TPTP Problem File: SLH0800^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : IMP_Compiler_Reuse/0005_Compiler/prob_00433_021819__5878708_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1646 ( 587 unt; 366 typ; 0 def)
% Number of atoms : 3491 (1504 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11350 ( 494 ~; 87 |; 293 &;8964 @)
% ( 0 <=>;1512 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 6 avg)
% Number of types : 69 ( 68 usr)
% Number of type conns : 820 ( 820 >; 0 *; 0 +; 0 <<)
% Number of symbols : 301 ( 298 usr; 18 con; 0-3 aty)
% Number of variables : 3649 ( 108 ^;3292 !; 249 ?;3649 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:42:53.417
%------------------------------------------------------------------------------
% Could-be-implicit typings (68)
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% Explicit typings (298)
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power_power_nat: nat > nat > nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Compiler__Oinstr_M_062_It__Compiler__Oinstr_M_Eo_J_J_001t__List__Olist_It__Compiler__Oinstr_J,type,
produc7818391707689388878_instr: ( instr > instr > $o ) > list_instr > produc7133927873007960030_instr ).
thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001t__List__Olist_It__Int__Oint_J,type,
produc8618682346314911123st_int: ( int > int > $o ) > list_int > produc5834231552977413017st_int ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_M_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
produc1436111165245395422t_char: ( list_char > list_char > $o ) > list_list_char > produc8794813868974875620t_char ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__Int__Oint,type,
produc5790713362662368625nt_int: ( list_char > int ) > int > produc4435102495419491129nt_int ).
thf(sy_c_Product__Type_OPair_001t__Compiler__Oinstr_001t__Compiler__Oinstr,type,
produc7105755196911784279_instr: instr > instr > produc6572865875912645095_instr ).
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__List__Olist_It__Compiler__Oinstr_J_001t__List__Olist_It__Compiler__Oinstr_J,type,
produc7750586281064920791_instr: list_instr > list_instr > produc8891168062546637159_instr ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_J,type,
produc1273992478785463195t_char: list_list_list_char > list_list_list_char > produc3331752281674800867t_char ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
produc9134825680149865339t_char: list_list_char > list_list_char > produc525958773753613763t_char ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_001t__List__Olist_It__String__Ochar_J,type,
produc9067971387720190315t_char: list_list_char > list_char > produc5482112689541410355t_char ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
produc3965054194175396271nt_int: list_list_char > produc4435102495419491129nt_int > produc4189061985984568957nt_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
produc2694321116511041643t_char: list_char > list_list_char > produc4406237598079604531t_char ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
produc1606739095562851931t_char: list_char > list_char > produc6859378861088059299t_char ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int2: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
insert1920755237625066145t_char: list_list_char > set_list_list_char > set_list_list_char ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__String__Ochar_J,type,
insert_list_char2: list_char > set_list_char > set_list_char ).
thf(sy_c_Set_Othe__elem_001t__Compiler__Oinstr,type,
the_elem_instr: set_instr > instr ).
thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
the_elem_int: set_int > int ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
the_el4834769204129890620t_char: set_list_list_char > list_list_char ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__String__Ochar_J,type,
the_elem_list_char: set_list_char > list_char ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J_J,type,
accp_P4562858270263085236nt_int: ( produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o ) > produc4189061985984568957nt_int > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J,type,
member_list_char_int: ( list_char > int ) > set_list_char_int > $o ).
thf(sy_c_member_001t__Compiler__Oinstr,type,
member_instr2: instr > set_instr > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int2: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Compiler__Oinstr_J,type,
member_list_instr: list_instr > set_list_instr > $o ).
thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
member_list_int: list_int > set_list_int > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_J,type,
member3246555595837571096t_char: list_list_list_char > set_li9190364784432779447t_char > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
member4762036990631661320t_char: list_list_char > set_list_list_char > $o ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J,type,
member5218582320954849874t_char: list_P8512142354091196201t_char > set_li2822776881692981001t_char > $o ).
thf(sy_c_member_001t__List__Olist_It__String__Ochar_J,type,
member_list_char2: list_char > set_list_char > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
member3599195040465536738nt_int: produc4435102495419491129nt_int > set_Pr5290250543098163609nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Compiler__Oinstr_Mt__Compiler__Oinstr_J,type,
member8320768277060436240_instr: produc6572865875912645095_instr > set_Pr3711236633289739975_instr > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Compiler__Oinstr_J_Mt__List__Olist_It__Compiler__Oinstr_J_J,type,
member7594433970877843344_instr: produc8891168062546637159_instr > set_Pr1149628334386456903_instr > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_Mt__List__Olist_It__List__Olist_It__String__Ochar_J_J_J,type,
member5558557582278090860t_char: produc525958773753613763t_char > set_Pr5266275940962171427t_char > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_Mt__List__Olist_It__String__Ochar_J_J,type,
member2188552656405064028t_char: produc5482112689541410355t_char > set_Pr5852146504233451795t_char > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J_J,type,
member7220921130490543508nt_int: produc4189061985984568957nt_int > set_Pr4358015327039922227nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__List__Olist_It__String__Ochar_J_J_J,type,
member1112677564943258204t_char: produc4406237598079604531t_char > set_Pr7541255050816019t_char > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
member6023471406134104908t_char: produc6859378861088059299t_char > set_Pr4480099295215233283t_char > $o ).
thf(sy_c_member_001t__String__Ochar,type,
member_char: char > set_char > $o ).
thf(sy_v_P,type,
p: list_instr ).
% Relevant facts (1268)
thf(fact_0_distinct__n__lists,axiom,
! [Xs: list_list_list_char,N: nat] :
( ( distin92326681159624120t_char @ Xs )
=> ( distin8725639686020792008t_char @ ( n_list6447629135212850403t_char @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_1_distinct__n__lists,axiom,
! [Xs: list_P8512142354091196201t_char,N: nat] :
( ( distin3640772593324101788t_char @ Xs )
=> ( distin2423292844557324194t_char @ ( n_list9037368511801905841t_char @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_2_distinct__n__lists,axiom,
! [Xs: list_char,N: nat] :
( ( distinct_char @ Xs )
=> ( distinct_list_char @ ( n_lists_char @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_3_distinct__n__lists,axiom,
! [Xs: list_list_char,N: nat] :
( ( distinct_list_char @ Xs )
=> ( distin92326681159624120t_char @ ( n_lists_list_char @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_4_distinct__union,axiom,
! [Xs: list_list_list_char,Ys: list_list_list_char] :
( ( distin92326681159624120t_char @ ( union_list_list_char @ Xs @ Ys ) )
= ( distin92326681159624120t_char @ Ys ) ) ).
% distinct_union
thf(fact_5_distinct__union,axiom,
! [Xs: list_char,Ys: list_char] :
( ( distinct_char @ ( union_char @ Xs @ Ys ) )
= ( distinct_char @ Ys ) ) ).
% distinct_union
thf(fact_6_distinct__union,axiom,
! [Xs: list_P8512142354091196201t_char,Ys: list_P8512142354091196201t_char] :
( ( distin3640772593324101788t_char @ ( union_8776386356374642845t_char @ Xs @ Ys ) )
= ( distin3640772593324101788t_char @ Ys ) ) ).
% distinct_union
thf(fact_7_distinct__union,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( distinct_list_char @ ( union_list_char @ Xs @ Ys ) )
= ( distinct_list_char @ Ys ) ) ).
% distinct_union
thf(fact_8_distinct__product__lists,axiom,
! [Xss: list_l2647756019783878417t_char] :
( ! [X: list_list_list_char] :
( ( member3246555595837571096t_char @ X @ ( set_li658947045408919724t_char @ Xss ) )
=> ( distin92326681159624120t_char @ X ) )
=> ( distin8725639686020792008t_char @ ( produc4788588374362985666t_char @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_9_distinct__product__lists,axiom,
! [Xss: list_l5897289931111006767t_char] :
( ! [X: list_P8512142354091196201t_char] :
( ( member5218582320954849874t_char @ X @ ( set_li4426379080490120382t_char @ Xss ) )
=> ( distin3640772593324101788t_char @ X ) )
=> ( distin2423292844557324194t_char @ ( produc6320113761674426130t_char @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_10_distinct__product__lists,axiom,
! [Xss: list_list_char] :
( ! [X: list_char] :
( ( member_list_char2 @ X @ ( set_list_char2 @ Xss ) )
=> ( distinct_char @ X ) )
=> ( distinct_list_char @ ( product_lists_char @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_11_distinct__product__lists,axiom,
! [Xss: list_list_list_char] :
( ! [X: list_list_char] :
( ( member4762036990631661320t_char @ X @ ( set_list_list_char2 @ Xss ) )
=> ( distinct_list_char @ X ) )
=> ( distin92326681159624120t_char @ ( produc4928866681682397746t_char @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_12_distinct__insert,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( distin92326681159624120t_char @ ( insert1002245835581533269t_char @ X2 @ Xs ) )
= ( distin92326681159624120t_char @ Xs ) ) ).
% distinct_insert
thf(fact_13_distinct__insert,axiom,
! [X2: char,Xs: list_char] :
( ( distinct_char @ ( insert_char @ X2 @ Xs ) )
= ( distinct_char @ Xs ) ) ).
% distinct_insert
thf(fact_14_distinct__insert,axiom,
! [X2: produc6859378861088059299t_char,Xs: list_P8512142354091196201t_char] :
( ( distin3640772593324101788t_char @ ( insert2518444870618099391t_char @ X2 @ Xs ) )
= ( distin3640772593324101788t_char @ Xs ) ) ).
% distinct_insert
thf(fact_15_distinct__insert,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( distinct_list_char @ ( insert_list_char @ X2 @ Xs ) )
= ( distinct_list_char @ Xs ) ) ).
% distinct_insert
thf(fact_16_distinct__product,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( distinct_list_char @ Xs )
=> ( ( distinct_list_char @ Ys )
=> ( distin3640772593324101788t_char @ ( produc6248063980778377481t_char @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_17_distinct__product,axiom,
! [Xs: list_char,Ys: list_char] :
( ( distinct_char @ Xs )
=> ( ( distinct_char @ Ys )
=> ( distin708547731832358524r_char @ ( product_char_char @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_18_distinct__product,axiom,
! [Xs: list_list_char,Ys: list_char] :
( ( distinct_list_char @ Xs )
=> ( ( distinct_char @ Ys )
=> ( distin4782212841183787788r_char @ ( produc6749299271248437113r_char @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_19_distinct__product,axiom,
! [Xs: list_char,Ys: list_list_char] :
( ( distinct_char @ Xs )
=> ( ( distinct_list_char @ Ys )
=> ( distin9195492875776795660t_char @ ( produc6022865740284284537t_char @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_20_distinct__product,axiom,
! [Xs: list_list_list_char,Ys: list_char] :
( ( distin92326681159624120t_char @ Xs )
=> ( ( distinct_char @ Ys )
=> ( distin6958075609223262492r_char @ ( produc5481708525379922825r_char @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_21_distinct__product,axiom,
! [Xs: list_char,Ys: list_list_list_char] :
( ( distinct_char @ Xs )
=> ( ( distin92326681159624120t_char @ Ys )
=> ( distin782115389035578140t_char @ ( produc7315003633597286793t_char @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_22_distinct__product,axiom,
! [Xs: list_list_char,Ys: list_list_list_char] :
( ( distinct_list_char @ Xs )
=> ( ( distin92326681159624120t_char @ Ys )
=> ( distin7540760125400508332t_char @ ( produc8617151401925608473t_char @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_23_distinct__product,axiom,
! [Xs: list_list_list_char,Ys: list_list_char] :
( ( distin92326681159624120t_char @ Xs )
=> ( ( distinct_list_char @ Ys )
=> ( distin8616635216862314156t_char @ ( produc5767429636279981337t_char @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_24_distinct__product,axiom,
! [Xs: list_list_list_char,Ys: list_list_list_char] :
( ( distin92326681159624120t_char @ Xs )
=> ( ( distin92326681159624120t_char @ Ys )
=> ( distin5147519316492193212t_char @ ( produc6233462945809023017t_char @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_25_distinct__product,axiom,
! [Xs: list_char,Ys: list_P8512142354091196201t_char] :
( ( distinct_char @ Xs )
=> ( ( distin3640772593324101788t_char @ Ys )
=> ( distin4585828291901285966t_char @ ( produc6426372462082274443t_char @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_26_distinct1__rotate,axiom,
! [Xs: list_list_list_char] :
( ( distin92326681159624120t_char @ ( rotate5166916878618229726t_char @ Xs ) )
= ( distin92326681159624120t_char @ Xs ) ) ).
% distinct1_rotate
thf(fact_27_distinct1__rotate,axiom,
! [Xs: list_char] :
( ( distinct_char @ ( rotate1_char @ Xs ) )
= ( distinct_char @ Xs ) ) ).
% distinct1_rotate
thf(fact_28_distinct1__rotate,axiom,
! [Xs: list_P8512142354091196201t_char] :
( ( distin3640772593324101788t_char @ ( rotate6387814990712887542t_char @ Xs ) )
= ( distin3640772593324101788t_char @ Xs ) ) ).
% distinct1_rotate
thf(fact_29_distinct1__rotate,axiom,
! [Xs: list_list_char] :
( ( distinct_list_char @ ( rotate1_list_char @ Xs ) )
= ( distinct_list_char @ Xs ) ) ).
% distinct1_rotate
thf(fact_30_distinct__rotate,axiom,
! [N: nat,Xs: list_list_list_char] :
( ( distin92326681159624120t_char @ ( rotate4570796657432407479t_char @ N @ Xs ) )
= ( distin92326681159624120t_char @ Xs ) ) ).
% distinct_rotate
thf(fact_31_distinct__rotate,axiom,
! [N: nat,Xs: list_char] :
( ( distinct_char @ ( rotate_char @ N @ Xs ) )
= ( distinct_char @ Xs ) ) ).
% distinct_rotate
thf(fact_32_distinct__rotate,axiom,
! [N: nat,Xs: list_P8512142354091196201t_char] :
( ( distin3640772593324101788t_char @ ( rotate8363826417023601885t_char @ N @ Xs ) )
= ( distin3640772593324101788t_char @ Xs ) ) ).
% distinct_rotate
thf(fact_33_distinct__rotate,axiom,
! [N: nat,Xs: list_list_char] :
( ( distinct_list_char @ ( rotate_list_char @ N @ Xs ) )
= ( distinct_list_char @ Xs ) ) ).
% distinct_rotate
thf(fact_34_distinct__remdups,axiom,
! [Xs: list_list_char] : ( distinct_list_char @ ( remdups_list_char @ Xs ) ) ).
% distinct_remdups
thf(fact_35_distinct__remdups,axiom,
! [Xs: list_list_list_char] : ( distin92326681159624120t_char @ ( remdup58496044459526934t_char @ Xs ) ) ).
% distinct_remdups
thf(fact_36_distinct__remdups,axiom,
! [Xs: list_char] : ( distinct_char @ ( remdups_char @ Xs ) ) ).
% distinct_remdups
thf(fact_37_distinct__remdups,axiom,
! [Xs: list_P8512142354091196201t_char] : ( distin3640772593324101788t_char @ ( remdup1424198276734426046t_char @ Xs ) ) ).
% distinct_remdups
thf(fact_38_remdups__id__iff__distinct,axiom,
! [Xs: list_list_char] :
( ( ( remdups_list_char @ Xs )
= Xs )
= ( distinct_list_char @ Xs ) ) ).
% remdups_id_iff_distinct
thf(fact_39_remdups__id__iff__distinct,axiom,
! [Xs: list_list_list_char] :
( ( ( remdup58496044459526934t_char @ Xs )
= Xs )
= ( distin92326681159624120t_char @ Xs ) ) ).
% remdups_id_iff_distinct
thf(fact_40_remdups__id__iff__distinct,axiom,
! [Xs: list_char] :
( ( ( remdups_char @ Xs )
= Xs )
= ( distinct_char @ Xs ) ) ).
% remdups_id_iff_distinct
thf(fact_41_remdups__id__iff__distinct,axiom,
! [Xs: list_P8512142354091196201t_char] :
( ( ( remdup1424198276734426046t_char @ Xs )
= Xs )
= ( distin3640772593324101788t_char @ Xs ) ) ).
% remdups_id_iff_distinct
thf(fact_42_set__remdups,axiom,
! [Xs: list_list_list_char] :
( ( set_list_list_char2 @ ( remdup58496044459526934t_char @ Xs ) )
= ( set_list_list_char2 @ Xs ) ) ).
% set_remdups
thf(fact_43_set__remdups,axiom,
! [Xs: list_list_char] :
( ( set_list_char2 @ ( remdups_list_char @ Xs ) )
= ( set_list_char2 @ Xs ) ) ).
% set_remdups
thf(fact_44_set__rotate,axiom,
! [N: nat,Xs: list_list_list_char] :
( ( set_list_list_char2 @ ( rotate4570796657432407479t_char @ N @ Xs ) )
= ( set_list_list_char2 @ Xs ) ) ).
% set_rotate
thf(fact_45_set__rotate,axiom,
! [N: nat,Xs: list_list_char] :
( ( set_list_char2 @ ( rotate_list_char @ N @ Xs ) )
= ( set_list_char2 @ Xs ) ) ).
% set_rotate
thf(fact_46_set__rotate1,axiom,
! [Xs: list_list_list_char] :
( ( set_list_list_char2 @ ( rotate5166916878618229726t_char @ Xs ) )
= ( set_list_list_char2 @ Xs ) ) ).
% set_rotate1
thf(fact_47_set__rotate1,axiom,
! [Xs: list_list_char] :
( ( set_list_char2 @ ( rotate1_list_char @ Xs ) )
= ( set_list_char2 @ Xs ) ) ).
% set_rotate1
thf(fact_48_in__set__insert,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ( ( insert1002245835581533269t_char @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_49_in__set__insert,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ( ( insert_list_char @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_50_insert__remdups,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( insert_list_char @ X2 @ ( remdups_list_char @ Xs ) )
= ( remdups_list_char @ ( insert_list_char @ X2 @ Xs ) ) ) ).
% insert_remdups
thf(fact_51_rotate1__rotate__swap,axiom,
! [N: nat,Xs: list_list_char] :
( ( rotate1_list_char @ ( rotate_list_char @ N @ Xs ) )
= ( rotate_list_char @ N @ ( rotate1_list_char @ Xs ) ) ) ).
% rotate1_rotate_swap
thf(fact_52_distinct__remdups__id,axiom,
! [Xs: list_list_char] :
( ( distinct_list_char @ Xs )
=> ( ( remdups_list_char @ Xs )
= Xs ) ) ).
% distinct_remdups_id
thf(fact_53_distinct__remdups__id,axiom,
! [Xs: list_list_list_char] :
( ( distin92326681159624120t_char @ Xs )
=> ( ( remdup58496044459526934t_char @ Xs )
= Xs ) ) ).
% distinct_remdups_id
thf(fact_54_distinct__remdups__id,axiom,
! [Xs: list_char] :
( ( distinct_char @ Xs )
=> ( ( remdups_char @ Xs )
= Xs ) ) ).
% distinct_remdups_id
thf(fact_55_distinct__remdups__id,axiom,
! [Xs: list_P8512142354091196201t_char] :
( ( distin3640772593324101788t_char @ Xs )
=> ( ( remdup1424198276734426046t_char @ Xs )
= Xs ) ) ).
% distinct_remdups_id
thf(fact_56_subseqs__distinctD,axiom,
! [Ys: list_list_list_char,Xs: list_list_list_char] :
( ( member3246555595837571096t_char @ Ys @ ( set_li658947045408919724t_char @ ( subseq1546420061081462748t_char @ Xs ) ) )
=> ( ( distin92326681159624120t_char @ Xs )
=> ( distin92326681159624120t_char @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_57_subseqs__distinctD,axiom,
! [Ys: list_P8512142354091196201t_char,Xs: list_P8512142354091196201t_char] :
( ( member5218582320954849874t_char @ Ys @ ( set_li4426379080490120382t_char @ ( subseq4653230730214077816t_char @ Xs ) ) )
=> ( ( distin3640772593324101788t_char @ Xs )
=> ( distin3640772593324101788t_char @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_58_subseqs__distinctD,axiom,
! [Ys: list_list_char,Xs: list_list_char] :
( ( member4762036990631661320t_char @ Ys @ ( set_list_list_char2 @ ( subseqs_list_char @ Xs ) ) )
=> ( ( distinct_list_char @ Xs )
=> ( distinct_list_char @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_59_subseqs__distinctD,axiom,
! [Ys: list_char,Xs: list_char] :
( ( member_list_char2 @ Ys @ ( set_list_char2 @ ( subseqs_char @ Xs ) ) )
=> ( ( distinct_char @ Xs )
=> ( distinct_char @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_60_set__union,axiom,
! [Xs: list_list_list_char,Ys: list_list_list_char] :
( ( set_list_list_char2 @ ( union_list_list_char @ Xs @ Ys ) )
= ( sup_su9123505219413647059t_char @ ( set_list_list_char2 @ Xs ) @ ( set_list_list_char2 @ Ys ) ) ) ).
% set_union
thf(fact_61_set__union,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( set_list_char2 @ ( union_list_char @ Xs @ Ys ) )
= ( sup_su1962856675925542723t_char @ ( set_list_char2 @ Xs ) @ ( set_list_char2 @ Ys ) ) ) ).
% set_union
thf(fact_62_List_Ounion__def,axiom,
( union_list_char
= ( fold_l256471224249224159t_char @ insert_list_char ) ) ).
% List.union_def
thf(fact_63_in__set__member,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
= ( member5037114757636781590t_char @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_64_in__set__member,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
= ( member_list_char @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_65_rotate__Suc,axiom,
! [N: nat,Xs: list_list_char] :
( ( rotate_list_char @ ( suc @ N ) @ Xs )
= ( rotate1_list_char @ ( rotate_list_char @ N @ Xs ) ) ) ).
% rotate_Suc
thf(fact_66_not__in__set__insert,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ( ( insert1002245835581533269t_char @ X2 @ Xs )
= ( cons_list_list_char @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_67_not__in__set__insert,axiom,
! [X2: list_char,Xs: list_list_char] :
( ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ( ( insert_list_char @ X2 @ Xs )
= ( cons_list_char @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_68_not__in__set__insert,axiom,
! [X2: int,Xs: list_int] :
( ~ ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
=> ( ( insert_int @ X2 @ Xs )
= ( cons_int @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_69_List_Oset__insert,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( set_list_list_char2 @ ( insert1002245835581533269t_char @ X2 @ Xs ) )
= ( insert1920755237625066145t_char @ X2 @ ( set_list_list_char2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_70_List_Oset__insert,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( set_list_char2 @ ( insert_list_char @ X2 @ Xs ) )
= ( insert_list_char2 @ X2 @ ( set_list_char2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_71_list__ex1__iff,axiom,
( list_e9192628380528740584t_char
= ( ^ [P: list_list_char > $o,Xs2: list_list_list_char] :
? [X3: list_list_char] :
( ( member4762036990631661320t_char @ X3 @ ( set_list_list_char2 @ Xs2 ) )
& ( P @ X3 )
& ! [Y: list_list_char] :
( ( ( member4762036990631661320t_char @ Y @ ( set_list_list_char2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_72_list__ex1__iff,axiom,
( list_ex1_list_char
= ( ^ [P: list_char > $o,Xs2: list_list_char] :
? [X3: list_char] :
( ( member_list_char2 @ X3 @ ( set_list_char2 @ Xs2 ) )
& ( P @ X3 )
& ! [Y: list_char] :
( ( ( member_list_char2 @ Y @ ( set_list_char2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_73_remove1__remdups,axiom,
! [Xs: list_list_char,X2: list_char] :
( ( distinct_list_char @ Xs )
=> ( ( remove1_list_char @ X2 @ ( remdups_list_char @ Xs ) )
= ( remdups_list_char @ ( remove1_list_char @ X2 @ Xs ) ) ) ) ).
% remove1_remdups
thf(fact_74_remove1__remdups,axiom,
! [Xs: list_list_list_char,X2: list_list_char] :
( ( distin92326681159624120t_char @ Xs )
=> ( ( remove4265589039595904469t_char @ X2 @ ( remdup58496044459526934t_char @ Xs ) )
= ( remdup58496044459526934t_char @ ( remove4265589039595904469t_char @ X2 @ Xs ) ) ) ) ).
% remove1_remdups
thf(fact_75_remove1__remdups,axiom,
! [Xs: list_char,X2: char] :
( ( distinct_char @ Xs )
=> ( ( remove1_char @ X2 @ ( remdups_char @ Xs ) )
= ( remdups_char @ ( remove1_char @ X2 @ Xs ) ) ) ) ).
% remove1_remdups
thf(fact_76_remove1__remdups,axiom,
! [Xs: list_P8512142354091196201t_char,X2: produc6859378861088059299t_char] :
( ( distin3640772593324101788t_char @ Xs )
=> ( ( remove5295477908717968703t_char @ X2 @ ( remdup1424198276734426046t_char @ Xs ) )
= ( remdup1424198276734426046t_char @ ( remove5295477908717968703t_char @ X2 @ Xs ) ) ) ) ).
% remove1_remdups
thf(fact_77_length__n__lists__elem,axiom,
! [Ys: list_char,N: nat,Xs: list_char] :
( ( member_list_char2 @ Ys @ ( set_list_char2 @ ( n_lists_char @ N @ Xs ) ) )
=> ( ( size_size_list_char @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_78_length__n__lists__elem,axiom,
! [Ys: list_list_char,N: nat,Xs: list_list_char] :
( ( member4762036990631661320t_char @ Ys @ ( set_list_list_char2 @ ( n_lists_list_char @ N @ Xs ) ) )
=> ( ( size_s356637359517785349t_char @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_79_distinct__zipI2,axiom,
! [Ys: list_list_char,Xs: list_list_char] :
( ( distinct_list_char @ Ys )
=> ( distin3640772593324101788t_char @ ( zip_li5382857322596760955t_char @ Xs @ Ys ) ) ) ).
% distinct_zipI2
thf(fact_80_list_Oinject,axiom,
! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X22 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_81_length__remdups__eq,axiom,
! [Xs: list_list_char] :
( ( ( size_s356637359517785349t_char @ ( remdups_list_char @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) )
= ( ( remdups_list_char @ Xs )
= Xs ) ) ).
% length_remdups_eq
thf(fact_82_in__set__remove1,axiom,
! [A: list_list_char,B: list_list_char,Xs: list_list_list_char] :
( ( A != B )
=> ( ( member4762036990631661320t_char @ A @ ( set_list_list_char2 @ ( remove4265589039595904469t_char @ B @ Xs ) ) )
= ( member4762036990631661320t_char @ A @ ( set_list_list_char2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_83_in__set__remove1,axiom,
! [A: list_char,B: list_char,Xs: list_list_char] :
( ( A != B )
=> ( ( member_list_char2 @ A @ ( set_list_char2 @ ( remove1_list_char @ B @ Xs ) ) )
= ( member_list_char2 @ A @ ( set_list_char2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_84_length__rotate,axiom,
! [N: nat,Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( rotate_list_char @ N @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_rotate
thf(fact_85_length__rotate1,axiom,
! [Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( rotate1_list_char @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_rotate1
thf(fact_86_list_Osimps_I15_J,axiom,
! [X21: list_list_char,X22: list_list_list_char] :
( ( set_list_list_char2 @ ( cons_list_list_char @ X21 @ X22 ) )
= ( insert1920755237625066145t_char @ X21 @ ( set_list_list_char2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_87_list_Osimps_I15_J,axiom,
! [X21: list_char,X22: list_list_char] :
( ( set_list_char2 @ ( cons_list_char @ X21 @ X22 ) )
= ( insert_list_char2 @ X21 @ ( set_list_char2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_88_list_Osimps_I15_J,axiom,
! [X21: int,X22: list_int] :
( ( set_int2 @ ( cons_int @ X21 @ X22 ) )
= ( insert_int2 @ X21 @ ( set_int2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_89_member__rec_I1_J,axiom,
! [X2: int,Xs: list_int,Y2: int] :
( ( member_int @ ( cons_int @ X2 @ Xs ) @ Y2 )
= ( ( X2 = Y2 )
| ( member_int @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_90_remove1_Osimps_I2_J,axiom,
! [X2: int,Y2: int,Xs: list_int] :
( ( ( X2 = Y2 )
=> ( ( remove1_int @ X2 @ ( cons_int @ Y2 @ Xs ) )
= Xs ) )
& ( ( X2 != Y2 )
=> ( ( remove1_int @ X2 @ ( cons_int @ Y2 @ Xs ) )
= ( cons_int @ Y2 @ ( remove1_int @ X2 @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_91_not__Cons__self2,axiom,
! [X2: int,Xs: list_int] :
( ( cons_int @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_92_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_93_Suc__length__conv,axiom,
! [N: nat,Xs: list_list_char] :
( ( ( suc @ N )
= ( size_s356637359517785349t_char @ Xs ) )
= ( ? [Y: list_char,Ys2: list_list_char] :
( ( Xs
= ( cons_list_char @ Y @ Ys2 ) )
& ( ( size_s356637359517785349t_char @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_94_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_95_length__Suc__conv,axiom,
! [Xs: list_list_char,N: nat] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( suc @ N ) )
= ( ? [Y: list_char,Ys2: list_list_char] :
( ( Xs
= ( cons_list_char @ Y @ Ys2 ) )
& ( ( size_s356637359517785349t_char @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_96_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_char] :
( ( size_s356637359517785349t_char @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_97_neq__if__length__neq,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
!= ( size_s356637359517785349t_char @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_98_Cons__in__subseqsD,axiom,
! [Y2: list_char,Ys: list_list_char,Xs: list_list_char] :
( ( member4762036990631661320t_char @ ( cons_list_char @ Y2 @ Ys ) @ ( set_list_list_char2 @ ( subseqs_list_char @ Xs ) ) )
=> ( member4762036990631661320t_char @ Ys @ ( set_list_list_char2 @ ( subseqs_list_char @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_99_Cons__in__subseqsD,axiom,
! [Y2: char,Ys: list_char,Xs: list_char] :
( ( member_list_char2 @ ( cons_char @ Y2 @ Ys ) @ ( set_list_char2 @ ( subseqs_char @ Xs ) ) )
=> ( member_list_char2 @ Ys @ ( set_list_char2 @ ( subseqs_char @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_100_Cons__in__subseqsD,axiom,
! [Y2: int,Ys: list_int,Xs: list_int] :
( ( member_list_int @ ( cons_int @ Y2 @ Ys ) @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) )
=> ( member_list_int @ Ys @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_101_set__ConsD,axiom,
! [Y2: list_list_char,X2: list_list_char,Xs: list_list_list_char] :
( ( member4762036990631661320t_char @ Y2 @ ( set_list_list_char2 @ ( cons_list_list_char @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member4762036990631661320t_char @ Y2 @ ( set_list_list_char2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_102_set__ConsD,axiom,
! [Y2: list_char,X2: list_char,Xs: list_list_char] :
( ( member_list_char2 @ Y2 @ ( set_list_char2 @ ( cons_list_char @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_list_char2 @ Y2 @ ( set_list_char2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_103_set__ConsD,axiom,
! [Y2: int,X2: int,Xs: list_int] :
( ( member_int2 @ Y2 @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_int2 @ Y2 @ ( set_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_104_list_Oset__cases,axiom,
! [E: list_list_char,A: list_list_list_char] :
( ( member4762036990631661320t_char @ E @ ( set_list_list_char2 @ A ) )
=> ( ! [Z2: list_list_list_char] :
( A
!= ( cons_list_list_char @ E @ Z2 ) )
=> ~ ! [Z1: list_list_char,Z2: list_list_list_char] :
( ( A
= ( cons_list_list_char @ Z1 @ Z2 ) )
=> ~ ( member4762036990631661320t_char @ E @ ( set_list_list_char2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_105_list_Oset__cases,axiom,
! [E: list_char,A: list_list_char] :
( ( member_list_char2 @ E @ ( set_list_char2 @ A ) )
=> ( ! [Z2: list_list_char] :
( A
!= ( cons_list_char @ E @ Z2 ) )
=> ~ ! [Z1: list_char,Z2: list_list_char] :
( ( A
= ( cons_list_char @ Z1 @ Z2 ) )
=> ~ ( member_list_char2 @ E @ ( set_list_char2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_106_list_Oset__cases,axiom,
! [E: int,A: list_int] :
( ( member_int2 @ E @ ( set_int2 @ A ) )
=> ( ! [Z2: list_int] :
( A
!= ( cons_int @ E @ Z2 ) )
=> ~ ! [Z1: int,Z2: list_int] :
( ( A
= ( cons_int @ Z1 @ Z2 ) )
=> ~ ( member_int2 @ E @ ( set_int2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_107_list_Oset__intros_I1_J,axiom,
! [X21: list_list_char,X22: list_list_list_char] : ( member4762036990631661320t_char @ X21 @ ( set_list_list_char2 @ ( cons_list_list_char @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_108_list_Oset__intros_I1_J,axiom,
! [X21: list_char,X22: list_list_char] : ( member_list_char2 @ X21 @ ( set_list_char2 @ ( cons_list_char @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_109_list_Oset__intros_I1_J,axiom,
! [X21: int,X22: list_int] : ( member_int2 @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_110_list_Oset__intros_I2_J,axiom,
! [Y2: list_list_char,X22: list_list_list_char,X21: list_list_char] :
( ( member4762036990631661320t_char @ Y2 @ ( set_list_list_char2 @ X22 ) )
=> ( member4762036990631661320t_char @ Y2 @ ( set_list_list_char2 @ ( cons_list_list_char @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_111_list_Oset__intros_I2_J,axiom,
! [Y2: list_char,X22: list_list_char,X21: list_char] :
( ( member_list_char2 @ Y2 @ ( set_list_char2 @ X22 ) )
=> ( member_list_char2 @ Y2 @ ( set_list_char2 @ ( cons_list_char @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_112_list_Oset__intros_I2_J,axiom,
! [Y2: int,X22: list_int,X21: int] :
( ( member_int2 @ Y2 @ ( set_int2 @ X22 ) )
=> ( member_int2 @ Y2 @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_113_distinct__length__2__or__more,axiom,
! [A: list_char,B: list_char,Xs: list_list_char] :
( ( distinct_list_char @ ( cons_list_char @ A @ ( cons_list_char @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_list_char @ ( cons_list_char @ A @ Xs ) )
& ( distinct_list_char @ ( cons_list_char @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_114_distinct__length__2__or__more,axiom,
! [A: list_list_char,B: list_list_char,Xs: list_list_list_char] :
( ( distin92326681159624120t_char @ ( cons_list_list_char @ A @ ( cons_list_list_char @ B @ Xs ) ) )
= ( ( A != B )
& ( distin92326681159624120t_char @ ( cons_list_list_char @ A @ Xs ) )
& ( distin92326681159624120t_char @ ( cons_list_list_char @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_115_distinct__length__2__or__more,axiom,
! [A: char,B: char,Xs: list_char] :
( ( distinct_char @ ( cons_char @ A @ ( cons_char @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_char @ ( cons_char @ A @ Xs ) )
& ( distinct_char @ ( cons_char @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_116_distinct__length__2__or__more,axiom,
! [A: produc6859378861088059299t_char,B: produc6859378861088059299t_char,Xs: list_P8512142354091196201t_char] :
( ( distin3640772593324101788t_char @ ( cons_P922464210127128153t_char @ A @ ( cons_P922464210127128153t_char @ B @ Xs ) ) )
= ( ( A != B )
& ( distin3640772593324101788t_char @ ( cons_P922464210127128153t_char @ A @ Xs ) )
& ( distin3640772593324101788t_char @ ( cons_P922464210127128153t_char @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_117_distinct__length__2__or__more,axiom,
! [A: int,B: int,Xs: list_int] :
( ( distinct_int @ ( cons_int @ A @ ( cons_int @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_int @ ( cons_int @ A @ Xs ) )
& ( distinct_int @ ( cons_int @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_118_in__set__product__lists__length,axiom,
! [Xs: list_char,Xss: list_list_char] :
( ( member_list_char2 @ Xs @ ( set_list_char2 @ ( product_lists_char @ Xss ) ) )
=> ( ( size_size_list_char @ Xs )
= ( size_s356637359517785349t_char @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_119_in__set__product__lists__length,axiom,
! [Xs: list_list_char,Xss: list_list_list_char] :
( ( member4762036990631661320t_char @ Xs @ ( set_list_list_char2 @ ( produc4928866681682397746t_char @ Xss ) ) )
=> ( ( size_s356637359517785349t_char @ Xs )
= ( size_s2562795203299699605t_char @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_120_notin__set__remove1,axiom,
! [X2: list_list_char,Xs: list_list_list_char,Y2: list_list_char] :
( ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ ( remove4265589039595904469t_char @ Y2 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_121_notin__set__remove1,axiom,
! [X2: list_char,Xs: list_list_char,Y2: list_char] :
( ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ ( remove1_list_char @ Y2 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_122_remove1__idem,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ( ( remove4265589039595904469t_char @ X2 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_123_remove1__idem,axiom,
! [X2: list_char,Xs: list_list_char] :
( ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ( ( remove1_list_char @ X2 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_124_distinct__remove1,axiom,
! [Xs: list_list_char,X2: list_char] :
( ( distinct_list_char @ Xs )
=> ( distinct_list_char @ ( remove1_list_char @ X2 @ Xs ) ) ) ).
% distinct_remove1
thf(fact_125_distinct__remove1,axiom,
! [Xs: list_list_list_char,X2: list_list_char] :
( ( distin92326681159624120t_char @ Xs )
=> ( distin92326681159624120t_char @ ( remove4265589039595904469t_char @ X2 @ Xs ) ) ) ).
% distinct_remove1
thf(fact_126_distinct__remove1,axiom,
! [Xs: list_char,X2: char] :
( ( distinct_char @ Xs )
=> ( distinct_char @ ( remove1_char @ X2 @ Xs ) ) ) ).
% distinct_remove1
thf(fact_127_distinct__remove1,axiom,
! [Xs: list_P8512142354091196201t_char,X2: produc6859378861088059299t_char] :
( ( distin3640772593324101788t_char @ Xs )
=> ( distin3640772593324101788t_char @ ( remove5295477908717968703t_char @ X2 @ Xs ) ) ) ).
% distinct_remove1
thf(fact_128_distinct_Osimps_I2_J,axiom,
! [X2: char,Xs: list_char] :
( ( distinct_char @ ( cons_char @ X2 @ Xs ) )
= ( ~ ( member_char @ X2 @ ( set_char2 @ Xs ) )
& ( distinct_char @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_129_distinct_Osimps_I2_J,axiom,
! [X2: produc6859378861088059299t_char,Xs: list_P8512142354091196201t_char] :
( ( distin3640772593324101788t_char @ ( cons_P922464210127128153t_char @ X2 @ Xs ) )
= ( ~ ( member6023471406134104908t_char @ X2 @ ( set_Pr2195431618410804152t_char @ Xs ) )
& ( distin3640772593324101788t_char @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_130_distinct_Osimps_I2_J,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( distin92326681159624120t_char @ ( cons_list_list_char @ X2 @ Xs ) )
= ( ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
& ( distin92326681159624120t_char @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_131_distinct_Osimps_I2_J,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( distinct_list_char @ ( cons_list_char @ X2 @ Xs ) )
= ( ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
& ( distinct_list_char @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_132_distinct_Osimps_I2_J,axiom,
! [X2: int,Xs: list_int] :
( ( distinct_int @ ( cons_int @ X2 @ Xs ) )
= ( ~ ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
& ( distinct_int @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_133_remdups_Osimps_I2_J,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ( ( remdup58496044459526934t_char @ ( cons_list_list_char @ X2 @ Xs ) )
= ( remdup58496044459526934t_char @ Xs ) ) )
& ( ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ( ( remdup58496044459526934t_char @ ( cons_list_list_char @ X2 @ Xs ) )
= ( cons_list_list_char @ X2 @ ( remdup58496044459526934t_char @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_134_remdups_Osimps_I2_J,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ( ( remdups_list_char @ ( cons_list_char @ X2 @ Xs ) )
= ( remdups_list_char @ Xs ) ) )
& ( ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ( ( remdups_list_char @ ( cons_list_char @ X2 @ Xs ) )
= ( cons_list_char @ X2 @ ( remdups_list_char @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_135_remdups_Osimps_I2_J,axiom,
! [X2: int,Xs: list_int] :
( ( ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
=> ( ( remdups_int @ ( cons_int @ X2 @ Xs ) )
= ( remdups_int @ Xs ) ) )
& ( ~ ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
=> ( ( remdups_int @ ( cons_int @ X2 @ Xs ) )
= ( cons_int @ X2 @ ( remdups_int @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_136_subseqs__refl,axiom,
! [Xs: list_list_char] : ( member4762036990631661320t_char @ Xs @ ( set_list_list_char2 @ ( subseqs_list_char @ Xs ) ) ) ).
% subseqs_refl
thf(fact_137_subseqs__refl,axiom,
! [Xs: list_char] : ( member_list_char2 @ Xs @ ( set_list_char2 @ ( subseqs_char @ Xs ) ) ) ).
% subseqs_refl
thf(fact_138_List_Oinsert__def,axiom,
( insert1002245835581533269t_char
= ( ^ [X3: list_list_char,Xs2: list_list_list_char] : ( if_lis8692959347405121979t_char @ ( member4762036990631661320t_char @ X3 @ ( set_list_list_char2 @ Xs2 ) ) @ Xs2 @ ( cons_list_list_char @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_139_List_Oinsert__def,axiom,
( insert_list_char
= ( ^ [X3: list_char,Xs2: list_list_char] : ( if_list_list_char @ ( member_list_char2 @ X3 @ ( set_list_char2 @ Xs2 ) ) @ Xs2 @ ( cons_list_char @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_140_List_Oinsert__def,axiom,
( insert_int
= ( ^ [X3: int,Xs2: list_int] : ( if_list_int @ ( member_int2 @ X3 @ ( set_int2 @ Xs2 ) ) @ Xs2 @ ( cons_int @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_141_distinct__zipI1,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( distinct_list_char @ Xs )
=> ( distin3640772593324101788t_char @ ( zip_li5382857322596760955t_char @ Xs @ Ys ) ) ) ).
% distinct_zipI1
thf(fact_142_can__select__set__list__ex1,axiom,
! [P2: list_list_char > $o,A2: list_list_list_char] :
( ( can_se5539732612864659124t_char @ P2 @ ( set_list_list_char2 @ A2 ) )
= ( list_e9192628380528740584t_char @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_143_can__select__set__list__ex1,axiom,
! [P2: list_char > $o,A2: list_list_char] :
( ( can_select_list_char @ P2 @ ( set_list_char2 @ A2 ) )
= ( list_ex1_list_char @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_144_length__Cons,axiom,
! [X2: int,Xs: list_int] :
( ( size_size_list_int @ ( cons_int @ X2 @ Xs ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_Cons
thf(fact_145_length__Cons,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( cons_list_char @ X2 @ Xs ) )
= ( suc @ ( size_s356637359517785349t_char @ Xs ) ) ) ).
% length_Cons
thf(fact_146_union__set__fold,axiom,
! [Xs: list_list_list_char,A2: set_list_list_char] :
( ( sup_su9123505219413647059t_char @ ( set_list_list_char2 @ Xs ) @ A2 )
= ( fold_l3769578904332300837t_char @ insert1920755237625066145t_char @ Xs @ A2 ) ) ).
% union_set_fold
thf(fact_147_union__set__fold,axiom,
! [Xs: list_list_char,A2: set_list_char] :
( ( sup_su1962856675925542723t_char @ ( set_list_char2 @ Xs ) @ A2 )
= ( fold_l6814815795985924741t_char @ insert_list_char2 @ Xs @ A2 ) ) ).
% union_set_fold
thf(fact_148_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_149_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_150_gen__length__code_I2_J,axiom,
! [N: nat,X2: int,Xs: list_int] :
( ( gen_length_int @ N @ ( cons_int @ X2 @ Xs ) )
= ( gen_length_int @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_151_Sup__fin_Oset__eq__fold,axiom,
! [X2: int,Xs: list_int] :
( ( lattic1091506334969745077in_int @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) )
= ( fold_int_int @ sup_sup_int @ Xs @ X2 ) ) ).
% Sup_fin.set_eq_fold
thf(fact_152_length__n__lists,axiom,
! [N: nat,Xs: list_list_char] :
( ( size_s2562795203299699605t_char @ ( n_lists_list_char @ N @ Xs ) )
= ( power_power_nat @ ( size_s356637359517785349t_char @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_153_length__n__lists,axiom,
! [N: nat,Xs: list_char] :
( ( size_s356637359517785349t_char @ ( n_lists_char @ N @ Xs ) )
= ( power_power_nat @ ( size_size_list_char @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_154_size__neq__size__imp__neq,axiom,
! [X2: char,Y2: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_155_size__neq__size__imp__neq,axiom,
! [X2: list_list_char,Y2: list_list_char] :
( ( ( size_s356637359517785349t_char @ X2 )
!= ( size_s356637359517785349t_char @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_156_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_157_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_158_ShiftD,axiom,
! [Kl: list_int,Kl2: set_list_int,K: int] :
( ( member_list_int @ Kl @ ( bNF_Gr1870224194279859149ft_int @ Kl2 @ K ) )
=> ( member_list_int @ ( cons_int @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_159_in__set__impl__in__set__zip1,axiom,
! [Xs: list_list_list_char,Ys: list_P98921565119350719nt_int,X2: list_list_char] :
( ( ( size_s2562795203299699605t_char @ Xs )
= ( size_s2729875514769589291nt_int @ Ys ) )
=> ( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ~ ! [Y3: produc4435102495419491129nt_int] :
~ ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X2 @ Y3 ) @ ( set_Pr8107714755537002024nt_int @ ( zip_li4478691307266062991nt_int @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_160_in__set__impl__in__set__zip1,axiom,
! [Xs: list_list_char_int,Ys: list_int,X2: list_char > int] :
( ( ( size_s3714362573911081296ar_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( member_list_char_int @ X2 @ ( set_list_char_int2 @ Xs ) )
=> ~ ! [Y3: int] :
~ ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y3 ) @ ( set_Pr8994527289597011790nt_int @ ( zip_li343459552841501841nt_int @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_161_in__set__impl__in__set__zip1,axiom,
! [Xs: list_list_list_char,Ys: list_list_char,X2: list_list_char] :
( ( ( size_s2562795203299699605t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ~ ! [Y3: list_char] :
~ ( member2188552656405064028t_char @ ( produc9067971387720190315t_char @ X2 @ Y3 ) @ ( set_Pr1396349415940334536t_char @ ( zip_li5988017610170933387t_char @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_162_in__set__impl__in__set__zip1,axiom,
! [Xs: list_list_char,Ys: list_list_char,X2: list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ~ ! [Y3: list_char] :
~ ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ X2 @ Y3 ) @ ( set_Pr2195431618410804152t_char @ ( zip_li5382857322596760955t_char @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_163_in__set__impl__in__set__zip2,axiom,
! [Xs: list_list_list_char,Ys: list_P98921565119350719nt_int,Y2: produc4435102495419491129nt_int] :
( ( ( size_s2562795203299699605t_char @ Xs )
= ( size_s2729875514769589291nt_int @ Ys ) )
=> ( ( member3599195040465536738nt_int @ Y2 @ ( set_Pr8994527289597011790nt_int @ Ys ) )
=> ~ ! [X: list_list_char] :
~ ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X @ Y2 ) @ ( set_Pr8107714755537002024nt_int @ ( zip_li4478691307266062991nt_int @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_164_in__set__impl__in__set__zip2,axiom,
! [Xs: list_list_char_int,Ys: list_int,Y2: int] :
( ( ( size_s3714362573911081296ar_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( member_int2 @ Y2 @ ( set_int2 @ Ys ) )
=> ~ ! [X: list_char > int] :
~ ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X @ Y2 ) @ ( set_Pr8994527289597011790nt_int @ ( zip_li343459552841501841nt_int @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_165_in__set__impl__in__set__zip2,axiom,
! [Xs: list_list_char,Ys: list_list_list_char,Y2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s2562795203299699605t_char @ Ys ) )
=> ( ( member4762036990631661320t_char @ Y2 @ ( set_list_list_char2 @ Ys ) )
=> ~ ! [X: list_char] :
~ ( member1112677564943258204t_char @ ( produc2694321116511041643t_char @ X @ Y2 ) @ ( set_Pr320474324478528712t_char @ ( zip_li8837739375816560523t_char @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_166_in__set__impl__in__set__zip2,axiom,
! [Xs: list_list_char,Ys: list_list_char,Y2: list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ( ( member_list_char2 @ Y2 @ ( set_list_char2 @ Ys ) )
=> ~ ! [X: list_char] :
~ ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ X @ Y2 ) @ ( set_Pr2195431618410804152t_char @ ( zip_li5382857322596760955t_char @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_167_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X3: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_168_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_list_char] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s356637359517785349t_char @ Xs ) )
= ( ? [X3: list_char,Ys2: list_list_char] :
( ( Xs
= ( cons_list_char @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_s356637359517785349t_char @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_169_distinct__remove1__removeAll,axiom,
! [Xs: list_list_char,X2: list_char] :
( ( distinct_list_char @ Xs )
=> ( ( remove1_list_char @ X2 @ Xs )
= ( removeAll_list_char @ X2 @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_170_distinct__remove1__removeAll,axiom,
! [Xs: list_list_list_char,X2: list_list_char] :
( ( distin92326681159624120t_char @ Xs )
=> ( ( remove4265589039595904469t_char @ X2 @ Xs )
= ( remove3141867459347406693t_char @ X2 @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_171_distinct__remove1__removeAll,axiom,
! [Xs: list_char,X2: char] :
( ( distinct_char @ Xs )
=> ( ( remove1_char @ X2 @ Xs )
= ( removeAll_char @ X2 @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_172_distinct__remove1__removeAll,axiom,
! [Xs: list_P8512142354091196201t_char,X2: produc6859378861088059299t_char] :
( ( distin3640772593324101788t_char @ Xs )
=> ( ( remove5295477908717968703t_char @ X2 @ Xs )
= ( remove442673217080424367t_char @ X2 @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_173_distinct__set__subseqs,axiom,
! [Xs: list_char] :
( ( distinct_char @ Xs )
=> ( distinct_set_char @ ( map_li8676856207254825134t_char @ set_char2 @ ( subseqs_char @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_174_distinct__set__subseqs,axiom,
! [Xs: list_P8512142354091196201t_char] :
( ( distin3640772593324101788t_char @ Xs )
=> ( distin2984542136165541372t_char @ ( map_li4259561042490804190t_char @ set_Pr2195431618410804152t_char @ ( subseq4653230730214077816t_char @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_175_distinct__set__subseqs,axiom,
! [Xs: list_list_list_char] :
( ( distin92326681159624120t_char @ Xs )
=> ( distin2701972443157563118t_char @ ( map_li2196609425031054830t_char @ set_list_list_char2 @ ( subseq1546420061081462748t_char @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_176_distinct__set__subseqs,axiom,
! [Xs: list_list_char] :
( ( distinct_list_char @ Xs )
=> ( distin8900834477317634270t_char @ ( map_li1475901098584161742t_char @ set_list_char2 @ ( subseqs_list_char @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_177_remove1__split,axiom,
! [A: list_list_char,Xs: list_list_list_char,Ys: list_list_list_char] :
( ( member4762036990631661320t_char @ A @ ( set_list_list_char2 @ Xs ) )
=> ( ( ( remove4265589039595904469t_char @ A @ Xs )
= Ys )
= ( ? [Ls: list_list_list_char,Rs: list_list_list_char] :
( ( Xs
= ( append3251196042929938198t_char @ Ls @ ( cons_list_list_char @ A @ Rs ) ) )
& ~ ( member4762036990631661320t_char @ A @ ( set_list_list_char2 @ Ls ) )
& ( Ys
= ( append3251196042929938198t_char @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_178_remove1__split,axiom,
! [A: list_char,Xs: list_list_char,Ys: list_list_char] :
( ( member_list_char2 @ A @ ( set_list_char2 @ Xs ) )
=> ( ( ( remove1_list_char @ A @ Xs )
= Ys )
= ( ? [Ls: list_list_char,Rs: list_list_char] :
( ( Xs
= ( append_list_char @ Ls @ ( cons_list_char @ A @ Rs ) ) )
& ~ ( member_list_char2 @ A @ ( set_list_char2 @ Ls ) )
& ( Ys
= ( append_list_char @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_179_remove1__split,axiom,
! [A: int,Xs: list_int,Ys: list_int] :
( ( member_int2 @ A @ ( set_int2 @ Xs ) )
=> ( ( ( remove1_int @ A @ Xs )
= Ys )
= ( ? [Ls: list_int,Rs: list_int] :
( ( Xs
= ( append_int @ Ls @ ( cons_int @ A @ Rs ) ) )
& ~ ( member_int2 @ A @ ( set_int2 @ Ls ) )
& ( Ys
= ( append_int @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_180_insert__Nil,axiom,
! [X2: instr] :
( ( insert_instr @ X2 @ nil_instr )
= ( cons_instr @ X2 @ nil_instr ) ) ).
% insert_Nil
thf(fact_181_insert__Nil,axiom,
! [X2: list_char] :
( ( insert_list_char @ X2 @ nil_list_char )
= ( cons_list_char @ X2 @ nil_list_char ) ) ).
% insert_Nil
thf(fact_182_insert__Nil,axiom,
! [X2: int] :
( ( insert_int @ X2 @ nil_int )
= ( cons_int @ X2 @ nil_int ) ) ).
% insert_Nil
thf(fact_183_length__product,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( size_s1919724266886658965t_char @ ( produc6248063980778377481t_char @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s356637359517785349t_char @ Xs ) @ ( size_s356637359517785349t_char @ Ys ) ) ) ).
% length_product
thf(fact_184_le__sup__iff,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_less_eq_nat @ X2 @ Z )
& ( ord_less_eq_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_185_le__sup__iff,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X2 @ Y2 ) @ Z )
= ( ( ord_less_eq_int @ X2 @ Z )
& ( ord_less_eq_int @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_186_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_187_sup_Obounded__iff,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_188_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_189_map__is__Nil__conv,axiom,
! [F: instr > instr,Xs: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= nil_instr )
= ( Xs = nil_instr ) ) ).
% map_is_Nil_conv
thf(fact_190_map__is__Nil__conv,axiom,
! [F: list_char > instr,Xs: list_list_char] :
( ( ( map_list_char_instr @ F @ Xs )
= nil_instr )
= ( Xs = nil_list_char ) ) ).
% map_is_Nil_conv
thf(fact_191_map__is__Nil__conv,axiom,
! [F: instr > list_char,Xs: list_instr] :
( ( ( map_instr_list_char @ F @ Xs )
= nil_list_char )
= ( Xs = nil_instr ) ) ).
% map_is_Nil_conv
thf(fact_192_map__is__Nil__conv,axiom,
! [F: list_char > list_char,Xs: list_list_char] :
( ( ( map_li116305933131242120t_char @ F @ Xs )
= nil_list_char )
= ( Xs = nil_list_char ) ) ).
% map_is_Nil_conv
thf(fact_193_Nil__is__map__conv,axiom,
! [F: instr > instr,Xs: list_instr] :
( ( nil_instr
= ( map_instr_instr @ F @ Xs ) )
= ( Xs = nil_instr ) ) ).
% Nil_is_map_conv
thf(fact_194_Nil__is__map__conv,axiom,
! [F: list_char > instr,Xs: list_list_char] :
( ( nil_instr
= ( map_list_char_instr @ F @ Xs ) )
= ( Xs = nil_list_char ) ) ).
% Nil_is_map_conv
thf(fact_195_Nil__is__map__conv,axiom,
! [F: instr > list_char,Xs: list_instr] :
( ( nil_list_char
= ( map_instr_list_char @ F @ Xs ) )
= ( Xs = nil_instr ) ) ).
% Nil_is_map_conv
thf(fact_196_Nil__is__map__conv,axiom,
! [F: list_char > list_char,Xs: list_list_char] :
( ( nil_list_char
= ( map_li116305933131242120t_char @ F @ Xs ) )
= ( Xs = nil_list_char ) ) ).
% Nil_is_map_conv
thf(fact_197_list_Omap__disc__iff,axiom,
! [F: instr > instr,A: list_instr] :
( ( ( map_instr_instr @ F @ A )
= nil_instr )
= ( A = nil_instr ) ) ).
% list.map_disc_iff
thf(fact_198_list_Omap__disc__iff,axiom,
! [F: list_char > instr,A: list_list_char] :
( ( ( map_list_char_instr @ F @ A )
= nil_instr )
= ( A = nil_list_char ) ) ).
% list.map_disc_iff
thf(fact_199_list_Omap__disc__iff,axiom,
! [F: instr > list_char,A: list_instr] :
( ( ( map_instr_list_char @ F @ A )
= nil_list_char )
= ( A = nil_instr ) ) ).
% list.map_disc_iff
thf(fact_200_list_Omap__disc__iff,axiom,
! [F: list_char > list_char,A: list_list_char] :
( ( ( map_li116305933131242120t_char @ F @ A )
= nil_list_char )
= ( A = nil_list_char ) ) ).
% list.map_disc_iff
thf(fact_201_append__is__Nil__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= nil_instr )
= ( ( Xs = nil_instr )
& ( Ys = nil_instr ) ) ) ).
% append_is_Nil_conv
thf(fact_202_append__is__Nil__conv,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( append_list_char @ Xs @ Ys )
= nil_list_char )
= ( ( Xs = nil_list_char )
& ( Ys = nil_list_char ) ) ) ).
% append_is_Nil_conv
thf(fact_203_Nil__is__append__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( nil_instr
= ( append_instr @ Xs @ Ys ) )
= ( ( Xs = nil_instr )
& ( Ys = nil_instr ) ) ) ).
% Nil_is_append_conv
thf(fact_204_Nil__is__append__conv,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( nil_list_char
= ( append_list_char @ Xs @ Ys ) )
= ( ( Xs = nil_list_char )
& ( Ys = nil_list_char ) ) ) ).
% Nil_is_append_conv
thf(fact_205_self__append__conv2,axiom,
! [Y2: list_instr,Xs: list_instr] :
( ( Y2
= ( append_instr @ Xs @ Y2 ) )
= ( Xs = nil_instr ) ) ).
% self_append_conv2
thf(fact_206_self__append__conv2,axiom,
! [Y2: list_list_char,Xs: list_list_char] :
( ( Y2
= ( append_list_char @ Xs @ Y2 ) )
= ( Xs = nil_list_char ) ) ).
% self_append_conv2
thf(fact_207_append__self__conv2,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= Ys )
= ( Xs = nil_instr ) ) ).
% append_self_conv2
thf(fact_208_append__self__conv2,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( append_list_char @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_char ) ) ).
% append_self_conv2
thf(fact_209_self__append__conv,axiom,
! [Y2: list_instr,Ys: list_instr] :
( ( Y2
= ( append_instr @ Y2 @ Ys ) )
= ( Ys = nil_instr ) ) ).
% self_append_conv
thf(fact_210_self__append__conv,axiom,
! [Y2: list_list_char,Ys: list_list_char] :
( ( Y2
= ( append_list_char @ Y2 @ Ys ) )
= ( Ys = nil_list_char ) ) ).
% self_append_conv
thf(fact_211_append__self__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= Xs )
= ( Ys = nil_instr ) ) ).
% append_self_conv
thf(fact_212_append__self__conv,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( append_list_char @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_char ) ) ).
% append_self_conv
thf(fact_213_append__Nil2,axiom,
! [Xs: list_instr] :
( ( append_instr @ Xs @ nil_instr )
= Xs ) ).
% append_Nil2
thf(fact_214_append__Nil2,axiom,
! [Xs: list_list_char] :
( ( append_list_char @ Xs @ nil_list_char )
= Xs ) ).
% append_Nil2
thf(fact_215_append_Oright__neutral,axiom,
! [A: list_instr] :
( ( append_instr @ A @ nil_instr )
= A ) ).
% append.right_neutral
thf(fact_216_append_Oright__neutral,axiom,
! [A: list_list_char] :
( ( append_list_char @ A @ nil_list_char )
= A ) ).
% append.right_neutral
thf(fact_217_length__map,axiom,
! [F: list_char > list_char,Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( map_li116305933131242120t_char @ F @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_map
thf(fact_218_append__eq__append__conv,axiom,
! [Xs: list_list_char,Ys: list_list_char,Us: list_list_char,Vs: list_list_char] :
( ( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
| ( ( size_s356637359517785349t_char @ Us )
= ( size_s356637359517785349t_char @ Vs ) ) )
=> ( ( ( append_list_char @ Xs @ Us )
= ( append_list_char @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_219_zip__eq__Nil__iff,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( zip_instr_instr @ Xs @ Ys )
= nil_Pr2203008264400890349_instr )
= ( ( Xs = nil_instr )
| ( Ys = nil_instr ) ) ) ).
% zip_eq_Nil_iff
thf(fact_220_zip__eq__Nil__iff,axiom,
! [Xs: list_instr,Ys: list_list_char] :
( ( ( zip_instr_list_char @ Xs @ Ys )
= nil_Pr8046467687620146281t_char )
= ( ( Xs = nil_instr )
| ( Ys = nil_list_char ) ) ) ).
% zip_eq_Nil_iff
thf(fact_221_zip__eq__Nil__iff,axiom,
! [Xs: list_list_char,Ys: list_instr] :
( ( ( zip_list_char_instr @ Xs @ Ys )
= nil_Pr4340628689008376749_instr )
= ( ( Xs = nil_list_char )
| ( Ys = nil_instr ) ) ) ).
% zip_eq_Nil_iff
thf(fact_222_zip__eq__Nil__iff,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( zip_li5382857322596760955t_char @ Xs @ Ys )
= nil_Pr3678812991771325097t_char )
= ( ( Xs = nil_list_char )
| ( Ys = nil_list_char ) ) ) ).
% zip_eq_Nil_iff
thf(fact_223_Nil__eq__zip__iff,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( nil_Pr2203008264400890349_instr
= ( zip_instr_instr @ Xs @ Ys ) )
= ( ( Xs = nil_instr )
| ( Ys = nil_instr ) ) ) ).
% Nil_eq_zip_iff
thf(fact_224_Nil__eq__zip__iff,axiom,
! [Xs: list_instr,Ys: list_list_char] :
( ( nil_Pr8046467687620146281t_char
= ( zip_instr_list_char @ Xs @ Ys ) )
= ( ( Xs = nil_instr )
| ( Ys = nil_list_char ) ) ) ).
% Nil_eq_zip_iff
thf(fact_225_Nil__eq__zip__iff,axiom,
! [Xs: list_list_char,Ys: list_instr] :
( ( nil_Pr4340628689008376749_instr
= ( zip_list_char_instr @ Xs @ Ys ) )
= ( ( Xs = nil_list_char )
| ( Ys = nil_instr ) ) ) ).
% Nil_eq_zip_iff
thf(fact_226_Nil__eq__zip__iff,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( nil_Pr3678812991771325097t_char
= ( zip_li5382857322596760955t_char @ Xs @ Ys ) )
= ( ( Xs = nil_list_char )
| ( Ys = nil_list_char ) ) ) ).
% Nil_eq_zip_iff
thf(fact_227_remdups__eq__nil__iff,axiom,
! [X2: list_instr] :
( ( ( remdups_instr @ X2 )
= nil_instr )
= ( X2 = nil_instr ) ) ).
% remdups_eq_nil_iff
thf(fact_228_remdups__eq__nil__iff,axiom,
! [X2: list_list_char] :
( ( ( remdups_list_char @ X2 )
= nil_list_char )
= ( X2 = nil_list_char ) ) ).
% remdups_eq_nil_iff
thf(fact_229_remdups__eq__nil__right__iff,axiom,
! [X2: list_instr] :
( ( nil_instr
= ( remdups_instr @ X2 ) )
= ( X2 = nil_instr ) ) ).
% remdups_eq_nil_right_iff
thf(fact_230_remdups__eq__nil__right__iff,axiom,
! [X2: list_list_char] :
( ( nil_list_char
= ( remdups_list_char @ X2 ) )
= ( X2 = nil_list_char ) ) ).
% remdups_eq_nil_right_iff
thf(fact_231_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_instr] :
( ( ( rotate_instr @ N @ Xs )
= nil_instr )
= ( Xs = nil_instr ) ) ).
% rotate_is_Nil_conv
thf(fact_232_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_list_char] :
( ( ( rotate_list_char @ N @ Xs )
= nil_list_char )
= ( Xs = nil_list_char ) ) ).
% rotate_is_Nil_conv
thf(fact_233_removeAll__id,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ( ( remove3141867459347406693t_char @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_234_removeAll__id,axiom,
! [X2: list_char,Xs: list_list_char] :
( ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ( ( removeAll_list_char @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_235_rotate1__is__Nil__conv,axiom,
! [Xs: list_instr] :
( ( ( rotate1_instr @ Xs )
= nil_instr )
= ( Xs = nil_instr ) ) ).
% rotate1_is_Nil_conv
thf(fact_236_rotate1__is__Nil__conv,axiom,
! [Xs: list_list_char] :
( ( ( rotate1_list_char @ Xs )
= nil_list_char )
= ( Xs = nil_list_char ) ) ).
% rotate1_is_Nil_conv
thf(fact_237_list__ex1__simps_I1_J,axiom,
! [P2: instr > $o] :
~ ( list_ex1_instr @ P2 @ nil_instr ) ).
% list_ex1_simps(1)
thf(fact_238_list__ex1__simps_I1_J,axiom,
! [P2: list_char > $o] :
~ ( list_ex1_list_char @ P2 @ nil_list_char ) ).
% list_ex1_simps(1)
thf(fact_239_append1__eq__conv,axiom,
! [Xs: list_instr,X2: instr,Ys: list_instr,Y2: instr] :
( ( ( append_instr @ Xs @ ( cons_instr @ X2 @ nil_instr ) )
= ( append_instr @ Ys @ ( cons_instr @ Y2 @ nil_instr ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_240_append1__eq__conv,axiom,
! [Xs: list_list_char,X2: list_char,Ys: list_list_char,Y2: list_char] :
( ( ( append_list_char @ Xs @ ( cons_list_char @ X2 @ nil_list_char ) )
= ( append_list_char @ Ys @ ( cons_list_char @ Y2 @ nil_list_char ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_241_append1__eq__conv,axiom,
! [Xs: list_int,X2: int,Ys: list_int,Y2: int] :
( ( ( append_int @ Xs @ ( cons_int @ X2 @ nil_int ) )
= ( append_int @ Ys @ ( cons_int @ Y2 @ nil_int ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_242_zip__Cons__Cons,axiom,
! [X2: int,Xs: list_int,Y2: int,Ys: list_int] :
( ( zip_int_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys ) )
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ ( zip_int_int @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_243_zip__Cons__Cons,axiom,
! [X2: list_list_char,Xs: list_list_list_char,Y2: produc4435102495419491129nt_int,Ys: list_P98921565119350719nt_int] :
( ( zip_li4478691307266062991nt_int @ ( cons_list_list_char @ X2 @ Xs ) @ ( cons_P7721559881313335791nt_int @ Y2 @ Ys ) )
= ( cons_P3979909588204300359nt_int @ ( produc3965054194175396271nt_int @ X2 @ Y2 ) @ ( zip_li4478691307266062991nt_int @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_244_zip__Cons__Cons,axiom,
! [X2: list_char > int,Xs: list_list_char_int,Y2: int,Ys: list_int] :
( ( zip_li343459552841501841nt_int @ ( cons_list_char_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys ) )
= ( cons_P7721559881313335791nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y2 ) @ ( zip_li343459552841501841nt_int @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_245_set__append,axiom,
! [Xs: list_list_list_char,Ys: list_list_list_char] :
( ( set_list_list_char2 @ ( append3251196042929938198t_char @ Xs @ Ys ) )
= ( sup_su9123505219413647059t_char @ ( set_list_list_char2 @ Xs ) @ ( set_list_list_char2 @ Ys ) ) ) ).
% set_append
thf(fact_246_set__append,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( set_list_char2 @ ( append_list_char @ Xs @ Ys ) )
= ( sup_su1962856675925542723t_char @ ( set_list_char2 @ Xs ) @ ( set_list_char2 @ Ys ) ) ) ).
% set_append
thf(fact_247_zip__append,axiom,
! [Xs: list_list_char,Us: list_list_char,Ys: list_list_char,Vs: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Us ) )
=> ( ( zip_li5382857322596760955t_char @ ( append_list_char @ Xs @ Ys ) @ ( append_list_char @ Us @ Vs ) )
= ( append655500003500722110t_char @ ( zip_li5382857322596760955t_char @ Xs @ Us ) @ ( zip_li5382857322596760955t_char @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_248_length__remdups__leq,axiom,
! [Xs: list_list_char] : ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ ( remdups_list_char @ Xs ) ) @ ( size_s356637359517785349t_char @ Xs ) ) ).
% length_remdups_leq
thf(fact_249_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X: nat] :
( ( P2 @ X )
& ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_250_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_251_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_252_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_253_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_254_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_255_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_256_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_257_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_258_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_259_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_260_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_261_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_262_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_263_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_264_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_265_splice_Ocases,axiom,
! [X2: produc8891168062546637159_instr] :
( ! [Ys3: list_instr] :
( X2
!= ( produc7750586281064920791_instr @ nil_instr @ Ys3 ) )
=> ~ ! [X: instr,Xs3: list_instr,Ys3: list_instr] :
( X2
!= ( produc7750586281064920791_instr @ ( cons_instr @ X @ Xs3 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_266_splice_Ocases,axiom,
! [X2: produc525958773753613763t_char] :
( ! [Ys3: list_list_char] :
( X2
!= ( produc9134825680149865339t_char @ nil_list_char @ Ys3 ) )
=> ~ ! [X: list_char,Xs3: list_list_char,Ys3: list_list_char] :
( X2
!= ( produc9134825680149865339t_char @ ( cons_list_char @ X @ Xs3 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_267_splice_Ocases,axiom,
! [X2: produc1186641810826059865st_int] :
( ! [Ys3: list_int] :
( X2
!= ( produc364263696895485585st_int @ nil_int @ Ys3 ) )
=> ~ ! [X: int,Xs3: list_int,Ys3: list_int] :
( X2
!= ( produc364263696895485585st_int @ ( cons_int @ X @ Xs3 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_268_shuffles_Ocases,axiom,
! [X2: produc8891168062546637159_instr] :
( ! [Ys3: list_instr] :
( X2
!= ( produc7750586281064920791_instr @ nil_instr @ Ys3 ) )
=> ( ! [Xs3: list_instr] :
( X2
!= ( produc7750586281064920791_instr @ Xs3 @ nil_instr ) )
=> ~ ! [X: instr,Xs3: list_instr,Y3: instr,Ys3: list_instr] :
( X2
!= ( produc7750586281064920791_instr @ ( cons_instr @ X @ Xs3 ) @ ( cons_instr @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_269_shuffles_Ocases,axiom,
! [X2: produc525958773753613763t_char] :
( ! [Ys3: list_list_char] :
( X2
!= ( produc9134825680149865339t_char @ nil_list_char @ Ys3 ) )
=> ( ! [Xs3: list_list_char] :
( X2
!= ( produc9134825680149865339t_char @ Xs3 @ nil_list_char ) )
=> ~ ! [X: list_char,Xs3: list_list_char,Y3: list_char,Ys3: list_list_char] :
( X2
!= ( produc9134825680149865339t_char @ ( cons_list_char @ X @ Xs3 ) @ ( cons_list_char @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_270_shuffles_Ocases,axiom,
! [X2: produc1186641810826059865st_int] :
( ! [Ys3: list_int] :
( X2
!= ( produc364263696895485585st_int @ nil_int @ Ys3 ) )
=> ( ! [Xs3: list_int] :
( X2
!= ( produc364263696895485585st_int @ Xs3 @ nil_int ) )
=> ~ ! [X: int,Xs3: list_int,Y3: int,Ys3: list_int] :
( X2
!= ( produc364263696895485585st_int @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_271_transpose_Ocases,axiom,
! [X2: list_list_instr] :
( ( X2 != nil_list_instr )
=> ( ! [Xss2: list_list_instr] :
( X2
!= ( cons_list_instr @ nil_instr @ Xss2 ) )
=> ~ ! [X: instr,Xs3: list_instr,Xss2: list_list_instr] :
( X2
!= ( cons_list_instr @ ( cons_instr @ X @ Xs3 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_272_transpose_Ocases,axiom,
! [X2: list_list_list_char] :
( ( X2 != nil_list_list_char )
=> ( ! [Xss2: list_list_list_char] :
( X2
!= ( cons_list_list_char @ nil_list_char @ Xss2 ) )
=> ~ ! [X: list_char,Xs3: list_list_char,Xss2: list_list_list_char] :
( X2
!= ( cons_list_list_char @ ( cons_list_char @ X @ Xs3 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_273_transpose_Ocases,axiom,
! [X2: list_list_char] :
( ( X2 != nil_list_char )
=> ( ! [Xss2: list_list_char] :
( X2
!= ( cons_list_char @ nil_char @ Xss2 ) )
=> ~ ! [X: char,Xs3: list_char,Xss2: list_list_char] :
( X2
!= ( cons_list_char @ ( cons_char @ X @ Xs3 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_274_transpose_Ocases,axiom,
! [X2: list_list_int] :
( ( X2 != nil_list_int )
=> ( ! [Xss2: list_list_int] :
( X2
!= ( cons_list_int @ nil_int @ Xss2 ) )
=> ~ ! [X: int,Xs3: list_int,Xss2: list_list_int] :
( X2
!= ( cons_list_int @ ( cons_int @ X @ Xs3 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_275_eq__Nil__appendI,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( Xs = Ys )
=> ( Xs
= ( append_instr @ nil_instr @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_276_eq__Nil__appendI,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_char @ nil_list_char @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_277_append_Oleft__neutral,axiom,
! [A: list_instr] :
( ( append_instr @ nil_instr @ A )
= A ) ).
% append.left_neutral
thf(fact_278_append_Oleft__neutral,axiom,
! [A: list_list_char] :
( ( append_list_char @ nil_list_char @ A )
= A ) ).
% append.left_neutral
thf(fact_279_append__Nil,axiom,
! [Ys: list_instr] :
( ( append_instr @ nil_instr @ Ys )
= Ys ) ).
% append_Nil
thf(fact_280_append__Nil,axiom,
! [Ys: list_list_char] :
( ( append_list_char @ nil_list_char @ Ys )
= Ys ) ).
% append_Nil
thf(fact_281_removeAll_Osimps_I1_J,axiom,
! [X2: instr] :
( ( removeAll_instr @ X2 @ nil_instr )
= nil_instr ) ).
% removeAll.simps(1)
thf(fact_282_removeAll_Osimps_I1_J,axiom,
! [X2: list_char] :
( ( removeAll_list_char @ X2 @ nil_list_char )
= nil_list_char ) ).
% removeAll.simps(1)
thf(fact_283_list_Osimps_I8_J,axiom,
! [F: instr > instr] :
( ( map_instr_instr @ F @ nil_instr )
= nil_instr ) ).
% list.simps(8)
thf(fact_284_list_Osimps_I8_J,axiom,
! [F: instr > list_char] :
( ( map_instr_list_char @ F @ nil_instr )
= nil_list_char ) ).
% list.simps(8)
thf(fact_285_list_Osimps_I8_J,axiom,
! [F: list_char > instr] :
( ( map_list_char_instr @ F @ nil_list_char )
= nil_instr ) ).
% list.simps(8)
thf(fact_286_list_Osimps_I8_J,axiom,
! [F: list_char > list_char] :
( ( map_li116305933131242120t_char @ F @ nil_list_char )
= nil_list_char ) ).
% list.simps(8)
thf(fact_287_rev__induct,axiom,
! [P2: list_instr > $o,Xs: list_instr] :
( ( P2 @ nil_instr )
=> ( ! [X: instr,Xs3: list_instr] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_instr @ Xs3 @ ( cons_instr @ X @ nil_instr ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_288_rev__induct,axiom,
! [P2: list_list_char > $o,Xs: list_list_char] :
( ( P2 @ nil_list_char )
=> ( ! [X: list_char,Xs3: list_list_char] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_list_char @ Xs3 @ ( cons_list_char @ X @ nil_list_char ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_289_rev__induct,axiom,
! [P2: list_int > $o,Xs: list_int] :
( ( P2 @ nil_int )
=> ( ! [X: int,Xs3: list_int] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_int @ Xs3 @ ( cons_int @ X @ nil_int ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_290_rev__exhaust,axiom,
! [Xs: list_instr] :
( ( Xs != nil_instr )
=> ~ ! [Ys3: list_instr,Y3: instr] :
( Xs
!= ( append_instr @ Ys3 @ ( cons_instr @ Y3 @ nil_instr ) ) ) ) ).
% rev_exhaust
thf(fact_291_rev__exhaust,axiom,
! [Xs: list_list_char] :
( ( Xs != nil_list_char )
=> ~ ! [Ys3: list_list_char,Y3: list_char] :
( Xs
!= ( append_list_char @ Ys3 @ ( cons_list_char @ Y3 @ nil_list_char ) ) ) ) ).
% rev_exhaust
thf(fact_292_rev__exhaust,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ~ ! [Ys3: list_int,Y3: int] :
( Xs
!= ( append_int @ Ys3 @ ( cons_int @ Y3 @ nil_int ) ) ) ) ).
% rev_exhaust
thf(fact_293_Cons__eq__append__conv,axiom,
! [X2: instr,Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( cons_instr @ X2 @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ( ( Ys = nil_instr )
& ( ( cons_instr @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_instr] :
( ( ( cons_instr @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_instr @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_294_Cons__eq__append__conv,axiom,
! [X2: list_char,Xs: list_list_char,Ys: list_list_char,Zs: list_list_char] :
( ( ( cons_list_char @ X2 @ Xs )
= ( append_list_char @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_char )
& ( ( cons_list_char @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_list_char] :
( ( ( cons_list_char @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_list_char @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_295_Cons__eq__append__conv,axiom,
! [X2: int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( cons_int @ X2 @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ( ( Ys = nil_int )
& ( ( cons_int @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_int] :
( ( ( cons_int @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_int @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_296_append__eq__Cons__conv,axiom,
! [Ys: list_instr,Zs: list_instr,X2: instr,Xs: list_instr] :
( ( ( append_instr @ Ys @ Zs )
= ( cons_instr @ X2 @ Xs ) )
= ( ( ( Ys = nil_instr )
& ( Zs
= ( cons_instr @ X2 @ Xs ) ) )
| ? [Ys4: list_instr] :
( ( Ys
= ( cons_instr @ X2 @ Ys4 ) )
& ( ( append_instr @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_297_append__eq__Cons__conv,axiom,
! [Ys: list_list_char,Zs: list_list_char,X2: list_char,Xs: list_list_char] :
( ( ( append_list_char @ Ys @ Zs )
= ( cons_list_char @ X2 @ Xs ) )
= ( ( ( Ys = nil_list_char )
& ( Zs
= ( cons_list_char @ X2 @ Xs ) ) )
| ? [Ys4: list_list_char] :
( ( Ys
= ( cons_list_char @ X2 @ Ys4 ) )
& ( ( append_list_char @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_298_append__eq__Cons__conv,axiom,
! [Ys: list_int,Zs: list_int,X2: int,Xs: list_int] :
( ( ( append_int @ Ys @ Zs )
= ( cons_int @ X2 @ Xs ) )
= ( ( ( Ys = nil_int )
& ( Zs
= ( cons_int @ X2 @ Xs ) ) )
| ? [Ys4: list_int] :
( ( Ys
= ( cons_int @ X2 @ Ys4 ) )
& ( ( append_int @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_299_rev__nonempty__induct,axiom,
! [Xs: list_instr,P2: list_instr > $o] :
( ( Xs != nil_instr )
=> ( ! [X: instr] : ( P2 @ ( cons_instr @ X @ nil_instr ) )
=> ( ! [X: instr,Xs3: list_instr] :
( ( Xs3 != nil_instr )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_instr @ Xs3 @ ( cons_instr @ X @ nil_instr ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_300_rev__nonempty__induct,axiom,
! [Xs: list_list_char,P2: list_list_char > $o] :
( ( Xs != nil_list_char )
=> ( ! [X: list_char] : ( P2 @ ( cons_list_char @ X @ nil_list_char ) )
=> ( ! [X: list_char,Xs3: list_list_char] :
( ( Xs3 != nil_list_char )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_list_char @ Xs3 @ ( cons_list_char @ X @ nil_list_char ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_301_rev__nonempty__induct,axiom,
! [Xs: list_int,P2: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X: int] : ( P2 @ ( cons_int @ X @ nil_int ) )
=> ( ! [X: int,Xs3: list_int] :
( ( Xs3 != nil_int )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_int @ Xs3 @ ( cons_int @ X @ nil_int ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_302_sorted__wrt_Ocases,axiom,
! [X2: produc7133927873007960030_instr] :
( ! [P3: instr > instr > $o] :
( X2
!= ( produc7818391707689388878_instr @ P3 @ nil_instr ) )
=> ~ ! [P3: instr > instr > $o,X: instr,Ys3: list_instr] :
( X2
!= ( produc7818391707689388878_instr @ P3 @ ( cons_instr @ X @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_303_sorted__wrt_Ocases,axiom,
! [X2: produc8794813868974875620t_char] :
( ! [P3: list_char > list_char > $o] :
( X2
!= ( produc1436111165245395422t_char @ P3 @ nil_list_char ) )
=> ~ ! [P3: list_char > list_char > $o,X: list_char,Ys3: list_list_char] :
( X2
!= ( produc1436111165245395422t_char @ P3 @ ( cons_list_char @ X @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_304_sorted__wrt_Ocases,axiom,
! [X2: produc5834231552977413017st_int] :
( ! [P3: int > int > $o] :
( X2
!= ( produc8618682346314911123st_int @ P3 @ nil_int ) )
=> ~ ! [P3: int > int > $o,X: int,Ys3: list_int] :
( X2
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_305_successively_Ocases,axiom,
! [X2: produc7133927873007960030_instr] :
( ! [P3: instr > instr > $o] :
( X2
!= ( produc7818391707689388878_instr @ P3 @ nil_instr ) )
=> ( ! [P3: instr > instr > $o,X: instr] :
( X2
!= ( produc7818391707689388878_instr @ P3 @ ( cons_instr @ X @ nil_instr ) ) )
=> ~ ! [P3: instr > instr > $o,X: instr,Y3: instr,Xs3: list_instr] :
( X2
!= ( produc7818391707689388878_instr @ P3 @ ( cons_instr @ X @ ( cons_instr @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_306_successively_Ocases,axiom,
! [X2: produc8794813868974875620t_char] :
( ! [P3: list_char > list_char > $o] :
( X2
!= ( produc1436111165245395422t_char @ P3 @ nil_list_char ) )
=> ( ! [P3: list_char > list_char > $o,X: list_char] :
( X2
!= ( produc1436111165245395422t_char @ P3 @ ( cons_list_char @ X @ nil_list_char ) ) )
=> ~ ! [P3: list_char > list_char > $o,X: list_char,Y3: list_char,Xs3: list_list_char] :
( X2
!= ( produc1436111165245395422t_char @ P3 @ ( cons_list_char @ X @ ( cons_list_char @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_307_successively_Ocases,axiom,
! [X2: produc5834231552977413017st_int] :
( ! [P3: int > int > $o] :
( X2
!= ( produc8618682346314911123st_int @ P3 @ nil_int ) )
=> ( ! [P3: int > int > $o,X: int] :
( X2
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X @ nil_int ) ) )
=> ~ ! [P3: int > int > $o,X: int,Y3: int,Xs3: list_int] :
( X2
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X @ ( cons_int @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_308_length__removeAll__less__eq,axiom,
! [X2: list_char,Xs: list_list_char] : ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ ( removeAll_list_char @ X2 @ Xs ) ) @ ( size_s356637359517785349t_char @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_309_subseqs_Osimps_I1_J,axiom,
( ( subseqs_char @ nil_char )
= ( cons_list_char @ nil_char @ nil_list_char ) ) ).
% subseqs.simps(1)
thf(fact_310_subseqs_Osimps_I1_J,axiom,
( ( subseqs_instr @ nil_instr )
= ( cons_list_instr @ nil_instr @ nil_list_instr ) ) ).
% subseqs.simps(1)
thf(fact_311_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_char @ nil_list_char )
= ( cons_list_list_char @ nil_list_char @ nil_list_list_char ) ) ).
% subseqs.simps(1)
thf(fact_312_product_Osimps_I1_J,axiom,
! [Uu: list_list_char] :
( ( produc6248063980778377481t_char @ nil_list_char @ Uu )
= nil_Pr3678812991771325097t_char ) ).
% product.simps(1)
thf(fact_313_product__lists_Osimps_I1_J,axiom,
( ( product_lists_instr @ nil_list_instr )
= ( cons_list_instr @ nil_instr @ nil_list_instr ) ) ).
% product_lists.simps(1)
thf(fact_314_product__lists_Osimps_I1_J,axiom,
( ( produc4928866681682397746t_char @ nil_list_list_char )
= ( cons_list_list_char @ nil_list_char @ nil_list_list_char ) ) ).
% product_lists.simps(1)
thf(fact_315_product__lists_Osimps_I1_J,axiom,
( ( product_lists_char @ nil_list_char )
= ( cons_list_char @ nil_char @ nil_list_char ) ) ).
% product_lists.simps(1)
thf(fact_316_vars_Osimps_I1_J,axiom,
( ( vars @ nil_instr )
= nil_list_char ) ).
% vars.simps(1)
thf(fact_317_same__length__different,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( Xs != Ys )
=> ( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ? [Pre: list_instr,X: instr,Xs4: list_instr,Y3: instr,Ys5: list_instr] :
( ( X != Y3 )
& ( Xs
= ( append_instr @ Pre @ ( append_instr @ ( cons_instr @ X @ nil_instr ) @ Xs4 ) ) )
& ( Ys
= ( append_instr @ Pre @ ( append_instr @ ( cons_instr @ Y3 @ nil_instr ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_318_same__length__different,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs != Ys )
=> ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ? [Pre: list_int,X: int,Xs4: list_int,Y3: int,Ys5: list_int] :
( ( X != Y3 )
& ( Xs
= ( append_int @ Pre @ ( append_int @ ( cons_int @ X @ nil_int ) @ Xs4 ) ) )
& ( Ys
= ( append_int @ Pre @ ( append_int @ ( cons_int @ Y3 @ nil_int ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_319_same__length__different,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( Xs != Ys )
=> ( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ? [Pre: list_list_char,X: list_char,Xs4: list_list_char,Y3: list_char,Ys5: list_list_char] :
( ( X != Y3 )
& ( Xs
= ( append_list_char @ Pre @ ( append_list_char @ ( cons_list_char @ X @ nil_list_char ) @ Xs4 ) ) )
& ( Ys
= ( append_list_char @ Pre @ ( append_list_char @ ( cons_list_char @ Y3 @ nil_list_char ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_320_not__distinct__decomp,axiom,
! [Ws: list_list_list_char] :
( ~ ( distin92326681159624120t_char @ Ws )
=> ? [Xs3: list_list_list_char,Ys3: list_list_list_char,Zs2: list_list_list_char,Y3: list_list_char] :
( Ws
= ( append3251196042929938198t_char @ Xs3 @ ( append3251196042929938198t_char @ ( cons_list_list_char @ Y3 @ nil_list_list_char ) @ ( append3251196042929938198t_char @ Ys3 @ ( append3251196042929938198t_char @ ( cons_list_list_char @ Y3 @ nil_list_list_char ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_321_not__distinct__decomp,axiom,
! [Ws: list_char] :
( ~ ( distinct_char @ Ws )
=> ? [Xs3: list_char,Ys3: list_char,Zs2: list_char,Y3: char] :
( Ws
= ( append_char @ Xs3 @ ( append_char @ ( cons_char @ Y3 @ nil_char ) @ ( append_char @ Ys3 @ ( append_char @ ( cons_char @ Y3 @ nil_char ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_322_not__distinct__decomp,axiom,
! [Ws: list_P8512142354091196201t_char] :
( ~ ( distin3640772593324101788t_char @ Ws )
=> ? [Xs3: list_P8512142354091196201t_char,Ys3: list_P8512142354091196201t_char,Zs2: list_P8512142354091196201t_char,Y3: produc6859378861088059299t_char] :
( Ws
= ( append655500003500722110t_char @ Xs3 @ ( append655500003500722110t_char @ ( cons_P922464210127128153t_char @ Y3 @ nil_Pr3678812991771325097t_char ) @ ( append655500003500722110t_char @ Ys3 @ ( append655500003500722110t_char @ ( cons_P922464210127128153t_char @ Y3 @ nil_Pr3678812991771325097t_char ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_323_not__distinct__decomp,axiom,
! [Ws: list_instr] :
( ~ ( distinct_instr @ Ws )
=> ? [Xs3: list_instr,Ys3: list_instr,Zs2: list_instr,Y3: instr] :
( Ws
= ( append_instr @ Xs3 @ ( append_instr @ ( cons_instr @ Y3 @ nil_instr ) @ ( append_instr @ Ys3 @ ( append_instr @ ( cons_instr @ Y3 @ nil_instr ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_324_not__distinct__decomp,axiom,
! [Ws: list_list_char] :
( ~ ( distinct_list_char @ Ws )
=> ? [Xs3: list_list_char,Ys3: list_list_char,Zs2: list_list_char,Y3: list_char] :
( Ws
= ( append_list_char @ Xs3 @ ( append_list_char @ ( cons_list_char @ Y3 @ nil_list_char ) @ ( append_list_char @ Ys3 @ ( append_list_char @ ( cons_list_char @ Y3 @ nil_list_char ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_325_not__distinct__decomp,axiom,
! [Ws: list_int] :
( ~ ( distinct_int @ Ws )
=> ? [Xs3: list_int,Ys3: list_int,Zs2: list_int,Y3: int] :
( Ws
= ( append_int @ Xs3 @ ( append_int @ ( cons_int @ Y3 @ nil_int ) @ ( append_int @ Ys3 @ ( append_int @ ( cons_int @ Y3 @ nil_int ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_326_map__eq__Cons__conv,axiom,
! [F: int > int,Xs: list_int,Y2: int,Ys: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( cons_int @ Y2 @ Ys ) )
= ( ? [Z3: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_int_int @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_327_Cons__eq__map__conv,axiom,
! [X2: int,Xs: list_int,F: int > int,Ys: list_int] :
( ( ( cons_int @ X2 @ Xs )
= ( map_int_int @ F @ Ys ) )
= ( ? [Z3: int,Zs3: list_int] :
( ( Ys
= ( cons_int @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_int_int @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_328_map__eq__Cons__D,axiom,
! [F: int > int,Xs: list_int,Y2: int,Ys: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( cons_int @ Y2 @ Ys ) )
=> ? [Z4: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_int_int @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_329_Cons__eq__map__D,axiom,
! [X2: int,Xs: list_int,F: int > int,Ys: list_int] :
( ( ( cons_int @ X2 @ Xs )
= ( map_int_int @ F @ Ys ) )
=> ? [Z4: int,Zs2: list_int] :
( ( Ys
= ( cons_int @ Z4 @ Zs2 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_int_int @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_330_list_Osimps_I9_J,axiom,
! [F: int > int,X21: int,X22: list_int] :
( ( map_int_int @ F @ ( cons_int @ X21 @ X22 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_int_int @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_331_list_Omap__ident__strong,axiom,
! [T: list_list_list_char,F: list_list_char > list_list_char] :
( ! [Z4: list_list_char] :
( ( member4762036990631661320t_char @ Z4 @ ( set_list_list_char2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_li6264137933337555624t_char @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_332_list_Omap__ident__strong,axiom,
! [T: list_list_char,F: list_char > list_char] :
( ! [Z4: list_char] :
( ( member_list_char2 @ Z4 @ ( set_list_char2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_li116305933131242120t_char @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_333_map__idI,axiom,
! [Xs: list_list_list_char,F: list_list_char > list_list_char] :
( ! [X: list_list_char] :
( ( member4762036990631661320t_char @ X @ ( set_list_list_char2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_li6264137933337555624t_char @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_334_map__idI,axiom,
! [Xs: list_list_char,F: list_char > list_char] :
( ! [X: list_char] :
( ( member_list_char2 @ X @ ( set_list_char2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_li116305933131242120t_char @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_335_rotate1_Osimps_I2_J,axiom,
! [X2: instr,Xs: list_instr] :
( ( rotate1_instr @ ( cons_instr @ X2 @ Xs ) )
= ( append_instr @ Xs @ ( cons_instr @ X2 @ nil_instr ) ) ) ).
% rotate1.simps(2)
thf(fact_336_rotate1_Osimps_I2_J,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( rotate1_list_char @ ( cons_list_char @ X2 @ Xs ) )
= ( append_list_char @ Xs @ ( cons_list_char @ X2 @ nil_list_char ) ) ) ).
% rotate1.simps(2)
thf(fact_337_rotate1_Osimps_I2_J,axiom,
! [X2: int,Xs: list_int] :
( ( rotate1_int @ ( cons_int @ X2 @ Xs ) )
= ( append_int @ Xs @ ( cons_int @ X2 @ nil_int ) ) ) ).
% rotate1.simps(2)
thf(fact_338_Cons__eq__appendI,axiom,
! [X2: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( cons_int @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_int @ Xs1 @ Zs ) )
=> ( ( cons_int @ X2 @ Xs )
= ( append_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_339_append__Cons,axiom,
! [X2: int,Xs: list_int,Ys: list_int] :
( ( append_int @ ( cons_int @ X2 @ Xs ) @ Ys )
= ( cons_int @ X2 @ ( append_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_340_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_341_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_342_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_343_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_344_Suc__le__D,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M2 )
=> ? [M3: nat] :
( M2
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_345_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_346_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_347_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_348_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
=> ( P2 @ M4 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_349_nat__induct__at__least,axiom,
! [M: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P2 @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_350_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X: nat] : ( R @ X @ X )
=> ( ! [X: nat,Y3: nat,Z4: nat] :
( ( R @ X @ Y3 )
=> ( ( R @ Y3 @ Z4 )
=> ( R @ X @ Z4 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_351_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_352_inf__sup__ord_I4_J,axiom,
! [Y2: int,X2: int] : ( ord_less_eq_int @ Y2 @ ( sup_sup_int @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_353_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_354_inf__sup__ord_I3_J,axiom,
! [X2: int,Y2: int] : ( ord_less_eq_int @ X2 @ ( sup_sup_int @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_355_le__supE,axiom,
! [A: nat,B: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A @ X2 )
=> ~ ( ord_less_eq_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_356_le__supE,axiom,
! [A: int,B: int,X2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_int @ A @ X2 )
=> ~ ( ord_less_eq_int @ B @ X2 ) ) ) ).
% le_supE
thf(fact_357_le__supI,axiom,
! [A: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X2 )
=> ( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_358_le__supI,axiom,
! [A: int,X2: int,B: int] :
( ( ord_less_eq_int @ A @ X2 )
=> ( ( ord_less_eq_int @ B @ X2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_359_sup__ge1,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_360_sup__ge1,axiom,
! [X2: int,Y2: int] : ( ord_less_eq_int @ X2 @ ( sup_sup_int @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_361_sup__ge2,axiom,
! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_362_sup__ge2,axiom,
! [Y2: int,X2: int] : ( ord_less_eq_int @ Y2 @ ( sup_sup_int @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_363_le__supI1,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_364_le__supI1,axiom,
! [X2: int,A: int,B: int] :
( ( ord_less_eq_int @ X2 @ A )
=> ( ord_less_eq_int @ X2 @ ( sup_sup_int @ A @ B ) ) ) ).
% le_supI1
thf(fact_365_le__supI2,axiom,
! [X2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_366_le__supI2,axiom,
! [X2: int,B: int,A: int] :
( ( ord_less_eq_int @ X2 @ B )
=> ( ord_less_eq_int @ X2 @ ( sup_sup_int @ A @ B ) ) ) ).
% le_supI2
thf(fact_367_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_368_sup_Omono,axiom,
! [C: int,A: int,D: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ( ord_less_eq_int @ D @ B )
=> ( ord_less_eq_int @ ( sup_sup_int @ C @ D ) @ ( sup_sup_int @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_369_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_370_sup__mono,axiom,
! [A: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ B @ D )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ ( sup_sup_int @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_371_sup__least,axiom,
! [Y2: nat,X2: nat,Z: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_372_sup__least,axiom,
! [Y2: int,X2: int,Z: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ( ( ord_less_eq_int @ Z @ X2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_373_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y: nat] :
( ( sup_sup_nat @ X3 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_374_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y: int] :
( ( sup_sup_int @ X3 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_375_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_376_sup_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( sup_sup_int @ A @ B ) ) ) ).
% sup.orderE
thf(fact_377_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_378_sup_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( sup_sup_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% sup.orderI
thf(fact_379_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y2: nat] :
( ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ X @ ( F @ X @ Y3 ) )
=> ( ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X @ Y3 ) )
=> ( ! [X: nat,Y3: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ( ( ord_less_eq_nat @ Z4 @ X )
=> ( ord_less_eq_nat @ ( F @ Y3 @ Z4 ) @ X ) ) )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_380_sup__unique,axiom,
! [F: int > int > int,X2: int,Y2: int] :
( ! [X: int,Y3: int] : ( ord_less_eq_int @ X @ ( F @ X @ Y3 ) )
=> ( ! [X: int,Y3: int] : ( ord_less_eq_int @ Y3 @ ( F @ X @ Y3 ) )
=> ( ! [X: int,Y3: int,Z4: int] :
( ( ord_less_eq_int @ Y3 @ X )
=> ( ( ord_less_eq_int @ Z4 @ X )
=> ( ord_less_eq_int @ ( F @ Y3 @ Z4 ) @ X ) ) )
=> ( ( sup_sup_int @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_381_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_382_sup_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( sup_sup_int @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_383_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_384_sup_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( sup_sup_int @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_385_sup__absorb1,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_386_sup__absorb1,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ( ( sup_sup_int @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_387_sup__absorb2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_388_sup__absorb2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( sup_sup_int @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_389_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_390_sup_OboundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_int @ B @ A )
=> ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_391_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_392_sup_OboundedI,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_393_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_394_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( A3
= ( sup_sup_int @ A3 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_395_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_396_sup_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B ) ) ).
% sup.cobounded1
thf(fact_397_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_398_sup_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A @ B ) ) ).
% sup.cobounded2
thf(fact_399_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_400_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( sup_sup_int @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_401_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( sup_sup_nat @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_402_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( sup_sup_int @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_403_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_404_sup_OcoboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_405_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_406_sup_OcoboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_407_list__nonempty__induct,axiom,
! [Xs: list_instr,P2: list_instr > $o] :
( ( Xs != nil_instr )
=> ( ! [X: instr] : ( P2 @ ( cons_instr @ X @ nil_instr ) )
=> ( ! [X: instr,Xs3: list_instr] :
( ( Xs3 != nil_instr )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_408_list__nonempty__induct,axiom,
! [Xs: list_list_char,P2: list_list_char > $o] :
( ( Xs != nil_list_char )
=> ( ! [X: list_char] : ( P2 @ ( cons_list_char @ X @ nil_list_char ) )
=> ( ! [X: list_char,Xs3: list_list_char] :
( ( Xs3 != nil_list_char )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_list_char @ X @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_409_list__nonempty__induct,axiom,
! [Xs: list_int,P2: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X: int] : ( P2 @ ( cons_int @ X @ nil_int ) )
=> ( ! [X: int,Xs3: list_int] :
( ( Xs3 != nil_int )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_410_list__induct2_H,axiom,
! [P2: list_instr > list_instr > $o,Xs: list_instr,Ys: list_instr] :
( ( P2 @ nil_instr @ nil_instr )
=> ( ! [X: instr,Xs3: list_instr] : ( P2 @ ( cons_instr @ X @ Xs3 ) @ nil_instr )
=> ( ! [Y3: instr,Ys3: list_instr] : ( P2 @ nil_instr @ ( cons_instr @ Y3 @ Ys3 ) )
=> ( ! [X: instr,Xs3: list_instr,Y3: instr,Ys3: list_instr] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) @ ( cons_instr @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_411_list__induct2_H,axiom,
! [P2: list_instr > list_list_char > $o,Xs: list_instr,Ys: list_list_char] :
( ( P2 @ nil_instr @ nil_list_char )
=> ( ! [X: instr,Xs3: list_instr] : ( P2 @ ( cons_instr @ X @ Xs3 ) @ nil_list_char )
=> ( ! [Y3: list_char,Ys3: list_list_char] : ( P2 @ nil_instr @ ( cons_list_char @ Y3 @ Ys3 ) )
=> ( ! [X: instr,Xs3: list_instr,Y3: list_char,Ys3: list_list_char] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) @ ( cons_list_char @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_412_list__induct2_H,axiom,
! [P2: list_list_char > list_instr > $o,Xs: list_list_char,Ys: list_instr] :
( ( P2 @ nil_list_char @ nil_instr )
=> ( ! [X: list_char,Xs3: list_list_char] : ( P2 @ ( cons_list_char @ X @ Xs3 ) @ nil_instr )
=> ( ! [Y3: instr,Ys3: list_instr] : ( P2 @ nil_list_char @ ( cons_instr @ Y3 @ Ys3 ) )
=> ( ! [X: list_char,Xs3: list_list_char,Y3: instr,Ys3: list_instr] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_list_char @ X @ Xs3 ) @ ( cons_instr @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_413_list__induct2_H,axiom,
! [P2: list_list_char > list_list_char > $o,Xs: list_list_char,Ys: list_list_char] :
( ( P2 @ nil_list_char @ nil_list_char )
=> ( ! [X: list_char,Xs3: list_list_char] : ( P2 @ ( cons_list_char @ X @ Xs3 ) @ nil_list_char )
=> ( ! [Y3: list_char,Ys3: list_list_char] : ( P2 @ nil_list_char @ ( cons_list_char @ Y3 @ Ys3 ) )
=> ( ! [X: list_char,Xs3: list_list_char,Y3: list_char,Ys3: list_list_char] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_list_char @ X @ Xs3 ) @ ( cons_list_char @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_414_list__induct2_H,axiom,
! [P2: list_instr > list_int > $o,Xs: list_instr,Ys: list_int] :
( ( P2 @ nil_instr @ nil_int )
=> ( ! [X: instr,Xs3: list_instr] : ( P2 @ ( cons_instr @ X @ Xs3 ) @ nil_int )
=> ( ! [Y3: int,Ys3: list_int] : ( P2 @ nil_instr @ ( cons_int @ Y3 @ Ys3 ) )
=> ( ! [X: instr,Xs3: list_instr,Y3: int,Ys3: list_int] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_415_list__induct2_H,axiom,
! [P2: list_list_char > list_int > $o,Xs: list_list_char,Ys: list_int] :
( ( P2 @ nil_list_char @ nil_int )
=> ( ! [X: list_char,Xs3: list_list_char] : ( P2 @ ( cons_list_char @ X @ Xs3 ) @ nil_int )
=> ( ! [Y3: int,Ys3: list_int] : ( P2 @ nil_list_char @ ( cons_int @ Y3 @ Ys3 ) )
=> ( ! [X: list_char,Xs3: list_list_char,Y3: int,Ys3: list_int] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_list_char @ X @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_416_list__induct2_H,axiom,
! [P2: list_int > list_instr > $o,Xs: list_int,Ys: list_instr] :
( ( P2 @ nil_int @ nil_instr )
=> ( ! [X: int,Xs3: list_int] : ( P2 @ ( cons_int @ X @ Xs3 ) @ nil_instr )
=> ( ! [Y3: instr,Ys3: list_instr] : ( P2 @ nil_int @ ( cons_instr @ Y3 @ Ys3 ) )
=> ( ! [X: int,Xs3: list_int,Y3: instr,Ys3: list_instr] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_instr @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_417_list__induct2_H,axiom,
! [P2: list_int > list_list_char > $o,Xs: list_int,Ys: list_list_char] :
( ( P2 @ nil_int @ nil_list_char )
=> ( ! [X: int,Xs3: list_int] : ( P2 @ ( cons_int @ X @ Xs3 ) @ nil_list_char )
=> ( ! [Y3: list_char,Ys3: list_list_char] : ( P2 @ nil_int @ ( cons_list_char @ Y3 @ Ys3 ) )
=> ( ! [X: int,Xs3: list_int,Y3: list_char,Ys3: list_list_char] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_list_char @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_418_list__induct2_H,axiom,
! [P2: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P2 @ nil_int @ nil_int )
=> ( ! [X: int,Xs3: list_int] : ( P2 @ ( cons_int @ X @ Xs3 ) @ nil_int )
=> ( ! [Y3: int,Ys3: list_int] : ( P2 @ nil_int @ ( cons_int @ Y3 @ Ys3 ) )
=> ( ! [X: int,Xs3: list_int,Y3: int,Ys3: list_int] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_419_neq__Nil__conv,axiom,
! [Xs: list_instr] :
( ( Xs != nil_instr )
= ( ? [Y: instr,Ys2: list_instr] :
( Xs
= ( cons_instr @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_420_neq__Nil__conv,axiom,
! [Xs: list_list_char] :
( ( Xs != nil_list_char )
= ( ? [Y: list_char,Ys2: list_list_char] :
( Xs
= ( cons_list_char @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_421_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y: int,Ys2: list_int] :
( Xs
= ( cons_int @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_422_remdups__adj_Ocases,axiom,
! [X2: list_instr] :
( ( X2 != nil_instr )
=> ( ! [X: instr] :
( X2
!= ( cons_instr @ X @ nil_instr ) )
=> ~ ! [X: instr,Y3: instr,Xs3: list_instr] :
( X2
!= ( cons_instr @ X @ ( cons_instr @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_423_remdups__adj_Ocases,axiom,
! [X2: list_list_char] :
( ( X2 != nil_list_char )
=> ( ! [X: list_char] :
( X2
!= ( cons_list_char @ X @ nil_list_char ) )
=> ~ ! [X: list_char,Y3: list_char,Xs3: list_list_char] :
( X2
!= ( cons_list_char @ X @ ( cons_list_char @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_424_remdups__adj_Ocases,axiom,
! [X2: list_int] :
( ( X2 != nil_int )
=> ( ! [X: int] :
( X2
!= ( cons_int @ X @ nil_int ) )
=> ~ ! [X: int,Y3: int,Xs3: list_int] :
( X2
!= ( cons_int @ X @ ( cons_int @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_425_min__list_Ocases,axiom,
! [X2: list_int] :
( ! [X: int,Xs3: list_int] :
( X2
!= ( cons_int @ X @ Xs3 ) )
=> ( X2 = nil_int ) ) ).
% min_list.cases
thf(fact_426_list_Oexhaust,axiom,
! [Y2: list_instr] :
( ( Y2 != nil_instr )
=> ~ ! [X212: instr,X222: list_instr] :
( Y2
!= ( cons_instr @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_427_list_Oexhaust,axiom,
! [Y2: list_list_char] :
( ( Y2 != nil_list_char )
=> ~ ! [X212: list_char,X222: list_list_char] :
( Y2
!= ( cons_list_char @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_428_list_Oexhaust,axiom,
! [Y2: list_int] :
( ( Y2 != nil_int )
=> ~ ! [X212: int,X222: list_int] :
( Y2
!= ( cons_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_429_list_OdiscI,axiom,
! [List: list_instr,X21: instr,X22: list_instr] :
( ( List
= ( cons_instr @ X21 @ X22 ) )
=> ( List != nil_instr ) ) ).
% list.discI
thf(fact_430_list_OdiscI,axiom,
! [List: list_list_char,X21: list_char,X22: list_list_char] :
( ( List
= ( cons_list_char @ X21 @ X22 ) )
=> ( List != nil_list_char ) ) ).
% list.discI
thf(fact_431_list_OdiscI,axiom,
! [List: list_int,X21: int,X22: list_int] :
( ( List
= ( cons_int @ X21 @ X22 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_432_list_Odistinct_I1_J,axiom,
! [X21: instr,X22: list_instr] :
( nil_instr
!= ( cons_instr @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_433_list_Odistinct_I1_J,axiom,
! [X21: list_char,X22: list_list_char] :
( nil_list_char
!= ( cons_list_char @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_434_list_Odistinct_I1_J,axiom,
! [X21: int,X22: list_int] :
( nil_int
!= ( cons_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_435_rotate__map,axiom,
! [N: nat,F: list_char > list_char,Xs: list_list_char] :
( ( rotate_list_char @ N @ ( map_li116305933131242120t_char @ F @ Xs ) )
= ( map_li116305933131242120t_char @ F @ ( rotate_list_char @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_436_rotate1__map,axiom,
! [F: list_char > list_char,Xs: list_list_char] :
( ( rotate1_list_char @ ( map_li116305933131242120t_char @ F @ Xs ) )
= ( map_li116305933131242120t_char @ F @ ( rotate1_list_char @ Xs ) ) ) ).
% rotate1_map
thf(fact_437_distinct_Osimps_I1_J,axiom,
distin92326681159624120t_char @ nil_list_list_char ).
% distinct.simps(1)
thf(fact_438_distinct_Osimps_I1_J,axiom,
distinct_char @ nil_char ).
% distinct.simps(1)
thf(fact_439_distinct_Osimps_I1_J,axiom,
distin3640772593324101788t_char @ nil_Pr3678812991771325097t_char ).
% distinct.simps(1)
thf(fact_440_distinct_Osimps_I1_J,axiom,
distinct_instr @ nil_instr ).
% distinct.simps(1)
thf(fact_441_distinct_Osimps_I1_J,axiom,
distinct_list_char @ nil_list_char ).
% distinct.simps(1)
thf(fact_442_remdups_Osimps_I1_J,axiom,
( ( remdups_instr @ nil_instr )
= nil_instr ) ).
% remdups.simps(1)
thf(fact_443_remdups_Osimps_I1_J,axiom,
( ( remdups_list_char @ nil_list_char )
= nil_list_char ) ).
% remdups.simps(1)
thf(fact_444_remove1_Osimps_I1_J,axiom,
! [X2: instr] :
( ( remove1_instr @ X2 @ nil_instr )
= nil_instr ) ).
% remove1.simps(1)
thf(fact_445_remove1_Osimps_I1_J,axiom,
! [X2: list_char] :
( ( remove1_list_char @ X2 @ nil_list_char )
= nil_list_char ) ).
% remove1.simps(1)
thf(fact_446_length__append__singleton,axiom,
! [Xs: list_instr,X2: instr] :
( ( size_size_list_instr @ ( append_instr @ Xs @ ( cons_instr @ X2 @ nil_instr ) ) )
= ( suc @ ( size_size_list_instr @ Xs ) ) ) ).
% length_append_singleton
thf(fact_447_length__append__singleton,axiom,
! [Xs: list_int,X2: int] :
( ( size_size_list_int @ ( append_int @ Xs @ ( cons_int @ X2 @ nil_int ) ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_append_singleton
thf(fact_448_length__append__singleton,axiom,
! [Xs: list_list_char,X2: list_char] :
( ( size_s356637359517785349t_char @ ( append_list_char @ Xs @ ( cons_list_char @ X2 @ nil_list_char ) ) )
= ( suc @ ( size_s356637359517785349t_char @ Xs ) ) ) ).
% length_append_singleton
thf(fact_449_length__Suc__conv__rev,axiom,
! [Xs: list_instr,N: nat] :
( ( ( size_size_list_instr @ Xs )
= ( suc @ N ) )
= ( ? [Y: instr,Ys2: list_instr] :
( ( Xs
= ( append_instr @ Ys2 @ ( cons_instr @ Y @ nil_instr ) ) )
& ( ( size_size_list_instr @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_450_length__Suc__conv__rev,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y: int,Ys2: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ Y @ nil_int ) ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_451_length__Suc__conv__rev,axiom,
! [Xs: list_list_char,N: nat] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( suc @ N ) )
= ( ? [Y: list_char,Ys2: list_list_char] :
( ( Xs
= ( append_list_char @ Ys2 @ ( cons_list_char @ Y @ nil_list_char ) ) )
& ( ( size_s356637359517785349t_char @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_452_rotate1_Osimps_I1_J,axiom,
( ( rotate1_instr @ nil_instr )
= nil_instr ) ).
% rotate1.simps(1)
thf(fact_453_rotate1_Osimps_I1_J,axiom,
( ( rotate1_list_char @ nil_list_char )
= nil_list_char ) ).
% rotate1.simps(1)
thf(fact_454_removeAll_Osimps_I2_J,axiom,
! [X2: int,Y2: int,Xs: list_int] :
( ( ( X2 = Y2 )
=> ( ( removeAll_int @ X2 @ ( cons_int @ Y2 @ Xs ) )
= ( removeAll_int @ X2 @ Xs ) ) )
& ( ( X2 != Y2 )
=> ( ( removeAll_int @ X2 @ ( cons_int @ Y2 @ Xs ) )
= ( cons_int @ Y2 @ ( removeAll_int @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_455_distinct__removeAll,axiom,
! [Xs: list_list_char,X2: list_char] :
( ( distinct_list_char @ Xs )
=> ( distinct_list_char @ ( removeAll_list_char @ X2 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_456_distinct__removeAll,axiom,
! [Xs: list_list_list_char,X2: list_list_char] :
( ( distin92326681159624120t_char @ Xs )
=> ( distin92326681159624120t_char @ ( remove3141867459347406693t_char @ X2 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_457_distinct__removeAll,axiom,
! [Xs: list_char,X2: char] :
( ( distinct_char @ Xs )
=> ( distinct_char @ ( removeAll_char @ X2 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_458_distinct__removeAll,axiom,
! [Xs: list_P8512142354091196201t_char,X2: produc6859378861088059299t_char] :
( ( distin3640772593324101788t_char @ Xs )
=> ( distin3640772593324101788t_char @ ( remove442673217080424367t_char @ X2 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_459_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_instr @ N @ nil_instr )
= N ) ).
% gen_length_code(1)
thf(fact_460_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_list_char @ N @ nil_list_char )
= N ) ).
% gen_length_code(1)
thf(fact_461_member__rec_I2_J,axiom,
! [Y2: instr] :
~ ( member_instr @ nil_instr @ Y2 ) ).
% member_rec(2)
thf(fact_462_member__rec_I2_J,axiom,
! [Y2: list_char] :
~ ( member_list_char @ nil_list_char @ Y2 ) ).
% member_rec(2)
thf(fact_463_split__list,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ? [Ys3: list_list_list_char,Zs2: list_list_list_char] :
( Xs
= ( append3251196042929938198t_char @ Ys3 @ ( cons_list_list_char @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_464_split__list,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ? [Ys3: list_list_char,Zs2: list_list_char] :
( Xs
= ( append_list_char @ Ys3 @ ( cons_list_char @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_465_split__list,axiom,
! [X2: int,Xs: list_int] :
( ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
=> ? [Ys3: list_int,Zs2: list_int] :
( Xs
= ( append_int @ Ys3 @ ( cons_int @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_466_split__list__last,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ? [Ys3: list_list_list_char,Zs2: list_list_list_char] :
( ( Xs
= ( append3251196042929938198t_char @ Ys3 @ ( cons_list_list_char @ X2 @ Zs2 ) ) )
& ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_467_split__list__last,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ? [Ys3: list_list_char,Zs2: list_list_char] :
( ( Xs
= ( append_list_char @ Ys3 @ ( cons_list_char @ X2 @ Zs2 ) ) )
& ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_468_split__list__last,axiom,
! [X2: int,Xs: list_int] :
( ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
=> ? [Ys3: list_int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys3 @ ( cons_int @ X2 @ Zs2 ) ) )
& ~ ( member_int2 @ X2 @ ( set_int2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_469_split__list__prop,axiom,
! [Xs: list_list_list_char,P2: list_list_char > $o] :
( ? [X4: list_list_char] :
( ( member4762036990631661320t_char @ X4 @ ( set_list_list_char2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys3: list_list_list_char,X: list_list_char] :
( ? [Zs2: list_list_list_char] :
( Xs
= ( append3251196042929938198t_char @ Ys3 @ ( cons_list_list_char @ X @ Zs2 ) ) )
& ( P2 @ X ) ) ) ).
% split_list_prop
thf(fact_470_split__list__prop,axiom,
! [Xs: list_list_char,P2: list_char > $o] :
( ? [X4: list_char] :
( ( member_list_char2 @ X4 @ ( set_list_char2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys3: list_list_char,X: list_char] :
( ? [Zs2: list_list_char] :
( Xs
= ( append_list_char @ Ys3 @ ( cons_list_char @ X @ Zs2 ) ) )
& ( P2 @ X ) ) ) ).
% split_list_prop
thf(fact_471_split__list__prop,axiom,
! [Xs: list_int,P2: int > $o] :
( ? [X4: int] :
( ( member_int2 @ X4 @ ( set_int2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys3: list_int,X: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys3 @ ( cons_int @ X @ Zs2 ) ) )
& ( P2 @ X ) ) ) ).
% split_list_prop
thf(fact_472_split__list__first,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ? [Ys3: list_list_list_char,Zs2: list_list_list_char] :
( ( Xs
= ( append3251196042929938198t_char @ Ys3 @ ( cons_list_list_char @ X2 @ Zs2 ) ) )
& ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_473_split__list__first,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ? [Ys3: list_list_char,Zs2: list_list_char] :
( ( Xs
= ( append_list_char @ Ys3 @ ( cons_list_char @ X2 @ Zs2 ) ) )
& ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_474_split__list__first,axiom,
! [X2: int,Xs: list_int] :
( ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
=> ? [Ys3: list_int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys3 @ ( cons_int @ X2 @ Zs2 ) ) )
& ~ ( member_int2 @ X2 @ ( set_int2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_475_split__list__propE,axiom,
! [Xs: list_list_list_char,P2: list_list_char > $o] :
( ? [X4: list_list_char] :
( ( member4762036990631661320t_char @ X4 @ ( set_list_list_char2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys3: list_list_list_char,X: list_list_char] :
( ? [Zs2: list_list_list_char] :
( Xs
= ( append3251196042929938198t_char @ Ys3 @ ( cons_list_list_char @ X @ Zs2 ) ) )
=> ~ ( P2 @ X ) ) ) ).
% split_list_propE
thf(fact_476_split__list__propE,axiom,
! [Xs: list_list_char,P2: list_char > $o] :
( ? [X4: list_char] :
( ( member_list_char2 @ X4 @ ( set_list_char2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys3: list_list_char,X: list_char] :
( ? [Zs2: list_list_char] :
( Xs
= ( append_list_char @ Ys3 @ ( cons_list_char @ X @ Zs2 ) ) )
=> ~ ( P2 @ X ) ) ) ).
% split_list_propE
thf(fact_477_split__list__propE,axiom,
! [Xs: list_int,P2: int > $o] :
( ? [X4: int] :
( ( member_int2 @ X4 @ ( set_int2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys3: list_int,X: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys3 @ ( cons_int @ X @ Zs2 ) ) )
=> ~ ( P2 @ X ) ) ) ).
% split_list_propE
thf(fact_478_append__Cons__eq__iff,axiom,
! [X2: list_list_char,Xs: list_list_list_char,Ys: list_list_list_char,Xs5: list_list_list_char,Ys6: list_list_list_char] :
( ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ( ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Ys ) )
=> ( ( ( append3251196042929938198t_char @ Xs @ ( cons_list_list_char @ X2 @ Ys ) )
= ( append3251196042929938198t_char @ Xs5 @ ( cons_list_list_char @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_479_append__Cons__eq__iff,axiom,
! [X2: list_char,Xs: list_list_char,Ys: list_list_char,Xs5: list_list_char,Ys6: list_list_char] :
( ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ( ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Ys ) )
=> ( ( ( append_list_char @ Xs @ ( cons_list_char @ X2 @ Ys ) )
= ( append_list_char @ Xs5 @ ( cons_list_char @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_480_append__Cons__eq__iff,axiom,
! [X2: int,Xs: list_int,Ys: list_int,Xs5: list_int,Ys6: list_int] :
( ~ ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
=> ( ~ ( member_int2 @ X2 @ ( set_int2 @ Ys ) )
=> ( ( ( append_int @ Xs @ ( cons_int @ X2 @ Ys ) )
= ( append_int @ Xs5 @ ( cons_int @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_481_in__set__conv__decomp,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
= ( ? [Ys2: list_list_list_char,Zs3: list_list_list_char] :
( Xs
= ( append3251196042929938198t_char @ Ys2 @ ( cons_list_list_char @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_482_in__set__conv__decomp,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
= ( ? [Ys2: list_list_char,Zs3: list_list_char] :
( Xs
= ( append_list_char @ Ys2 @ ( cons_list_char @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_483_in__set__conv__decomp,axiom,
! [X2: int,Xs: list_int] :
( ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
= ( ? [Ys2: list_int,Zs3: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_484_split__list__last__prop,axiom,
! [Xs: list_list_list_char,P2: list_list_char > $o] :
( ? [X4: list_list_char] :
( ( member4762036990631661320t_char @ X4 @ ( set_list_list_char2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys3: list_list_list_char,X: list_list_char,Zs2: list_list_list_char] :
( ( Xs
= ( append3251196042929938198t_char @ Ys3 @ ( cons_list_list_char @ X @ Zs2 ) ) )
& ( P2 @ X )
& ! [Xa: list_list_char] :
( ( member4762036990631661320t_char @ Xa @ ( set_list_list_char2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_485_split__list__last__prop,axiom,
! [Xs: list_list_char,P2: list_char > $o] :
( ? [X4: list_char] :
( ( member_list_char2 @ X4 @ ( set_list_char2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys3: list_list_char,X: list_char,Zs2: list_list_char] :
( ( Xs
= ( append_list_char @ Ys3 @ ( cons_list_char @ X @ Zs2 ) ) )
& ( P2 @ X )
& ! [Xa: list_char] :
( ( member_list_char2 @ Xa @ ( set_list_char2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_486_split__list__last__prop,axiom,
! [Xs: list_int,P2: int > $o] :
( ? [X4: int] :
( ( member_int2 @ X4 @ ( set_int2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys3: list_int,X: int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys3 @ ( cons_int @ X @ Zs2 ) ) )
& ( P2 @ X )
& ! [Xa: int] :
( ( member_int2 @ Xa @ ( set_int2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_487_split__list__first__prop,axiom,
! [Xs: list_list_list_char,P2: list_list_char > $o] :
( ? [X4: list_list_char] :
( ( member4762036990631661320t_char @ X4 @ ( set_list_list_char2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys3: list_list_list_char,X: list_list_char] :
( ? [Zs2: list_list_list_char] :
( Xs
= ( append3251196042929938198t_char @ Ys3 @ ( cons_list_list_char @ X @ Zs2 ) ) )
& ( P2 @ X )
& ! [Xa: list_list_char] :
( ( member4762036990631661320t_char @ Xa @ ( set_list_list_char2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_488_split__list__first__prop,axiom,
! [Xs: list_list_char,P2: list_char > $o] :
( ? [X4: list_char] :
( ( member_list_char2 @ X4 @ ( set_list_char2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys3: list_list_char,X: list_char] :
( ? [Zs2: list_list_char] :
( Xs
= ( append_list_char @ Ys3 @ ( cons_list_char @ X @ Zs2 ) ) )
& ( P2 @ X )
& ! [Xa: list_char] :
( ( member_list_char2 @ Xa @ ( set_list_char2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_489_split__list__first__prop,axiom,
! [Xs: list_int,P2: int > $o] :
( ? [X4: int] :
( ( member_int2 @ X4 @ ( set_int2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys3: list_int,X: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys3 @ ( cons_int @ X @ Zs2 ) ) )
& ( P2 @ X )
& ! [Xa: int] :
( ( member_int2 @ Xa @ ( set_int2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_490_split__list__last__propE,axiom,
! [Xs: list_list_list_char,P2: list_list_char > $o] :
( ? [X4: list_list_char] :
( ( member4762036990631661320t_char @ X4 @ ( set_list_list_char2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys3: list_list_list_char,X: list_list_char,Zs2: list_list_list_char] :
( ( Xs
= ( append3251196042929938198t_char @ Ys3 @ ( cons_list_list_char @ X @ Zs2 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa: list_list_char] :
( ( member4762036990631661320t_char @ Xa @ ( set_list_list_char2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_491_split__list__last__propE,axiom,
! [Xs: list_list_char,P2: list_char > $o] :
( ? [X4: list_char] :
( ( member_list_char2 @ X4 @ ( set_list_char2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys3: list_list_char,X: list_char,Zs2: list_list_char] :
( ( Xs
= ( append_list_char @ Ys3 @ ( cons_list_char @ X @ Zs2 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa: list_char] :
( ( member_list_char2 @ Xa @ ( set_list_char2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_492_split__list__last__propE,axiom,
! [Xs: list_int,P2: int > $o] :
( ? [X4: int] :
( ( member_int2 @ X4 @ ( set_int2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys3: list_int,X: int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys3 @ ( cons_int @ X @ Zs2 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa: int] :
( ( member_int2 @ Xa @ ( set_int2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_493_split__list__first__propE,axiom,
! [Xs: list_list_list_char,P2: list_list_char > $o] :
( ? [X4: list_list_char] :
( ( member4762036990631661320t_char @ X4 @ ( set_list_list_char2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys3: list_list_list_char,X: list_list_char] :
( ? [Zs2: list_list_list_char] :
( Xs
= ( append3251196042929938198t_char @ Ys3 @ ( cons_list_list_char @ X @ Zs2 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa: list_list_char] :
( ( member4762036990631661320t_char @ Xa @ ( set_list_list_char2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_494_split__list__first__propE,axiom,
! [Xs: list_list_char,P2: list_char > $o] :
( ? [X4: list_char] :
( ( member_list_char2 @ X4 @ ( set_list_char2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys3: list_list_char,X: list_char] :
( ? [Zs2: list_list_char] :
( Xs
= ( append_list_char @ Ys3 @ ( cons_list_char @ X @ Zs2 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa: list_char] :
( ( member_list_char2 @ Xa @ ( set_list_char2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_495_split__list__first__propE,axiom,
! [Xs: list_int,P2: int > $o] :
( ? [X4: int] :
( ( member_int2 @ X4 @ ( set_int2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys3: list_int,X: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys3 @ ( cons_int @ X @ Zs2 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa: int] :
( ( member_int2 @ Xa @ ( set_int2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_496_in__set__conv__decomp__last,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
= ( ? [Ys2: list_list_list_char,Zs3: list_list_list_char] :
( ( Xs
= ( append3251196042929938198t_char @ Ys2 @ ( cons_list_list_char @ X2 @ Zs3 ) ) )
& ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_497_in__set__conv__decomp__last,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
= ( ? [Ys2: list_list_char,Zs3: list_list_char] :
( ( Xs
= ( append_list_char @ Ys2 @ ( cons_list_char @ X2 @ Zs3 ) ) )
& ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_498_in__set__conv__decomp__last,axiom,
! [X2: int,Xs: list_int] :
( ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
= ( ? [Ys2: list_int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X2 @ Zs3 ) ) )
& ~ ( member_int2 @ X2 @ ( set_int2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_499_in__set__conv__decomp__first,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
= ( ? [Ys2: list_list_list_char,Zs3: list_list_list_char] :
( ( Xs
= ( append3251196042929938198t_char @ Ys2 @ ( cons_list_list_char @ X2 @ Zs3 ) ) )
& ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_500_in__set__conv__decomp__first,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
= ( ? [Ys2: list_list_char,Zs3: list_list_char] :
( ( Xs
= ( append_list_char @ Ys2 @ ( cons_list_char @ X2 @ Zs3 ) ) )
& ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_501_in__set__conv__decomp__first,axiom,
! [X2: int,Xs: list_int] :
( ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
= ( ? [Ys2: list_int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X2 @ Zs3 ) ) )
& ~ ( member_int2 @ X2 @ ( set_int2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_502_split__list__last__prop__iff,axiom,
! [Xs: list_list_list_char,P2: list_list_char > $o] :
( ( ? [X3: list_list_char] :
( ( member4762036990631661320t_char @ X3 @ ( set_list_list_char2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys2: list_list_list_char,X3: list_list_char,Zs3: list_list_list_char] :
( ( Xs
= ( append3251196042929938198t_char @ Ys2 @ ( cons_list_list_char @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Y: list_list_char] :
( ( member4762036990631661320t_char @ Y @ ( set_list_list_char2 @ Zs3 ) )
=> ~ ( P2 @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_503_split__list__last__prop__iff,axiom,
! [Xs: list_list_char,P2: list_char > $o] :
( ( ? [X3: list_char] :
( ( member_list_char2 @ X3 @ ( set_list_char2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys2: list_list_char,X3: list_char,Zs3: list_list_char] :
( ( Xs
= ( append_list_char @ Ys2 @ ( cons_list_char @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Y: list_char] :
( ( member_list_char2 @ Y @ ( set_list_char2 @ Zs3 ) )
=> ~ ( P2 @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_504_split__list__last__prop__iff,axiom,
! [Xs: list_int,P2: int > $o] :
( ( ? [X3: int] :
( ( member_int2 @ X3 @ ( set_int2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys2: list_int,X3: int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Y: int] :
( ( member_int2 @ Y @ ( set_int2 @ Zs3 ) )
=> ~ ( P2 @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_505_split__list__first__prop__iff,axiom,
! [Xs: list_list_list_char,P2: list_list_char > $o] :
( ( ? [X3: list_list_char] :
( ( member4762036990631661320t_char @ X3 @ ( set_list_list_char2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys2: list_list_list_char,X3: list_list_char] :
( ? [Zs3: list_list_list_char] :
( Xs
= ( append3251196042929938198t_char @ Ys2 @ ( cons_list_list_char @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Y: list_list_char] :
( ( member4762036990631661320t_char @ Y @ ( set_list_list_char2 @ Ys2 ) )
=> ~ ( P2 @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_506_split__list__first__prop__iff,axiom,
! [Xs: list_list_char,P2: list_char > $o] :
( ( ? [X3: list_char] :
( ( member_list_char2 @ X3 @ ( set_list_char2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys2: list_list_char,X3: list_char] :
( ? [Zs3: list_list_char] :
( Xs
= ( append_list_char @ Ys2 @ ( cons_list_char @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Y: list_char] :
( ( member_list_char2 @ Y @ ( set_list_char2 @ Ys2 ) )
=> ~ ( P2 @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_507_split__list__first__prop__iff,axiom,
! [Xs: list_int,P2: int > $o] :
( ( ? [X3: int] :
( ( member_int2 @ X3 @ ( set_int2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys2: list_int,X3: int] :
( ? [Zs3: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Y: int] :
( ( member_int2 @ Y @ ( set_int2 @ Ys2 ) )
=> ~ ( P2 @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_508_impossible__Cons,axiom,
! [Xs: list_int,Ys: list_int,X2: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
=> ( Xs
!= ( cons_int @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_509_impossible__Cons,axiom,
! [Xs: list_list_char,Ys: list_list_char,X2: list_char] :
( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ ( size_s356637359517785349t_char @ Ys ) )
=> ( Xs
!= ( cons_list_char @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_510_list__induct3,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr,P2: list_instr > list_instr > list_instr > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( ( size_size_list_instr @ Ys )
= ( size_size_list_instr @ Zs ) )
=> ( ( P2 @ nil_instr @ nil_instr @ nil_instr )
=> ( ! [X: instr,Xs3: list_instr,Y3: instr,Ys3: list_instr,Z4: instr,Zs2: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_instr @ Ys3 ) )
=> ( ( ( size_size_list_instr @ Ys3 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) @ ( cons_instr @ Y3 @ Ys3 ) @ ( cons_instr @ Z4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_511_list__induct3,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_int,P2: list_instr > list_instr > list_int > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( ( size_size_list_instr @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P2 @ nil_instr @ nil_instr @ nil_int )
=> ( ! [X: instr,Xs3: list_instr,Y3: instr,Ys3: list_instr,Z4: int,Zs2: list_int] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_instr @ Ys3 ) )
=> ( ( ( size_size_list_instr @ Ys3 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) @ ( cons_instr @ Y3 @ Ys3 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_512_list__induct3,axiom,
! [Xs: list_instr,Ys: list_int,Zs: list_instr,P2: list_instr > list_int > list_instr > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_instr @ Zs ) )
=> ( ( P2 @ nil_instr @ nil_int @ nil_instr )
=> ( ! [X: instr,Xs3: list_instr,Y3: int,Ys3: list_int,Z4: instr,Zs2: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
=> ( ( ( size_size_list_int @ Ys3 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_instr @ Z4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_513_list__induct3,axiom,
! [Xs: list_instr,Ys: list_int,Zs: list_int,P2: list_instr > list_int > list_int > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P2 @ nil_instr @ nil_int @ nil_int )
=> ( ! [X: instr,Xs3: list_instr,Y3: int,Ys3: list_int,Z4: int,Zs2: list_int] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
=> ( ( ( size_size_list_int @ Ys3 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_514_list__induct3,axiom,
! [Xs: list_int,Ys: list_instr,Zs: list_instr,P2: list_int > list_instr > list_instr > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( ( size_size_list_instr @ Ys )
= ( size_size_list_instr @ Zs ) )
=> ( ( P2 @ nil_int @ nil_instr @ nil_instr )
=> ( ! [X: int,Xs3: list_int,Y3: instr,Ys3: list_instr,Z4: instr,Zs2: list_instr] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_instr @ Ys3 ) )
=> ( ( ( size_size_list_instr @ Ys3 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_instr @ Y3 @ Ys3 ) @ ( cons_instr @ Z4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_515_list__induct3,axiom,
! [Xs: list_int,Ys: list_instr,Zs: list_int,P2: list_int > list_instr > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( ( size_size_list_instr @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P2 @ nil_int @ nil_instr @ nil_int )
=> ( ! [X: int,Xs3: list_int,Y3: instr,Ys3: list_instr,Z4: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_instr @ Ys3 ) )
=> ( ( ( size_size_list_instr @ Ys3 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_instr @ Y3 @ Ys3 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_516_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_instr,P2: list_int > list_int > list_instr > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_instr @ Zs ) )
=> ( ( P2 @ nil_int @ nil_int @ nil_instr )
=> ( ! [X: int,Xs3: list_int,Y3: int,Ys3: list_int,Z4: instr,Zs2: list_instr] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
=> ( ( ( size_size_list_int @ Ys3 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_instr @ Z4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_517_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,P2: list_int > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P2 @ nil_int @ nil_int @ nil_int )
=> ( ! [X: int,Xs3: list_int,Y3: int,Ys3: list_int,Z4: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
=> ( ( ( size_size_list_int @ Ys3 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_518_list__induct3,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_list_char,P2: list_instr > list_instr > list_list_char > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( ( size_size_list_instr @ Ys )
= ( size_s356637359517785349t_char @ Zs ) )
=> ( ( P2 @ nil_instr @ nil_instr @ nil_list_char )
=> ( ! [X: instr,Xs3: list_instr,Y3: instr,Ys3: list_instr,Z4: list_char,Zs2: list_list_char] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_instr @ Ys3 ) )
=> ( ( ( size_size_list_instr @ Ys3 )
= ( size_s356637359517785349t_char @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) @ ( cons_instr @ Y3 @ Ys3 ) @ ( cons_list_char @ Z4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_519_list__induct3,axiom,
! [Xs: list_instr,Ys: list_int,Zs: list_list_char,P2: list_instr > list_int > list_list_char > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_s356637359517785349t_char @ Zs ) )
=> ( ( P2 @ nil_instr @ nil_int @ nil_list_char )
=> ( ! [X: instr,Xs3: list_instr,Y3: int,Ys3: list_int,Z4: list_char,Zs2: list_list_char] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
=> ( ( ( size_size_list_int @ Ys3 )
= ( size_s356637359517785349t_char @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_list_char @ Z4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_520_list__induct2,axiom,
! [Xs: list_instr,Ys: list_instr,P2: list_instr > list_instr > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( P2 @ nil_instr @ nil_instr )
=> ( ! [X: instr,Xs3: list_instr,Y3: instr,Ys3: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_instr @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) @ ( cons_instr @ Y3 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_521_list__induct2,axiom,
! [Xs: list_instr,Ys: list_int,P2: list_instr > list_int > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P2 @ nil_instr @ nil_int )
=> ( ! [X: instr,Xs3: list_instr,Y3: int,Ys3: list_int] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_522_list__induct2,axiom,
! [Xs: list_int,Ys: list_instr,P2: list_int > list_instr > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( P2 @ nil_int @ nil_instr )
=> ( ! [X: int,Xs3: list_int,Y3: instr,Ys3: list_instr] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_instr @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_instr @ Y3 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_523_list__induct2,axiom,
! [Xs: list_int,Ys: list_int,P2: list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P2 @ nil_int @ nil_int )
=> ( ! [X: int,Xs3: list_int,Y3: int,Ys3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_524_list__induct2,axiom,
! [Xs: list_instr,Ys: list_list_char,P2: list_instr > list_list_char > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ( ( P2 @ nil_instr @ nil_list_char )
=> ( ! [X: instr,Xs3: list_instr,Y3: list_char,Ys3: list_list_char] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_s356637359517785349t_char @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_instr @ X @ Xs3 ) @ ( cons_list_char @ Y3 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_525_list__induct2,axiom,
! [Xs: list_int,Ys: list_list_char,P2: list_int > list_list_char > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ( ( P2 @ nil_int @ nil_list_char )
=> ( ! [X: int,Xs3: list_int,Y3: list_char,Ys3: list_list_char] :
( ( ( size_size_list_int @ Xs3 )
= ( size_s356637359517785349t_char @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_int @ X @ Xs3 ) @ ( cons_list_char @ Y3 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_526_list__induct2,axiom,
! [Xs: list_list_char,Ys: list_instr,P2: list_list_char > list_instr > $o] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( P2 @ nil_list_char @ nil_instr )
=> ( ! [X: list_char,Xs3: list_list_char,Y3: instr,Ys3: list_instr] :
( ( ( size_s356637359517785349t_char @ Xs3 )
= ( size_size_list_instr @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_list_char @ X @ Xs3 ) @ ( cons_instr @ Y3 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_527_list__induct2,axiom,
! [Xs: list_list_char,Ys: list_int,P2: list_list_char > list_int > $o] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P2 @ nil_list_char @ nil_int )
=> ( ! [X: list_char,Xs3: list_list_char,Y3: int,Ys3: list_int] :
( ( ( size_s356637359517785349t_char @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_list_char @ X @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_528_list__induct2,axiom,
! [Xs: list_list_char,Ys: list_list_char,P2: list_list_char > list_list_char > $o] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ( ( P2 @ nil_list_char @ nil_list_char )
=> ( ! [X: list_char,Xs3: list_list_char,Y3: list_char,Ys3: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs3 )
= ( size_s356637359517785349t_char @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_list_char @ X @ Xs3 ) @ ( cons_list_char @ Y3 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_529_zip__eq__ConsE,axiom,
! [Xs: list_int,Ys: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
( ( ( zip_int_int @ Xs @ Ys )
= ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
=> ~ ! [X: int,Xs4: list_int] :
( ( Xs
= ( cons_int @ X @ Xs4 ) )
=> ! [Y3: int,Ys5: list_int] :
( ( Ys
= ( cons_int @ Y3 @ Ys5 ) )
=> ( ( Xy
= ( product_Pair_int_int @ X @ Y3 ) )
=> ( Xys
!= ( zip_int_int @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_530_zip__eq__ConsE,axiom,
! [Xs: list_list_list_char,Ys: list_P98921565119350719nt_int,Xy: produc4189061985984568957nt_int,Xys: list_P7813408393781968525nt_int] :
( ( ( zip_li4478691307266062991nt_int @ Xs @ Ys )
= ( cons_P3979909588204300359nt_int @ Xy @ Xys ) )
=> ~ ! [X: list_list_char,Xs4: list_list_list_char] :
( ( Xs
= ( cons_list_list_char @ X @ Xs4 ) )
=> ! [Y3: produc4435102495419491129nt_int,Ys5: list_P98921565119350719nt_int] :
( ( Ys
= ( cons_P7721559881313335791nt_int @ Y3 @ Ys5 ) )
=> ( ( Xy
= ( produc3965054194175396271nt_int @ X @ Y3 ) )
=> ( Xys
!= ( zip_li4478691307266062991nt_int @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_531_zip__eq__ConsE,axiom,
! [Xs: list_list_char_int,Ys: list_int,Xy: produc4435102495419491129nt_int,Xys: list_P98921565119350719nt_int] :
( ( ( zip_li343459552841501841nt_int @ Xs @ Ys )
= ( cons_P7721559881313335791nt_int @ Xy @ Xys ) )
=> ~ ! [X: list_char > int,Xs4: list_list_char_int] :
( ( Xs
= ( cons_list_char_int @ X @ Xs4 ) )
=> ! [Y3: int,Ys5: list_int] :
( ( Ys
= ( cons_int @ Y3 @ Ys5 ) )
=> ( ( Xy
= ( produc5790713362662368625nt_int @ X @ Y3 ) )
=> ( Xys
!= ( zip_li343459552841501841nt_int @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_532_zip__same,axiom,
! [A: list_list_char,B: list_list_char,Xs: list_list_list_char] :
( ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ A @ B ) @ ( set_Pr565814602432443608t_char @ ( zip_li6982233095522658971t_char @ Xs @ Xs ) ) )
= ( ( member4762036990631661320t_char @ A @ ( set_list_list_char2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_533_zip__same,axiom,
! [A: list_char,B: list_char,Xs: list_list_char] :
( ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ A @ B ) @ ( set_Pr2195431618410804152t_char @ ( zip_li5382857322596760955t_char @ Xs @ Xs ) ) )
= ( ( member_list_char2 @ A @ ( set_list_char2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_534_in__set__zipE,axiom,
! [X2: list_list_char,Y2: list_list_char,Xs: list_list_list_char,Ys: list_list_list_char] :
( ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ X2 @ Y2 ) @ ( set_Pr565814602432443608t_char @ ( zip_li6982233095522658971t_char @ Xs @ Ys ) ) )
=> ~ ( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ~ ( member4762036990631661320t_char @ Y2 @ ( set_list_list_char2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_535_in__set__zipE,axiom,
! [X2: list_list_char,Y2: list_char,Xs: list_list_list_char,Ys: list_list_char] :
( ( member2188552656405064028t_char @ ( produc9067971387720190315t_char @ X2 @ Y2 ) @ ( set_Pr1396349415940334536t_char @ ( zip_li5988017610170933387t_char @ Xs @ Ys ) ) )
=> ~ ( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ~ ( member_list_char2 @ Y2 @ ( set_list_char2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_536_in__set__zipE,axiom,
! [X2: list_char,Y2: list_list_char,Xs: list_list_char,Ys: list_list_list_char] :
( ( member1112677564943258204t_char @ ( produc2694321116511041643t_char @ X2 @ Y2 ) @ ( set_Pr320474324478528712t_char @ ( zip_li8837739375816560523t_char @ Xs @ Ys ) ) )
=> ~ ( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ~ ( member4762036990631661320t_char @ Y2 @ ( set_list_list_char2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_537_in__set__zipE,axiom,
! [X2: list_char,Y2: list_char,Xs: list_list_char,Ys: list_list_char] :
( ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ X2 @ Y2 ) @ ( set_Pr2195431618410804152t_char @ ( zip_li5382857322596760955t_char @ Xs @ Ys ) ) )
=> ~ ( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ~ ( member_list_char2 @ Y2 @ ( set_list_char2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_538_in__set__zipE,axiom,
! [X2: list_list_char,Y2: produc4435102495419491129nt_int,Xs: list_list_list_char,Ys: list_P98921565119350719nt_int] :
( ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X2 @ Y2 ) @ ( set_Pr8107714755537002024nt_int @ ( zip_li4478691307266062991nt_int @ Xs @ Ys ) ) )
=> ~ ( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ~ ( member3599195040465536738nt_int @ Y2 @ ( set_Pr8994527289597011790nt_int @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_539_in__set__zipE,axiom,
! [X2: list_char > int,Y2: int,Xs: list_list_char_int,Ys: list_int] :
( ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y2 ) @ ( set_Pr8994527289597011790nt_int @ ( zip_li343459552841501841nt_int @ Xs @ Ys ) ) )
=> ~ ( ( member_list_char_int @ X2 @ ( set_list_char_int2 @ Xs ) )
=> ~ ( member_int2 @ Y2 @ ( set_int2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_540_set__zip__leftD,axiom,
! [X2: list_list_char,Y2: produc4435102495419491129nt_int,Xs: list_list_list_char,Ys: list_P98921565119350719nt_int] :
( ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X2 @ Y2 ) @ ( set_Pr8107714755537002024nt_int @ ( zip_li4478691307266062991nt_int @ Xs @ Ys ) ) )
=> ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_541_set__zip__leftD,axiom,
! [X2: list_char > int,Y2: int,Xs: list_list_char_int,Ys: list_int] :
( ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y2 ) @ ( set_Pr8994527289597011790nt_int @ ( zip_li343459552841501841nt_int @ Xs @ Ys ) ) )
=> ( member_list_char_int @ X2 @ ( set_list_char_int2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_542_set__zip__rightD,axiom,
! [X2: list_list_char,Y2: produc4435102495419491129nt_int,Xs: list_list_list_char,Ys: list_P98921565119350719nt_int] :
( ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X2 @ Y2 ) @ ( set_Pr8107714755537002024nt_int @ ( zip_li4478691307266062991nt_int @ Xs @ Ys ) ) )
=> ( member3599195040465536738nt_int @ Y2 @ ( set_Pr8994527289597011790nt_int @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_543_set__zip__rightD,axiom,
! [X2: list_char > int,Y2: int,Xs: list_list_char_int,Ys: list_int] :
( ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y2 ) @ ( set_Pr8994527289597011790nt_int @ ( zip_li343459552841501841nt_int @ Xs @ Ys ) ) )
=> ( member_int2 @ Y2 @ ( set_int2 @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_544_remove1__append,axiom,
! [X2: list_list_char,Xs: list_list_list_char,Ys: list_list_list_char] :
( ( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ( ( remove4265589039595904469t_char @ X2 @ ( append3251196042929938198t_char @ Xs @ Ys ) )
= ( append3251196042929938198t_char @ ( remove4265589039595904469t_char @ X2 @ Xs ) @ Ys ) ) )
& ( ~ ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ( ( remove4265589039595904469t_char @ X2 @ ( append3251196042929938198t_char @ Xs @ Ys ) )
= ( append3251196042929938198t_char @ Xs @ ( remove4265589039595904469t_char @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_545_remove1__append,axiom,
! [X2: list_char,Xs: list_list_char,Ys: list_list_char] :
( ( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ( ( remove1_list_char @ X2 @ ( append_list_char @ Xs @ Ys ) )
= ( append_list_char @ ( remove1_list_char @ X2 @ Xs ) @ Ys ) ) )
& ( ~ ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ( ( remove1_list_char @ X2 @ ( append_list_char @ Xs @ Ys ) )
= ( append_list_char @ Xs @ ( remove1_list_char @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_546_rotate__append,axiom,
! [L: list_list_char,Q: list_list_char] :
( ( rotate_list_char @ ( size_s356637359517785349t_char @ L ) @ ( append_list_char @ L @ Q ) )
= ( append_list_char @ Q @ L ) ) ).
% rotate_append
thf(fact_547_distinct__singleton,axiom,
! [X2: list_list_char] : ( distin92326681159624120t_char @ ( cons_list_list_char @ X2 @ nil_list_list_char ) ) ).
% distinct_singleton
thf(fact_548_distinct__singleton,axiom,
! [X2: char] : ( distinct_char @ ( cons_char @ X2 @ nil_char ) ) ).
% distinct_singleton
thf(fact_549_distinct__singleton,axiom,
! [X2: produc6859378861088059299t_char] : ( distin3640772593324101788t_char @ ( cons_P922464210127128153t_char @ X2 @ nil_Pr3678812991771325097t_char ) ) ).
% distinct_singleton
thf(fact_550_distinct__singleton,axiom,
! [X2: instr] : ( distinct_instr @ ( cons_instr @ X2 @ nil_instr ) ) ).
% distinct_singleton
thf(fact_551_distinct__singleton,axiom,
! [X2: list_char] : ( distinct_list_char @ ( cons_list_char @ X2 @ nil_list_char ) ) ).
% distinct_singleton
thf(fact_552_distinct__singleton,axiom,
! [X2: int] : ( distinct_int @ ( cons_int @ X2 @ nil_int ) ) ).
% distinct_singleton
thf(fact_553_not__distinct__conv__prefix,axiom,
! [As: list_char] :
( ( ~ ( distinct_char @ As ) )
= ( ? [Xs2: list_char,Y: char,Ys2: list_char] :
( ( member_char @ Y @ ( set_char2 @ Xs2 ) )
& ( distinct_char @ Xs2 )
& ( As
= ( append_char @ Xs2 @ ( cons_char @ Y @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_554_not__distinct__conv__prefix,axiom,
! [As: list_P8512142354091196201t_char] :
( ( ~ ( distin3640772593324101788t_char @ As ) )
= ( ? [Xs2: list_P8512142354091196201t_char,Y: produc6859378861088059299t_char,Ys2: list_P8512142354091196201t_char] :
( ( member6023471406134104908t_char @ Y @ ( set_Pr2195431618410804152t_char @ Xs2 ) )
& ( distin3640772593324101788t_char @ Xs2 )
& ( As
= ( append655500003500722110t_char @ Xs2 @ ( cons_P922464210127128153t_char @ Y @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_555_not__distinct__conv__prefix,axiom,
! [As: list_list_list_char] :
( ( ~ ( distin92326681159624120t_char @ As ) )
= ( ? [Xs2: list_list_list_char,Y: list_list_char,Ys2: list_list_list_char] :
( ( member4762036990631661320t_char @ Y @ ( set_list_list_char2 @ Xs2 ) )
& ( distin92326681159624120t_char @ Xs2 )
& ( As
= ( append3251196042929938198t_char @ Xs2 @ ( cons_list_list_char @ Y @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_556_not__distinct__conv__prefix,axiom,
! [As: list_list_char] :
( ( ~ ( distinct_list_char @ As ) )
= ( ? [Xs2: list_list_char,Y: list_char,Ys2: list_list_char] :
( ( member_list_char2 @ Y @ ( set_list_char2 @ Xs2 ) )
& ( distinct_list_char @ Xs2 )
& ( As
= ( append_list_char @ Xs2 @ ( cons_list_char @ Y @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_557_not__distinct__conv__prefix,axiom,
! [As: list_int] :
( ( ~ ( distinct_int @ As ) )
= ( ? [Xs2: list_int,Y: int,Ys2: list_int] :
( ( member_int2 @ Y @ ( set_int2 @ Xs2 ) )
& ( distinct_int @ Xs2 )
& ( As
= ( append_int @ Xs2 @ ( cons_int @ Y @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_558_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_559_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_560_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_561_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_562_prod_Oinject,axiom,
! [X1: list_list_char,X23: produc4435102495419491129nt_int,Y1: list_list_char,Y23: produc4435102495419491129nt_int] :
( ( ( produc3965054194175396271nt_int @ X1 @ X23 )
= ( produc3965054194175396271nt_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_563_prod_Oinject,axiom,
! [X1: list_char > int,X23: int,Y1: list_char > int,Y23: int] :
( ( ( produc5790713362662368625nt_int @ X1 @ X23 )
= ( produc5790713362662368625nt_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_564_old_Oprod_Oinject,axiom,
! [A: list_list_char,B: produc4435102495419491129nt_int,A4: list_list_char,B3: produc4435102495419491129nt_int] :
( ( ( produc3965054194175396271nt_int @ A @ B )
= ( produc3965054194175396271nt_int @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_565_old_Oprod_Oinject,axiom,
! [A: list_char > int,B: int,A4: list_char > int,B3: int] :
( ( ( produc5790713362662368625nt_int @ A @ B )
= ( produc5790713362662368625nt_int @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_566_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_567_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_568_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_569_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_570_the__elem__set,axiom,
! [X2: instr] :
( ( the_elem_instr @ ( set_instr2 @ ( cons_instr @ X2 @ nil_instr ) ) )
= X2 ) ).
% the_elem_set
thf(fact_571_the__elem__set,axiom,
! [X2: list_list_char] :
( ( the_el4834769204129890620t_char @ ( set_list_list_char2 @ ( cons_list_list_char @ X2 @ nil_list_list_char ) ) )
= X2 ) ).
% the_elem_set
thf(fact_572_the__elem__set,axiom,
! [X2: list_char] :
( ( the_elem_list_char @ ( set_list_char2 @ ( cons_list_char @ X2 @ nil_list_char ) ) )
= X2 ) ).
% the_elem_set
thf(fact_573_the__elem__set,axiom,
! [X2: int] :
( ( the_elem_int @ ( set_int2 @ ( cons_int @ X2 @ nil_int ) ) )
= X2 ) ).
% the_elem_set
thf(fact_574_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_575_SuccD,axiom,
! [K: instr,Kl2: set_list_instr,Kl: list_instr] :
( ( member_instr2 @ K @ ( bNF_Gr2000454363642481146_instr @ Kl2 @ Kl ) )
=> ( member_list_instr @ ( append_instr @ Kl @ ( cons_instr @ K @ nil_instr ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_576_SuccD,axiom,
! [K: list_char,Kl2: set_list_list_char,Kl: list_list_char] :
( ( member_list_char2 @ K @ ( bNF_Gr8360121328854636294t_char @ Kl2 @ Kl ) )
=> ( member4762036990631661320t_char @ ( append_list_char @ Kl @ ( cons_list_char @ K @ nil_list_char ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_577_SuccD,axiom,
! [K: int,Kl2: set_list_int,Kl: list_int] :
( ( member_int2 @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl2 @ Kl ) )
=> ( member_list_int @ ( append_int @ Kl @ ( cons_int @ K @ nil_int ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_578_subset__code_I1_J,axiom,
! [Xs: list_list_list_char,B4: set_list_list_char] :
( ( ord_le9029460191049200135t_char @ ( set_list_list_char2 @ Xs ) @ B4 )
= ( ! [X3: list_list_char] :
( ( member4762036990631661320t_char @ X3 @ ( set_list_list_char2 @ Xs ) )
=> ( member4762036990631661320t_char @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_579_subset__code_I1_J,axiom,
! [Xs: list_list_char,B4: set_list_char] :
( ( ord_le1207710660481778807t_char @ ( set_list_char2 @ Xs ) @ B4 )
= ( ! [X3: list_char] :
( ( member_list_char2 @ X3 @ ( set_list_char2 @ Xs ) )
=> ( member_list_char2 @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_580_set__subset__Cons,axiom,
! [Xs: list_list_list_char,X2: list_list_char] : ( ord_le9029460191049200135t_char @ ( set_list_list_char2 @ Xs ) @ ( set_list_list_char2 @ ( cons_list_list_char @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_581_set__subset__Cons,axiom,
! [Xs: list_list_char,X2: list_char] : ( ord_le1207710660481778807t_char @ ( set_list_char2 @ Xs ) @ ( set_list_char2 @ ( cons_list_char @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_582_set__subset__Cons,axiom,
! [Xs: list_int,X2: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_583_set__remove1__subset,axiom,
! [X2: list_list_char,Xs: list_list_list_char] : ( ord_le9029460191049200135t_char @ ( set_list_list_char2 @ ( remove4265589039595904469t_char @ X2 @ Xs ) ) @ ( set_list_list_char2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_584_set__remove1__subset,axiom,
! [X2: list_char,Xs: list_list_char] : ( ord_le1207710660481778807t_char @ ( set_list_char2 @ ( remove1_list_char @ X2 @ Xs ) ) @ ( set_list_char2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_585_product_Osimps_I2_J,axiom,
! [X2: list_char,Xs: list_list_char,Ys: list_list_char] :
( ( produc6248063980778377481t_char @ ( cons_list_char @ X2 @ Xs ) @ Ys )
= ( append655500003500722110t_char @ ( map_li5260691853286448188t_char @ ( produc1606739095562851931t_char @ X2 ) @ Ys ) @ ( produc6248063980778377481t_char @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_586_product_Osimps_I2_J,axiom,
! [X2: list_list_char,Xs: list_list_list_char,Ys: list_P98921565119350719nt_int] :
( ( produc1288524852212238721nt_int @ ( cons_list_list_char @ X2 @ Xs ) @ Ys )
= ( append4417635439285053090nt_int @ ( map_Pr6590262102874458482nt_int @ ( produc3965054194175396271nt_int @ X2 ) @ Ys ) @ ( produc1288524852212238721nt_int @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_587_product_Osimps_I2_J,axiom,
! [X2: list_char > int,Xs: list_list_char_int,Ys: list_int] :
( ( produc1208666211023118367nt_int @ ( cons_list_char_int @ X2 @ Xs ) @ Ys )
= ( append7454595674686929748nt_int @ ( map_in6126364964427645639nt_int @ ( produc5790713362662368625nt_int @ X2 ) @ Ys ) @ ( produc1208666211023118367nt_int @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_588_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_589_order__antisym__conv,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_590_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_591_linorder__le__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_592_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_593_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_594_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_595_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_596_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_597_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_598_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_599_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_600_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_601_linorder__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_602_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_603_order__eq__refl,axiom,
! [X2: int,Y2: int] :
( ( X2 = Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_604_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_605_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_606_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_607_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_608_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_609_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_610_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_611_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_612_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_613_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_614_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_615_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_616_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_617_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_618_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_619_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_620_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_621_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_622_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_623_linorder__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: int,B5: int] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_624_order__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_625_order__trans,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_626_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_627_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_628_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_629_order__antisym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_630_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_631_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_632_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_633_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_634_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ( ord_less_eq_nat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_635_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
& ( ord_less_eq_int @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_636_le__cases3,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_637_le__cases3,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ( ord_less_eq_int @ X2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_638_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_639_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_640_prod__induct3,axiom,
! [P2: produc4189061985984568957nt_int > $o,X2: produc4189061985984568957nt_int] :
( ! [A5: list_list_char,B5: list_char > int,C2: int] : ( P2 @ ( produc3965054194175396271nt_int @ A5 @ ( produc5790713362662368625nt_int @ B5 @ C2 ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct3
thf(fact_641_prod__cases3,axiom,
! [Y2: produc4189061985984568957nt_int] :
~ ! [A5: list_list_char,B5: list_char > int,C2: int] :
( Y2
!= ( produc3965054194175396271nt_int @ A5 @ ( produc5790713362662368625nt_int @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_642_Pair__inject,axiom,
! [A: list_list_char,B: produc4435102495419491129nt_int,A4: list_list_char,B3: produc4435102495419491129nt_int] :
( ( ( produc3965054194175396271nt_int @ A @ B )
= ( produc3965054194175396271nt_int @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_643_Pair__inject,axiom,
! [A: list_char > int,B: int,A4: list_char > int,B3: int] :
( ( ( produc5790713362662368625nt_int @ A @ B )
= ( produc5790713362662368625nt_int @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_644_prod__cases,axiom,
! [P2: produc4189061985984568957nt_int > $o,P4: produc4189061985984568957nt_int] :
( ! [A5: list_list_char,B5: produc4435102495419491129nt_int] : ( P2 @ ( produc3965054194175396271nt_int @ A5 @ B5 ) )
=> ( P2 @ P4 ) ) ).
% prod_cases
thf(fact_645_prod__cases,axiom,
! [P2: produc4435102495419491129nt_int > $o,P4: produc4435102495419491129nt_int] :
( ! [A5: list_char > int,B5: int] : ( P2 @ ( produc5790713362662368625nt_int @ A5 @ B5 ) )
=> ( P2 @ P4 ) ) ).
% prod_cases
thf(fact_646_surj__pair,axiom,
! [P4: produc4189061985984568957nt_int] :
? [X: list_list_char,Y3: produc4435102495419491129nt_int] :
( P4
= ( produc3965054194175396271nt_int @ X @ Y3 ) ) ).
% surj_pair
thf(fact_647_surj__pair,axiom,
! [P4: produc4435102495419491129nt_int] :
? [X: list_char > int,Y3: int] :
( P4
= ( produc5790713362662368625nt_int @ X @ Y3 ) ) ).
% surj_pair
thf(fact_648_old_Oprod_Oexhaust,axiom,
! [Y2: produc4189061985984568957nt_int] :
~ ! [A5: list_list_char,B5: produc4435102495419491129nt_int] :
( Y2
!= ( produc3965054194175396271nt_int @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_649_old_Oprod_Oexhaust,axiom,
! [Y2: produc4435102495419491129nt_int] :
~ ! [A5: list_char > int,B5: int] :
( Y2
!= ( produc5790713362662368625nt_int @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_650_Succ__Shift,axiom,
! [Kl2: set_list_int,K: int,Kl: list_int] :
( ( bNF_Gr6350390219475566417cc_int @ ( bNF_Gr1870224194279859149ft_int @ Kl2 @ K ) @ Kl )
= ( bNF_Gr6350390219475566417cc_int @ Kl2 @ ( cons_int @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_651_empty__Shift,axiom,
! [Kl2: set_list_instr,K: instr] :
( ( member_list_instr @ nil_instr @ Kl2 )
=> ( ( member_instr2 @ K @ ( bNF_Gr2000454363642481146_instr @ Kl2 @ nil_instr ) )
=> ( member_list_instr @ nil_instr @ ( bNF_Gr3707929385946037886_instr @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_652_empty__Shift,axiom,
! [Kl2: set_list_list_char,K: list_char] :
( ( member4762036990631661320t_char @ nil_list_char @ Kl2 )
=> ( ( member_list_char2 @ K @ ( bNF_Gr8360121328854636294t_char @ Kl2 @ nil_list_char ) )
=> ( member4762036990631661320t_char @ nil_list_char @ ( bNF_Gr5181636479692657026t_char @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_653_power__commuting__commutes,axiom,
! [X2: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X2 @ Y2 )
= ( times_times_nat @ Y2 @ X2 ) )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ Y2 )
= ( times_times_nat @ Y2 @ ( power_power_nat @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_654_power__commuting__commutes,axiom,
! [X2: int,Y2: int,N: nat] :
( ( ( times_times_int @ X2 @ Y2 )
= ( times_times_int @ Y2 @ X2 ) )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ Y2 )
= ( times_times_int @ Y2 @ ( power_power_int @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_655_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_656_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_657_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_658_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_659_SuccI,axiom,
! [Kl: list_instr,K: instr,Kl2: set_list_instr] :
( ( member_list_instr @ ( append_instr @ Kl @ ( cons_instr @ K @ nil_instr ) ) @ Kl2 )
=> ( member_instr2 @ K @ ( bNF_Gr2000454363642481146_instr @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_660_SuccI,axiom,
! [Kl: list_list_char,K: list_char,Kl2: set_list_list_char] :
( ( member4762036990631661320t_char @ ( append_list_char @ Kl @ ( cons_list_char @ K @ nil_list_char ) ) @ Kl2 )
=> ( member_list_char2 @ K @ ( bNF_Gr8360121328854636294t_char @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_661_SuccI,axiom,
! [Kl: list_int,K: int,Kl2: set_list_int] :
( ( member_list_int @ ( append_int @ Kl @ ( cons_int @ K @ nil_int ) ) @ Kl2 )
=> ( member_int2 @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_662_Cons__in__lex,axiom,
! [X2: int,Xs: list_int,Y2: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys ) ) @ ( lex_int @ R2 ) )
= ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R2 )
& ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) ) )
| ( ( X2 = Y2 )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lex_int @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_663_Cons__in__lex,axiom,
! [X2: list_char,Xs: list_list_char,Y2: list_char,Ys: list_list_char,R2: set_Pr4480099295215233283t_char] :
( ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ ( cons_list_char @ X2 @ Xs ) @ ( cons_list_char @ Y2 @ Ys ) ) @ ( lex_list_char @ R2 ) )
= ( ( ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ X2 @ Y2 ) @ R2 )
& ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) ) )
| ( ( X2 = Y2 )
& ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ Xs @ Ys ) @ ( lex_list_char @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_664_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_char @ N @ nil_char )
= ( cons_list_char @ nil_char @ nil_list_char ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_char @ N @ nil_char )
= nil_list_char ) ) ) ).
% n_lists_Nil
thf(fact_665_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_instr @ N @ nil_instr )
= ( cons_list_instr @ nil_instr @ nil_list_instr ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_instr @ N @ nil_instr )
= nil_list_instr ) ) ) ).
% n_lists_Nil
thf(fact_666_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_list_char @ N @ nil_list_char )
= ( cons_list_list_char @ nil_list_char @ nil_list_list_char ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_list_char @ N @ nil_list_char )
= nil_list_list_char ) ) ) ).
% n_lists_Nil
thf(fact_667_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_instr,X2: instr,Ys: list_instr,Y2: instr,R2: set_Pr3711236633289739975_instr] :
( ( member7594433970877843344_instr @ ( produc7750586281064920791_instr @ ( append_instr @ Xs @ ( cons_instr @ X2 @ nil_instr ) ) @ ( append_instr @ Ys @ ( cons_instr @ Y2 @ nil_instr ) ) ) @ ( listrel1_instr @ R2 ) )
= ( ( ( member7594433970877843344_instr @ ( produc7750586281064920791_instr @ Xs @ Ys ) @ ( listrel1_instr @ R2 ) )
& ( X2 = Y2 ) )
| ( ( Xs = Ys )
& ( member8320768277060436240_instr @ ( produc7105755196911784279_instr @ X2 @ Y2 ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_668_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_list_char,X2: list_char,Ys: list_list_char,Y2: list_char,R2: set_Pr4480099295215233283t_char] :
( ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ ( append_list_char @ Xs @ ( cons_list_char @ X2 @ nil_list_char ) ) @ ( append_list_char @ Ys @ ( cons_list_char @ Y2 @ nil_list_char ) ) ) @ ( listrel1_list_char @ R2 ) )
= ( ( ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ Xs @ Ys ) @ ( listrel1_list_char @ R2 ) )
& ( X2 = Y2 ) )
| ( ( Xs = Ys )
& ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ X2 @ Y2 ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_669_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_int,X2: int,Ys: list_int,Y2: int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ ( cons_int @ X2 @ nil_int ) ) @ ( append_int @ Ys @ ( cons_int @ Y2 @ nil_int ) ) ) @ ( listrel1_int @ R2 ) )
= ( ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
& ( X2 = Y2 ) )
| ( ( Xs = Ys )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_670_n__lists_Osimps_I1_J,axiom,
! [Xs: list_char] :
( ( n_lists_char @ zero_zero_nat @ Xs )
= ( cons_list_char @ nil_char @ nil_list_char ) ) ).
% n_lists.simps(1)
thf(fact_671_n__lists_Osimps_I1_J,axiom,
! [Xs: list_instr] :
( ( n_lists_instr @ zero_zero_nat @ Xs )
= ( cons_list_instr @ nil_instr @ nil_list_instr ) ) ).
% n_lists.simps(1)
thf(fact_672_n__lists_Osimps_I1_J,axiom,
! [Xs: list_list_char] :
( ( n_lists_list_char @ zero_zero_nat @ Xs )
= ( cons_list_list_char @ nil_list_char @ nil_list_list_char ) ) ).
% n_lists.simps(1)
thf(fact_673_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_674_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_675_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_676_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_677_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_678_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_679_bind__simps_I1_J,axiom,
! [F: instr > list_instr] :
( ( bind_instr_instr @ nil_instr @ F )
= nil_instr ) ).
% bind_simps(1)
thf(fact_680_bind__simps_I1_J,axiom,
! [F: instr > list_list_char] :
( ( bind_instr_list_char @ nil_instr @ F )
= nil_list_char ) ).
% bind_simps(1)
thf(fact_681_bind__simps_I1_J,axiom,
! [F: list_char > list_instr] :
( ( bind_list_char_instr @ nil_list_char @ F )
= nil_instr ) ).
% bind_simps(1)
thf(fact_682_bind__simps_I1_J,axiom,
! [F: list_char > list_list_char] :
( ( bind_l8408037539174948939t_char @ nil_list_char @ F )
= nil_list_char ) ).
% bind_simps(1)
thf(fact_683_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_684_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_685_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_686_length__0__conv,axiom,
! [Xs: list_instr] :
( ( ( size_size_list_instr @ Xs )
= zero_zero_nat )
= ( Xs = nil_instr ) ) ).
% length_0_conv
thf(fact_687_length__0__conv,axiom,
! [Xs: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_char ) ) ).
% length_0_conv
thf(fact_688_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_689_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_690_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M: nat] :
( ( ( power_power_nat @ X2 @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X2
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_691_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_692_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_693_Cons__listrel1__Cons,axiom,
! [X2: int,Xs: list_int,Y2: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys ) ) @ ( listrel1_int @ R2 ) )
= ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R2 )
& ( Xs = Ys ) )
| ( ( X2 = Y2 )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_694_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_695_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_696_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_697_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_698_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_699_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_700_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_701_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_702_diff__induct,axiom,
! [P2: nat > nat > $o,M: nat,N: nat] :
( ! [X: nat] : ( P2 @ X @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X: nat,Y3: nat] :
( ( P2 @ X @ Y3 )
=> ( P2 @ ( suc @ X ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_703_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_704_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_705_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_706_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_707_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_708_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_709_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_710_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_711_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_712_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_713_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_714_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_715_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_716_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_717_list_Osize_I3_J,axiom,
( ( size_size_list_instr @ nil_instr )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_718_list_Osize_I3_J,axiom,
( ( size_s356637359517785349t_char @ nil_list_char )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_719_listrel1I2,axiom,
! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int,X2: int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ X2 @ Ys ) ) @ ( listrel1_int @ R2 ) ) ) ).
% listrel1I2
thf(fact_720_not__listrel1__Nil,axiom,
! [Xs: list_instr,R2: set_Pr3711236633289739975_instr] :
~ ( member7594433970877843344_instr @ ( produc7750586281064920791_instr @ Xs @ nil_instr ) @ ( listrel1_instr @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_721_not__listrel1__Nil,axiom,
! [Xs: list_list_char,R2: set_Pr4480099295215233283t_char] :
~ ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ Xs @ nil_list_char ) @ ( listrel1_list_char @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_722_not__Nil__listrel1,axiom,
! [Xs: list_instr,R2: set_Pr3711236633289739975_instr] :
~ ( member7594433970877843344_instr @ ( produc7750586281064920791_instr @ nil_instr @ Xs ) @ ( listrel1_instr @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_723_not__Nil__listrel1,axiom,
! [Xs: list_list_char,R2: set_Pr4480099295215233283t_char] :
~ ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ nil_list_char @ Xs ) @ ( listrel1_list_char @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_724_listrel1__eq__len,axiom,
! [Xs: list_list_char,Ys: list_list_char,R2: set_Pr4480099295215233283t_char] :
( ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ Xs @ Ys ) @ ( listrel1_list_char @ R2 ) )
=> ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_725_Nil2__notin__lex,axiom,
! [Xs: list_instr,R2: set_Pr3711236633289739975_instr] :
~ ( member7594433970877843344_instr @ ( produc7750586281064920791_instr @ Xs @ nil_instr ) @ ( lex_instr @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_726_Nil2__notin__lex,axiom,
! [Xs: list_list_char,R2: set_Pr4480099295215233283t_char] :
~ ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ Xs @ nil_list_char ) @ ( lex_list_char @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_727_Nil__notin__lex,axiom,
! [Ys: list_instr,R2: set_Pr3711236633289739975_instr] :
~ ( member7594433970877843344_instr @ ( produc7750586281064920791_instr @ nil_instr @ Ys ) @ ( lex_instr @ R2 ) ) ).
% Nil_notin_lex
thf(fact_728_Nil__notin__lex,axiom,
! [Ys: list_list_char,R2: set_Pr4480099295215233283t_char] :
~ ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ nil_list_char @ Ys ) @ ( lex_list_char @ R2 ) ) ).
% Nil_notin_lex
thf(fact_729_length__code,axiom,
( size_s356637359517785349t_char
= ( gen_length_list_char @ zero_zero_nat ) ) ).
% length_code
thf(fact_730_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_731_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_732_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_733_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_734_listrel1I1,axiom,
! [X2: int,Y2: int,R2: set_Pr958786334691620121nt_int,Xs: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R2 )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Xs ) ) @ ( listrel1_int @ R2 ) ) ) ).
% listrel1I1
thf(fact_735_Cons__listrel1E1,axiom,
! [X2: int,Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ Ys ) @ ( listrel1_int @ R2 ) )
=> ( ! [Y3: int] :
( ( Ys
= ( cons_int @ Y3 @ Xs ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R2 ) )
=> ~ ! [Zs2: list_int] :
( ( Ys
= ( cons_int @ X2 @ Zs2 ) )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Zs2 ) @ ( listrel1_int @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_736_Cons__listrel1E2,axiom,
! [Xs: list_int,Y2: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y2 @ Ys ) ) @ ( listrel1_int @ R2 ) )
=> ( ! [X: int] :
( ( Xs
= ( cons_int @ X @ Ys ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ R2 ) )
=> ~ ! [Zs2: list_int] :
( ( Xs
= ( cons_int @ Y2 @ Zs2 ) )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Zs2 @ Ys ) @ ( listrel1_int @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_737_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_738_lex__append__rightI,axiom,
! [Xs: list_list_char,Ys: list_list_char,R2: set_Pr4480099295215233283t_char,Vs: list_list_char,Us: list_list_char] :
( ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ Xs @ Ys ) @ ( lex_list_char @ R2 ) )
=> ( ( ( size_s356637359517785349t_char @ Vs )
= ( size_s356637359517785349t_char @ Us ) )
=> ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ ( append_list_char @ Xs @ Us ) @ ( append_list_char @ Ys @ Vs ) ) @ ( lex_list_char @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_739_listrel1E,axiom,
! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
=> ~ ! [X: int,Y3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ R2 )
=> ! [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ ( cons_int @ X @ Vs2 ) ) )
=> ( Ys
!= ( append_int @ Us2 @ ( cons_int @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_740_listrel1I,axiom,
! [X2: int,Y2: int,R2: set_Pr958786334691620121nt_int,Xs: list_int,Us: list_int,Vs: list_int,Ys: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R2 )
=> ( ( Xs
= ( append_int @ Us @ ( cons_int @ X2 @ Vs ) ) )
=> ( ( Ys
= ( append_int @ Us @ ( cons_int @ Y2 @ Vs ) ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_741_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_742_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_743_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_744_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_745_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_746_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_747_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_748_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_749_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_750_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_751_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_752_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_753_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_754_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_755_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_756_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_757_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_758_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_759_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_760_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_761_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_762_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_763_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_764_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_765_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_766_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_767_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_768_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_769_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_770_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_771_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_772_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_773_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_774_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_775_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_776_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_777_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_778_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_779_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_780_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_781_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_782_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_783_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_784_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_785_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_786_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_787_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_788_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_789_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_790_subrelI,axiom,
! [R2: set_Pr4358015327039922227nt_int,S: set_Pr4358015327039922227nt_int] :
( ! [X: list_list_char,Y3: produc4435102495419491129nt_int] :
( ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X @ Y3 ) @ R2 )
=> ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X @ Y3 ) @ S ) )
=> ( ord_le5912571515218079891nt_int @ R2 @ S ) ) ).
% subrelI
thf(fact_791_subrelI,axiom,
! [R2: set_Pr5290250543098163609nt_int,S: set_Pr5290250543098163609nt_int] :
( ! [X: list_char > int,Y3: int] :
( ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X @ Y3 ) @ R2 )
=> ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X @ Y3 ) @ S ) )
=> ( ord_le5108646760900237625nt_int @ R2 @ S ) ) ).
% subrelI
thf(fact_792_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_793_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_794_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_795_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_796_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_797_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_798_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_799_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_800_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_801_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_802_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_803_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_804_partition__set,axiom,
! [P2: list_list_char > $o,Xs: list_list_list_char,Yes: list_list_list_char,No: list_list_list_char] :
( ( ( partit1434900279125871154t_char @ P2 @ Xs )
= ( produc1273992478785463195t_char @ Yes @ No ) )
=> ( ( sup_su9123505219413647059t_char @ ( set_list_list_char2 @ Yes ) @ ( set_list_list_char2 @ No ) )
= ( set_list_list_char2 @ Xs ) ) ) ).
% partition_set
thf(fact_805_partition__set,axiom,
! [P2: list_char > $o,Xs: list_list_char,Yes: list_list_char,No: list_list_char] :
( ( ( partition_list_char @ P2 @ Xs )
= ( produc9134825680149865339t_char @ Yes @ No ) )
=> ( ( sup_su1962856675925542723t_char @ ( set_list_char2 @ Yes ) @ ( set_list_char2 @ No ) )
= ( set_list_char2 @ Xs ) ) ) ).
% partition_set
thf(fact_806_ssubst__Pair__rhs,axiom,
! [R2: list_list_char,S: produc4435102495419491129nt_int,R: set_Pr4358015327039922227nt_int,S2: produc4435102495419491129nt_int] :
( ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ R2 @ S ) @ R )
=> ( ( S2 = S )
=> ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ R2 @ S2 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_807_ssubst__Pair__rhs,axiom,
! [R2: list_char > int,S: int,R: set_Pr5290250543098163609nt_int,S2: int] :
( ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ R2 @ S ) @ R )
=> ( ( S2 = S )
=> ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ R2 @ S2 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_808_partition_Osimps_I1_J,axiom,
! [P2: instr > $o] :
( ( partition_instr @ P2 @ nil_instr )
= ( produc7750586281064920791_instr @ nil_instr @ nil_instr ) ) ).
% partition.simps(1)
thf(fact_809_partition_Osimps_I1_J,axiom,
! [P2: list_char > $o] :
( ( partition_list_char @ P2 @ nil_list_char )
= ( produc9134825680149865339t_char @ nil_list_char @ nil_list_char ) ) ).
% partition.simps(1)
thf(fact_810_partition__P,axiom,
! [P2: list_list_char > $o,Xs: list_list_list_char,Yes: list_list_list_char,No: list_list_list_char] :
( ( ( partit1434900279125871154t_char @ P2 @ Xs )
= ( produc1273992478785463195t_char @ Yes @ No ) )
=> ( ! [X4: list_list_char] :
( ( member4762036990631661320t_char @ X4 @ ( set_list_list_char2 @ Yes ) )
=> ( P2 @ X4 ) )
& ! [X4: list_list_char] :
( ( member4762036990631661320t_char @ X4 @ ( set_list_list_char2 @ No ) )
=> ~ ( P2 @ X4 ) ) ) ) ).
% partition_P
thf(fact_811_partition__P,axiom,
! [P2: list_char > $o,Xs: list_list_char,Yes: list_list_char,No: list_list_char] :
( ( ( partition_list_char @ P2 @ Xs )
= ( produc9134825680149865339t_char @ Yes @ No ) )
=> ( ! [X4: list_char] :
( ( member_list_char2 @ X4 @ ( set_list_char2 @ Yes ) )
=> ( P2 @ X4 ) )
& ! [X4: list_char] :
( ( member_list_char2 @ X4 @ ( set_list_char2 @ No ) )
=> ~ ( P2 @ X4 ) ) ) ) ).
% partition_P
thf(fact_812_remdups__adj__length__ge1,axiom,
! [Xs: list_instr] :
( ( Xs != nil_instr )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_instr @ ( remdups_adj_instr @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_813_remdups__adj__length__ge1,axiom,
! [Xs: list_list_char] :
( ( Xs != nil_list_char )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_s356637359517785349t_char @ ( remdup1386677393433641509t_char @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_814_subset__code_I3_J,axiom,
~ ( ord_le4956778034350794079_instr @ ( coset_instr @ nil_instr ) @ ( set_instr2 @ nil_instr ) ) ).
% subset_code(3)
thf(fact_815_subset__code_I3_J,axiom,
~ ( ord_le9029460191049200135t_char @ ( coset_list_list_char @ nil_list_list_char ) @ ( set_list_list_char2 @ nil_list_list_char ) ) ).
% subset_code(3)
thf(fact_816_subset__code_I3_J,axiom,
~ ( ord_le1207710660481778807t_char @ ( coset_list_char @ nil_list_char ) @ ( set_list_char2 @ nil_list_char ) ) ).
% subset_code(3)
thf(fact_817_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_818_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_819_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_820_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_821_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_822_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_823_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_824_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_825_remdups__adj__Nil__iff,axiom,
! [Xs: list_instr] :
( ( ( remdups_adj_instr @ Xs )
= nil_instr )
= ( Xs = nil_instr ) ) ).
% remdups_adj_Nil_iff
thf(fact_826_remdups__adj__Nil__iff,axiom,
! [Xs: list_list_char] :
( ( ( remdup1386677393433641509t_char @ Xs )
= nil_list_char )
= ( Xs = nil_list_char ) ) ).
% remdups_adj_Nil_iff
thf(fact_827_remdups__adj__set,axiom,
! [Xs: list_list_list_char] :
( ( set_list_list_char2 @ ( remdup6342444477279876917t_char @ Xs ) )
= ( set_list_list_char2 @ Xs ) ) ).
% remdups_adj_set
thf(fact_828_remdups__adj__set,axiom,
! [Xs: list_list_char] :
( ( set_list_char2 @ ( remdup1386677393433641509t_char @ Xs ) )
= ( set_list_char2 @ Xs ) ) ).
% remdups_adj_set
thf(fact_829_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_830_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_831_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_832_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_833_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_834_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_835_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_836_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_837_length__greater__0__conv,axiom,
! [Xs: list_instr] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_instr @ Xs ) )
= ( Xs != nil_instr ) ) ).
% length_greater_0_conv
thf(fact_838_length__greater__0__conv,axiom,
! [Xs: list_list_char] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s356637359517785349t_char @ Xs ) )
= ( Xs != nil_list_char ) ) ).
% length_greater_0_conv
thf(fact_839_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_840_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_841_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P2 @ M4 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_842_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_843_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_844_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_845_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_846_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_847_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_848_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_849_order__le__imp__less__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_850_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_851_linorder__le__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_852_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_853_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_854_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_855_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_856_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_857_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_858_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_859_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_860_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_861_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_862_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_863_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_864_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_865_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_866_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_867_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_868_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_869_order__less__le__trans,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_870_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_871_order__le__less__trans,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_872_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_873_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_874_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_875_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_876_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_877_order__less__imp__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_878_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_879_linorder__not__less,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_880_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_881_linorder__not__le,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y2 ) )
= ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_882_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_883_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_884_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_885_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_886_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_887_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_888_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_889_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_890_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_891_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_892_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_893_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_894_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_895_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_896_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_897_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_898_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_899_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_900_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_901_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_902_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_903_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_904_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_905_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_906_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_907_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_908_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_909_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_910_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_911_not__le__imp__less,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_eq_int @ Y2 @ X2 )
=> ( ord_less_int @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_912_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ~ ( ord_less_eq_nat @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_913_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
& ~ ( ord_less_eq_int @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_914_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_915_antisym__conv2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_916_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_917_antisym__conv1,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_918_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_919_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_920_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_921_leI,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% leI
thf(fact_922_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_923_leD,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_int @ X2 @ Y2 ) ) ).
% leD
thf(fact_924_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_instr @ nil_instr )
= nil_instr ) ).
% remdups_adj.simps(1)
thf(fact_925_remdups__adj_Osimps_I1_J,axiom,
( ( remdup1386677393433641509t_char @ nil_list_char )
= nil_list_char ) ).
% remdups_adj.simps(1)
thf(fact_926_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_927_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_928_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_929_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_930_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_931_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_932_remdups__adj__distinct,axiom,
! [Xs: list_list_char] :
( ( distinct_list_char @ Xs )
=> ( ( remdup1386677393433641509t_char @ Xs )
= Xs ) ) ).
% remdups_adj_distinct
thf(fact_933_remdups__adj__distinct,axiom,
! [Xs: list_list_list_char] :
( ( distin92326681159624120t_char @ Xs )
=> ( ( remdup6342444477279876917t_char @ Xs )
= Xs ) ) ).
% remdups_adj_distinct
thf(fact_934_remdups__adj__distinct,axiom,
! [Xs: list_char] :
( ( distinct_char @ Xs )
=> ( ( remdups_adj_char @ Xs )
= Xs ) ) ).
% remdups_adj_distinct
thf(fact_935_remdups__adj__distinct,axiom,
! [Xs: list_P8512142354091196201t_char] :
( ( distin3640772593324101788t_char @ Xs )
=> ( ( remdup829983452295490527t_char @ Xs )
= Xs ) ) ).
% remdups_adj_distinct
thf(fact_936_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_937_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P2 @ M4 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_938_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P2 @ M4 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_939_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_940_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_941_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_942_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_943_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_944_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_945_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_946_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_947_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_948_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_949_strict__inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P2 @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P2 @ ( suc @ I2 ) )
=> ( P2 @ I2 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_950_less__Suc__induct,axiom,
! [I: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P2 @ I2 @ J2 )
=> ( ( P2 @ J2 @ K2 )
=> ( P2 @ I2 @ K2 ) ) ) ) )
=> ( P2 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_951_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_952_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_953_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_954_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_955_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P2 @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_956_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_957_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_958_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P2 @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_959_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_960_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_961_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_962_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_963_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_964_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_965_remdups__adj_Osimps_I3_J,axiom,
! [X2: int,Y2: int,Xs: list_int] :
( ( ( X2 = Y2 )
=> ( ( remdups_adj_int @ ( cons_int @ X2 @ ( cons_int @ Y2 @ Xs ) ) )
= ( remdups_adj_int @ ( cons_int @ X2 @ Xs ) ) ) )
& ( ( X2 != Y2 )
=> ( ( remdups_adj_int @ ( cons_int @ X2 @ ( cons_int @ Y2 @ Xs ) ) )
= ( cons_int @ X2 @ ( remdups_adj_int @ ( cons_int @ Y2 @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_966_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_967_sup_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_968_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_969_sup_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ A )
=> ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_970_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_971_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( A3
= ( sup_sup_int @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_972_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_973_sup_Ostrict__boundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ ( sup_sup_int @ B @ C ) @ A )
=> ~ ( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_974_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_975_sup_Oabsorb4,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( sup_sup_int @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_976_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_977_sup_Oabsorb3,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( sup_sup_int @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_978_less__supI2,axiom,
! [X2: nat,B: nat,A: nat] :
( ( ord_less_nat @ X2 @ B )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_979_less__supI2,axiom,
! [X2: int,B: int,A: int] :
( ( ord_less_int @ X2 @ B )
=> ( ord_less_int @ X2 @ ( sup_sup_int @ A @ B ) ) ) ).
% less_supI2
thf(fact_980_less__supI1,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_nat @ X2 @ A )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_981_less__supI1,axiom,
! [X2: int,A: int,B: int] :
( ( ord_less_int @ X2 @ A )
=> ( ord_less_int @ X2 @ ( sup_sup_int @ A @ B ) ) ) ).
% less_supI1
thf(fact_982_length__induct,axiom,
! [P2: list_list_char > $o,Xs: list_list_char] :
( ! [Xs3: list_list_char] :
( ! [Ys7: list_list_char] :
( ( ord_less_nat @ ( size_s356637359517785349t_char @ Ys7 ) @ ( size_s356637359517785349t_char @ Xs3 ) )
=> ( P2 @ Ys7 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_983_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_984_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_985_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_986_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_987_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_988_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_989_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_990_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_991_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_992_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_993_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_994_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_995_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_996_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_997_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_998_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_999_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1000_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1001_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1002_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1003_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1004_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1005_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1006_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1007_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1008_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1009_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1010_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1011_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1012_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1013_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1014_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1015_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1016_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1017_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1018_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1019_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1020_remdups__adj_Osimps_I2_J,axiom,
! [X2: instr] :
( ( remdups_adj_instr @ ( cons_instr @ X2 @ nil_instr ) )
= ( cons_instr @ X2 @ nil_instr ) ) ).
% remdups_adj.simps(2)
thf(fact_1021_remdups__adj_Osimps_I2_J,axiom,
! [X2: list_char] :
( ( remdup1386677393433641509t_char @ ( cons_list_char @ X2 @ nil_list_char ) )
= ( cons_list_char @ X2 @ nil_list_char ) ) ).
% remdups_adj.simps(2)
thf(fact_1022_remdups__adj_Osimps_I2_J,axiom,
! [X2: int] :
( ( remdups_adj_int @ ( cons_int @ X2 @ nil_int ) )
= ( cons_int @ X2 @ nil_int ) ) ).
% remdups_adj.simps(2)
thf(fact_1023_remdups__adj_Oelims,axiom,
! [X2: list_instr,Y2: list_instr] :
( ( ( remdups_adj_instr @ X2 )
= Y2 )
=> ( ( ( X2 = nil_instr )
=> ( Y2 != nil_instr ) )
=> ( ! [X: instr] :
( ( X2
= ( cons_instr @ X @ nil_instr ) )
=> ( Y2
!= ( cons_instr @ X @ nil_instr ) ) )
=> ~ ! [X: instr,Y3: instr,Xs3: list_instr] :
( ( X2
= ( cons_instr @ X @ ( cons_instr @ Y3 @ Xs3 ) ) )
=> ~ ( ( ( X = Y3 )
=> ( Y2
= ( remdups_adj_instr @ ( cons_instr @ X @ Xs3 ) ) ) )
& ( ( X != Y3 )
=> ( Y2
= ( cons_instr @ X @ ( remdups_adj_instr @ ( cons_instr @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_1024_remdups__adj_Oelims,axiom,
! [X2: list_list_char,Y2: list_list_char] :
( ( ( remdup1386677393433641509t_char @ X2 )
= Y2 )
=> ( ( ( X2 = nil_list_char )
=> ( Y2 != nil_list_char ) )
=> ( ! [X: list_char] :
( ( X2
= ( cons_list_char @ X @ nil_list_char ) )
=> ( Y2
!= ( cons_list_char @ X @ nil_list_char ) ) )
=> ~ ! [X: list_char,Y3: list_char,Xs3: list_list_char] :
( ( X2
= ( cons_list_char @ X @ ( cons_list_char @ Y3 @ Xs3 ) ) )
=> ~ ( ( ( X = Y3 )
=> ( Y2
= ( remdup1386677393433641509t_char @ ( cons_list_char @ X @ Xs3 ) ) ) )
& ( ( X != Y3 )
=> ( Y2
= ( cons_list_char @ X @ ( remdup1386677393433641509t_char @ ( cons_list_char @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_1025_remdups__adj_Oelims,axiom,
! [X2: list_int,Y2: list_int] :
( ( ( remdups_adj_int @ X2 )
= Y2 )
=> ( ( ( X2 = nil_int )
=> ( Y2 != nil_int ) )
=> ( ! [X: int] :
( ( X2
= ( cons_int @ X @ nil_int ) )
=> ( Y2
!= ( cons_int @ X @ nil_int ) ) )
=> ~ ! [X: int,Y3: int,Xs3: list_int] :
( ( X2
= ( cons_int @ X @ ( cons_int @ Y3 @ Xs3 ) ) )
=> ~ ( ( ( X = Y3 )
=> ( Y2
= ( remdups_adj_int @ ( cons_int @ X @ Xs3 ) ) ) )
& ( ( X != Y3 )
=> ( Y2
= ( cons_int @ X @ ( remdups_adj_int @ ( cons_int @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_1026_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1027_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1028_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1029_dec__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_1030_inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% inc_induct
thf(fact_1031_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1032_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1033_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1034_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1035_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1036_remdups__adj__length,axiom,
! [Xs: list_list_char] : ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ ( remdup1386677393433641509t_char @ Xs ) ) @ ( size_s356637359517785349t_char @ Xs ) ) ).
% remdups_adj_length
thf(fact_1037_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1038_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1039_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1040_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1041_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1042_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1043_maps__simps_I2_J,axiom,
! [F: instr > list_instr] :
( ( maps_instr_instr @ F @ nil_instr )
= nil_instr ) ).
% maps_simps(2)
thf(fact_1044_maps__simps_I2_J,axiom,
! [F: instr > list_list_char] :
( ( maps_instr_list_char @ F @ nil_instr )
= nil_list_char ) ).
% maps_simps(2)
thf(fact_1045_maps__simps_I2_J,axiom,
! [F: list_char > list_instr] :
( ( maps_list_char_instr @ F @ nil_list_char )
= nil_instr ) ).
% maps_simps(2)
thf(fact_1046_maps__simps_I2_J,axiom,
! [F: list_char > list_list_char] :
( ( maps_l1725862226396057349t_char @ F @ nil_list_char )
= nil_list_char ) ).
% maps_simps(2)
thf(fact_1047_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1048_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1049_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1050_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1051_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1052_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1053_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1054_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1055_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1056_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1057_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1058_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1059_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1060_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1061_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1062_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1063_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1064_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1065_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1066_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1067_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1068_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1069_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1070_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1071_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_1072_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_1073_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1074_length__pos__if__in__set,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s2562795203299699605t_char @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1075_length__pos__if__in__set,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s356637359517785349t_char @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1076_remdups__adj__append__two,axiom,
! [Xs: list_instr,X2: instr,Y2: instr] :
( ( remdups_adj_instr @ ( append_instr @ Xs @ ( cons_instr @ X2 @ ( cons_instr @ Y2 @ nil_instr ) ) ) )
= ( append_instr @ ( remdups_adj_instr @ ( append_instr @ Xs @ ( cons_instr @ X2 @ nil_instr ) ) ) @ ( if_list_instr @ ( X2 = Y2 ) @ nil_instr @ ( cons_instr @ Y2 @ nil_instr ) ) ) ) ).
% remdups_adj_append_two
thf(fact_1077_remdups__adj__append__two,axiom,
! [Xs: list_list_char,X2: list_char,Y2: list_char] :
( ( remdup1386677393433641509t_char @ ( append_list_char @ Xs @ ( cons_list_char @ X2 @ ( cons_list_char @ Y2 @ nil_list_char ) ) ) )
= ( append_list_char @ ( remdup1386677393433641509t_char @ ( append_list_char @ Xs @ ( cons_list_char @ X2 @ nil_list_char ) ) ) @ ( if_list_list_char @ ( X2 = Y2 ) @ nil_list_char @ ( cons_list_char @ Y2 @ nil_list_char ) ) ) ) ).
% remdups_adj_append_two
thf(fact_1078_remdups__adj__append__two,axiom,
! [Xs: list_int,X2: int,Y2: int] :
( ( remdups_adj_int @ ( append_int @ Xs @ ( cons_int @ X2 @ ( cons_int @ Y2 @ nil_int ) ) ) )
= ( append_int @ ( remdups_adj_int @ ( append_int @ Xs @ ( cons_int @ X2 @ nil_int ) ) ) @ ( if_list_int @ ( X2 = Y2 ) @ nil_int @ ( cons_int @ Y2 @ nil_int ) ) ) ) ).
% remdups_adj_append_two
thf(fact_1079_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1080_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1081_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1082_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1083_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1084_length__removeAll__less,axiom,
! [X2: list_list_char,Xs: list_list_list_char] :
( ( member4762036990631661320t_char @ X2 @ ( set_list_list_char2 @ Xs ) )
=> ( ord_less_nat @ ( size_s2562795203299699605t_char @ ( remove3141867459347406693t_char @ X2 @ Xs ) ) @ ( size_s2562795203299699605t_char @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_1085_length__removeAll__less,axiom,
! [X2: list_char,Xs: list_list_char] :
( ( member_list_char2 @ X2 @ ( set_list_char2 @ Xs ) )
=> ( ord_less_nat @ ( size_s356637359517785349t_char @ ( removeAll_list_char @ X2 @ Xs ) ) @ ( size_s356637359517785349t_char @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_1086_subset__code_I2_J,axiom,
! [A2: set_list_list_char,Ys: list_list_list_char] :
( ( ord_le9029460191049200135t_char @ A2 @ ( coset_list_list_char @ Ys ) )
= ( ! [X3: list_list_char] :
( ( member4762036990631661320t_char @ X3 @ ( set_list_list_char2 @ Ys ) )
=> ~ ( member4762036990631661320t_char @ X3 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_1087_subset__code_I2_J,axiom,
! [A2: set_list_char,Ys: list_list_char] :
( ( ord_le1207710660481778807t_char @ A2 @ ( coset_list_char @ Ys ) )
= ( ! [X3: list_char] :
( ( member_list_char2 @ X3 @ ( set_list_char2 @ Ys ) )
=> ~ ( member_list_char2 @ X3 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_1088_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1089_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1090_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1091_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1092_mult__le__cancel__iff2,axiom,
! [Z: int,X2: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X2 ) @ ( times_times_int @ Z @ Y2 ) )
= ( ord_less_eq_int @ X2 @ Y2 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1093_mult__le__cancel__iff1,axiom,
! [Z: int,X2: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y2 @ Z ) )
= ( ord_less_eq_int @ X2 @ Y2 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1094_mult__less__iff1,axiom,
! [Z: int,X2: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y2 @ Z ) )
= ( ord_less_int @ X2 @ Y2 ) ) ) ).
% mult_less_iff1
thf(fact_1095_Cons__lenlex__iff,axiom,
! [M: int,Ms: list_int,N: int,Ns: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M @ Ms ) @ ( cons_int @ N @ Ns ) ) @ ( lenlex_int @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) )
| ( ( ( size_size_list_int @ Ms )
= ( size_size_list_int @ Ns ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1096_Cons__lenlex__iff,axiom,
! [M: list_char,Ms: list_list_char,N: list_char,Ns: list_list_char,R2: set_Pr4480099295215233283t_char] :
( ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ ( cons_list_char @ M @ Ms ) @ ( cons_list_char @ N @ Ns ) ) @ ( lenlex_list_char @ R2 ) )
= ( ( ord_less_nat @ ( size_s356637359517785349t_char @ Ms ) @ ( size_s356637359517785349t_char @ Ns ) )
| ( ( ( size_s356637359517785349t_char @ Ms )
= ( size_s356637359517785349t_char @ Ns ) )
& ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ Ms @ Ns ) @ ( lenlex_list_char @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1097_Nil__lenlex__iff1,axiom,
! [Ns: list_instr,R2: set_Pr3711236633289739975_instr] :
( ( member7594433970877843344_instr @ ( produc7750586281064920791_instr @ nil_instr @ Ns ) @ ( lenlex_instr @ R2 ) )
= ( Ns != nil_instr ) ) ).
% Nil_lenlex_iff1
thf(fact_1098_Nil__lenlex__iff1,axiom,
! [Ns: list_list_char,R2: set_Pr4480099295215233283t_char] :
( ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ nil_list_char @ Ns ) @ ( lenlex_list_char @ R2 ) )
= ( Ns != nil_list_char ) ) ).
% Nil_lenlex_iff1
thf(fact_1099_Nil__lenlex__iff2,axiom,
! [Ns: list_instr,R2: set_Pr3711236633289739975_instr] :
~ ( member7594433970877843344_instr @ ( produc7750586281064920791_instr @ Ns @ nil_instr ) @ ( lenlex_instr @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_1100_Nil__lenlex__iff2,axiom,
! [Ns: list_list_char,R2: set_Pr4480099295215233283t_char] :
~ ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ Ns @ nil_list_char ) @ ( lenlex_list_char @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_1101_lenlex__length,axiom,
! [Ms: list_list_char,Ns: list_list_char,R2: set_Pr4480099295215233283t_char] :
( ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ Ms @ Ns ) @ ( lenlex_list_char @ R2 ) )
=> ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Ms ) @ ( size_s356637359517785349t_char @ Ns ) ) ) ).
% lenlex_length
thf(fact_1102_lenlex__append1,axiom,
! [Us: list_list_char,Xs: list_list_char,R: set_Pr4480099295215233283t_char,Vs: list_list_char,Ys: list_list_char] :
( ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ Us @ Xs ) @ ( lenlex_list_char @ R ) )
=> ( ( ( size_s356637359517785349t_char @ Vs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ( member5558557582278090860t_char @ ( produc9134825680149865339t_char @ ( append_list_char @ Us @ Vs ) @ ( append_list_char @ Xs @ Ys ) ) @ ( lenlex_list_char @ R ) ) ) ) ).
% lenlex_append1
thf(fact_1103_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P2 @ I4 ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M ) ) ) ).
% nat_descend_induct
thf(fact_1104_complete__interval,axiom,
! [A: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C2 ) )
=> ( P2 @ X4 ) )
& ! [D2: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ D2 ) )
=> ( P2 @ X ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1105_complete__interval,axiom,
! [A: int,B: int,P2: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ C2 ) )
=> ( P2 @ X4 ) )
& ! [D2: int] :
( ! [X: int] :
( ( ( ord_less_eq_int @ A @ X )
& ( ord_less_int @ X @ D2 ) )
=> ( P2 @ X ) )
=> ( ord_less_eq_int @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1106_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_1107_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% minf(8)
thf(fact_1108_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_1109_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ord_less_eq_int @ X4 @ T ) ) ).
% minf(6)
thf(fact_1110_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_1111_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ord_less_eq_int @ T @ X4 ) ) ).
% pinf(8)
thf(fact_1112_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_1113_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% pinf(6)
thf(fact_1114_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A4 ) )
= ( ord_less_nat @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1115_verit__comp__simplify1_I3_J,axiom,
! [B3: int,A4: int] :
( ( ~ ( ord_less_eq_int @ B3 @ A4 ) )
= ( ord_less_int @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1116_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1117_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1118_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1119_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1120_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1121_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1122_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1123_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1124_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1125_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_1126_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_1127_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1128_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_1129_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_1130_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_1131_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_1132_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1133_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1134_rotate__length01,axiom,
! [Xs: list_list_char,N: nat] :
( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ one_one_nat )
=> ( ( rotate_list_char @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_1135_rotate1__length01,axiom,
! [Xs: list_list_char] :
( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ one_one_nat )
=> ( ( rotate1_list_char @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_1136_power__strict__increasing__iff,axiom,
! [B: nat,X2: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_nat @ X2 @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_1137_power__strict__increasing__iff,axiom,
! [B: int,X2: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_nat @ X2 @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_1138_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1139_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1140_power__increasing__iff,axiom,
! [B: nat,X2: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_1141_power__increasing__iff,axiom,
! [B: int,X2: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_1142_left__right__inverse__power,axiom,
! [X2: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X2 @ Y2 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y2 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_1143_left__right__inverse__power,axiom,
! [X2: int,Y2: int,N: nat] :
( ( ( times_times_int @ X2 @ Y2 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y2 @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_1144_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_1145_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_1146_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1147_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1148_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1149_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1150_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1151_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1152_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1153_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1154_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1155_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1156_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1157_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1158_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_1159_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_1160_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_1161_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_1162_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_1163_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1164_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1165_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1166_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1167_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1168_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1169_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_1170_mult__right__le__one__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X2 @ Y2 ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1171_mult__left__le__one__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y2 @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1172_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1173_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1174_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1175_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1176_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1177_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_1178_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1179_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1180_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1181_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1182_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_1183_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_1184_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1185_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1186_power__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1187_power__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1188_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1189_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1190_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_1191_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_1192_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1193_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1194_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1195_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1196_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1197_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1198_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1199_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1200_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1201_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1202_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1203_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1204_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1205_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1206_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1207_verit__la__generic,axiom,
! [A: int,X2: int] :
( ( ord_less_eq_int @ A @ X2 )
| ( A = X2 )
| ( ord_less_eq_int @ X2 @ A ) ) ).
% verit_la_generic
thf(fact_1208_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1209_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1210_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1211_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1212_imp__le__cong,axiom,
! [X2: int,X5: int,P2: $o,P5: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P2 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_1213_conj__le__cong,axiom,
! [X2: int,X5: int,P2: $o,P5: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P2 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_1214_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1215_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1216_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1217_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1218_int__distrib_I3_J,axiom,
! [Z12: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z12 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z12 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1219_int__distrib_I4_J,axiom,
! [W: int,Z12: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z12 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z12 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1220_int__le__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P2 @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_le_induct
thf(fact_1221_plusinfinity,axiom,
! [D: int,P5: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X: int,K2: int] :
( ( P5 @ X )
= ( P5 @ ( minus_minus_int @ X @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z6: int] :
! [X: int] :
( ( ord_less_int @ Z6 @ X )
=> ( ( P2 @ X )
= ( P5 @ X ) ) )
=> ( ? [X_1: int] : ( P5 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_1222_minusinfinity,axiom,
! [D: int,P1: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X: int,K2: int] :
( ( P1 @ X )
= ( P1 @ ( minus_minus_int @ X @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z6: int] :
! [X: int] :
( ( ord_less_int @ X @ Z6 )
=> ( ( P2 @ X )
= ( P1 @ X ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1223_decr__mult__lemma,axiom,
! [D: int,P2: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X: int] :
( ( P2 @ X )
=> ( P2 @ ( minus_minus_int @ X @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P2 @ X4 )
=> ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1224_to__m__state_Oelims,axiom,
! [X2: list_list_char,Xa2: list_char > int,Xb: int,Y2: int] :
( ( ( to_m_state @ X2 @ Xa2 @ Xb )
= Y2 )
=> ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X2 ) ) ) )
=> ( Y2
= ( Xa2 @ ( inth_list_char @ X2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X2 ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X2 ) ) ) )
=> ( Y2 = zero_zero_int ) ) ) ) ).
% to_m_state.elims
thf(fact_1225_to__m__state_Osimps,axiom,
( to_m_state
= ( ^ [Xs2: list_list_char,S3: list_char > int,A3: int] :
( if_int
@ ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ A3 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) ) )
@ ( S3 @ ( inth_list_char @ Xs2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) @ A3 ) ) )
@ zero_zero_int ) ) ) ).
% to_m_state.simps
thf(fact_1226_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1227_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1228_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1229_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1230_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1231_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1232_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1233_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1234_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1235_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1236_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1237_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1238_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1239_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1240_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1241_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1242_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1243_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1244_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1245_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1246_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1247_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1248_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1249_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1250_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1251_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1252_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1253_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1254_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1255_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1256_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1257_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1258_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1259_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1260_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1261_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1262_zdiff__int__split,axiom,
! [P2: int > $o,X2: nat,Y2: nat] :
( ( P2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y2 ) ) )
= ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( P2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
& ( ( ord_less_nat @ X2 @ Y2 )
=> ( P2 @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1263_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1264_to__m__state_Opelims,axiom,
! [X2: list_list_char,Xa2: list_char > int,Xb: int,Y2: int] :
( ( ( to_m_state @ X2 @ Xa2 @ Xb )
= Y2 )
=> ( ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X2 @ ( produc5790713362662368625nt_int @ Xa2 @ Xb ) ) )
=> ~ ( ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X2 ) ) ) )
=> ( Y2
= ( Xa2 @ ( inth_list_char @ X2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X2 ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X2 ) ) ) )
=> ( Y2 = zero_zero_int ) ) )
=> ~ ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X2 @ ( produc5790713362662368625nt_int @ Xa2 @ Xb ) ) ) ) ) ) ).
% to_m_state.pelims
thf(fact_1265_to__m__state_Ocases,axiom,
! [X2: produc4189061985984568957nt_int] :
~ ! [Xs3: list_list_char,S4: list_char > int,A5: int] :
( X2
!= ( produc3965054194175396271nt_int @ Xs3 @ ( produc5790713362662368625nt_int @ S4 @ A5 ) ) ) ).
% to_m_state.cases
thf(fact_1266_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1267_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
% Helper facts (11)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y2: int] :
( ( if_int @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y2: int] :
( ( if_int @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X2: list_int,Y2: list_int] :
( ( if_list_int @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X2: list_int,Y2: list_int] :
( ( if_list_int @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Compiler__Oinstr_J_T,axiom,
! [X2: list_instr,Y2: list_instr] :
( ( if_list_instr @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Compiler__Oinstr_J_T,axiom,
! [X2: list_instr,Y2: list_instr] :
( ( if_list_instr @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_T,axiom,
! [X2: list_list_char,Y2: list_list_char] :
( ( if_list_list_char @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_T,axiom,
! [X2: list_list_char,Y2: list_list_char] :
( ( if_list_list_char @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_J_T,axiom,
! [X2: list_list_list_char,Y2: list_list_list_char] :
( ( if_lis8692959347405121979t_char @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_J_T,axiom,
! [X2: list_list_list_char,Y2: list_list_list_char] :
( ( if_lis8692959347405121979t_char @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
distinct_list_char @ ( vars @ p ) ).
%------------------------------------------------------------------------------