C @ A )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_606_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_607_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_608_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_609_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ( times_times_real @ A @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_610_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_611_mult__eq__1__iff,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( times_times_nat @ M3 @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M3
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_612_one__eq__mult__iff,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M3 @ N ) )
      = ( ( M3
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_613_mult__Suc__right,axiom,
    ! [M3: nat,N: nat] :
      ( ( times_times_nat @ M3 @ ( suc @ N ) )
      = ( plus_plus_nat @ M3 @ ( times_times_nat @ M3 @ N ) ) ) ).

% mult_Suc_right
thf(fact_614_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_615_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_616_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_617_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_618_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_619_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_620_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_621_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_622_lambda__one,axiom,
    ( ( ^ [X: real] : X )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_623_lambda__one,axiom,
    ( ( ^ [X: nat] : X )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_624_lambda__one,axiom,
    ( ( ^ [X: int] : X )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_625_nat__distrib_I2_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% nat_distrib(2)
thf(fact_626_nat__distrib_I2_J,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 ) ) ) ).

% nat_distrib(2)
thf(fact_627_nat__distrib_I2_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 ) ) ) ).

% nat_distrib(2)
thf(fact_628_nat__distrib_I1_J,axiom,
    ! [M3: nat,N: nat,K2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M3 @ N ) @ K2 )
      = ( plus_plus_nat @ ( times_times_nat @ M3 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).

% nat_distrib(1)
thf(fact_629_add__mult__distrib2,axiom,
    ! [K2: nat,M3: nat,N: nat] :
      ( ( times_times_nat @ K2 @ ( plus_plus_nat @ M3 @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N ) ) ) ).

% add_mult_distrib2
thf(fact_630_mult__delta__left,axiom,
    ! [B: $o,X2: nat,Y: nat] :
      ( ( B
       => ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
          = ( times_times_nat @ X2 @ Y ) ) )
      & ( ~ B
       => ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
          = zero_zero_nat ) ) ) ).

% mult_delta_left
thf(fact_631_mult__delta__left,axiom,
    ! [B: $o,X2: int,Y: int] :
      ( ( B
       => ( ( times_times_int @ ( if_int @ B @ X2 @ zero_zero_int ) @ Y )
          = ( times_times_int @ X2 @ Y ) ) )
      & ( ~ B
       => ( ( times_times_int @ ( if_int @ B @ X2 @ zero_zero_int ) @ Y )
          = zero_zero_int ) ) ) ).

% mult_delta_left
thf(fact_632_mult__delta__right,axiom,
    ! [B: $o,X2: nat,Y: nat] :
      ( ( B
       => ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
          = ( times_times_nat @ X2 @ Y ) ) )
      & ( ~ B
       => ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
          = zero_zero_nat ) ) ) ).

% mult_delta_right
thf(fact_633_mult__delta__right,axiom,
    ! [B: $o,X2: int,Y: int] :
      ( ( B
       => ( ( times_times_int @ X2 @ ( if_int @ B @ Y @ zero_zero_int ) )
          = ( times_times_int @ X2 @ Y ) ) )
      & ( ~ B
       => ( ( times_times_int @ X2 @ ( if_int @ B @ Y @ zero_zero_int ) )
          = zero_zero_int ) ) ) ).

% mult_delta_right
thf(fact_634_times__nat_Osimps_I1_J,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% times_nat.simps(1)
thf(fact_635_Suc__mult__cancel1,axiom,
    ! [K2: nat,M3: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K2 ) @ M3 )
        = ( times_times_nat @ ( suc @ K2 ) @ N ) )
      = ( M3 = N ) ) ).

% Suc_mult_cancel1
thf(fact_636_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_637_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_638_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A6: nat,B6: nat] : ( times_times_nat @ B6 @ A6 ) ) ) ).

% mult.commute
thf(fact_639_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A6: int,B6: int] : ( times_times_int @ B6 @ A6 ) ) ) ).

% mult.commute
thf(fact_640_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_641_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_642_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_643_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_644_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_645_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_646_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_647_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_648_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_649_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_650_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_651_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_652_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_653_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_654_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_655_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_656_nat__mult__eq__cancel__disj,axiom,
    ! [K2: nat,M3: nat,N: nat] :
      ( ( ( times_times_nat @ K2 @ M3 )
        = ( times_times_nat @ K2 @ N ) )
      = ( ( K2 = zero_zero_nat )
        | ( M3 = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_657_gen__fib_Osimps_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( gen_fib @ A @ B @ zero_zero_nat )
      = A ) ).

% gen_fib.simps(1)
thf(fact_658_add__scale__eq__noteq,axiom,
    ! [R: real,A: real,B: real,C: real,D: real] :
      ( ( R != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_659_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_660_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_661_mult__eq__self__implies__10,axiom,
    ! [M3: nat,N: nat] :
      ( ( M3
        = ( times_times_nat @ M3 @ N ) )
     => ( ( N = one_one_nat )
        | ( M3 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_662_times__nat_Osimps_I2_J,axiom,
    ! [M3: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M3 ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M3 @ N ) ) ) ).

% times_nat.simps(2)
thf(fact_663_gen__fib_Osimps_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( gen_fib @ A @ B @ ( suc @ zero_zero_nat ) )
      = B ) ).

% gen_fib.simps(2)
thf(fact_664_gen__fib_Osimps_I3_J,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( gen_fib @ A @ B @ ( suc @ ( suc @ N ) ) )
      = ( gen_fib @ B @ ( plus_plus_nat @ A @ B ) @ ( suc @ N ) ) ) ).

% gen_fib.simps(3)
thf(fact_665_gen__fib__recurrence,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( gen_fib @ A @ B @ ( suc @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( gen_fib @ A @ B @ N ) @ ( gen_fib @ A @ B @ ( suc @ N ) ) ) ) ).

% gen_fib_recurrence
thf(fact_666_sum__squares__eq__zero__iff,axiom,
    ! [X2: real,Y: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) )
        = zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_667_sum__squares__eq__zero__iff,axiom,
    ! [X2: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_668_mult__if__delta,axiom,
    ! [P: $o,Q: real] :
      ( ( P
       => ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
          = Q ) )
      & ( ~ P
       => ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
          = zero_zero_real ) ) ) ).

% mult_if_delta
thf(fact_669_mult__if__delta,axiom,
    ! [P: $o,Q: nat] :
      ( ( P
       => ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
          = Q ) )
      & ( ~ P
       => ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
          = zero_zero_nat ) ) ) ).

% mult_if_delta
thf(fact_670_mult__if__delta,axiom,
    ! [P: $o,Q: int] :
      ( ( P
       => ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q )
          = Q ) )
      & ( ~ P
       => ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q )
          = zero_zero_int ) ) ) ).

% mult_if_delta
thf(fact_671_forall__vector__1,axiom,
    ( ( ^ [P5: finite6921312364749396332l_num1 > $o] :
        ! [X6: finite6921312364749396332l_num1] : ( P5 @ X6 ) )
    = ( ^ [P6: finite6921312364749396332l_num1 > $o] :
        ! [X: a] : ( P6 @ ( cartes8384502946292686932l_num1 @ ( cons_a @ X @ nil_a ) ) ) ) ) ).

% forall_vector_1
thf(fact_672_vector__3_I1_J,axiom,
    ! [X2: a,Y: a,Z3: a] :
      ( ( finite5778773563382317831l_num1 @ ( cartes2935649508342116554l_num1 @ ( cons_a @ X2 @ ( cons_a @ Y @ ( cons_a @ Z3 @ nil_a ) ) ) ) @ one_on7819281148064737470l_num1 )
      = X2 ) ).

% vector_3(1)
thf(fact_673_forall__vector__3,axiom,
    ( ( ^ [P5: finite4021129948254137928l_num1 > $o] :
        ! [X6: finite4021129948254137928l_num1] : ( P5 @ X6 ) )
    = ( ^ [P6: finite4021129948254137928l_num1 > $o] :
        ! [X: a,Y3: a,Z2: a] : ( P6 @ ( cartes2935649508342116554l_num1 @ ( cons_a @ X @ ( cons_a @ Y3 @ ( cons_a @ Z2 @ nil_a ) ) ) ) ) ) ) ).

% forall_vector_3
thf(fact_674_forall__vector__2,axiom,
    ( ( ^ [P5: finite7106886190244132489l_num1 > $o] :
        ! [X6: finite7106886190244132489l_num1] : ( P5 @ X6 ) )
    = ( ^ [P6: finite7106886190244132489l_num1 > $o] :
        ! [X: a,Y3: a] : ( P6 @ ( cartes8208129909578303883l_num1 @ ( cons_a @ X @ ( cons_a @ Y3 @ nil_a ) ) ) ) ) ) ).

% forall_vector_2
thf(fact_675_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_676_insert__Nil,axiom,
    ! [X2: a] :
      ( ( insert_a @ X2 @ nil_a )
      = ( cons_a @ X2 @ nil_a ) ) ).

% insert_Nil
thf(fact_677_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_678_fps__mult__nth__1_H,axiom,
    ! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real] :
      ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ G ) @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_real @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ ( suc @ zero_zero_nat ) ) ) @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ ( suc @ zero_zero_nat ) ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) ) ).

% fps_mult_nth_1'
thf(fact_679_fps__mult__nth__1_H,axiom,
    ! [F: formal_Power_fps_nat,G: formal_Power_fps_nat] :
      ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ G ) @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ zero_zero_nat ) @ ( formal3720337525774269570th_nat @ G @ ( suc @ zero_zero_nat ) ) ) @ ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ ( suc @ zero_zero_nat ) ) @ ( formal3720337525774269570th_nat @ G @ zero_zero_nat ) ) ) ) ).

% fps_mult_nth_1'
thf(fact_680_fps__mult__nth__1_H,axiom,
    ! [F: formal_Power_fps_int,G: formal_Power_fps_int] :
      ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ G ) @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_int @ ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ ( formal3717847055265219294th_int @ G @ ( suc @ zero_zero_nat ) ) ) @ ( times_times_int @ ( formal3717847055265219294th_int @ F @ ( suc @ zero_zero_nat ) ) @ ( formal3717847055265219294th_int @ G @ zero_zero_nat ) ) ) ) ).

% fps_mult_nth_1'
thf(fact_681_fps__is__right__unit__iff__zeroth__is__right__unit,axiom,
    ! [F: formal3361831859752904756s_real] :
      ( ( ? [G2: formal3361831859752904756s_real] :
            ( one_on8598947968683843321s_real
            = ( times_7561426564079326009s_real @ G2 @ F ) ) )
      = ( ? [K: real] :
            ( one_one_real
            = ( times_times_real @ K @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) ) ) ) ) ).

% fps_is_right_unit_iff_zeroth_is_right_unit
thf(fact_682_fps__is__right__unit__iff__zeroth__is__right__unit,axiom,
    ! [F: formal_Power_fps_int] :
      ( ( ? [G2: formal_Power_fps_int] :
            ( one_on8395608022581818233ps_int
            = ( times_3091854549176928185ps_int @ G2 @ F ) ) )
      = ( ? [K: int] :
            ( one_one_int
            = ( times_times_int @ K @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) ) ) ) ) ).

% fps_is_right_unit_iff_zeroth_is_right_unit
thf(fact_683_fps__is__left__unit__iff__zeroth__is__left__unit,axiom,
    ! [F: formal3361831859752904756s_real] :
      ( ( ? [G2: formal3361831859752904756s_real] :
            ( one_on8598947968683843321s_real
            = ( times_7561426564079326009s_real @ F @ G2 ) ) )
      = ( ? [K: real] :
            ( one_one_real
            = ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ K ) ) ) ) ).

% fps_is_left_unit_iff_zeroth_is_left_unit
thf(fact_684_fps__is__left__unit__iff__zeroth__is__left__unit,axiom,
    ! [F: formal_Power_fps_int] :
      ( ( ? [G2: formal_Power_fps_int] :
            ( one_on8395608022581818233ps_int
            = ( times_3091854549176928185ps_int @ F @ G2 ) ) )
      = ( ? [K: int] :
            ( one_one_int
            = ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ K ) ) ) ) ).

% fps_is_left_unit_iff_zeroth_is_left_unit
thf(fact_685_fps__add__nth,axiom,
    ! [F: formal_Power_fps_nat,G: formal_Power_fps_nat,N: nat] :
      ( ( formal3720337525774269570th_nat @ ( plus_p6043471806551771617ps_nat @ F @ G ) @ N )
      = ( plus_plus_nat @ ( formal3720337525774269570th_nat @ F @ N ) @ ( formal3720337525774269570th_nat @ G @ N ) ) ) ).

% fps_add_nth
thf(fact_686_fps__add__nth,axiom,
    ! [F: formal_Power_fps_int,G: formal_Power_fps_int,N: nat] :
      ( ( formal3717847055265219294th_int @ ( plus_p1865620787042574909ps_int @ F @ G ) @ N )
      = ( plus_plus_int @ ( formal3717847055265219294th_int @ F @ N ) @ ( formal3717847055265219294th_int @ G @ N ) ) ) ).

% fps_add_nth
thf(fact_687_fps__add__nth,axiom,
    ! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real,N: nat] :
      ( ( formal2580924720334399070h_real @ ( plus_p6008488439947570109s_real @ F @ G ) @ N )
      = ( plus_plus_real @ ( formal2580924720334399070h_real @ F @ N ) @ ( formal2580924720334399070h_real @ G @ N ) ) ) ).

% fps_add_nth
thf(fact_688_fps__zero__nth,axiom,
    ! [N: nat] :
      ( ( formal3720337525774269570th_nat @ zero_z8531573698755551073ps_nat @ N )
      = zero_zero_nat ) ).

% fps_zero_nth
thf(fact_689_fps__zero__nth,axiom,
    ! [N: nat] :
      ( ( formal506555069972267724_nth_a @ zero_z2379532579395251523_fps_a @ N )
      = zero_zero_a ) ).

% fps_zero_nth
thf(fact_690_fps__zero__nth,axiom,
    ! [N: nat] :
      ( ( formal3717847055265219294th_int @ zero_z4353722679246354365ps_int @ N )
      = zero_zero_int ) ).

% fps_zero_nth
thf(fact_691_fps__mult__nth__0,axiom,
    ! [F: formal_Power_fps_nat,G: formal_Power_fps_nat] :
      ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ G ) @ zero_zero_nat )
      = ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ zero_zero_nat ) @ ( formal3720337525774269570th_nat @ G @ zero_zero_nat ) ) ) ).

% fps_mult_nth_0
thf(fact_692_fps__mult__nth__0,axiom,
    ! [F: formal_Power_fps_int,G: formal_Power_fps_int] :
      ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ G ) @ zero_zero_nat )
      = ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ ( formal3717847055265219294th_int @ G @ zero_zero_nat ) ) ) ).

% fps_mult_nth_0
thf(fact_693_fps__one__nth,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal2580924720334399070h_real @ one_on8598947968683843321s_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal2580924720334399070h_real @ one_on8598947968683843321s_real @ N )
          = zero_zero_real ) ) ) ).

% fps_one_nth
thf(fact_694_fps__one__nth,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ one_on3350087005236239133ps_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ one_on3350087005236239133ps_nat @ N )
          = zero_zero_nat ) ) ) ).

% fps_one_nth
thf(fact_695_fps__one__nth,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ one_on8395608022581818233ps_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ one_on8395608022581818233ps_int @ N )
          = zero_zero_int ) ) ) ).

% fps_one_nth
thf(fact_696_fps__nonzero__nth,axiom,
    ! [F: formal_Power_fps_nat] :
      ( ( F != zero_z8531573698755551073ps_nat )
      = ( ? [N4: nat] :
            ( ( formal3720337525774269570th_nat @ F @ N4 )
           != zero_zero_nat ) ) ) ).

% fps_nonzero_nth
thf(fact_697_fps__nonzero__nth,axiom,
    ! [F: formal_Power_fps_a] :
      ( ( F != zero_z2379532579395251523_fps_a )
      = ( ? [N4: nat] :
            ( ( formal506555069972267724_nth_a @ F @ N4 )
           != zero_zero_a ) ) ) ).

% fps_nonzero_nth
thf(fact_698_fps__nonzero__nth,axiom,
    ! [F: formal_Power_fps_int] :
      ( ( F != zero_z4353722679246354365ps_int )
      = ( ? [N4: nat] :
            ( ( formal3717847055265219294th_int @ F @ N4 )
           != zero_zero_int ) ) ) ).

% fps_nonzero_nth
thf(fact_699_fps__nonzeroI,axiom,
    ! [F: formal_Power_fps_nat,N: nat] :
      ( ( ( formal3720337525774269570th_nat @ F @ N )
       != zero_zero_nat )
     => ( F != zero_z8531573698755551073ps_nat ) ) ).

% fps_nonzeroI
thf(fact_700_fps__nonzeroI,axiom,
    ! [F: formal_Power_fps_a,N: nat] :
      ( ( ( formal506555069972267724_nth_a @ F @ N )
       != zero_zero_a )
     => ( F != zero_z2379532579395251523_fps_a ) ) ).

% fps_nonzeroI
thf(fact_701_fps__nonzeroI,axiom,
    ! [F: formal_Power_fps_int,N: nat] :
      ( ( ( formal3717847055265219294th_int @ F @ N )
       != zero_zero_int )
     => ( F != zero_z4353722679246354365ps_int ) ) ).

% fps_nonzeroI
thf(fact_702_fps__mult__nth__1,axiom,
    ! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real] :
      ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ G ) @ one_one_nat )
      = ( plus_plus_real @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ one_one_nat ) ) @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ one_one_nat ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) ) ).

% fps_mult_nth_1
thf(fact_703_fps__mult__nth__1,axiom,
    ! [F: formal_Power_fps_nat,G: formal_Power_fps_nat] :
      ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ G ) @ one_one_nat )
      = ( plus_plus_nat @ ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ zero_zero_nat ) @ ( formal3720337525774269570th_nat @ G @ one_one_nat ) ) @ ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ one_one_nat ) @ ( formal3720337525774269570th_nat @ G @ zero_zero_nat ) ) ) ) ).

% fps_mult_nth_1
thf(fact_704_fps__mult__nth__1,axiom,
    ! [F: formal_Power_fps_int,G: formal_Power_fps_int] :
      ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ G ) @ one_one_nat )
      = ( plus_plus_int @ ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ ( formal3717847055265219294th_int @ G @ one_one_nat ) ) @ ( times_times_int @ ( formal3717847055265219294th_int @ F @ one_one_nat ) @ ( formal3717847055265219294th_int @ G @ zero_zero_nat ) ) ) ) ).

% fps_mult_nth_1
thf(fact_705_fps__hypergeo__0,axiom,
    ! [As: list_real,Bs: list_real,C: real] :
      ( ( formal2580924720334399070h_real @ ( formal6618874005373735610o_real @ As @ Bs @ C ) @ zero_zero_nat )
      = one_one_real ) ).

% fps_hypergeo_0
thf(fact_706_fps__cos__nth__0,axiom,
    ! [C: real] :
      ( ( formal2580924720334399070h_real @ ( formal461277676486907980s_real @ C ) @ zero_zero_nat )
      = one_one_real ) ).

% fps_cos_nth_0
thf(fact_707_fps__ln__mult__add,axiom,
    ! [C: real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( D != zero_zero_real )
       => ( ( plus_p6008488439947570109s_real @ ( formal8688746759596762231n_real @ C ) @ ( formal8688746759596762231n_real @ D ) )
          = ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ ( plus_plus_real @ C @ D ) ) @ ( formal8688746759596762231n_real @ ( times_times_real @ C @ D ) ) ) ) ) ) ).

% fps_ln_mult_add
thf(fact_708_fps__const__mult,axiom,
    ! [C: nat,D: nat] :
      ( ( times_7269705568686124893ps_nat @ ( formal5286749789737391404st_nat @ C ) @ ( formal5286749789737391404st_nat @ D ) )
      = ( formal5286749789737391404st_nat @ ( times_times_nat @ C @ D ) ) ) ).

% fps_const_mult
thf(fact_709_fps__const__mult,axiom,
    ! [C: int,D: int] :
      ( ( times_3091854549176928185ps_int @ ( formal5284259319228341128st_int @ C ) @ ( formal5284259319228341128st_int @ D ) )
      = ( formal5284259319228341128st_int @ ( times_times_int @ C @ D ) ) ) ).

% fps_const_mult
thf(fact_710_fps__const__0__eq__0,axiom,
    ( ( formal5286749789737391404st_nat @ zero_zero_nat )
    = zero_z8531573698755551073ps_nat ) ).

% fps_const_0_eq_0
thf(fact_711_fps__const__0__eq__0,axiom,
    ( ( formal2476588372563181730onst_a @ zero_zero_a )
    = zero_z2379532579395251523_fps_a ) ).

% fps_const_0_eq_0
thf(fact_712_fps__const__0__eq__0,axiom,
    ( ( formal5284259319228341128st_int @ zero_zero_int )
    = zero_z4353722679246354365ps_int ) ).

% fps_const_0_eq_0
thf(fact_713_fps__const__1__eq__1,axiom,
    ( ( formal5286749789737391404st_nat @ one_one_nat )
    = one_on3350087005236239133ps_nat ) ).

% fps_const_1_eq_1
thf(fact_714_fps__const__1__eq__1,axiom,
    ( ( formal5284259319228341128st_int @ one_one_int )
    = one_on8395608022581818233ps_int ) ).

% fps_const_1_eq_1
thf(fact_715_fps__const__1__eq__1,axiom,
    ( ( formal2098867297714113032t_real @ one_one_real )
    = one_on8598947968683843321s_real ) ).

% fps_const_1_eq_1
thf(fact_716_fps__const__add,axiom,
    ! [C: nat,D: nat] :
      ( ( plus_p6043471806551771617ps_nat @ ( formal5286749789737391404st_nat @ C ) @ ( formal5286749789737391404st_nat @ D ) )
      = ( formal5286749789737391404st_nat @ ( plus_plus_nat @ C @ D ) ) ) ).

% fps_const_add
thf(fact_717_fps__const__add,axiom,
    ! [C: int,D: int] :
      ( ( plus_p1865620787042574909ps_int @ ( formal5284259319228341128st_int @ C ) @ ( formal5284259319228341128st_int @ D ) )
      = ( formal5284259319228341128st_int @ ( plus_plus_int @ C @ D ) ) ) ).

% fps_const_add
thf(fact_718_fps__const__add,axiom,
    ! [C: real,D: real] :
      ( ( plus_p6008488439947570109s_real @ ( formal2098867297714113032t_real @ C ) @ ( formal2098867297714113032t_real @ D ) )
      = ( formal2098867297714113032t_real @ ( plus_plus_real @ C @ D ) ) ) ).

% fps_const_add
thf(fact_719_fps__nth__fps__const,axiom,
    ! [N: nat,C: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ ( formal5286749789737391404st_nat @ C ) @ N )
          = C ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ ( formal5286749789737391404st_nat @ C ) @ N )
          = zero_zero_nat ) ) ) ).

% fps_nth_fps_const
thf(fact_720_fps__nth__fps__const,axiom,
    ! [N: nat,C: a] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal506555069972267724_nth_a @ ( formal2476588372563181730onst_a @ C ) @ N )
          = C ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal506555069972267724_nth_a @ ( formal2476588372563181730onst_a @ C ) @ N )
          = zero_zero_a ) ) ) ).

% fps_nth_fps_const
thf(fact_721_fps__nth__fps__const,axiom,
    ! [N: nat,C: int] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ ( formal5284259319228341128st_int @ C ) @ N )
          = C ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ ( formal5284259319228341128st_int @ C ) @ N )
          = zero_zero_int ) ) ) ).

% fps_nth_fps_const
thf(fact_722_fps__mult__left__const__nth,axiom,
    ! [C: nat,F: formal_Power_fps_nat,N: nat] :
      ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ ( formal5286749789737391404st_nat @ C ) @ F ) @ N )
      = ( times_times_nat @ C @ ( formal3720337525774269570th_nat @ F @ N ) ) ) ).

% fps_mult_left_const_nth
thf(fact_723_fps__mult__left__const__nth,axiom,
    ! [C: int,F: formal_Power_fps_int,N: nat] :
      ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ ( formal5284259319228341128st_int @ C ) @ F ) @ N )
      = ( times_times_int @ C @ ( formal3717847055265219294th_int @ F @ N ) ) ) ).

% fps_mult_left_const_nth
thf(fact_724_fps__mult__right__const__nth,axiom,
    ! [F: formal_Power_fps_nat,C: nat,N: nat] :
      ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ ( formal5286749789737391404st_nat @ C ) ) @ N )
      = ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ N ) @ C ) ) ).

% fps_mult_right_const_nth
thf(fact_725_fps__mult__right__const__nth,axiom,
    ! [F: formal_Power_fps_int,C: int,N: nat] :
      ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ ( formal5284259319228341128st_int @ C ) ) @ N )
      = ( times_times_int @ ( formal3717847055265219294th_int @ F @ N ) @ C ) ) ).

% fps_mult_right_const_nth
thf(fact_726_fps__inverse__one_H,axiom,
    ( ( ( inverse_inverse_real @ one_one_real )
      = one_one_real )
   => ( ( invers68952373231134600s_real @ one_on8598947968683843321s_real )
      = one_on8598947968683843321s_real ) ) ).

% fps_inverse_one'
thf(fact_727_fps__const__nonzero__eq__nonzero,axiom,
    ! [C: nat] :
      ( ( C != zero_zero_nat )
     => ( ( formal5286749789737391404st_nat @ C )
       != zero_z8531573698755551073ps_nat ) ) ).

% fps_const_nonzero_eq_nonzero
thf(fact_728_fps__const__nonzero__eq__nonzero,axiom,
    ! [C: a] :
      ( ( C != zero_zero_a )
     => ( ( formal2476588372563181730onst_a @ C )
       != zero_z2379532579395251523_fps_a ) ) ).

% fps_const_nonzero_eq_nonzero
thf(fact_729_fps__const__nonzero__eq__nonzero,axiom,
    ! [C: int] :
      ( ( C != zero_zero_int )
     => ( ( formal5284259319228341128st_int @ C )
       != zero_z4353722679246354365ps_int ) ) ).

% fps_const_nonzero_eq_nonzero
thf(fact_730_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
        = one_one_real ) ) ).

% right_inverse
thf(fact_731_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
        = one_one_real ) ) ).

% left_inverse
thf(fact_732_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
        = one_one_real ) ) ).

% field_class.field_inverse
thf(fact_733_division__ring__inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_734_inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).

% inverse_add
thf(fact_735_fps__sin__add,axiom,
    ! [A: real,B: real] :
      ( ( formal6437758938379178589n_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_p6008488439947570109s_real @ ( times_7561426564079326009s_real @ ( formal6437758938379178589n_real @ A ) @ ( formal461277676486907980s_real @ B ) ) @ ( times_7561426564079326009s_real @ ( formal461277676486907980s_real @ A ) @ ( formal6437758938379178589n_real @ B ) ) ) ) ).

% fps_sin_add
thf(fact_736_fps__XD__0th,axiom,
    ! [A: formal_Power_fps_nat] :
      ( ( formal3720337525774269570th_nat @ ( formal814923487339530757XD_nat @ A ) @ zero_zero_nat )
      = zero_zero_nat ) ).

% fps_XD_0th
thf(fact_737_fps__XD__0th,axiom,
    ! [A: formal_Power_fps_int] :
      ( ( formal3717847055265219294th_int @ ( formal812433016830480481XD_int @ A ) @ zero_zero_nat )
      = zero_zero_int ) ).

% fps_XD_0th
thf(fact_738_fps__XDp0,axiom,
    ( ( formal9197787955091086413Dp_nat @ zero_zero_nat )
    = formal814923487339530757XD_nat ) ).

% fps_XDp0
thf(fact_739_fps__XDp0,axiom,
    ( ( formal9195297484582036137Dp_int @ zero_zero_int )
    = formal812433016830480481XD_int ) ).

% fps_XDp0
thf(fact_740_fps__integral__linear,axiom,
    ! [A: real,F: formal3361831859752904756s_real,B: real,G: formal3361831859752904756s_real,A0: real,B0: real] :
      ( ( formal8984515926053063617l_real @ ( plus_p6008488439947570109s_real @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ A ) @ F ) @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ B ) @ G ) ) @ ( plus_plus_real @ ( times_times_real @ A @ A0 ) @ ( times_times_real @ B @ B0 ) ) )
      = ( plus_p6008488439947570109s_real @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ A ) @ ( formal8984515926053063617l_real @ F @ A0 ) ) @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ B ) @ ( formal8984515926053063617l_real @ G @ B0 ) ) ) ) ).

% fps_integral_linear
thf(fact_741_fps__integral__nth__0__Suc_I2_J,axiom,
    ! [A: formal3361831859752904756s_real,A0: real,N: nat] :
      ( ( formal2580924720334399070h_real @ ( formal8984515926053063617l_real @ A @ A0 ) @ ( suc @ N ) )
      = ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( formal2580924720334399070h_real @ A @ N ) ) ) ).

% fps_integral_nth_0_Suc(2)
thf(fact_742_fps__integral0__fps__const_H,axiom,
    ! [C: real] :
      ( ( ( inverse_inverse_real @ one_one_real )
        = one_one_real )
     => ( ( formal8984515926053063617l_real @ ( formal2098867297714113032t_real @ C ) @ zero_zero_real )
        = ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ C ) @ formal4708490801539276157X_real ) ) ) ).

% fps_integral0_fps_const'
thf(fact_743_of__nat__eq__iff,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M3 )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M3 = N ) ) ).

% of_nat_eq_iff
thf(fact_744_of__nat__eq__iff,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M3 )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M3 = N ) ) ).

% of_nat_eq_iff
thf(fact_745_of__nat__eq__0__iff,axiom,
    ! [M3: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M3 )
        = zero_zero_nat )
      = ( M3 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_746_of__nat__eq__0__iff,axiom,
    ! [M3: nat] :
      ( ( ( semiri1314217659103216013at_int @ M3 )
        = zero_zero_int )
      = ( M3 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_747_of__nat__eq__0__iff,axiom,
    ! [M3: nat] :
      ( ( ( semiri5074537144036343181t_real @ M3 )
        = zero_zero_real )
      = ( M3 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_748_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_749_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_750_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_751_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_752_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_753_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_754_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N )
        = one_one_nat )
      = ( N = one_one_nat ) ) ).

% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_755_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1314217659103216013at_int @ N )
        = one_one_int )
      = ( N = one_one_nat ) ) ).

% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_756_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri5074537144036343181t_real @ N )
        = one_one_real )
      = ( N = one_one_nat ) ) ).

% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_757_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_758_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_759_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_760_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_761_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_762_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_763_of__nat__add,axiom,
    ! [M3: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M3 @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_764_of__nat__add,axiom,
    ! [M3: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M3 @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_765_of__nat__add,axiom,
    ! [M3: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M3 @ N ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_add
thf(fact_766_of__nat__mult,axiom,
    ! [M3: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M3 @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_767_of__nat__mult,axiom,
    ! [M3: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M3 @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_768_of__nat__mult,axiom,
    ! [M3: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M3 @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_mult
thf(fact_769_Num_Oof__nat__simps_I3_J,axiom,
    ! [M3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M3 ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M3 ) ) ) ).

% Num.of_nat_simps(3)
thf(fact_770_Num_Oof__nat__simps_I3_J,axiom,
    ! [M3: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M3 ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M3 ) ) ) ).

% Num.of_nat_simps(3)
thf(fact_771_Num_Oof__nat__simps_I3_J,axiom,
    ! [M3: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M3 ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M3 ) ) ) ).

% Num.of_nat_simps(3)
thf(fact_772_fps__nth__of__nat,axiom,
    ! [N: nat,C: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ ( semiri1524631719018205113ps_nat @ C ) @ N )
          = ( semiri1316708129612266289at_nat @ C ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ ( semiri1524631719018205113ps_nat @ C ) @ N )
          = zero_zero_nat ) ) ) ).

% fps_nth_of_nat
thf(fact_773_fps__nth__of__nat,axiom,
    ! [N: nat,C: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ ( semiri6570152736363784213ps_int @ C ) @ N )
          = ( semiri1314217659103216013at_int @ C ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ ( semiri6570152736363784213ps_int @ C ) @ N )
          = zero_zero_int ) ) ) ).

% fps_nth_of_nat
thf(fact_774_fps__nth__of__nat,axiom,
    ! [N: nat,C: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal2580924720334399070h_real @ ( semiri2475410149736220053s_real @ C ) @ N )
          = ( semiri5074537144036343181t_real @ C ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal2580924720334399070h_real @ ( semiri2475410149736220053s_real @ C ) @ N )
          = zero_zero_real ) ) ) ).

% fps_nth_of_nat
thf(fact_775_fps__mult__of__nat__nth_I1_J,axiom,
    ! [K2: nat,F: formal_Power_fps_nat,N: nat] :
      ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ ( semiri1524631719018205113ps_nat @ K2 ) @ F ) @ N )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ K2 ) @ ( formal3720337525774269570th_nat @ F @ N ) ) ) ).

% fps_mult_of_nat_nth(1)
thf(fact_776_fps__mult__of__nat__nth_I1_J,axiom,
    ! [K2: nat,F: formal_Power_fps_int,N: nat] :
      ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ ( semiri6570152736363784213ps_int @ K2 ) @ F ) @ N )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ ( formal3717847055265219294th_int @ F @ N ) ) ) ).

% fps_mult_of_nat_nth(1)
thf(fact_777_fps__mult__of__nat__nth_I1_J,axiom,
    ! [K2: nat,F: formal3361831859752904756s_real,N: nat] :
      ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ ( semiri2475410149736220053s_real @ K2 ) @ F ) @ N )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( formal2580924720334399070h_real @ F @ N ) ) ) ).

% fps_mult_of_nat_nth(1)
thf(fact_778_fps__mult__of__nat__nth_I2_J,axiom,
    ! [F: formal_Power_fps_nat,K2: nat,N: nat] :
      ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ ( semiri1524631719018205113ps_nat @ K2 ) ) @ N )
      = ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ N ) @ ( semiri1316708129612266289at_nat @ K2 ) ) ) ).

% fps_mult_of_nat_nth(2)
thf(fact_779_fps__mult__of__nat__nth_I2_J,axiom,
    ! [F: formal_Power_fps_int,K2: nat,N: nat] :
      ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ ( semiri6570152736363784213ps_int @ K2 ) ) @ N )
      = ( times_times_int @ ( formal3717847055265219294th_int @ F @ N ) @ ( semiri1314217659103216013at_int @ K2 ) ) ) ).

% fps_mult_of_nat_nth(2)
thf(fact_780_fps__mult__of__nat__nth_I2_J,axiom,
    ! [F: formal3361831859752904756s_real,K2: nat,N: nat] :
      ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ ( semiri2475410149736220053s_real @ K2 ) ) @ N )
      = ( times_times_real @ ( formal2580924720334399070h_real @ F @ N ) @ ( semiri5074537144036343181t_real @ K2 ) ) ) ).

% fps_mult_of_nat_nth(2)
thf(fact_781_fps__XD__nth,axiom,
    ! [A: formal_Power_fps_nat,N: nat] :
      ( ( formal3720337525774269570th_nat @ ( formal814923487339530757XD_nat @ A ) @ N )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( formal3720337525774269570th_nat @ A @ N ) ) ) ).

% fps_XD_nth
thf(fact_782_fps__XD__nth,axiom,
    ! [A: formal_Power_fps_int,N: nat] :
      ( ( formal3717847055265219294th_int @ ( formal812433016830480481XD_int @ A ) @ N )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( formal3717847055265219294th_int @ A @ N ) ) ) ).

% fps_XD_nth
thf(fact_783_fps__XD__nth,axiom,
    ! [A: formal3361831859752904756s_real,N: nat] :
      ( ( formal2580924720334399070h_real @ ( formal4292469010823635553D_real @ A ) @ N )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( formal2580924720334399070h_real @ A @ N ) ) ) ).

% fps_XD_nth
thf(fact_784_fps__X__nth,axiom,
    ! [N: nat] :
      ( ( ( N = one_one_nat )
       => ( ( formal2580924720334399070h_real @ formal4708490801539276157X_real @ N )
          = one_one_real ) )
      & ( ( N != one_one_nat )
       => ( ( formal2580924720334399070h_real @ formal4708490801539276157X_real @ N )
          = zero_zero_real ) ) ) ).

% fps_X_nth
thf(fact_785_fps__X__nth,axiom,
    ! [N: nat] :
      ( ( ( N = one_one_nat )
       => ( ( formal3720337525774269570th_nat @ formal1744162128437646113_X_nat @ N )
          = one_one_nat ) )
      & ( ( N != one_one_nat )
       => ( ( formal3720337525774269570th_nat @ formal1744162128437646113_X_nat @ N )
          = zero_zero_nat ) ) ) ).

% fps_X_nth
thf(fact_786_fps__X__nth,axiom,
    ! [N: nat] :
      ( ( ( N = one_one_nat )
       => ( ( formal3717847055265219294th_int @ formal1741671657928595837_X_int @ N )
          = one_one_int ) )
      & ( ( N != one_one_nat )
       => ( ( formal3717847055265219294th_int @ formal1741671657928595837_X_int @ N )
          = zero_zero_int ) ) ) ).

% fps_X_nth
thf(fact_787_fps__XD__Suc,axiom,
    ! [A: formal_Power_fps_nat,N: nat] :
      ( ( formal3720337525774269570th_nat @ ( formal814923487339530757XD_nat @ A ) @ ( suc @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( formal3720337525774269570th_nat @ A @ ( suc @ N ) ) ) ) ).

% fps_XD_Suc
thf(fact_788_fps__XD__Suc,axiom,
    ! [A: formal_Power_fps_int,N: nat] :
      ( ( formal3717847055265219294th_int @ ( formal812433016830480481XD_int @ A ) @ ( suc @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( formal3717847055265219294th_int @ A @ ( suc @ N ) ) ) ) ).

% fps_XD_Suc
thf(fact_789_fps__XD__Suc,axiom,
    ! [A: formal3361831859752904756s_real,N: nat] :
      ( ( formal2580924720334399070h_real @ ( formal4292469010823635553D_real @ A ) @ ( suc @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( formal2580924720334399070h_real @ A @ ( suc @ N ) ) ) ) ).

% fps_XD_Suc
thf(fact_790_fps__XDp__nth,axiom,
    ! [C: nat,A: formal_Power_fps_nat,N: nat] :
      ( ( formal3720337525774269570th_nat @ ( formal9197787955091086413Dp_nat @ C @ A ) @ N )
      = ( times_times_nat @ ( plus_plus_nat @ C @ ( semiri1316708129612266289at_nat @ N ) ) @ ( formal3720337525774269570th_nat @ A @ N ) ) ) ).

% fps_XDp_nth
thf(fact_791_fps__XDp__nth,axiom,
    ! [C: int,A: formal_Power_fps_int,N: nat] :
      ( ( formal3717847055265219294th_int @ ( formal9195297484582036137Dp_int @ C @ A ) @ N )
      = ( times_times_int @ ( plus_plus_int @ C @ ( semiri1314217659103216013at_int @ N ) ) @ ( formal3717847055265219294th_int @ A @ N ) ) ) ).

% fps_XDp_nth
thf(fact_792_fps__XDp__nth,axiom,
    ! [C: real,A: formal3361831859752904756s_real,N: nat] :
      ( ( formal2580924720334399070h_real @ ( formal2839450981996073129p_real @ C @ A ) @ N )
      = ( times_times_real @ ( plus_plus_real @ C @ ( semiri5074537144036343181t_real @ N ) ) @ ( formal2580924720334399070h_real @ A @ N ) ) ) ).

% fps_XDp_nth
thf(fact_793_inverse__fps__of__nat,axiom,
    ! [N: nat] :
      ( ( invers68952373231134600s_real @ ( semiri2475410149736220053s_real @ N ) )
      = ( formal2098867297714113032t_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% inverse_fps_of_nat
thf(fact_794_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_795_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_796_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_797_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_798_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_799_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_800_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_801_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_802_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_803_fps__of__nat,axiom,
    ! [C: nat] :
      ( ( formal5284259319228341128st_int @ ( semiri1314217659103216013at_int @ C ) )
      = ( semiri6570152736363784213ps_int @ C ) ) ).

% fps_of_nat
thf(fact_804_fps__of__nat,axiom,
    ! [C: nat] :
      ( ( formal2098867297714113032t_real @ ( semiri5074537144036343181t_real @ C ) )
      = ( semiri2475410149736220053s_real @ C ) ) ).

% fps_of_nat
thf(fact_805_fps__integral0__one_H,axiom,
    ( ( ( inverse_inverse_real @ one_one_real )
      = one_one_real )
   => ( ( formal8984515926053063617l_real @ one_on8598947968683843321s_real @ zero_zero_real )
      = formal4708490801539276157X_real ) ) ).

% fps_integral0_one'
thf(fact_806_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N4: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
          @ N4
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_807_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N4: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
          @ N4
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_808_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N4: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
          @ N4
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_809_fps__mult__right__fps__X__plus__1__nth,axiom,
    ! [N: nat,A: formal_Power_fps_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ A @ ( plus_p6043471806551771617ps_nat @ one_on3350087005236239133ps_nat @ formal1744162128437646113_X_nat ) ) @ N )
          = ( formal3720337525774269570th_nat @ A @ N ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ A @ ( plus_p6043471806551771617ps_nat @ one_on3350087005236239133ps_nat @ formal1744162128437646113_X_nat ) ) @ N )
          = ( plus_plus_nat @ ( formal3720337525774269570th_nat @ A @ N ) @ ( formal3720337525774269570th_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% fps_mult_right_fps_X_plus_1_nth
thf(fact_810_fps__mult__right__fps__X__plus__1__nth,axiom,
    ! [N: nat,A: formal_Power_fps_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ A @ ( plus_p1865620787042574909ps_int @ one_on8395608022581818233ps_int @ formal1741671657928595837_X_int ) ) @ N )
          = ( formal3717847055265219294th_int @ A @ N ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ A @ ( plus_p1865620787042574909ps_int @ one_on8395608022581818233ps_int @ formal1741671657928595837_X_int ) ) @ N )
          = ( plus_plus_int @ ( formal3717847055265219294th_int @ A @ N ) @ ( formal3717847055265219294th_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% fps_mult_right_fps_X_plus_1_nth
thf(fact_811_fps__mult__right__fps__X__plus__1__nth,axiom,
    ! [N: nat,A: formal3361831859752904756s_real] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ A @ ( plus_p6008488439947570109s_real @ one_on8598947968683843321s_real @ formal4708490801539276157X_real ) ) @ N )
          = ( formal2580924720334399070h_real @ A @ N ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ A @ ( plus_p6008488439947570109s_real @ one_on8598947968683843321s_real @ formal4708490801539276157X_real ) ) @ N )
          = ( plus_plus_real @ ( formal2580924720334399070h_real @ A @ N ) @ ( formal2580924720334399070h_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% fps_mult_right_fps_X_plus_1_nth
thf(fact_812_fps__mult__fps__X__plus__1__nth,axiom,
    ! [N: nat,A: formal_Power_fps_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ ( plus_p6043471806551771617ps_nat @ one_on3350087005236239133ps_nat @ formal1744162128437646113_X_nat ) @ A ) @ N )
          = ( formal3720337525774269570th_nat @ A @ N ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ ( plus_p6043471806551771617ps_nat @ one_on3350087005236239133ps_nat @ formal1744162128437646113_X_nat ) @ A ) @ N )
          = ( plus_plus_nat @ ( formal3720337525774269570th_nat @ A @ N ) @ ( formal3720337525774269570th_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% fps_mult_fps_X_plus_1_nth
thf(fact_813_fps__mult__fps__X__plus__1__nth,axiom,
    ! [N: nat,A: formal_Power_fps_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ ( plus_p1865620787042574909ps_int @ one_on8395608022581818233ps_int @ formal1741671657928595837_X_int ) @ A ) @ N )
          = ( formal3717847055265219294th_int @ A @ N ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ ( plus_p1865620787042574909ps_int @ one_on8395608022581818233ps_int @ formal1741671657928595837_X_int ) @ A ) @ N )
          = ( plus_plus_int @ ( formal3717847055265219294th_int @ A @ N ) @ ( formal3717847055265219294th_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% fps_mult_fps_X_plus_1_nth
thf(fact_814_fps__mult__fps__X__plus__1__nth,axiom,
    ! [N: nat,A: formal3361831859752904756s_real] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ ( plus_p6008488439947570109s_real @ one_on8598947968683843321s_real @ formal4708490801539276157X_real ) @ A ) @ N )
          = ( formal2580924720334399070h_real @ A @ N ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ ( plus_p6008488439947570109s_real @ one_on8598947968683843321s_real @ formal4708490801539276157X_real ) @ A ) @ N )
          = ( plus_plus_real @ ( formal2580924720334399070h_real @ A @ N ) @ ( formal2580924720334399070h_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% fps_mult_fps_X_plus_1_nth
thf(fact_815_arith__simps_I57_J,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% arith_simps(57)
thf(fact_816_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_817_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_818_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_819_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_820_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_821_right__minus__eq,axiom,
    ! [A: int,B: int] :
      ( ( ( minus_minus_int @ A @ B )
        = zero_zero_int )
      = ( A = B ) ) ).

% right_minus_eq
thf(fact_822_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_823_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_824_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_825_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_826_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_827_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_828_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_829_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_830_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_831_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_832_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_833_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_834_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_835_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_836_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_837_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_838_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_839_diff__Pair,axiom,
    ! [A: nat,B: interval_float,C: nat,D: interval_float] :
      ( ( minus_272754428694525614_float @ ( produc6067866782486577919_float @ A @ B ) @ ( produc6067866782486577919_float @ C @ D ) )
      = ( produc6067866782486577919_float @ ( minus_minus_nat @ A @ C ) @ ( minus_8777630837827617647_float @ B @ D ) ) ) ).

% diff_Pair
thf(fact_840_diff__Pair,axiom,
    ! [A: nat,B: float,C: nat,D: float] :
      ( ( minus_4346062978694385630_float @ ( produc518625033508411951_float @ A @ B ) @ ( produc518625033508411951_float @ C @ D ) )
      = ( produc518625033508411951_float @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_float @ B @ D ) ) ) ).

% diff_Pair
thf(fact_841_diff__Pair,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( minus_4365393887724441320at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( product_Pair_nat_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ D ) ) ) ).

% diff_Pair
thf(fact_842_diff__Pair,axiom,
    ! [A: nat,B: int,C: nat,D: int] :
      ( ( minus_187542868215244612at_int @ ( product_Pair_nat_int @ A @ B ) @ ( product_Pair_nat_int @ C @ D ) )
      = ( product_Pair_nat_int @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ).

% diff_Pair
thf(fact_843_diff__Pair,axiom,
    ! [A: int,B: nat,C: int,D: nat] :
      ( ( minus_5364747599625495876nt_nat @ ( product_Pair_int_nat @ A @ B ) @ ( product_Pair_int_nat @ C @ D ) )
      = ( product_Pair_int_nat @ ( minus_minus_int @ A @ C ) @ ( minus_minus_nat @ B @ D ) ) ) ).

% diff_Pair
thf(fact_844_diff__Pair,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( minus_1186896580116299168nt_int @ ( product_Pair_int_int @ A @ B ) @ ( product_Pair_int_int @ C @ D ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ).

% diff_Pair
thf(fact_845_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_846_diff__self__eq__0,axiom,
    ! [M3: nat] :
      ( ( minus_minus_nat @ M3 @ M3 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_847_diff__Suc__Suc,axiom,
    ! [M3: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M3 ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M3 @ N ) ) ).

% diff_Suc_Suc
thf(fact_848_Suc__diff__diff,axiom,
    ! [M3: nat,N: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M3 ) @ N ) @ ( suc @ K2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M3 @ N ) @ K2 ) ) ).

% Suc_diff_diff
thf(fact_849_fps__sub__nth,axiom,
    ! [F: formal_Power_fps_nat,G: formal_Power_fps_nat,N: nat] :
      ( ( formal3720337525774269570th_nat @ ( minus_1563896255634514737ps_nat @ F @ G ) @ N )
      = ( minus_minus_nat @ ( formal3720337525774269570th_nat @ F @ N ) @ ( formal3720337525774269570th_nat @ G @ N ) ) ) ).

% fps_sub_nth
thf(fact_850_fps__sub__nth,axiom,
    ! [F: formal_Power_fps_int,G: formal_Power_fps_int,N: nat] :
      ( ( formal3717847055265219294th_int @ ( minus_6609417272980093837ps_int @ F @ G ) @ N )
      = ( minus_minus_int @ ( formal3717847055265219294th_int @ F @ N ) @ ( formal3717847055265219294th_int @ G @ N ) ) ) ).

% fps_sub_nth
thf(fact_851_diff__diff__left,axiom,
    ! [I3: nat,J2: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K2 )
      = ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).

% diff_diff_left
thf(fact_852_fps__const__minus,axiom,
    ! [C: int,D: int] :
      ( ( minus_6609417272980093837ps_int @ ( formal5284259319228341128st_int @ C ) @ ( formal5284259319228341128st_int @ D ) )
      = ( formal5284259319228341128st_int @ ( minus_minus_int @ C @ D ) ) ) ).

% fps_const_minus
thf(fact_853_arith__special_I21_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% arith_special(21)
thf(fact_854_arith__special_I21_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% arith_special(21)
thf(fact_855_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_856_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_857_fps__X__mult__right__nth,axiom,
    ! [N: nat,A: formal_Power_fps_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ A @ formal1744162128437646113_X_nat ) @ N )
          = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ A @ formal1744162128437646113_X_nat ) @ N )
          = ( formal3720337525774269570th_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fps_X_mult_right_nth
thf(fact_858_fps__X__mult__right__nth,axiom,
    ! [N: nat,A: formal_Power_fps_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ A @ formal1741671657928595837_X_int ) @ N )
          = zero_zero_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ A @ formal1741671657928595837_X_int ) @ N )
          = ( formal3717847055265219294th_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fps_X_mult_right_nth
thf(fact_859_fps__X__mult__nth,axiom,
    ! [N: nat,F: formal_Power_fps_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ formal1744162128437646113_X_nat @ F ) @ N )
          = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ formal1744162128437646113_X_nat @ F ) @ N )
          = ( formal3720337525774269570th_nat @ F @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fps_X_mult_nth
thf(fact_860_fps__X__mult__nth,axiom,
    ! [N: nat,F: formal_Power_fps_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ formal1741671657928595837_X_int @ F ) @ N )
          = zero_zero_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ formal1741671657928595837_X_int @ F ) @ N )
          = ( formal3717847055265219294th_int @ F @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fps_X_mult_nth
thf(fact_861_diff__Suc__eq__diff__pred,axiom,
    ! [M3: nat,N: nat] :
      ( ( minus_minus_nat @ M3 @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_862_diff__add__0,axiom,
    ! [N: nat,M3: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M3 ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_863_minus__nat_Osimps_I1_J,axiom,
    ! [M3: nat] :
      ( ( minus_minus_nat @ M3 @ zero_zero_nat )
      = M3 ) ).

% minus_nat.simps(1)
thf(fact_864_diffs0__imp__equal,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( minus_minus_nat @ M3 @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M3 )
          = zero_zero_nat )
       => ( M3 = N ) ) ) ).

% diffs0_imp_equal
thf(fact_865_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_866_cancel__ab__semigroup__add__class_Odiff__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 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_867_cancel__ab__semigroup__add__class_Odiff__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 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_868_diff__commute,axiom,
    ! [I3: nat,J2: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K2 ) @ J2 ) ) ).

% diff_commute
thf(fact_869_zero__induct__lemma,axiom,
    ! [P: nat > $o,K2: nat,I3: nat] :
      ( ( P @ K2 )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ ( minus_minus_nat @ K2 @ I3 ) ) ) ) ).

% zero_induct_lemma
thf(fact_870_diff__mult__distrib2,axiom,
    ! [K2: nat,M3: nat,N: nat] :
      ( ( times_times_nat @ K2 @ ( minus_minus_nat @ M3 @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_871_diff__mult__distrib,axiom,
    ! [M3: nat,N: nat,K2: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M3 @ N ) @ K2 )
      = ( minus_minus_nat @ ( times_times_nat @ M3 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).

% diff_mult_distrib
thf(fact_872_times__int__code_I2_J,axiom,
    ! [L2: int] :
      ( ( times_times_int @ zero_zero_int @ L2 )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_873_times__int__code_I1_J,axiom,
    ! [K2: int] :
      ( ( times_times_int @ K2 @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_874_group__cancel_Osub1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4
        = ( plus_plus_real @ K2 @ A ) )
     => ( ( minus_minus_real @ A4 @ B )
        = ( plus_plus_real @ K2 @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_875_group__cancel_Osub1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4
        = ( plus_plus_int @ K2 @ A ) )
     => ( ( minus_minus_int @ A4 @ B )
        = ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_876_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_877_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_878_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_879_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_880_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_881_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_882_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_883_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_884_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_885_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_886_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_887_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_888_diff__diff__add,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% diff_diff_add
thf(fact_889_diff__diff__add,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_add
thf(fact_890_diff__diff__add,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_add
thf(fact_891_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C @ B )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_892_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_893_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_894_Nat_Odiff__cancel,axiom,
    ! [K2: nat,M3: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( minus_minus_nat @ M3 @ N ) ) ).

% Nat.diff_cancel
thf(fact_895_diff__cancel2,axiom,
    ! [M3: nat,K2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
      = ( minus_minus_nat @ M3 @ N ) ) ).

% diff_cancel2
thf(fact_896_diff__add__inverse,axiom,
    ! [N: nat,M3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M3 ) @ N )
      = M3 ) ).

% diff_add_inverse
thf(fact_897_diff__add__inverse2,axiom,
    ! [M3: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N ) @ N )
      = M3 ) ).

% diff_add_inverse2
thf(fact_898_odd__nonzero,axiom,
    ! [Z3: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_899_plus__int__code_I2_J,axiom,
    ! [L2: int] :
      ( ( plus_plus_int @ zero_zero_int @ L2 )
      = L2 ) ).

% plus_int_code(2)
thf(fact_900_plus__int__code_I1_J,axiom,
    ! [K2: int] :
      ( ( plus_plus_int @ K2 @ zero_zero_int )
      = K2 ) ).

% plus_int_code(1)
thf(fact_901_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M4: nat,N4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% add_eq_if
thf(fact_902_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M4: nat,N4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% mult_eq_if
thf(fact_903_fps__integral0__fps__X__power,axiom,
    ! [N: nat] :
      ( ( formal8984515926053063617l_real @ ( power_1846127563762588094s_real @ formal4708490801539276157X_real @ N ) @ zero_zero_real )
      = ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) @ ( power_1846127563762588094s_real @ formal4708490801539276157X_real @ ( suc @ N ) ) ) ) ).

% fps_integral0_fps_X_power
thf(fact_904_Diff__empty,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ bot_bot_set_nat )
      = A4 ) ).

% Diff_empty
thf(fact_905_empty__Diff,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A4 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_906_Diff__cancel,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ A4 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_907_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_908_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_909_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_910_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_911_fps__const__power,axiom,
    ! [C: nat,N: nat] :
      ( ( power_568658719666546786ps_nat @ ( formal5286749789737391404st_nat @ C ) @ N )
      = ( formal5286749789737391404st_nat @ ( power_power_nat @ C @ N ) ) ) ).

% fps_const_power
thf(fact_912_fps__const__power,axiom,
    ! [C: int,N: nat] :
      ( ( power_5614179737012125886ps_int @ ( formal5284259319228341128st_int @ C ) @ N )
      = ( formal5284259319228341128st_int @ ( power_power_int @ C @ N ) ) ) ).

% fps_const_power
thf(fact_913_startsby__zero__power__iff,axiom,
    ! [A: formal_Power_fps_nat,N: nat] :
      ( ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ A @ N ) @ zero_zero_nat )
        = zero_zero_nat )
      = ( ( N != zero_zero_nat )
        & ( ( formal3720337525774269570th_nat @ A @ zero_zero_nat )
          = zero_zero_nat ) ) ) ).

% startsby_zero_power_iff
thf(fact_914_startsby__zero__power__iff,axiom,
    ! [A: formal_Power_fps_int,N: nat] :
      ( ( ( formal3717847055265219294th_int @ ( power_5614179737012125886ps_int @ A @ N ) @ zero_zero_nat )
        = zero_zero_int )
      = ( ( N != zero_zero_nat )
        & ( ( formal3717847055265219294th_int @ A @ zero_zero_nat )
          = zero_zero_int ) ) ) ).

% startsby_zero_power_iff
thf(fact_915_fps__X__power__nth,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( N = K2 )
       => ( ( formal2580924720334399070h_real @ ( power_1846127563762588094s_real @ formal4708490801539276157X_real @ K2 ) @ N )
          = one_one_real ) )
      & ( ( N != K2 )
       => ( ( formal2580924720334399070h_real @ ( power_1846127563762588094s_real @ formal4708490801539276157X_real @ K2 ) @ N )
          = zero_zero_real ) ) ) ).

% fps_X_power_nth
thf(fact_916_fps__X__power__nth,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( N = K2 )
       => ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ formal1744162128437646113_X_nat @ K2 ) @ N )
          = one_one_nat ) )
      & ( ( N != K2 )
       => ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ formal1744162128437646113_X_nat @ K2 ) @ N )
          = zero_zero_nat ) ) ) ).

% fps_X_power_nth
thf(fact_917_fps__X__power__nth,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( N = K2 )
       => ( ( formal3717847055265219294th_int @ ( power_5614179737012125886ps_int @ formal1741671657928595837_X_int @ K2 ) @ N )
          = one_one_int ) )
      & ( ( N != K2 )
       => ( ( formal3717847055265219294th_int @ ( power_5614179737012125886ps_int @ formal1741671657928595837_X_int @ K2 ) @ N )
          = zero_zero_int ) ) ) ).

% fps_X_power_nth
thf(fact_918_startsby__power,axiom,
    ! [A: formal_Power_fps_nat,V2: nat,N: nat] :
      ( ( ( formal3720337525774269570th_nat @ A @ zero_zero_nat )
        = V2 )
     => ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ A @ N ) @ zero_zero_nat )
        = ( power_power_nat @ V2 @ N ) ) ) ).

% startsby_power
thf(fact_919_startsby__power,axiom,
    ! [A: formal_Power_fps_int,V2: int,N: nat] :
      ( ( ( formal3717847055265219294th_int @ A @ zero_zero_nat )
        = V2 )
     => ( ( formal3717847055265219294th_int @ ( power_5614179737012125886ps_int @ A @ N ) @ zero_zero_nat )
        = ( power_power_int @ V2 @ N ) ) ) ).

% startsby_power
thf(fact_920_fps__nth__power__0,axiom,
    ! [A: formal_Power_fps_nat,N: nat] :
      ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ A @ N ) @ zero_zero_nat )
      = ( power_power_nat @ ( formal3720337525774269570th_nat @ A @ zero_zero_nat ) @ N ) ) ).

% fps_nth_power_0
thf(fact_921_fps__nth__power__0,axiom,
    ! [A: formal_Power_fps_int,N: nat] :
      ( ( formal3717847055265219294th_int @ ( power_5614179737012125886ps_int @ A @ N ) @ zero_zero_nat )
      = ( power_power_int @ ( formal3717847055265219294th_int @ A @ zero_zero_nat ) @ N ) ) ).

% fps_nth_power_0
thf(fact_922_power__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N )
       != zero_zero_nat ) ) ).

% power_not_zero
thf(fact_923_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N )
       != zero_zero_int ) ) ).

% power_not_zero
thf(fact_924_minus__int__code_I1_J,axiom,
    ! [K2: int] :
      ( ( minus_minus_int @ K2 @ zero_zero_int )
      = K2 ) ).

% minus_int_code(1)
thf(fact_925_startsby__zero__power__nth__same,axiom,
    ! [A: formal_Power_fps_nat,N: nat] :
      ( ( ( formal3720337525774269570th_nat @ A @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ A @ N ) @ N )
        = ( power_power_nat @ ( formal3720337525774269570th_nat @ A @ one_one_nat ) @ N ) ) ) ).

% startsby_zero_power_nth_same
thf(fact_926_startsby__zero__power__nth__same,axiom,
    ! [A: formal_Power_fps_int,N: nat] :
      ( ( ( formal3717847055265219294th_int @ A @ zero_zero_nat )
        = zero_zero_int )
     => ( ( formal3717847055265219294th_int @ ( power_5614179737012125886ps_int @ A @ N ) @ N )
        = ( power_power_int @ ( formal3717847055265219294th_int @ A @ one_one_nat ) @ N ) ) ) ).

% startsby_zero_power_nth_same
thf(fact_927_power_Osimps_I1_J,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power.simps(1)
thf(fact_928_power_Osimps_I1_J,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power.simps(1)
thf(fact_929_power_Osimps_I1_J,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power.simps(1)
thf(fact_930_power__Suc2,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_931_power__Suc2,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_932_power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_Suc
thf(fact_933_power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_Suc
thf(fact_934_fps__power__first,axiom,
    ! [A: formal_Power_fps_nat,N: nat] :
      ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ A @ N ) @ one_one_nat )
      = ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( power_power_nat @ ( formal3720337525774269570th_nat @ A @ zero_zero_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ ( formal3720337525774269570th_nat @ A @ one_one_nat ) ) ) ).

% fps_power_first
thf(fact_935_fps__power__first,axiom,
    ! [A: formal_Power_fps_int,N: nat] :
      ( ( formal3717847055265219294th_int @ ( power_5614179737012125886ps_int @ A @ N ) @ one_one_nat )
      = ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( formal3717847055265219294th_int @ A @ zero_zero_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ ( formal3717847055265219294th_int @ A @ one_one_nat ) ) ) ).

% fps_power_first
thf(fact_936_fps__power__first,axiom,
    ! [A: formal3361831859752904756s_real,N: nat] :
      ( ( formal2580924720334399070h_real @ ( power_1846127563762588094s_real @ A @ N ) @ one_one_nat )
      = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( formal2580924720334399070h_real @ A @ zero_zero_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ ( formal2580924720334399070h_real @ A @ one_one_nat ) ) ) ).

% fps_power_first
thf(fact_937_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_938_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_939_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_940_startsby__nonzero__power,axiom,
    ! [A: formal_Power_fps_nat,N: nat] :
      ( ( ( formal3720337525774269570th_nat @ A @ zero_zero_nat )
       != zero_zero_nat )
     => ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ A @ N ) @ zero_zero_nat )
       != zero_zero_nat ) ) ).

% startsby_nonzero_power
thf(fact_941_startsby__nonzero__power,axiom,
    ! [A: formal_Power_fps_int,N: nat] :
      ( ( ( formal3717847055265219294th_int @ A @ zero_zero_nat )
       != zero_zero_int )
     => ( ( formal3717847055265219294th_int @ ( power_5614179737012125886ps_int @ A @ N ) @ zero_zero_nat )
       != zero_zero_int ) ) ).

% startsby_nonzero_power
thf(fact_942_startsby__one__power,axiom,
    ! [A: formal_Power_fps_nat,N: nat] :
      ( ( ( formal3720337525774269570th_nat @ A @ zero_zero_nat )
        = one_one_nat )
     => ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ A @ N ) @ zero_zero_nat )
        = one_one_nat ) ) ).

% startsby_one_power
thf(fact_943_startsby__one__power,axiom,
    ! [A: formal_Power_fps_int,N: nat] :
      ( ( ( formal3717847055265219294th_int @ A @ zero_zero_nat )
        = one_one_int )
     => ( ( formal3717847055265219294th_int @ ( power_5614179737012125886ps_int @ A @ N ) @ zero_zero_nat )
        = one_one_int ) ) ).

% startsby_one_power
thf(fact_944_startsby__one__power,axiom,
    ! [A: formal3361831859752904756s_real,N: nat] :
      ( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
        = one_one_real )
     => ( ( formal2580924720334399070h_real @ ( power_1846127563762588094s_real @ A @ N ) @ zero_zero_nat )
        = one_one_real ) ) ).

% startsby_one_power
thf(fact_945_fps__power__first__eq_H,axiom,
    ! [A: formal_Power_fps_nat,N: nat] :
      ( ( ( formal3720337525774269570th_nat @ A @ one_one_nat )
        = one_one_nat )
     => ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ A @ N ) @ one_one_nat )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( power_power_nat @ ( formal3720337525774269570th_nat @ A @ zero_zero_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fps_power_first_eq'
thf(fact_946_fps__power__first__eq_H,axiom,
    ! [A: formal_Power_fps_int,N: nat] :
      ( ( ( formal3717847055265219294th_int @ A @ one_one_nat )
        = one_one_int )
     => ( ( formal3717847055265219294th_int @ ( power_5614179737012125886ps_int @ A @ N ) @ one_one_nat )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( formal3717847055265219294th_int @ A @ zero_zero_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fps_power_first_eq'
thf(fact_947_fps__power__first__eq_H,axiom,
    ! [A: formal3361831859752904756s_real,N: nat] :
      ( ( ( formal2580924720334399070h_real @ A @ one_one_nat )
        = one_one_real )
     => ( ( formal2580924720334399070h_real @ ( power_1846127563762588094s_real @ A @ N ) @ one_one_nat )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( formal2580924720334399070h_real @ A @ zero_zero_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fps_power_first_eq'
thf(fact_948_fps__power__Suc__eqD,axiom,
    ! [F: formal_Power_fps_int,M3: nat,G: formal_Power_fps_int] :
      ( ( ( power_5614179737012125886ps_int @ F @ ( suc @ M3 ) )
        = ( power_5614179737012125886ps_int @ G @ ( suc @ M3 ) ) )
     => ( ( ( formal3717847055265219294th_int @ F @ zero_zero_nat )
          = ( formal3717847055265219294th_int @ G @ zero_zero_nat ) )
       => ( ( ( formal3717847055265219294th_int @ F @ zero_zero_nat )
           != zero_zero_int )
         => ( F = G ) ) ) ) ).

% fps_power_Suc_eqD
thf(fact_949_fps__cos__add,axiom,
    ! [A: real,B: real] :
      ( ( formal461277676486907980s_real @ ( plus_plus_real @ A @ B ) )
      = ( minus_6791916864952032525s_real @ ( times_7561426564079326009s_real @ ( formal461277676486907980s_real @ A ) @ ( formal461277676486907980s_real @ B ) ) @ ( times_7561426564079326009s_real @ ( formal6437758938379178589n_real @ A ) @ ( formal6437758938379178589n_real @ B ) ) ) ) ).

% fps_cos_add
thf(fact_950_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P7: real,M4: nat] : ( if_real @ ( M4 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P7 @ ( power_power_real @ P7 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_951_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P7: nat,M4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P7 @ ( power_power_nat @ P7 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_952_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P7: int,M4: nat] : ( if_int @ ( M4 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P7 @ ( power_power_int @ P7 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_953_fps__power__first__eq,axiom,
    ! [A: formal_Power_fps_nat,N: nat] :
      ( ( ( formal3720337525774269570th_nat @ A @ zero_zero_nat )
        = one_one_nat )
     => ( ( formal3720337525774269570th_nat @ ( power_568658719666546786ps_nat @ A @ N ) @ one_one_nat )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( formal3720337525774269570th_nat @ A @ one_one_nat ) ) ) ) ).

% fps_power_first_eq
thf(fact_954_fps__power__first__eq,axiom,
    ! [A: formal_Power_fps_int,N: nat] :
      ( ( ( formal3717847055265219294th_int @ A @ zero_zero_nat )
        = one_one_int )
     => ( ( formal3717847055265219294th_int @ ( power_5614179737012125886ps_int @ A @ N ) @ one_one_nat )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( formal3717847055265219294th_int @ A @ one_one_nat ) ) ) ) ).

% fps_power_first_eq
thf(fact_955_fps__power__first__eq,axiom,
    ! [A: formal3361831859752904756s_real,N: nat] :
      ( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
        = one_one_real )
     => ( ( formal2580924720334399070h_real @ ( power_1846127563762588094s_real @ A @ N ) @ one_one_nat )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( formal2580924720334399070h_real @ A @ one_one_nat ) ) ) ) ).

% fps_power_first_eq
thf(fact_956_fps__hypergeo__1__0,axiom,
    ! [C: real] :
      ( ( formal6618874005373735610o_real @ ( cons_real @ one_one_real @ nil_real ) @ nil_real @ C )
      = ( divide1155267253282662278s_real @ one_on8598947968683843321s_real @ ( minus_6791916864952032525s_real @ one_on8598947968683843321s_real @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ C ) @ formal4708490801539276157X_real ) ) ) ) ).

% fps_hypergeo_1_0
thf(fact_957_fps__radical__power__nth,axiom,
    ! [R: nat > real > real,K2: nat,A: formal3361831859752904756s_real] :
      ( ( ( power_power_real @ ( R @ K2 @ ( formal2580924720334399070h_real @ A @ zero_zero_nat ) ) @ K2 )
        = ( formal2580924720334399070h_real @ A @ zero_zero_nat ) )
     => ( ( ( K2 = zero_zero_nat )
         => ( ( formal2580924720334399070h_real @ ( power_1846127563762588094s_real @ ( formal8604817403481219167l_real @ R @ K2 @ A ) @ K2 ) @ zero_zero_nat )
            = one_one_real ) )
        & ( ( K2 != zero_zero_nat )
         => ( ( formal2580924720334399070h_real @ ( power_1846127563762588094s_real @ ( formal8604817403481219167l_real @ R @ K2 @ A ) @ K2 ) @ zero_zero_nat )
            = ( formal2580924720334399070h_real @ A @ zero_zero_nat ) ) ) ) ) ).

% fps_radical_power_nth
thf(fact_958_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_959_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_960_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( divide_divide_real @ C @ A )
        = ( divide_divide_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_961_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_962_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_963_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_964_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_965_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_966_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_967_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_968_nat__power__eq__Suc__0__iff,axiom,
    ! [X2: nat,M3: nat] :
      ( ( ( power_power_nat @ X2 @ M3 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M3 = zero_zero_nat )
        | ( X2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_969_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_970_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_971_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_972_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_973_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_974_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_975_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_976_mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% mult_divide_mult_cancel_left
thf(fact_977_mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% mult_divide_mult_cancel_right
thf(fact_978_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = ( divide_divide_real @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_979_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_980_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_981_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_982_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_983_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_984_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = one_one_real )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_985_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A @ B ) )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_986_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_987_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = one_one_real ) ) ) ).

% divide_self_if
thf(fact_988_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ( divide_divide_real @ B @ A )
        = one_one_real )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_989_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B @ A ) )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_990_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( divide_divide_real @ one_one_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
thf(fact_991_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( divide_divide_real @ one_one_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
thf(fact_992_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_993_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_994_fps__radical__nth__0,axiom,
    ! [N: nat,R: nat > real > real,A: formal3361831859752904756s_real] :
      ( ( ( N = zero_zero_nat )
       => ( ( formal2580924720334399070h_real @ ( formal8604817403481219167l_real @ R @ N @ A ) @ zero_zero_nat )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( formal2580924720334399070h_real @ ( formal8604817403481219167l_real @ R @ N @ A ) @ zero_zero_nat )
          = ( R @ N @ ( formal2580924720334399070h_real @ A @ zero_zero_nat ) ) ) ) ) ).

% fps_radical_nth_0
thf(fact_995_fps__divide__nth__0,axiom,
    ! [G: formal3361831859752904756s_real,F: formal3361831859752904756s_real] :
      ( ( ( formal2580924720334399070h_real @ G @ zero_zero_nat )
       != zero_zero_real )
     => ( ( formal2580924720334399070h_real @ ( divide1155267253282662278s_real @ F @ G ) @ zero_zero_nat )
        = ( divide_divide_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) ) ).

% fps_divide_nth_0
thf(fact_996_frac__eq__eq,axiom,
    ! [Y: real,Z3: real,X2: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( ( divide_divide_real @ X2 @ Y )
            = ( divide_divide_real @ W2 @ Z3 ) )
          = ( ( times_times_real @ X2 @ Z3 )
            = ( times_times_real @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_997_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( divide_divide_real @ B @ C )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_998_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_999_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B
          = ( times_times_real @ A @ C ) )
       => ( ( divide_divide_real @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_1000_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = B )
       => ( A
          = ( divide_divide_real @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_1001_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B @ C )
          = A )
        = ( B
          = ( times_times_real @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_1002_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A
          = ( divide_divide_real @ B @ C ) )
        = ( ( times_times_real @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_1003_fps__const__divide,axiom,
    ! [X2: real,Y: real] :
      ( ( divide1155267253282662278s_real @ ( formal2098867297714113032t_real @ X2 ) @ ( formal2098867297714113032t_real @ Y ) )
      = ( formal2098867297714113032t_real @ ( divide_divide_real @ X2 @ Y ) ) ) ).

% fps_const_divide
thf(fact_1004_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( ( divide_divide_real @ A @ B )
          = one_one_real )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_1005_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z3: real,A: real,B: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z3 ) @ B )
          = B ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z3 ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_1006_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z3: real,A: real,B: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_1007_add__frac__eq,axiom,
    ! [Y: real,Z3: real,X2: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ W2 @ Z3 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z3 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) ) ) ) ).

% add_frac_eq
thf(fact_1008_add__frac__num,axiom,
    ! [Y: real,X2: real,Z3: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y ) @ Z3 )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_1009_add__num__frac,axiom,
    ! [Y: real,Z3: real,X2: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z3 @ ( divide_divide_real @ X2 @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_1010_add__divide__eq__iff,axiom,
    ! [Z3: real,X2: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y @ Z3 ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% add_divide_eq_iff
thf(fact_1011_divide__add__eq__iff,axiom,
    ! [Z3: real,X2: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z3 ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_add_eq_iff
thf(fact_1012_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z3: real,A: real,B: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_1013_diff__frac__eq,axiom,
    ! [Y: real,Z3: real,X2: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ W2 @ Z3 ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z3 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_1014_diff__divide__eq__iff,axiom,
    ! [Z3: real,X2: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( minus_minus_real @ X2 @ ( divide_divide_real @ Y @ Z3 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% diff_divide_eq_iff
thf(fact_1015_divide__diff__eq__iff,axiom,
    ! [Z3: real,X2: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Z3 ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( times_times_real @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_diff_eq_iff
thf(fact_1016_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ A )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_1017_fps__divide__1_H,axiom,
    ! [A: formal3361831859752904756s_real] :
      ( ( ( inverse_inverse_real @ one_one_real )
        = one_one_real )
     => ( ( divide1155267253282662278s_real @ A @ one_on8598947968683843321s_real )
        = A ) ) ).

% fps_divide_1'
thf(fact_1018_fps__div__fps__X__nth_H,axiom,
    ! [F: formal3361831859752904756s_real,K2: nat] :
      ( ( ( inverse_inverse_real @ one_one_real )
        = one_one_real )
     => ( ( formal2580924720334399070h_real @ ( divide1155267253282662278s_real @ F @ formal4708490801539276157X_real ) @ K2 )
        = ( formal2580924720334399070h_real @ F @ ( suc @ K2 ) ) ) ) ).

% fps_div_fps_X_nth'
thf(fact_1019_fps__div__fps__X__power__nth_H,axiom,
    ! [F: formal3361831859752904756s_real,N: nat,K2: nat] :
      ( ( ( inverse_inverse_real @ one_one_real )
        = one_one_real )
     => ( ( formal2580924720334399070h_real @ ( divide1155267253282662278s_real @ F @ ( power_1846127563762588094s_real @ formal4708490801539276157X_real @ N ) ) @ K2 )
        = ( formal2580924720334399070h_real @ F @ ( plus_plus_nat @ K2 @ N ) ) ) ) ).

% fps_div_fps_X_power_nth'
thf(fact_1020_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_1021_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_1022_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_1023_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_1024_div__by__Suc__0,axiom,
    ! [M3: nat] :
      ( ( divide_divide_nat @ M3 @ ( suc @ zero_zero_nat ) )
      = M3 ) ).

% div_by_Suc_0
thf(fact_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_nat__mult__div__cancel__disj,axiom,
    ! [K2: nat,M3: nat,N: nat] :
      ( ( ( K2 = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N ) )
          = zero_zero_nat ) )
      & ( ( K2 != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N ) )
          = ( divide_divide_nat @ M3 @ N ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_1036_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_1037_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_1038_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_1039_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_1040__092_060open_062bernoulli__pmf_A_I1_A_P_A_Ireal_A_Ilength_Axs_J_A_L_A1_J_J_A_092_060bind_062_A_I_092_060lambda_062y_O_Apmf__of__set_A_123_O_O_060length_Axs_125_A_092_060bind_062_A_I_092_060lambda_062k_O_Areturn__pmf_A_Iif_Ay_Athen_Alength_Axs_Aelse_Ak_J_J_J_A_061_Apmf__of__set_A_123_O_O_060length_Axs_A_L_A1_125_092_060close_062,axiom,
    ( ( probab4549177196568532785_o_nat @ ( probab6844364797682710202li_pmf @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( size_size_list_a @ xs ) ) @ one_one_real ) ) )
      @ ^ [Y3: $o] :
          ( probab5865774939380454169at_nat @ ( probab1830274953030043784et_nat @ ( set_ord_lessThan_nat @ ( size_size_list_a @ xs ) ) )
          @ ^ [K: nat] : ( probab4138676067695765237mf_nat @ ( if_nat @ Y3 @ ( size_size_list_a @ xs ) @ K ) ) ) )
    = ( probab1830274953030043784et_nat @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( size_size_list_a @ xs ) @ one_one_nat ) ) ) ) ).

% \<open>bernoulli_pmf (1 / (real (length xs) + 1)) \<bind> (\<lambda>y. pmf_of_set {..<length xs} \<bind> (\<lambda>k. return_pmf (if y then length xs else k))) = pmf_of_set {..<length xs + 1}\<close>
thf(fact_1041_b,axiom,
    ( ( probab5865774939380454169at_nat @ ( probab1830274953030043784et_nat @ ( set_ord_lessThan_nat @ ( size_size_list_a @ xs ) ) )
      @ ^ [K: nat] :
          ( probab4549177196568532785_o_nat @ ( probab6844364797682710202li_pmf @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( size_size_list_a @ xs ) ) @ one_one_real ) ) )
          @ ^ [Coin: $o] : ( probab4138676067695765237mf_nat @ ( if_nat @ Coin @ ( size_size_list_a @ xs ) @ K ) ) ) )
    = ( probab1830274953030043784et_nat @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( size_size_list_a @ xs ) @ one_one_nat ) ) ) ) ).

% b
thf(fact_1042__092_060open_062pmf__of__set_A_123_O_O_060length_Axs_125_A_092_060bind_062_A_I_092_060lambda_062k_O_Abernoulli__pmf_A_I1_A_P_A_Ireal_A_Ilength_Axs_J_A_L_A1_J_J_A_092_060bind_062_A_I_092_060lambda_062coin_O_Areturn__pmf_A_Iif_Acoin_Athen_Alength_Axs_Aelse_Ak_J_J_J_A_061_Abernoulli__pmf_A_I1_A_P_A_Ireal_A_Ilength_Axs_J_A_L_A1_J_J_A_092_060bind_062_A_I_092_060lambda_062y_O_Apmf__of__set_A_123_O_O_060length_Axs_125_A_092_060bind_062_A_I_092_060lambda_062k_O_Areturn__pmf_A_Iif_Ay_Athen_Alength_Axs_Aelse_Ak_J_J_J_092_060close_062,axiom,
    ( ( probab5865774939380454169at_nat @ ( probab1830274953030043784et_nat @ ( set_ord_lessThan_nat @ ( size_size_list_a @ xs ) ) )
      @ ^ [K: nat] :
          ( probab4549177196568532785_o_nat @ ( probab6844364797682710202li_pmf @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( size_size_list_a @ xs ) ) @ one_one_real ) ) )
          @ ^ [Coin: $o] : ( probab4138676067695765237mf_nat @ ( if_nat @ Coin @ ( size_size_list_a @ xs ) @ K ) ) ) )
    = ( probab4549177196568532785_o_nat @ ( probab6844364797682710202li_pmf @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( size_size_list_a @ xs ) ) @ one_one_real ) ) )
      @ ^ [Y3: $o] :
          ( probab5865774939380454169at_nat @ ( probab1830274953030043784et_nat @ ( set_ord_lessThan_nat @ ( size_size_list_a @ xs ) ) )
          @ ^ [K: nat] : ( probab4138676067695765237mf_nat @ ( if_nat @ Y3 @ ( size_size_list_a @ xs ) @ K ) ) ) ) ) ).

% \<open>pmf_of_set {..<length xs} \<bind> (\<lambda>k. bernoulli_pmf (1 / (real (length xs) + 1)) \<bind> (\<lambda>coin. return_pmf (if coin then length xs else k))) = bernoulli_pmf (1 / (real (length xs) + 1)) \<bind> (\<lambda>y. pmf_of_set {..<length xs} \<bind> (\<lambda>k. return_pmf (if y then length xs else k)))\<close>
thf(fact_1043_bernoulli__pmf__1,axiom,
    ( ( probab6844364797682710202li_pmf @ one_one_real )
    = ( probab7739007152833094963_pmf_o @ $true ) ) ).

% bernoulli_pmf_1
thf(fact_1044_int_Onat__pow__Suc,axiom,
    ! [X2: int,N: nat] :
      ( ( power_power_int @ X2 @ ( suc @ N ) )
      = ( times_times_int @ ( power_power_int @ X2 @ N ) @ X2 ) ) ).

% int.nat_pow_Suc
thf(fact_1045_int_Onat__pow__0,axiom,
    ! [X2: int] :
      ( ( power_power_int @ X2 @ zero_zero_nat )
      = one_one_int ) ).

% int.nat_pow_0
thf(fact_1046_int_Onat__pow__zero,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ( power_power_int @ zero_zero_int @ N )
        = zero_zero_int ) ) ).

% int.nat_pow_zero
thf(fact_1047_int_Ozero__not__one,axiom,
    zero_zero_int != one_one_int ).

% int.zero_not_one
thf(fact_1048_length__n__lists,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( size_s349497388124573686list_a @ ( n_lists_a @ N @ Xs ) )
      = ( power_power_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).

% length_n_lists
thf(fact_1049_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_1050_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_1051_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_1052_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_1053_fk__update__2_Oelims,axiom,
    ! [X2: nat,Xa: produc7248412053542808358at_nat,Y: probab692842508561698786at_nat] :
      ( ( ( freque7609560310271430554_2_nat @ X2 @ Xa )
        = Y )
     => ~ ! [M2: nat,X3: nat,L: nat] :
            ( ( Xa
              = ( produc487386426758144856at_nat @ M2 @ ( product_Pair_nat_nat @ X3 @ L ) ) )
           => ( Y
             != ( probab196166181871008007at_nat @ ( probab6844364797682710202li_pmf @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) )
                @ ^ [Coin: $o] : ( probab6260846384068219595at_nat @ ( produc487386426758144856at_nat @ ( plus_plus_nat @ M2 @ one_one_nat ) @ ( if_Pro6206227464963214023at_nat @ Coin @ ( product_Pair_nat_nat @ X2 @ zero_zero_nat ) @ ( product_Pair_nat_nat @ X3 @ ( plus_plus_nat @ L @ ( zero_n2687167440665602831ol_nat @ ( X3 = X2 ) ) ) ) ) ) ) ) ) ) ) ).

% fk_update_2.elims
thf(fact_1054_fk__update__2_Oelims,axiom,
    ! [X2: a,Xa: produc6774132644148096814_a_nat,Y: probab4139393509369344520_a_nat] :
      ( ( ( freque5687656950882125236te_2_a @ X2 @ Xa )
        = Y )
     => ~ ! [M2: nat,X3: a,L: nat] :
            ( ( Xa
              = ( produc148073511828866022_a_nat @ M2 @ ( product_Pair_a_nat @ X3 @ L ) ) )
           => ( Y
             != ( probab7780334331721979789_a_nat @ ( probab6844364797682710202li_pmf @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) )
                @ ^ [Coin: $o] : ( probab6296365750099732169_a_nat @ ( produc148073511828866022_a_nat @ ( plus_plus_nat @ M2 @ one_one_nat ) @ ( if_Pro2324015151519390263_a_nat @ Coin @ ( product_Pair_a_nat @ X2 @ zero_zero_nat ) @ ( product_Pair_a_nat @ X3 @ ( plus_plus_nat @ L @ ( zero_n2687167440665602831ol_nat @ ( X3 = X2 ) ) ) ) ) ) ) ) ) ) ) ).

% fk_update_2.elims
thf(fact_1055_fk__update__2_Osimps,axiom,
    ! [A: nat,M3: nat,X2: nat,L2: nat] :
      ( ( freque7609560310271430554_2_nat @ A @ ( produc487386426758144856at_nat @ M3 @ ( product_Pair_nat_nat @ X2 @ L2 ) ) )
      = ( probab196166181871008007at_nat @ ( probab6844364797682710202li_pmf @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) )
        @ ^ [Coin: $o] : ( probab6260846384068219595at_nat @ ( produc487386426758144856at_nat @ ( plus_plus_nat @ M3 @ one_one_nat ) @ ( if_Pro6206227464963214023at_nat @ Coin @ ( product_Pair_nat_nat @ A @ zero_zero_nat ) @ ( product_Pair_nat_nat @ X2 @ ( plus_plus_nat @ L2 @ ( zero_n2687167440665602831ol_nat @ ( X2 = A ) ) ) ) ) ) ) ) ) ).

% fk_update_2.simps
thf(fact_1056_fk__update__2_Osimps,axiom,
    ! [A: a,M3: nat,X2: a,L2: nat] :
      ( ( freque5687656950882125236te_2_a @ A @ ( produc148073511828866022_a_nat @ M3 @ ( product_Pair_a_nat @ X2 @ L2 ) ) )
      = ( probab7780334331721979789_a_nat @ ( probab6844364797682710202li_pmf @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) )
        @ ^ [Coin: $o] : ( probab6296365750099732169_a_nat @ ( produc148073511828866022_a_nat @ ( plus_plus_nat @ M3 @ one_one_nat ) @ ( if_Pro2324015151519390263_a_nat @ Coin @ ( product_Pair_a_nat @ A @ zero_zero_nat ) @ ( product_Pair_a_nat @ X2 @ ( plus_plus_nat @ L2 @ ( zero_n2687167440665602831ol_nat @ ( X2 = A ) ) ) ) ) ) ) ) ) ).

% fk_update_2.simps
thf(fact_1057_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1058_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1059_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_1060_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_1061_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_1062_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_1063_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P7: $o] : ( if_real @ P7 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_1064_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P7: $o] : ( if_nat @ P7 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_1065_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P7: $o] : ( if_int @ P7 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_1066_split__of__bool,axiom,
    ! [P: real > $o,P3: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P3 ) )
      = ( ( P3
         => ( P @ one_one_real ) )
        & ( ~ P3
         => ( P @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_1067_split__of__bool,axiom,
    ! [P: nat > $o,P3: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P3 ) )
      = ( ( P3
         => ( P @ one_one_nat ) )
        & ( ~ P3
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_1068_split__of__bool,axiom,
    ! [P: int > $o,P3: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P3 ) )
      = ( ( P3
         => ( P @ one_one_int ) )
        & ( ~ P3
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_1069_split__of__bool__asm,axiom,
    ! [P: real > $o,P3: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P3 ) )
      = ( ~ ( ( P3
              & ~ ( P @ one_one_real ) )
            | ( ~ P3
              & ~ ( P @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_1070_split__of__bool__asm,axiom,
    ! [P: nat > $o,P3: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P3 ) )
      = ( ~ ( ( P3
              & ~ ( P @ one_one_nat ) )
            | ( ~ P3
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_1071_split__of__bool__asm,axiom,
    ! [P: int > $o,P3: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P3 ) )
      = ( ~ ( ( P3
              & ~ ( P @ one_one_int ) )
            | ( ~ P3
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_1072_fk__update__2_Opelims,axiom,
    ! [X2: nat,Xa: produc7248412053542808358at_nat,Y: probab692842508561698786at_nat] :
      ( ( ( freque7609560310271430554_2_nat @ X2 @ Xa )
        = Y )
     => ( ( accp_P9051044500959620704at_nat @ freque8067986811854330241el_nat @ ( produc6385450045882626063at_nat @ X2 @ Xa ) )
       => ~ ! [M2: nat,X3: nat,L: nat] :
              ( ( Xa
                = ( produc487386426758144856at_nat @ M2 @ ( product_Pair_nat_nat @ X3 @ L ) ) )
             => ( ( Y
                  = ( probab196166181871008007at_nat @ ( probab6844364797682710202li_pmf @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) )
                    @ ^ [Coin: $o] : ( probab6260846384068219595at_nat @ ( produc487386426758144856at_nat @ ( plus_plus_nat @ M2 @ one_one_nat ) @ ( if_Pro6206227464963214023at_nat @ Coin @ ( product_Pair_nat_nat @ X2 @ zero_zero_nat ) @ ( product_Pair_nat_nat @ X3 @ ( plus_plus_nat @ L @ ( zero_n2687167440665602831ol_nat @ ( X3 = X2 ) ) ) ) ) ) ) ) )
               => ~ ( accp_P9051044500959620704at_nat @ freque8067986811854330241el_nat @ ( produc6385450045882626063at_nat @ X2 @ ( produc487386426758144856at_nat @ M2 @ ( product_Pair_nat_nat @ X3 @ L ) ) ) ) ) ) ) ) ).

% fk_update_2.pelims
thf(fact_1073_fk__update__2_Opelims,axiom,
    ! [X2: a,Xa: produc6774132644148096814_a_nat,Y: probab4139393509369344520_a_nat] :
      ( ( ( freque5687656950882125236te_2_a @ X2 @ Xa )
        = Y )
     => ( ( accp_P5245135987034508768_a_nat @ freque8292879874172675981_rel_a @ ( produc3776435479397687751_a_nat @ X2 @ Xa ) )
       => ~ ! [M2: nat,X3: a,L: nat] :
              ( ( Xa
                = ( produc148073511828866022_a_nat @ M2 @ ( product_Pair_a_nat @ X3 @ L ) ) )
             => ( ( Y
                  = ( probab7780334331721979789_a_nat @ ( probab6844364797682710202li_pmf @ ( divide_divide_real @ one_one_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) )
                    @ ^ [Coin: $o] : ( probab6296365750099732169_a_nat @ ( produc148073511828866022_a_nat @ ( plus_plus_nat @ M2 @ one_one_nat ) @ ( if_Pro2324015151519390263_a_nat @ Coin @ ( product_Pair_a_nat @ X2 @ zero_zero_nat ) @ ( product_Pair_a_nat @ X3 @ ( plus_plus_nat @ L @ ( zero_n2687167440665602831ol_nat @ ( X3 = X2 ) ) ) ) ) ) ) ) )
               => ~ ( accp_P5245135987034508768_a_nat @ freque8292879874172675981_rel_a @ ( produc3776435479397687751_a_nat @ X2 @ ( produc148073511828866022_a_nat @ M2 @ ( product_Pair_a_nat @ X3 @ L ) ) ) ) ) ) ) ) ).

% fk_update_2.pelims
thf(fact_1074_radical__inverse,axiom,
    ! [K2: nat,R: nat > real > real,A: formal3361831859752904756s_real] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ( power_power_real @ ( R @ K2 @ ( formal2580924720334399070h_real @ A @ zero_zero_nat ) ) @ K2 )
          = ( formal2580924720334399070h_real @ A @ zero_zero_nat ) )
       => ( ( ( power_power_real @ ( R @ K2 @ one_one_real ) @ K2 )
            = one_one_real )
         => ( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
             != zero_zero_real )
           => ( ( ( R @ K2 @ ( inverse_inverse_real @ ( formal2580924720334399070h_real @ A @ zero_zero_nat ) ) )
                = ( divide_divide_real @ ( R @ K2 @ one_one_real ) @ ( R @ K2 @ ( formal2580924720334399070h_real @ A @ zero_zero_nat ) ) ) )
              = ( ( formal8604817403481219167l_real @ R @ K2 @ ( invers68952373231134600s_real @ A ) )
                = ( divide1155267253282662278s_real @ ( formal8604817403481219167l_real @ R @ K2 @ one_on8598947968683843321s_real ) @ ( formal8604817403481219167l_real @ R @ K2 @ A ) ) ) ) ) ) ) ) ).

% radical_inverse
thf(fact_1075_zero__order_I5_J,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% zero_order(5)
thf(fact_1076_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_1077_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_1078_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_1079_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_1080_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_1081_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_1082_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1083_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_1084_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_1085_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1086_Suc__mono,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ N )
     => ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1087_Suc__less__eq,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) )
      = ( ord_less_nat @ M3 @ N ) ) ).

% Suc_less_eq
thf(fact_1088_lessThan__iff,axiom,
    ! [I3: nat,K2: nat] :
      ( ( member_nat @ I3 @ ( set_ord_lessThan_nat @ K2 ) )
      = ( ord_less_nat @ I3 @ K2 ) ) ).

% lessThan_iff
thf(fact_1089_nat__add__left__cancel__less,axiom,
    ! [K2: nat,M3: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( ord_less_nat @ M3 @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_1090_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_1091_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_1092_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_1093_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_1094_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_1095_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_1096_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel1
thf(fact_1097_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_1098_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_1099_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel2
thf(fact_1100_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_1101_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_1102_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1103_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_1104_linordered__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% linordered_ab_group_add_class.zero_less_double_add_iff_zero_less_single_add
thf(fact_1105_linordered__ab__group__add__class_Ozero__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 ) ) ).

% linordered_ab_group_add_class.zero_less_double_add_iff_zero_less_single_add
thf(fact_1106_double__add__less__zero__iff__single__less__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% double_add_less_zero_iff_single_less_zero
thf(fact_1107_double__add__less__zero__iff__single__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_less_zero
thf(fact_1108_lattice__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% lattice_ab_group_add_class.zero_less_double_add_iff_zero_less_single_add
thf(fact_1109_lattice__ab__group__add__class_Ozero__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 ) ) ).

% lattice_ab_group_add_class.zero_less_double_add_iff_zero_less_single_add
thf(fact_1110_diff__less__0__iff__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ zero_zero_int )
      = ( ord_less_int @ A @ B ) ) ).

% diff_less_0_iff_less
thf(fact_1111_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_1112_of__nat__less__iff,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M3 @ N ) ) ).

% of_nat_less_iff
thf(fact_1113_of__nat__less__iff,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M3 @ N ) ) ).

% of_nat_less_iff
thf(fact_1114_of__nat__less__iff,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ M3 @ N ) ) ).

% of_nat_less_iff
thf(fact_1115_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0

% Helper facts (13)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y: int] :
      ( ( if_int @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y: int] :
      ( ( if_int @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y: nat] :
      ( ( if_nat @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y: nat] :
      ( ( if_nat @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y: real] :
      ( ( if_real @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y: real] :
      ( ( if_real @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( if_list_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( if_list_a @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_a_nat,Y: product_prod_a_nat] :
      ( ( if_Pro2324015151519390263_a_nat @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_a_nat,Y: product_prod_a_nat] :
      ( ( if_Pro2324015151519390263_a_nat @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( fold_a7750079645758252821_a_nat
      @ ^ [X: a,S: probab4139393509369344520_a_nat] : ( probab5175300157245341303_a_nat @ S @ ( freque5687656950882125236te_2_a @ X ) )
      @ ( append_a @ xs @ ( cons_a @ x @ nil_a ) )
      @ ( probab6296365750099732169_a_nat @ ( produc148073511828866022_a_nat @ zero_zero_nat @ ( product_Pair_a_nat @ zero_zero_a @ zero_zero_nat ) ) ) )
    = ( probab5175300157245341303_a_nat
      @ ( probab1029661319033422245_a_nat @ ( probab1830274953030043784et_nat @ ( set_ord_lessThan_nat @ ( size_size_list_a @ xs ) ) )
        @ ^ [K: nat] : ( probab6296365750099732169_a_nat @ ( produc148073511828866022_a_nat @ ( size_size_list_a @ xs ) @ ( freque6565899915408260854etch_a @ xs @ K ) ) ) )
      @ ( freque5687656950882125236te_2_a @ x ) ) ) ).

%------------------------------------------------------------------------------