TPTP Problem File: SLH0235^1.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_00743_033685__5961070_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1621 ( 616 unt; 347 typ; 0 def)
% Number of atoms : 3519 (1560 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11220 ( 303 ~; 56 |; 236 &;9241 @)
% ( 0 <=>;1384 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Number of types : 69 ( 68 usr)
% Number of type conns : 1198 (1198 >; 0 *; 0 +; 0 <<)
% Number of symbols : 282 ( 279 usr; 12 con; 0-4 aty)
% Number of variables : 3849 ( 243 ^;3435 !; 171 ?;3849 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:48:52.447
%------------------------------------------------------------------------------
% Could-be-implicit typings (68)
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% Explicit typings (279)
thf(sy_c_BExp_Obexp_OAnd,type,
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accp_list_instr: ( list_instr > list_instr > $o ) > list_instr > $o ).
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accp_P6759666029860677300t_char: ( produc6385869745582161021t_char > produc6385869745582161021t_char > $o ) > produc6385869745582161021t_char > $o ).
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_001t__Compiler__Oinstr,type,
member_instr: instr > set_instr > $o ).
thf(sy_c_member_001t__Compiler__Om____instr,type,
member_m_instr: m_instr > set_m_instr > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: 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__Compiler__Om____instr_J,type,
member_list_m_instr: list_m_instr > set_list_m_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__String__Ochar_J_J,type,
member4762036990631661320t_char: list_list_char > set_list_list_char > $o ).
thf(sy_c_member_001t__List__Olist_It__String__Ochar_J,type,
member_list_char: list_char > set_list_char > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Int__Oint_J_Mt__List__Olist_It__String__Ochar_J_J,type,
member9086947928420754658t_char: produc699483346519933241t_char > set_Pr7487058302695755673t_char > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Int__Oint_J_Mt__Nat__Onat_J,type,
member561943782142090449nt_nat: produc1534919314292393896nt_nat > set_Pr9097919650187362824nt_nat > $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_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
member6299553290643319410st_int: produc369741286924889929st_int > set_Pr5580431329626101545st_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J,type,
member2095410489637970758ar_int: produc2931317944591925149ar_int > set_Pr6231067571260677117ar_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Int__Oint_J_Mt__Nat__Onat_J_J,type,
member7326949433154712456nt_nat: produc1019184382078353393nt_nat > set_Pr6426575109479981607nt_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
member9100552558438320297st_int: produc6425607678544837394st_int > set_Pr2116125220558876360st_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Compiler__Oinstr_J,type,
member6409878718942995611_instr: produc2078578717123253124_instr > set_Pr7323812789485128634_instr > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Compiler__Om____instr_J,type,
member1049328020672055910_instr: produc3073582426511957693_instr > set_Pr9167465429053182749_instr > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
member4262671552274231302at_int: product_prod_nat_int > set_Pr7995236796853374141at_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
member484235747325421115t_char: produc1457211279475724562t_char > set_Pr2507339680178222706t_char > $o ).
thf(sy_v_c_092_060_094sub_0621____,type,
c_1: com ).
thf(sy_v_c_092_060_094sub_0622____,type,
c_2: com ).
thf(sy_v_s_092_060_094sub_0621____,type,
s_1: list_char > int ).
thf(sy_v_s_092_060_094sub_0622____,type,
s_2: list_char > int ).
thf(sy_v_s_092_060_094sub_0623____,type,
s_3: list_char > int ).
thf(sy_v_stka____,type,
stka: list_int ).
% Relevant facts (1266)
thf(fact_0_exec__induct,axiom,
! [Pa: list_instr,X1a: int,X1b: list_char > int,X1c: list_int,X2a: int,X2b: list_char > int,X2c: list_int,P: int > ( list_char > int ) > list_int > int > ( list_char > int ) > list_int > $o] :
( ( star_P707599355569300323st_int @ ( exec1 @ Pa ) @ ( produc5086643055186798020st_int @ X1a @ ( produc8650753666468850689st_int @ X1b @ X1c ) ) @ ( produc5086643055186798020st_int @ X2a @ ( produc8650753666468850689st_int @ X2b @ X2c ) ) )
=> ( ! [A: int,Aa: list_char > int,B: list_int] : ( P @ A @ Aa @ B @ A @ Aa @ B )
=> ( ! [A: int,Aa: list_char > int,B: list_int,Ab: int,Ac: list_char > int,Ba: list_int,Ad: int,Ae: list_char > int,Bb: list_int] :
( ( exec1 @ Pa @ ( produc5086643055186798020st_int @ A @ ( produc8650753666468850689st_int @ Aa @ B ) ) @ ( produc5086643055186798020st_int @ Ab @ ( produc8650753666468850689st_int @ Ac @ Ba ) ) )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ Pa ) @ ( produc5086643055186798020st_int @ Ab @ ( produc8650753666468850689st_int @ Ac @ Ba ) ) @ ( produc5086643055186798020st_int @ Ad @ ( produc8650753666468850689st_int @ Ae @ Bb ) ) )
=> ( ( P @ Ab @ Ac @ Ba @ Ad @ Ae @ Bb )
=> ( P @ A @ Aa @ B @ Ad @ Ae @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X1c @ X2a @ X2b @ X2c ) ) ) ) ).
% exec_induct
thf(fact_1_exec__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C2: produc6425607678544837394st_int,P2: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ C @ C2 )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P2 ) ) @ C @ C2 ) ) ).
% exec_appendR
thf(fact_2_exec1__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C2: produc6425607678544837394st_int,P2: list_instr] :
( ( exec1 @ P @ C @ C2 )
=> ( exec1 @ ( append_instr @ P @ P2 ) @ C @ C2 ) ) ).
% exec1_appendR
thf(fact_3_append__eq__append__conv,axiom,
! [Xs: list_m_instr,Ys: list_m_instr,Us: list_m_instr,Vs: list_m_instr] :
( ( ( ( size_s5914166952900237872_instr @ Xs )
= ( size_s5914166952900237872_instr @ Ys ) )
| ( ( size_s5914166952900237872_instr @ Us )
= ( size_s5914166952900237872_instr @ Vs ) ) )
=> ( ( ( append_m_instr @ Xs @ Us )
= ( append_m_instr @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_4_append__eq__append__conv,axiom,
! [Xs: list_instr,Ys: list_instr,Us: list_instr,Vs: list_instr] :
( ( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
| ( ( size_size_list_instr @ Us )
= ( size_size_list_instr @ Vs ) ) )
=> ( ( ( append_instr @ Xs @ Us )
= ( append_instr @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_5_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_6_append__eq__append__conv,axiom,
! [Xs: list_int,Ys: list_int,Us: list_int,Vs: list_int] :
( ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
| ( ( size_size_list_int @ Us )
= ( size_size_list_int @ Vs ) ) )
=> ( ( ( append_int @ Xs @ Us )
= ( append_int @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_7_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_8_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_9_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_10_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_11_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_12_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_13_star__step1,axiom,
! [R: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int,Y: produc6425607678544837394st_int] :
( ( R @ X @ Y )
=> ( star_P707599355569300323st_int @ R @ X @ Y ) ) ).
% star_step1
thf(fact_14_star__step1,axiom,
! [R: produc1019184382078353393nt_nat > produc1019184382078353393nt_nat > $o,X: produc1019184382078353393nt_nat,Y: produc1019184382078353393nt_nat] :
( ( R @ X @ Y )
=> ( star_P5461765526710342210nt_nat @ R @ X @ Y ) ) ).
% star_step1
thf(fact_15_Seq_Ohyps_I1_J,axiom,
big_big_step @ ( produc5595214716300948949ar_int @ c_1 @ s_1 ) @ s_2 ).
% Seq.hyps(1)
thf(fact_16_append_Oassoc,axiom,
! [A2: list_list_char,B2: list_list_char,C: list_list_char] :
( ( append_list_char @ ( append_list_char @ A2 @ B2 ) @ C )
= ( append_list_char @ A2 @ ( append_list_char @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_17_append_Oassoc,axiom,
! [A2: list_m_instr,B2: list_m_instr,C: list_m_instr] :
( ( append_m_instr @ ( append_m_instr @ A2 @ B2 ) @ C )
= ( append_m_instr @ A2 @ ( append_m_instr @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_18_append_Oassoc,axiom,
! [A2: list_instr,B2: list_instr,C: list_instr] :
( ( append_instr @ ( append_instr @ A2 @ B2 ) @ C )
= ( append_instr @ A2 @ ( append_instr @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_19_append_Oassoc,axiom,
! [A2: list_int,B2: list_int,C: list_int] :
( ( append_int @ ( append_int @ A2 @ B2 ) @ C )
= ( append_int @ A2 @ ( append_int @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_20_append__assoc,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( append_instr @ ( append_instr @ Xs @ Ys ) @ Zs )
= ( append_instr @ Xs @ ( append_instr @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_21_append__assoc,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int] :
( ( append_int @ ( append_int @ Xs @ Ys ) @ Zs )
= ( append_int @ Xs @ ( append_int @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_22_append__assoc,axiom,
! [Xs: list_list_char,Ys: list_list_char,Zs: list_list_char] :
( ( append_list_char @ ( append_list_char @ Xs @ Ys ) @ Zs )
= ( append_list_char @ Xs @ ( append_list_char @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_23_append__assoc,axiom,
! [Xs: list_m_instr,Ys: list_m_instr,Zs: list_m_instr] :
( ( append_m_instr @ ( append_m_instr @ Xs @ Ys ) @ Zs )
= ( append_m_instr @ Xs @ ( append_m_instr @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_24_append__same__eq,axiom,
! [Ys: list_instr,Xs: list_instr,Zs: list_instr] :
( ( ( append_instr @ Ys @ Xs )
= ( append_instr @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_25_append__same__eq,axiom,
! [Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( append_int @ Ys @ Xs )
= ( append_int @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_26_append__same__eq,axiom,
! [Ys: list_list_char,Xs: list_list_char,Zs: list_list_char] :
( ( ( append_list_char @ Ys @ Xs )
= ( append_list_char @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_27_append__same__eq,axiom,
! [Ys: list_m_instr,Xs: list_m_instr,Zs: list_m_instr] :
( ( ( append_m_instr @ Ys @ Xs )
= ( append_m_instr @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_28_same__append__eq,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_29_same__append__eq,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_30_same__append__eq,axiom,
! [Xs: list_list_char,Ys: list_list_char,Zs: list_list_char] :
( ( ( append_list_char @ Xs @ Ys )
= ( append_list_char @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_31_same__append__eq,axiom,
! [Xs: list_m_instr,Ys: list_m_instr,Zs: list_m_instr] :
( ( ( append_m_instr @ Xs @ Ys )
= ( append_m_instr @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_32_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_33_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_34_exec__appendL,axiom,
! [P: list_instr,I: int,S: list_char > int,Stk: list_int,I2: int,S2: list_char > int,Stk2: list_int,P2: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P2 @ P ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P2 ) ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P2 ) ) @ I2 ) @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) ) ) ).
% exec_appendL
thf(fact_35_Seq_Ohyps_I2_J,axiom,
big_big_step @ ( produc5595214716300948949ar_int @ c_2 @ s_2 ) @ s_3 ).
% Seq.hyps(2)
thf(fact_36_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_37_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_38_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_39_size__neq__size__imp__neq,axiom,
! [X: list_m_instr,Y: list_m_instr] :
( ( ( size_s5914166952900237872_instr @ X )
!= ( size_s5914166952900237872_instr @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_40_size__neq__size__imp__neq,axiom,
! [X: list_instr,Y: list_instr] :
( ( ( size_size_list_instr @ X )
!= ( size_size_list_instr @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_41_size__neq__size__imp__neq,axiom,
! [X: list_list_char,Y: list_list_char] :
( ( ( size_s356637359517785349t_char @ X )
!= ( size_s356637359517785349t_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_42_size__neq__size__imp__neq,axiom,
! [X: list_int,Y: list_int] :
( ( ( size_size_list_int @ X )
!= ( size_size_list_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_43_neq__if__length__neq,axiom,
! [Xs: list_m_instr,Ys: list_m_instr] :
( ( ( size_s5914166952900237872_instr @ Xs )
!= ( size_s5914166952900237872_instr @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_44_neq__if__length__neq,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( size_size_list_instr @ Xs )
!= ( size_size_list_instr @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_45_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_46_neq__if__length__neq,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
!= ( size_size_list_int @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_47_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_m_instr] :
( ( size_s5914166952900237872_instr @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_48_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_instr] :
( ( size_size_list_instr @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_49_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_char] :
( ( size_s356637359517785349t_char @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_50_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_int] :
( ( size_size_list_int @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_51_append__eq__append__conv2,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr,Ts: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Zs @ Ts ) )
= ( ? [Us2: list_instr] :
( ( ( Xs
= ( append_instr @ Zs @ Us2 ) )
& ( ( append_instr @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_instr @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_instr @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_52_append__eq__append__conv2,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,Ts: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Zs @ Ts ) )
= ( ? [Us2: list_int] :
( ( ( Xs
= ( append_int @ Zs @ Us2 ) )
& ( ( append_int @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_int @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_int @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_53_append__eq__append__conv2,axiom,
! [Xs: list_list_char,Ys: list_list_char,Zs: list_list_char,Ts: list_list_char] :
( ( ( append_list_char @ Xs @ Ys )
= ( append_list_char @ Zs @ Ts ) )
= ( ? [Us2: list_list_char] :
( ( ( Xs
= ( append_list_char @ Zs @ Us2 ) )
& ( ( append_list_char @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_list_char @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_list_char @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_54_append__eq__append__conv2,axiom,
! [Xs: list_m_instr,Ys: list_m_instr,Zs: list_m_instr,Ts: list_m_instr] :
( ( ( append_m_instr @ Xs @ Ys )
= ( append_m_instr @ Zs @ Ts ) )
= ( ? [Us2: list_m_instr] :
( ( ( Xs
= ( append_m_instr @ Zs @ Us2 ) )
& ( ( append_m_instr @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_m_instr @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_m_instr @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_55_append__eq__appendI,axiom,
! [Xs: list_instr,Xs1: list_instr,Zs: list_instr,Ys: list_instr,Us: list_instr] :
( ( ( append_instr @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_instr @ Xs1 @ Us ) )
=> ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_56_append__eq__appendI,axiom,
! [Xs: list_int,Xs1: list_int,Zs: list_int,Ys: list_int,Us: list_int] :
( ( ( append_int @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_int @ Xs1 @ Us ) )
=> ( ( append_int @ Xs @ Ys )
= ( append_int @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_57_append__eq__appendI,axiom,
! [Xs: list_list_char,Xs1: list_list_char,Zs: list_list_char,Ys: list_list_char,Us: list_list_char] :
( ( ( append_list_char @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_char @ Xs1 @ Us ) )
=> ( ( append_list_char @ Xs @ Ys )
= ( append_list_char @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_58_append__eq__appendI,axiom,
! [Xs: list_m_instr,Xs1: list_m_instr,Zs: list_m_instr,Ys: list_m_instr,Us: list_m_instr] :
( ( ( append_m_instr @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_m_instr @ Xs1 @ Us ) )
=> ( ( append_m_instr @ Xs @ Ys )
= ( append_m_instr @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_59_star__trans,axiom,
! [R: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int,Y: produc6425607678544837394st_int,Z: produc6425607678544837394st_int] :
( ( star_P707599355569300323st_int @ R @ X @ Y )
=> ( ( star_P707599355569300323st_int @ R @ Y @ Z )
=> ( star_P707599355569300323st_int @ R @ X @ Z ) ) ) ).
% star_trans
thf(fact_60_star__trans,axiom,
! [R: produc1019184382078353393nt_nat > produc1019184382078353393nt_nat > $o,X: produc1019184382078353393nt_nat,Y: produc1019184382078353393nt_nat,Z: produc1019184382078353393nt_nat] :
( ( star_P5461765526710342210nt_nat @ R @ X @ Y )
=> ( ( star_P5461765526710342210nt_nat @ R @ Y @ Z )
=> ( star_P5461765526710342210nt_nat @ R @ X @ Z ) ) ) ).
% star_trans
thf(fact_61_star_Osimps,axiom,
( star_P707599355569300323st_int
= ( ^ [R2: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,A1: produc6425607678544837394st_int,A22: produc6425607678544837394st_int] :
( ? [X2: produc6425607678544837394st_int] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: produc6425607678544837394st_int,Y2: produc6425607678544837394st_int,Z2: produc6425607678544837394st_int] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( R2 @ X2 @ Y2 )
& ( star_P707599355569300323st_int @ R2 @ Y2 @ Z2 ) ) ) ) ) ).
% star.simps
thf(fact_62_star_Osimps,axiom,
( star_P5461765526710342210nt_nat
= ( ^ [R2: produc1019184382078353393nt_nat > produc1019184382078353393nt_nat > $o,A1: produc1019184382078353393nt_nat,A22: produc1019184382078353393nt_nat] :
( ? [X2: produc1019184382078353393nt_nat] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: produc1019184382078353393nt_nat,Y2: produc1019184382078353393nt_nat,Z2: produc1019184382078353393nt_nat] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( R2 @ X2 @ Y2 )
& ( star_P5461765526710342210nt_nat @ R2 @ Y2 @ Z2 ) ) ) ) ) ).
% star.simps
thf(fact_63_star_Ocases,axiom,
! [R: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,A12: produc6425607678544837394st_int,A23: produc6425607678544837394st_int] :
( ( star_P707599355569300323st_int @ R @ A12 @ A23 )
=> ( ( A23 != A12 )
=> ~ ! [Y3: produc6425607678544837394st_int] :
( ( R @ A12 @ Y3 )
=> ~ ( star_P707599355569300323st_int @ R @ Y3 @ A23 ) ) ) ) ).
% star.cases
thf(fact_64_star_Ocases,axiom,
! [R: produc1019184382078353393nt_nat > produc1019184382078353393nt_nat > $o,A12: produc1019184382078353393nt_nat,A23: produc1019184382078353393nt_nat] :
( ( star_P5461765526710342210nt_nat @ R @ A12 @ A23 )
=> ( ( A23 != A12 )
=> ~ ! [Y3: produc1019184382078353393nt_nat] :
( ( R @ A12 @ Y3 )
=> ~ ( star_P5461765526710342210nt_nat @ R @ Y3 @ A23 ) ) ) ) ).
% star.cases
thf(fact_65_star_Ostep,axiom,
! [R: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int,Y: produc6425607678544837394st_int,Z: produc6425607678544837394st_int] :
( ( R @ X @ Y )
=> ( ( star_P707599355569300323st_int @ R @ Y @ Z )
=> ( star_P707599355569300323st_int @ R @ X @ Z ) ) ) ).
% star.step
thf(fact_66_star_Ostep,axiom,
! [R: produc1019184382078353393nt_nat > produc1019184382078353393nt_nat > $o,X: produc1019184382078353393nt_nat,Y: produc1019184382078353393nt_nat,Z: produc1019184382078353393nt_nat] :
( ( R @ X @ Y )
=> ( ( star_P5461765526710342210nt_nat @ R @ Y @ Z )
=> ( star_P5461765526710342210nt_nat @ R @ X @ Z ) ) ) ).
% star.step
thf(fact_67_star_Orefl,axiom,
! [R: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int] : ( star_P707599355569300323st_int @ R @ X @ X ) ).
% star.refl
thf(fact_68_star_Orefl,axiom,
! [R: produc1019184382078353393nt_nat > produc1019184382078353393nt_nat > $o,X: produc1019184382078353393nt_nat] : ( star_P5461765526710342210nt_nat @ R @ X @ X ) ).
% star.refl
thf(fact_69_exec1__appendL,axiom,
! [P: list_instr,I: int,S: list_char > int,Stk: list_int,I2: int,S2: list_char > int,Stk2: list_int,P2: list_instr] :
( ( exec1 @ P @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) )
=> ( exec1 @ ( append_instr @ P2 @ P ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P2 ) ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P2 ) ) @ I2 ) @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) ) ) ).
% exec1_appendL
thf(fact_70_star__induct,axiom,
! [R: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X1a: int,X1b: produc369741286924889929st_int,X2a: int,X2b: produc369741286924889929st_int,P: int > produc369741286924889929st_int > int > produc369741286924889929st_int > $o] :
( ( star_P707599355569300323st_int @ R @ ( produc5086643055186798020st_int @ X1a @ X1b ) @ ( produc5086643055186798020st_int @ X2a @ X2b ) )
=> ( ! [A: int,B: produc369741286924889929st_int] : ( P @ A @ B @ A @ B )
=> ( ! [A: int,B: produc369741286924889929st_int,Aa: int,Ba: produc369741286924889929st_int,Ab: int,Bb: produc369741286924889929st_int] :
( ( R @ ( produc5086643055186798020st_int @ A @ B ) @ ( produc5086643055186798020st_int @ Aa @ Ba ) )
=> ( ( star_P707599355569300323st_int @ R @ ( produc5086643055186798020st_int @ Aa @ Ba ) @ ( produc5086643055186798020st_int @ Ab @ Bb ) )
=> ( ( P @ Aa @ Ba @ Ab @ Bb )
=> ( P @ A @ B @ Ab @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X2a @ X2b ) ) ) ) ).
% star_induct
thf(fact_71_star__induct,axiom,
! [R: produc1019184382078353393nt_nat > produc1019184382078353393nt_nat > $o,X1a: int,X1b: produc1534919314292393896nt_nat,X2a: int,X2b: produc1534919314292393896nt_nat,P: int > produc1534919314292393896nt_nat > int > produc1534919314292393896nt_nat > $o] :
( ( star_P5461765526710342210nt_nat @ R @ ( produc8342123264692915235nt_nat @ X1a @ X1b ) @ ( produc8342123264692915235nt_nat @ X2a @ X2b ) )
=> ( ! [A: int,B: produc1534919314292393896nt_nat] : ( P @ A @ B @ A @ B )
=> ( ! [A: int,B: produc1534919314292393896nt_nat,Aa: int,Ba: produc1534919314292393896nt_nat,Ab: int,Bb: produc1534919314292393896nt_nat] :
( ( R @ ( produc8342123264692915235nt_nat @ A @ B ) @ ( produc8342123264692915235nt_nat @ Aa @ Ba ) )
=> ( ( star_P5461765526710342210nt_nat @ R @ ( produc8342123264692915235nt_nat @ Aa @ Ba ) @ ( produc8342123264692915235nt_nat @ Ab @ Bb ) )
=> ( ( P @ Aa @ Ba @ Ab @ Bb )
=> ( P @ A @ B @ Ab @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X2a @ X2b ) ) ) ) ).
% star_induct
thf(fact_72_star__induct,axiom,
! [R: produc2931317944591925149ar_int > produc2931317944591925149ar_int > $o,X1a: com,X1b: list_char > int,X2a: com,X2b: list_char > int,P: com > ( list_char > int ) > com > ( list_char > int ) > $o] :
( ( star_P702767028171920396ar_int @ R @ ( produc5595214716300948949ar_int @ X1a @ X1b ) @ ( produc5595214716300948949ar_int @ X2a @ X2b ) )
=> ( ! [A: com,B: list_char > int] : ( P @ A @ B @ A @ B )
=> ( ! [A: com,B: list_char > int,Aa: com,Ba: list_char > int,Ab: com,Bb: list_char > int] :
( ( R @ ( produc5595214716300948949ar_int @ A @ B ) @ ( produc5595214716300948949ar_int @ Aa @ Ba ) )
=> ( ( star_P702767028171920396ar_int @ R @ ( produc5595214716300948949ar_int @ Aa @ Ba ) @ ( produc5595214716300948949ar_int @ Ab @ Bb ) )
=> ( ( P @ Aa @ Ba @ Ab @ Bb )
=> ( P @ A @ B @ Ab @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X2a @ X2b ) ) ) ) ).
% star_induct
thf(fact_73_star__induct,axiom,
! [R: produc369741286924889929st_int > produc369741286924889929st_int > $o,X1a: list_char > int,X1b: list_int,X2a: list_char > int,X2b: list_int,P: ( list_char > int ) > list_int > ( list_char > int ) > list_int > $o] :
( ( star_P8562542470885916728st_int @ R @ ( produc8650753666468850689st_int @ X1a @ X1b ) @ ( produc8650753666468850689st_int @ X2a @ X2b ) )
=> ( ! [A: list_char > int,B: list_int] : ( P @ A @ B @ A @ B )
=> ( ! [A: list_char > int,B: list_int,Aa: list_char > int,Ba: list_int,Ab: list_char > int,Bb: list_int] :
( ( R @ ( produc8650753666468850689st_int @ A @ B ) @ ( produc8650753666468850689st_int @ Aa @ Ba ) )
=> ( ( star_P8562542470885916728st_int @ R @ ( produc8650753666468850689st_int @ Aa @ Ba ) @ ( produc8650753666468850689st_int @ Ab @ Bb ) )
=> ( ( P @ Aa @ Ba @ Ab @ Bb )
=> ( P @ A @ B @ Ab @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X2a @ X2b ) ) ) ) ).
% star_induct
thf(fact_74_star__induct,axiom,
! [R: produc4435102495419491129nt_int > produc4435102495419491129nt_int > $o,X1a: list_char > int,X1b: int,X2a: list_char > int,X2b: int,P: ( list_char > int ) > int > ( list_char > int ) > int > $o] :
( ( star_P2206551578999486376nt_int @ R @ ( produc5790713362662368625nt_int @ X1a @ X1b ) @ ( produc5790713362662368625nt_int @ X2a @ X2b ) )
=> ( ! [A: list_char > int,B: int] : ( P @ A @ B @ A @ B )
=> ( ! [A: list_char > int,B: int,Aa: list_char > int,Ba: int,Ab: list_char > int,Bb: int] :
( ( R @ ( produc5790713362662368625nt_int @ A @ B ) @ ( produc5790713362662368625nt_int @ Aa @ Ba ) )
=> ( ( star_P2206551578999486376nt_int @ R @ ( produc5790713362662368625nt_int @ Aa @ Ba ) @ ( produc5790713362662368625nt_int @ Ab @ Bb ) )
=> ( ( P @ Aa @ Ba @ Ab @ Bb )
=> ( P @ A @ B @ Ab @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X2a @ X2b ) ) ) ) ).
% star_induct
thf(fact_75_star__induct,axiom,
! [R: produc699483346519933241t_char > produc699483346519933241t_char > $o,X1a: int > int,X1b: list_char,X2a: int > int,X2b: list_char,P: ( int > int ) > list_char > ( int > int ) > list_char > $o] :
( ( star_P7694304466954704296t_char @ R @ ( produc4670215617849501681t_char @ X1a @ X1b ) @ ( produc4670215617849501681t_char @ X2a @ X2b ) )
=> ( ! [A: int > int,B: list_char] : ( P @ A @ B @ A @ B )
=> ( ! [A: int > int,B: list_char,Aa: int > int,Ba: list_char,Ab: int > int,Bb: list_char] :
( ( R @ ( produc4670215617849501681t_char @ A @ B ) @ ( produc4670215617849501681t_char @ Aa @ Ba ) )
=> ( ( star_P7694304466954704296t_char @ R @ ( produc4670215617849501681t_char @ Aa @ Ba ) @ ( produc4670215617849501681t_char @ Ab @ Bb ) )
=> ( ( P @ Aa @ Ba @ Ab @ Bb )
=> ( P @ A @ B @ Ab @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X2a @ X2b ) ) ) ) ).
% star_induct
thf(fact_76_star__induct,axiom,
! [R: produc1534919314292393896nt_nat > produc1534919314292393896nt_nat > $o,X1a: int > int,X1b: nat,X2a: int > int,X2b: nat,P: ( int > int ) > nat > ( int > int ) > nat > $o] :
( ( star_P2443116637854396055nt_nat @ R @ ( produc2630796075472406112nt_nat @ X1a @ X1b ) @ ( produc2630796075472406112nt_nat @ X2a @ X2b ) )
=> ( ! [A: int > int,B: nat] : ( P @ A @ B @ A @ B )
=> ( ! [A: int > int,B: nat,Aa: int > int,Ba: nat,Ab: int > int,Bb: nat] :
( ( R @ ( produc2630796075472406112nt_nat @ A @ B ) @ ( produc2630796075472406112nt_nat @ Aa @ Ba ) )
=> ( ( star_P2443116637854396055nt_nat @ R @ ( produc2630796075472406112nt_nat @ Aa @ Ba ) @ ( produc2630796075472406112nt_nat @ Ab @ Bb ) )
=> ( ( P @ Aa @ Ba @ Ab @ Bb )
=> ( P @ A @ B @ Ab @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X2a @ X2b ) ) ) ) ).
% star_induct
thf(fact_77_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_78_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_79_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_80_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_81_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_82_add__cancel__right__right,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ A2 @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_83_add__cancel__right__right,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_84_add__cancel__right__left,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ B2 @ A2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_85_add__cancel__right__left,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_86_add__cancel__left__right,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_87_add__cancel__left__right,axiom,
! [A2: nat,B2: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_88_add__cancel__left__left,axiom,
! [B2: int,A2: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_89_add__cancel__left__left,axiom,
! [B2: nat,A2: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_90_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_91_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_92_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_93_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_94_add__left__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_95_add__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_96_mem__Collect__eq,axiom,
! [A2: list_char,P: list_char > $o] :
( ( member_list_char @ A2 @ ( collect_list_char @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_97_Collect__mem__eq,axiom,
! [A3: set_list_char] :
( ( collect_list_char
@ ^ [X2: list_char] : ( member_list_char @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_98_add__right__cancel,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_99_add__right__cancel,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_100_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_101_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_102_length__append,axiom,
! [Xs: list_m_instr,Ys: list_m_instr] :
( ( size_s5914166952900237872_instr @ ( append_m_instr @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s5914166952900237872_instr @ Xs ) @ ( size_s5914166952900237872_instr @ Ys ) ) ) ).
% length_append
thf(fact_103_length__append,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( size_size_list_instr @ ( append_instr @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_instr @ Xs ) @ ( size_size_list_instr @ Ys ) ) ) ).
% length_append
thf(fact_104_length__append,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( size_s356637359517785349t_char @ ( append_list_char @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s356637359517785349t_char @ Xs ) @ ( size_s356637359517785349t_char @ Ys ) ) ) ).
% length_append
thf(fact_105_length__append,axiom,
! [Xs: list_int,Ys: list_int] :
( ( size_size_list_int @ ( append_int @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_append
thf(fact_106_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_107_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_108_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_109_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_110_int__ops_I5_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_111_iexec_Ocases,axiom,
! [X: produc5995290525303592096st_int] :
~ ! [Ins: instr,I3: int,S3: list_char > int,Stk3: list_int] :
( X
!= ( produc1484136438817787418st_int @ Ins @ ( produc5086643055186798020st_int @ I3 @ ( produc8650753666468850689st_int @ S3 @ Stk3 ) ) ) ) ).
% iexec.cases
thf(fact_112_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_113_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_114_add__right__imp__eq,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_115_add__right__imp__eq,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_116_add__left__imp__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_117_add__left__imp__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_118_add_Oleft__commute,axiom,
! [B2: int,A2: int,C: int] :
( ( plus_plus_int @ B2 @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_119_add_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_120_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_121_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_122_add_Oright__cancel,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_123_add_Oleft__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_124_add_Oassoc,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_125_add_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_126_group__cancel_Oadd2,axiom,
! [B4: int,K: int,B2: int,A2: int] :
( ( B4
= ( plus_plus_int @ K @ B2 ) )
=> ( ( plus_plus_int @ A2 @ B4 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_127_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B2: nat,A2: nat] :
( ( B4
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A2 @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_128_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A2: int,B2: int] :
( ( A3
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A3 @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_129_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A2: nat,B2: nat] :
( ( A3
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_130_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_131_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_132_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_133_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_134_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_135_int__if,axiom,
! [P: $o,A2: nat,B2: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_136_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_137_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_138_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_139_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_140_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_141_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_142_verit__sum__simplify,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% verit_sum_simplify
thf(fact_143_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_144_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_145_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_146_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_147_prod_Oinject,axiom,
! [X1: com,X22: list_char > int,Y1: com,Y22: list_char > int] :
( ( ( produc5595214716300948949ar_int @ X1 @ X22 )
= ( produc5595214716300948949ar_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_148_prod_Oinject,axiom,
! [X1: list_char > int,X22: list_int,Y1: list_char > int,Y22: list_int] :
( ( ( produc8650753666468850689st_int @ X1 @ X22 )
= ( produc8650753666468850689st_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_149_prod_Oinject,axiom,
! [X1: list_char > int,X22: int,Y1: list_char > int,Y22: int] :
( ( ( produc5790713362662368625nt_int @ X1 @ X22 )
= ( produc5790713362662368625nt_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_150_prod_Oinject,axiom,
! [X1: int > int,X22: list_char,Y1: int > int,Y22: list_char] :
( ( ( produc4670215617849501681t_char @ X1 @ X22 )
= ( produc4670215617849501681t_char @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_151_prod_Oinject,axiom,
! [X1: int > int,X22: nat,Y1: int > int,Y22: nat] :
( ( ( produc2630796075472406112nt_nat @ X1 @ X22 )
= ( produc2630796075472406112nt_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_152_old_Oprod_Oinject,axiom,
! [A2: com,B2: list_char > int,A5: com,B5: list_char > int] :
( ( ( produc5595214716300948949ar_int @ A2 @ B2 )
= ( produc5595214716300948949ar_int @ A5 @ B5 ) )
= ( ( A2 = A5 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
thf(fact_153_old_Oprod_Oinject,axiom,
! [A2: list_char > int,B2: list_int,A5: list_char > int,B5: list_int] :
( ( ( produc8650753666468850689st_int @ A2 @ B2 )
= ( produc8650753666468850689st_int @ A5 @ B5 ) )
= ( ( A2 = A5 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
thf(fact_154_old_Oprod_Oinject,axiom,
! [A2: list_char > int,B2: int,A5: list_char > int,B5: int] :
( ( ( produc5790713362662368625nt_int @ A2 @ B2 )
= ( produc5790713362662368625nt_int @ A5 @ B5 ) )
= ( ( A2 = A5 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
thf(fact_155_old_Oprod_Oinject,axiom,
! [A2: int > int,B2: list_char,A5: int > int,B5: list_char] :
( ( ( produc4670215617849501681t_char @ A2 @ B2 )
= ( produc4670215617849501681t_char @ A5 @ B5 ) )
= ( ( A2 = A5 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
thf(fact_156_old_Oprod_Oinject,axiom,
! [A2: int > int,B2: nat,A5: int > int,B5: nat] :
( ( ( produc2630796075472406112nt_nat @ A2 @ B2 )
= ( produc2630796075472406112nt_nat @ A5 @ B5 ) )
= ( ( A2 = A5 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
thf(fact_157_iexec__shift,axiom,
! [N: int,I2: int,S2: list_char > int,Stk2: list_int,Ins2: instr,I: int,S: list_char > int,Stk: list_int] :
( ( ( produc5086643055186798020st_int @ ( plus_plus_int @ N @ I2 ) @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) )
= ( iexec @ Ins2 @ ( produc5086643055186798020st_int @ ( plus_plus_int @ N @ I ) @ ( produc8650753666468850689st_int @ S @ Stk ) ) ) )
= ( ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) )
= ( iexec @ Ins2 @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk ) ) ) ) ) ).
% iexec_shift
thf(fact_158_big__step__determ,axiom,
! [C: com,S: list_char > int,T: list_char > int,U: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S ) @ T )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S ) @ U )
=> ( U = T ) ) ) ).
% big_step_determ
thf(fact_159_sim__trans,axiom,
! [C: com,C2: com,C4: com] :
( ! [S3: list_char > int,T2: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S3 ) @ T2 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C2 @ S3 ) @ T2 ) )
=> ( ! [S3: list_char > int,T2: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C2 @ S3 ) @ T2 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C4 @ S3 ) @ T2 ) )
=> ! [S4: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S4 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C4 @ S4 ) @ T3 ) ) ) ) ).
% sim_trans
thf(fact_160_sim__refl,axiom,
! [C: com,S4: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S4 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S4 ) @ T3 ) ) ).
% sim_refl
thf(fact_161_sim__sym,axiom,
! [C: com,C2: com] :
( ( ! [S5: list_char > int,T4: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S5 ) @ T4 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C2 @ S5 ) @ T4 ) ) )
= ( ! [S5: list_char > int,T4: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C2 @ S5 ) @ T4 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S5 ) @ T4 ) ) ) ) ).
% sim_sym
thf(fact_162_Pair__inject,axiom,
! [A2: com,B2: list_char > int,A5: com,B5: list_char > int] :
( ( ( produc5595214716300948949ar_int @ A2 @ B2 )
= ( produc5595214716300948949ar_int @ A5 @ B5 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
thf(fact_163_Pair__inject,axiom,
! [A2: list_char > int,B2: list_int,A5: list_char > int,B5: list_int] :
( ( ( produc8650753666468850689st_int @ A2 @ B2 )
= ( produc8650753666468850689st_int @ A5 @ B5 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
thf(fact_164_Pair__inject,axiom,
! [A2: list_char > int,B2: int,A5: list_char > int,B5: int] :
( ( ( produc5790713362662368625nt_int @ A2 @ B2 )
= ( produc5790713362662368625nt_int @ A5 @ B5 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
thf(fact_165_Pair__inject,axiom,
! [A2: int > int,B2: list_char,A5: int > int,B5: list_char] :
( ( ( produc4670215617849501681t_char @ A2 @ B2 )
= ( produc4670215617849501681t_char @ A5 @ B5 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
thf(fact_166_Pair__inject,axiom,
! [A2: int > int,B2: nat,A5: int > int,B5: nat] :
( ( ( produc2630796075472406112nt_nat @ A2 @ B2 )
= ( produc2630796075472406112nt_nat @ A5 @ B5 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
thf(fact_167_prod__cases,axiom,
! [P: produc2931317944591925149ar_int > $o,P3: produc2931317944591925149ar_int] :
( ! [A: com,B: list_char > int] : ( P @ ( produc5595214716300948949ar_int @ A @ B ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_168_prod__cases,axiom,
! [P: produc369741286924889929st_int > $o,P3: produc369741286924889929st_int] :
( ! [A: list_char > int,B: list_int] : ( P @ ( produc8650753666468850689st_int @ A @ B ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_169_prod__cases,axiom,
! [P: produc4435102495419491129nt_int > $o,P3: produc4435102495419491129nt_int] :
( ! [A: list_char > int,B: int] : ( P @ ( produc5790713362662368625nt_int @ A @ B ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_170_prod__cases,axiom,
! [P: produc699483346519933241t_char > $o,P3: produc699483346519933241t_char] :
( ! [A: int > int,B: list_char] : ( P @ ( produc4670215617849501681t_char @ A @ B ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_171_prod__cases,axiom,
! [P: produc1534919314292393896nt_nat > $o,P3: produc1534919314292393896nt_nat] :
( ! [A: int > int,B: nat] : ( P @ ( produc2630796075472406112nt_nat @ A @ B ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_172_surj__pair,axiom,
! [P3: produc2931317944591925149ar_int] :
? [X3: com,Y3: list_char > int] :
( P3
= ( produc5595214716300948949ar_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_173_surj__pair,axiom,
! [P3: produc369741286924889929st_int] :
? [X3: list_char > int,Y3: list_int] :
( P3
= ( produc8650753666468850689st_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_174_surj__pair,axiom,
! [P3: produc4435102495419491129nt_int] :
? [X3: list_char > int,Y3: int] :
( P3
= ( produc5790713362662368625nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_175_surj__pair,axiom,
! [P3: produc699483346519933241t_char] :
? [X3: int > int,Y3: list_char] :
( P3
= ( produc4670215617849501681t_char @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_176_surj__pair,axiom,
! [P3: produc1534919314292393896nt_nat] :
? [X3: int > int,Y3: nat] :
( P3
= ( produc2630796075472406112nt_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_177_old_Oprod_Oexhaust,axiom,
! [Y: produc2931317944591925149ar_int] :
~ ! [A: com,B: list_char > int] :
( Y
!= ( produc5595214716300948949ar_int @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_178_old_Oprod_Oexhaust,axiom,
! [Y: produc369741286924889929st_int] :
~ ! [A: list_char > int,B: list_int] :
( Y
!= ( produc8650753666468850689st_int @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_179_old_Oprod_Oexhaust,axiom,
! [Y: produc4435102495419491129nt_int] :
~ ! [A: list_char > int,B: int] :
( Y
!= ( produc5790713362662368625nt_int @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_180_old_Oprod_Oexhaust,axiom,
! [Y: produc699483346519933241t_char] :
~ ! [A: int > int,B: list_char] :
( Y
!= ( produc4670215617849501681t_char @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_181_old_Oprod_Oexhaust,axiom,
! [Y: produc1534919314292393896nt_nat] :
~ ! [A: int > int,B: nat] :
( Y
!= ( produc2630796075472406112nt_nat @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_182_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_183_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A: nat,B: nat] :
( ( P @ A @ B )
= ( P @ B @ A ) )
=> ( ! [A: nat] : ( P @ A @ zero_zero_nat )
=> ( ! [A: nat,B: nat] :
( ( P @ A @ B )
=> ( P @ A @ ( plus_plus_nat @ A @ B ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_184_add__0__iff,axiom,
! [B2: int,A2: int] :
( ( B2
= ( plus_plus_int @ B2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_185_add__0__iff,axiom,
! [B2: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_186_Seq_OIH_I1_J,axiom,
! [Stk: list_int] : ( star_P707599355569300323st_int @ ( exec1 @ ( ccomp @ c_1 ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ s_1 @ Stk ) ) @ ( produc5086643055186798020st_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c_1 ) ) ) @ ( produc8650753666468850689st_int @ s_2 @ Stk ) ) ) ).
% Seq.IH(1)
thf(fact_187_exec__append__trans,axiom,
! [P: list_instr,S: list_char > int,Stk: list_int,I2: int,S2: list_char > int,Stk2: list_int,P2: list_instr,I4: int,S6: list_char > int,Stk4: list_int,J2: int] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) )
=> ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I2 )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P2 ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I4 @ ( produc8650753666468850689st_int @ S6 @ Stk4 ) ) )
=> ( ( J2
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I4 ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P2 ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ J2 @ ( produc8650753666468850689st_int @ S6 @ Stk4 ) ) ) ) ) ) ) ).
% exec_append_trans
thf(fact_188_gen__length__def,axiom,
( gen_length_m_instr
= ( ^ [N2: nat,Xs3: list_m_instr] : ( plus_plus_nat @ N2 @ ( size_s5914166952900237872_instr @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_189_gen__length__def,axiom,
( gen_length_instr
= ( ^ [N2: nat,Xs3: list_instr] : ( plus_plus_nat @ N2 @ ( size_size_list_instr @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_190_gen__length__def,axiom,
( gen_length_list_char
= ( ^ [N2: nat,Xs3: list_list_char] : ( plus_plus_nat @ N2 @ ( size_s356637359517785349t_char @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_191_gen__length__def,axiom,
( gen_length_int
= ( ^ [N2: nat,Xs3: list_int] : ( plus_plus_nat @ N2 @ ( size_size_list_int @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_192_length__code,axiom,
( size_s5914166952900237872_instr
= ( gen_length_m_instr @ zero_zero_nat ) ) ).
% length_code
thf(fact_193_length__code,axiom,
( size_size_list_instr
= ( gen_length_instr @ zero_zero_nat ) ) ).
% length_code
thf(fact_194_length__code,axiom,
( size_s356637359517785349t_char
= ( gen_length_list_char @ zero_zero_nat ) ) ).
% length_code
thf(fact_195_length__code,axiom,
( size_size_list_int
= ( gen_length_int @ zero_zero_nat ) ) ).
% length_code
thf(fact_196_exec__appendL__if,axiom,
! [P2: list_instr,I: int,P: list_instr,S: list_char > int,Stk: list_int,J: int,S2: list_char > int,Stk2: list_int,I2: int] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P2 ) ) @ I )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P2 ) ) ) @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) )
=> ( ( I2
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P2 ) ) @ J ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P2 @ P ) ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) ) ) ) ) ).
% exec_appendL_if
thf(fact_197_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_198_add__le__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_199_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_200_add__le__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_201_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_202_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_203_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_204_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_205_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_206_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_207_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_208_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_209_add__diff__cancel__right_H,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_210_add__diff__cancel__right_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_211_add__diff__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_212_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_213_add__diff__cancel__left_H,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_214_add__diff__cancel__left_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_215_add__diff__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( minus_minus_int @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_216_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_217_diff__add__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% diff_add_cancel
thf(fact_218_add__diff__cancel,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel
thf(fact_219_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_220_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_221_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_222_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_223_le__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_224_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_225_le__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_226_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_227_add__le__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_228_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_229_add__le__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_230_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_231_diff__ge__0__iff__ge,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ B2 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_232_diff__add__zero,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_233_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_234_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_235_Seq_OIH_I2_J,axiom,
! [Stk: list_int] : ( star_P707599355569300323st_int @ ( exec1 @ ( ccomp @ c_2 ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ s_2 @ Stk ) ) @ ( produc5086643055186798020st_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c_2 ) ) ) @ ( produc8650753666468850689st_int @ s_3 @ Stk ) ) ) ).
% Seq.IH(2)
thf(fact_236_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_237_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A2 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_238_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_239_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_240_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_241_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_242_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_243_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_244_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_245_le__add__diff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_246_diff__add,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% diff_add
thf(fact_247_le__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% le_diff_eq
thf(fact_248_diff__le__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_le_eq
thf(fact_249_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_250_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_251_diff__eq__diff__eq,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( A2 = B2 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_252_diff__mono,axiom,
! [A2: int,B2: int,D: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_253_diff__left__mono,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_254_diff__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_255_diff__eq__diff__less__eq,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A2 @ B2 )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_256_diff__right__commute,axiom,
! [A2: int,C: int,B2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_257_diff__right__commute,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_258_verit__la__generic,axiom,
! [A2: int,X: int] :
( ( ord_less_eq_int @ A2 @ X )
| ( A2 = X )
| ( ord_less_eq_int @ X @ A2 ) ) ).
% verit_la_generic
thf(fact_259_verit__la__disequality,axiom,
! [A2: int,B2: int] :
( ( A2 = B2 )
| ~ ( ord_less_eq_int @ A2 @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_260_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_261_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_262_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_263_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( minus_minus_int @ A4 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_264_diff__diff__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_265_diff__diff__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_266_add__implies__diff,axiom,
! [C: int,B2: int,A2: int] :
( ( ( plus_plus_int @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus_int @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_267_add__implies__diff,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_268_diff__add__eq__diff__diff__swap,axiom,
! [A2: int,B2: int,C: int] :
( ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_269_diff__add__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ).
% diff_add_eq
thf(fact_270_diff__diff__eq2,axiom,
! [A2: int,B2: int,C: int] :
( ( minus_minus_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ).
% diff_diff_eq2
thf(fact_271_add__diff__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% add_diff_eq
thf(fact_272_eq__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( A2
= ( minus_minus_int @ C @ B2 ) )
= ( ( plus_plus_int @ A2 @ B2 )
= C ) ) ).
% eq_diff_eq
thf(fact_273_diff__eq__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= C )
= ( A2
= ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_eq_eq
thf(fact_274_group__cancel_Osub1,axiom,
! [A3: int,K: int,A2: int,B2: int] :
( ( A3
= ( plus_plus_int @ K @ A2 ) )
=> ( ( minus_minus_int @ A3 @ B2 )
= ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.sub1
thf(fact_275_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_276_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_277_add__le__imp__le__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_278_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_279_add__le__imp__le__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_280_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_281_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_282_add__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_283_add__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_284_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C5: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C5 ) ) ) ).
% less_eqE
thf(fact_285_add__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_286_add__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_287_add__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_288_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_289_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_290_add__mono__thms__linordered__semiring_I1_J,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 @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_291_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_292_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_293_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_294_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_295_int__diff__cases,axiom,
! [Z: int] :
~ ! [M2: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_296_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_297_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_298_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_299_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_300_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_301_add__nonpos__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_302_add__nonpos__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_303_add__nonneg__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_304_add__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_305_add__increasing2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_306_add__increasing2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_307_add__decreasing2,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_308_add__decreasing2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_309_add__increasing,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_310_add__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_311_add__decreasing,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_312_add__decreasing,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_313_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_314_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_315_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_316_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_317_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W: int,Z2: int] :
? [N2: nat] :
( Z2
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_318_le__add__diff__inverse2,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_319_le__add__diff__inverse2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_320_le__add__diff__inverse,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_321_le__add__diff__inverse,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_322_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_323_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_324_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_325_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_326_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_327_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_328_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_329_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_330_imp__le__cong,axiom,
! [X: int,X4: int,P: $o,P2: $o] :
( ( X = X4 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_331_conj__le__cong,axiom,
! [X: int,X4: int,P: $o,P2: $o] :
( ( X = X4 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X4 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_332_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_333_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_334_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_335_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_336_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_337_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_338_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_339_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_340_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_341_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_342_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_343_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_344_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_345_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_346_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_347_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_348_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_349_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_350_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_351_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_352_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_353_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_354_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_355_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_356_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_357_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_358_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_359_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_360_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_361_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_362_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_363_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_364_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_365_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_366_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_367_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_368_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_369_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_370_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_371_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_372_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_373_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_374_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_375_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_376_add__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 @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_377_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_378_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_379_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_380_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_381_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_382_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_383_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_384_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_385_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_386_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_387_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_388_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_389_nle__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_390_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_391_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_392_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_393_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_394_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_395_ord__eq__le__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_396_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_397_ord__le__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_398_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_399_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_400_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_401_order_Otrans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_402_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_403_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_404_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_405_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( P @ A @ B ) )
=> ( ! [A: int,B: int] :
( ( P @ B @ A )
=> ( P @ A @ B ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_406_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( P @ A @ B ) )
=> ( ! [A: nat,B: nat] :
( ( P @ B @ A )
=> ( P @ A @ B ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_407_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_408_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_409_dual__order_Oantisym,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_410_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_411_dual__order_Otrans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_412_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_413_antisym,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_414_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_415_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_416_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_417_order__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_418_order__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_419_order__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_420_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_421_order__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_422_order__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_423_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_424_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_425_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_426_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_427_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_428_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_429_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_430_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_431_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_432_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_433_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_434_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_435_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_436_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_437_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_438_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_439_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_440_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_441_bcomp__correct,axiom,
! [I: int,B2: bexp,F: $o,S: list_char > int,Stk: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( bcomp @ ( produc4047900504771817624_o_int @ B2 @ ( product_Pair_o_int @ F @ I ) ) ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk ) )
@ ( produc5086643055186798020st_int
@ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B2 @ ( product_Pair_o_int @ F @ I ) ) ) ) )
@ ( if_int
@ ( F
= ( bval @ B2 @ S ) )
@ I
@ zero_zero_int ) )
@ ( produc8650753666468850689st_int @ S @ Stk ) ) ) ) ).
% bcomp_correct
thf(fact_442_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_443_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_444_exec1I,axiom,
! [C2: produc6425607678544837394st_int,P: list_instr,I: int,S: list_char > int,Stk: list_int] :
( ( C2
= ( iexec @ ( inth_instr @ P @ I ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) )
=> ( exec1 @ P @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ C2 ) ) ) ) ).
% exec1I
thf(fact_445_inth__append,axiom,
! [I: int,Xs: list_m_instr,Ys: list_m_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s5914166952900237872_instr @ Xs ) ) )
=> ( ( inth_m_instr @ ( append_m_instr @ Xs @ Ys ) @ I )
= ( inth_m_instr @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s5914166952900237872_instr @ Xs ) ) )
=> ( ( inth_m_instr @ ( append_m_instr @ Xs @ Ys ) @ I )
= ( inth_m_instr @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s5914166952900237872_instr @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_446_inth__append,axiom,
! [I: int,Xs: list_instr,Ys: list_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I )
= ( inth_instr @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I )
= ( inth_instr @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_447_inth__append,axiom,
! [I: int,Xs: list_list_char,Ys: list_list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I )
= ( inth_list_char @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I )
= ( inth_list_char @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_448_inth__append,axiom,
! [I: int,Xs: list_int,Ys: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys ) @ I )
= ( inth_int @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys ) @ I )
= ( inth_int @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_449_pred__subset__eq2,axiom,
! [R3: set_Pr6231067571260677117ar_int,S7: set_Pr6231067571260677117ar_int] :
( ( ord_le4000858847371626730_int_o
@ ^ [X2: com,Y2: list_char > int] : ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ X2 @ Y2 ) @ R3 )
@ ^ [X2: com,Y2: list_char > int] : ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ X2 @ Y2 ) @ S7 ) )
= ( ord_le6049463789062751133ar_int @ R3 @ S7 ) ) ).
% pred_subset_eq2
thf(fact_450_pred__subset__eq2,axiom,
! [R3: set_Pr5580431329626101545st_int,S7: set_Pr5580431329626101545st_int] :
( ( ord_le4781006829409452478_int_o
@ ^ [X2: list_char > int,Y2: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y2 ) @ R3 )
@ ^ [X2: list_char > int,Y2: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y2 ) @ S7 ) )
= ( ord_le5738032636065246921st_int @ R3 @ S7 ) ) ).
% pred_subset_eq2
thf(fact_451_pred__subset__eq2,axiom,
! [R3: set_Pr5290250543098163609nt_int,S7: set_Pr5290250543098163609nt_int] :
( ( ord_le8207994782381187918_int_o
@ ^ [X2: list_char > int,Y2: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y2 ) @ R3 )
@ ^ [X2: list_char > int,Y2: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y2 ) @ S7 ) )
= ( ord_le5108646760900237625nt_int @ R3 @ S7 ) ) ).
% pred_subset_eq2
thf(fact_452_pred__subset__eq2,axiom,
! [R3: set_Pr7487058302695755673t_char,S7: set_Pr7487058302695755673t_char] :
( ( ord_le7307311761407050830char_o
@ ^ [X2: int > int,Y2: list_char] : ( member9086947928420754658t_char @ ( produc4670215617849501681t_char @ X2 @ Y2 ) @ R3 )
@ ^ [X2: int > int,Y2: list_char] : ( member9086947928420754658t_char @ ( produc4670215617849501681t_char @ X2 @ Y2 ) @ S7 ) )
= ( ord_le7305454520497829689t_char @ R3 @ S7 ) ) ).
% pred_subset_eq2
thf(fact_453_pred__subset__eq2,axiom,
! [R3: set_Pr9097919650187362824nt_nat,S7: set_Pr9097919650187362824nt_nat] :
( ( ord_le2139789345526271135_nat_o
@ ^ [X2: int > int,Y2: nat] : ( member561943782142090449nt_nat @ ( produc2630796075472406112nt_nat @ X2 @ Y2 ) @ R3 )
@ ^ [X2: int > int,Y2: nat] : ( member561943782142090449nt_nat @ ( produc2630796075472406112nt_nat @ X2 @ Y2 ) @ S7 ) )
= ( ord_le3682719578369841576nt_nat @ R3 @ S7 ) ) ).
% pred_subset_eq2
thf(fact_454_pred__equals__eq2,axiom,
! [R3: set_Pr6231067571260677117ar_int,S7: set_Pr6231067571260677117ar_int] :
( ( ( ^ [X2: com,Y2: list_char > int] : ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: com,Y2: list_char > int] : ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ X2 @ Y2 ) @ S7 ) ) )
= ( R3 = S7 ) ) ).
% pred_equals_eq2
thf(fact_455_pred__equals__eq2,axiom,
! [R3: set_Pr5580431329626101545st_int,S7: set_Pr5580431329626101545st_int] :
( ( ( ^ [X2: list_char > int,Y2: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: list_char > int,Y2: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y2 ) @ S7 ) ) )
= ( R3 = S7 ) ) ).
% pred_equals_eq2
thf(fact_456_pred__equals__eq2,axiom,
! [R3: set_Pr5290250543098163609nt_int,S7: set_Pr5290250543098163609nt_int] :
( ( ( ^ [X2: list_char > int,Y2: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: list_char > int,Y2: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y2 ) @ S7 ) ) )
= ( R3 = S7 ) ) ).
% pred_equals_eq2
thf(fact_457_pred__equals__eq2,axiom,
! [R3: set_Pr7487058302695755673t_char,S7: set_Pr7487058302695755673t_char] :
( ( ( ^ [X2: int > int,Y2: list_char] : ( member9086947928420754658t_char @ ( produc4670215617849501681t_char @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: int > int,Y2: list_char] : ( member9086947928420754658t_char @ ( produc4670215617849501681t_char @ X2 @ Y2 ) @ S7 ) ) )
= ( R3 = S7 ) ) ).
% pred_equals_eq2
thf(fact_458_pred__equals__eq2,axiom,
! [R3: set_Pr9097919650187362824nt_nat,S7: set_Pr9097919650187362824nt_nat] :
( ( ( ^ [X2: int > int,Y2: nat] : ( member561943782142090449nt_nat @ ( produc2630796075472406112nt_nat @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: int > int,Y2: nat] : ( member561943782142090449nt_nat @ ( produc2630796075472406112nt_nat @ X2 @ Y2 ) @ S7 ) ) )
= ( R3 = S7 ) ) ).
% pred_equals_eq2
thf(fact_459_enumerate__append__eq,axiom,
! [N: nat,Xs: list_m_instr,Ys: list_m_instr] :
( ( enumerate_m_instr @ N @ ( append_m_instr @ Xs @ Ys ) )
= ( append8321214904294969048_instr @ ( enumerate_m_instr @ N @ Xs ) @ ( enumerate_m_instr @ ( plus_plus_nat @ N @ ( size_s5914166952900237872_instr @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_460_enumerate__append__eq,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( enumerate_instr @ N @ ( append_instr @ Xs @ Ys ) )
= ( append227978101926261161_instr @ ( enumerate_instr @ N @ Xs ) @ ( enumerate_instr @ ( plus_plus_nat @ N @ ( size_size_list_instr @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_461_enumerate__append__eq,axiom,
! [N: nat,Xs: list_list_char,Ys: list_list_char] :
( ( enumerate_list_char @ N @ ( append_list_char @ Xs @ Ys ) )
= ( append2138297074218479021t_char @ ( enumerate_list_char @ N @ Xs ) @ ( enumerate_list_char @ ( plus_plus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_462_enumerate__append__eq,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( enumerate_int @ N @ ( append_int @ Xs @ Ys ) )
= ( append6031344391939132024at_int @ ( enumerate_int @ N @ Xs ) @ ( enumerate_int @ ( plus_plus_nat @ N @ ( size_size_list_int @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_463_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_464_add__less__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_465_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_466_add__less__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_467_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_468_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_469_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_470_length__enumerate,axiom,
! [N: nat,Xs: list_m_instr] :
( ( size_s5049043960463095215_instr @ ( enumerate_m_instr @ N @ Xs ) )
= ( size_s5914166952900237872_instr @ Xs ) ) ).
% length_enumerate
thf(fact_471_length__enumerate,axiom,
! [N: nat,Xs: list_instr] :
( ( size_s142077898034696872_instr @ ( enumerate_instr @ N @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_enumerate
thf(fact_472_length__enumerate,axiom,
! [N: nat,Xs: list_list_char] :
( ( size_s3402568042334145540t_char @ ( enumerate_list_char @ N @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_enumerate
thf(fact_473_length__enumerate,axiom,
! [N: nat,Xs: list_int] :
( ( size_s2970893825323803983at_int @ ( enumerate_int @ N @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_enumerate
thf(fact_474_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_475_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_476_less__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_477_less__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_478_less__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_479_less__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_480_add__less__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_481_add__less__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_482_add__less__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_483_add__less__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_484_diff__gt__0__iff__gt,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( ord_less_int @ B2 @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_485_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_486_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_487_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_488_pred__subset__eq,axiom,
! [R3: set_list_char,S7: set_list_char] :
( ( ord_le4796328588573674190char_o
@ ^ [X2: list_char] : ( member_list_char @ X2 @ R3 )
@ ^ [X2: list_char] : ( member_list_char @ X2 @ S7 ) )
= ( ord_le1207710660481778807t_char @ R3 @ S7 ) ) ).
% pred_subset_eq
thf(fact_489_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_490_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_491_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_492_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_493_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_494_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_495_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_496_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_497_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_498_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_499_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_500_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_501_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_502_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_503_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_504_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_505_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_506_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_507_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_508_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_509_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_510_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_511_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_512_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_513_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_514_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_515_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_516_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_517_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_518_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_519_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_520_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_521_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_522_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_523_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_524_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_525_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_526_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_527_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_528_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_529_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_530_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_531_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_532_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_533_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P5: nat > $o] :
? [N2: nat] :
( ( P5 @ N2 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ~ ( P5 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_534_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( P @ A @ B ) )
=> ( ! [A: int] : ( P @ A @ A )
=> ( ! [A: int,B: int] :
( ( P @ B @ A )
=> ( P @ A @ B ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_535_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( P @ A @ B ) )
=> ( ! [A: nat] : ( P @ A @ A )
=> ( ! [A: nat,B: nat] :
( ( P @ B @ A )
=> ( P @ A @ B ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_536_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_537_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_538_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_539_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_540_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_541_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_542_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_543_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_544_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_545_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_546_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_547_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_548_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_549_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_550_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_551_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_552_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_553_order__less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_554_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_555_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_556_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_557_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_558_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_559_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_560_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_561_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_562_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_563_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_564_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_565_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_566_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_567_order__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_568_order__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_569_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_570_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_571_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_572_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_573_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_574_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_575_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_576_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_577_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_578_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_579_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_580_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_581_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_582_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_583_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_584_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_585_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_586_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_587_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_588_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_589_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_590_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_591_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_592_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_593_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_594_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_595_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_596_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_597_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_598_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_599_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_600_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_601_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_602_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_603_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_604_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_605_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_606_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_607_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_608_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_609_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_610_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_611_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_612_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_613_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_614_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_615_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_616_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_617_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_618_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_619_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_620_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_621_order__neq__le__trans,axiom,
! [A2: int,B2: int] :
( ( A2 != B2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_622_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_623_order__le__neq__trans,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_624_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_625_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_626_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_627_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_628_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_629_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_630_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_631_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_632_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_633_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_634_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_635_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_636_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_637_order_Ostrict__implies__order,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_638_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_639_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_640_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_641_dual__order_Ostrict__trans2,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_642_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_643_dual__order_Ostrict__trans1,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_644_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_645_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_646_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_647_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_648_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_649_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_650_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_651_order_Ostrict__trans2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_652_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_653_order_Ostrict__trans1,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_654_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_655_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_656_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_657_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_658_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_659_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_660_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_661_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ~ ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_662_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_663_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_664_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_665_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_666_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_667_nless__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_int @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_668_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_669_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_670_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_671_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_672_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_673_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_674_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_675_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_676_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_677_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_678_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_679_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_680_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_681_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_682_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_683_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_684_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_685_add__less__imp__less__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_686_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_687_add__less__imp__less__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_688_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_689_add__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_690_add__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_691_add__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_692_add__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_693_add__strict__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_694_add__strict__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_695_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_696_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_697_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_698_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_699_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_700_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_701_diff__strict__mono,axiom,
! [A2: int,B2: int,D: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_702_diff__eq__diff__less,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A2 @ B2 )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_703_diff__strict__left__mono,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_704_diff__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_705_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_706_add__less__le__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_707_add__less__le__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_708_add__le__less__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_709_add__le__less__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_710_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_711_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_712_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_713_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_714_pos__add__strict,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_715_pos__add__strict,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_716_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ! [C5: nat] :
( ( B2
= ( plus_plus_nat @ A2 @ C5 ) )
=> ( C5 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_717_add__pos__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_718_add__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_719_add__neg__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_720_add__neg__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_721_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_722_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_723_less__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% less_diff_eq
thf(fact_724_diff__less__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_less_eq
thf(fact_725_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: int,B2: int] :
( ~ ( ord_less_int @ A2 @ B2 )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_726_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ~ ( ord_less_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_727_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_728_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_729_add__strict__increasing2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_730_add__strict__increasing2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_731_add__strict__increasing,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_732_add__strict__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_733_add__pos__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_734_add__pos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_735_add__nonpos__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_736_add__nonpos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_737_add__nonneg__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_738_add__nonneg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_739_add__neg__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_740_add__neg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_741_subrelI,axiom,
! [R: set_Pr6231067571260677117ar_int,S: set_Pr6231067571260677117ar_int] :
( ! [X3: com,Y3: list_char > int] :
( ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ X3 @ Y3 ) @ R )
=> ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ X3 @ Y3 ) @ S ) )
=> ( ord_le6049463789062751133ar_int @ R @ S ) ) ).
% subrelI
thf(fact_742_subrelI,axiom,
! [R: set_Pr5580431329626101545st_int,S: set_Pr5580431329626101545st_int] :
( ! [X3: list_char > int,Y3: list_int] :
( ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X3 @ Y3 ) @ R )
=> ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X3 @ Y3 ) @ S ) )
=> ( ord_le5738032636065246921st_int @ R @ S ) ) ).
% subrelI
thf(fact_743_subrelI,axiom,
! [R: set_Pr5290250543098163609nt_int,S: set_Pr5290250543098163609nt_int] :
( ! [X3: list_char > int,Y3: int] :
( ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X3 @ Y3 ) @ R )
=> ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X3 @ Y3 ) @ S ) )
=> ( ord_le5108646760900237625nt_int @ R @ S ) ) ).
% subrelI
thf(fact_744_subrelI,axiom,
! [R: set_Pr7487058302695755673t_char,S: set_Pr7487058302695755673t_char] :
( ! [X3: int > int,Y3: list_char] :
( ( member9086947928420754658t_char @ ( produc4670215617849501681t_char @ X3 @ Y3 ) @ R )
=> ( member9086947928420754658t_char @ ( produc4670215617849501681t_char @ X3 @ Y3 ) @ S ) )
=> ( ord_le7305454520497829689t_char @ R @ S ) ) ).
% subrelI
thf(fact_745_subrelI,axiom,
! [R: set_Pr9097919650187362824nt_nat,S: set_Pr9097919650187362824nt_nat] :
( ! [X3: int > int,Y3: nat] :
( ( member561943782142090449nt_nat @ ( produc2630796075472406112nt_nat @ X3 @ Y3 ) @ R )
=> ( member561943782142090449nt_nat @ ( produc2630796075472406112nt_nat @ X3 @ Y3 ) @ S ) )
=> ( ord_le3682719578369841576nt_nat @ R @ S ) ) ).
% subrelI
thf(fact_746_is__num__normalize_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_747_int__ops_I6_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_748_to__m__state_Osimps,axiom,
( to_m_state
= ( ^ [Xs3: list_list_char,S5: list_char > int,A4: int] :
( if_int
@ ( ( ord_less_int @ zero_zero_int @ A4 )
& ( ord_less_eq_int @ A4 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs3 ) ) ) )
@ ( S5 @ ( inth_list_char @ Xs3 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs3 ) ) @ A4 ) ) )
@ zero_zero_int ) ) ) ).
% to_m_state.simps
thf(fact_749_to__m__state_Oelims,axiom,
! [X: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X @ Xa @ Xb )
= Y )
=> ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y = zero_zero_int ) ) ) ) ).
% to_m_state.elims
thf(fact_750_bcomp_Osimps_I3_J,axiom,
! [B_1: bexp,B_2: bexp,F: $o,I: int] :
( ( bcomp @ ( produc4047900504771817624_o_int @ ( and @ B_1 @ B_2 ) @ ( product_Pair_o_int @ F @ I ) ) )
= ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B_1 @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B_2 @ ( product_Pair_o_int @ F @ I ) ) ) ) ) @ ( if_int @ F @ zero_zero_int @ I ) ) ) ) ) @ ( bcomp @ ( produc4047900504771817624_o_int @ B_2 @ ( product_Pair_o_int @ F @ I ) ) ) ) ) ).
% bcomp.simps(3)
thf(fact_751_subsetI,axiom,
! [A3: set_list_char,B4: set_list_char] :
( ! [X3: list_char] :
( ( member_list_char @ X3 @ A3 )
=> ( member_list_char @ X3 @ B4 ) )
=> ( ord_le1207710660481778807t_char @ A3 @ B4 ) ) ).
% subsetI
thf(fact_752_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_753_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_754_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_755_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_756_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_757_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_758_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_759_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_760_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_761_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_762_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_763_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_764_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_765_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_766_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_767_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_768_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_769_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_770_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_771_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_772_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_773_length__induct,axiom,
! [P: list_m_instr > $o,Xs: list_m_instr] :
( ! [Xs2: list_m_instr] :
( ! [Ys2: list_m_instr] :
( ( ord_less_nat @ ( size_s5914166952900237872_instr @ Ys2 ) @ ( size_s5914166952900237872_instr @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_774_length__induct,axiom,
! [P: list_instr > $o,Xs: list_instr] :
( ! [Xs2: list_instr] :
( ! [Ys2: list_instr] :
( ( ord_less_nat @ ( size_size_list_instr @ Ys2 ) @ ( size_size_list_instr @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_775_length__induct,axiom,
! [P: list_list_char > $o,Xs: list_list_char] :
( ! [Xs2: list_list_char] :
( ! [Ys2: list_list_char] :
( ( ord_less_nat @ ( size_s356637359517785349t_char @ Ys2 ) @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_776_length__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ! [Xs2: list_int] :
( ! [Ys2: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_777_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_778_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_779_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_780_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_781_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_782_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_783_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_784_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_785_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_786_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_787_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_788_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_789_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_790_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_791_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_792_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_793_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_794_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_795_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_796_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_797_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_798_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_799_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_800_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_801_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_802_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_803_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_804_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ( ( ord_less_nat @ A2 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_805_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_806_in__mono,axiom,
! [A3: set_list_char,B4: set_list_char,X: list_char] :
( ( ord_le1207710660481778807t_char @ A3 @ B4 )
=> ( ( member_list_char @ X @ A3 )
=> ( member_list_char @ X @ B4 ) ) ) ).
% in_mono
thf(fact_807_subsetD,axiom,
! [A3: set_list_char,B4: set_list_char,C: list_char] :
( ( ord_le1207710660481778807t_char @ A3 @ B4 )
=> ( ( member_list_char @ C @ A3 )
=> ( member_list_char @ C @ B4 ) ) ) ).
% subsetD
thf(fact_808_subset__eq,axiom,
( ord_le1207710660481778807t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
! [X2: list_char] :
( ( member_list_char @ X2 @ A6 )
=> ( member_list_char @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_809_subset__iff,axiom,
( ord_le1207710660481778807t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
! [T4: list_char] :
( ( member_list_char @ T4 @ A6 )
=> ( member_list_char @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_810_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_811_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_812_Collect__subset,axiom,
! [A3: set_list_char,P: list_char > $o] :
( ord_le1207710660481778807t_char
@ ( collect_list_char
@ ^ [X2: list_char] :
( ( member_list_char @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_813_less__eq__set__def,axiom,
( ord_le1207710660481778807t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
( ord_le4796328588573674190char_o
@ ^ [X2: list_char] : ( member_list_char @ X2 @ A6 )
@ ^ [X2: list_char] : ( member_list_char @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_814_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_815_to__m__state_Opelims,axiom,
! [X: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) )
=> ~ ( ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y = zero_zero_int ) ) )
=> ~ ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) ) ) ) ) ).
% to_m_state.pelims
thf(fact_816_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K3 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_817_Diff__iff,axiom,
! [C: list_char,A3: set_list_char,B4: set_list_char] :
( ( member_list_char @ C @ ( minus_269717477326218608t_char @ A3 @ B4 ) )
= ( ( member_list_char @ C @ A3 )
& ~ ( member_list_char @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_818_DiffI,axiom,
! [C: list_char,A3: set_list_char,B4: set_list_char] :
( ( member_list_char @ C @ A3 )
=> ( ~ ( member_list_char @ C @ B4 )
=> ( member_list_char @ C @ ( minus_269717477326218608t_char @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_819_psubset__imp__ex__mem,axiom,
! [A3: set_list_char,B4: set_list_char] :
( ( ord_le5286566838938544235t_char @ A3 @ B4 )
=> ? [B: list_char] : ( member_list_char @ B @ ( minus_269717477326218608t_char @ B4 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_820_minus__set__def,axiom,
( minus_269717477326218608t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
( collect_list_char
@ ( minus_6821098565838606101char_o
@ ^ [X2: list_char] : ( member_list_char @ X2 @ A6 )
@ ^ [X2: list_char] : ( member_list_char @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_821_set__diff__eq,axiom,
( minus_269717477326218608t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
( collect_list_char
@ ^ [X2: list_char] :
( ( member_list_char @ X2 @ A6 )
& ~ ( member_list_char @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_822_DiffD2,axiom,
! [C: list_char,A3: set_list_char,B4: set_list_char] :
( ( member_list_char @ C @ ( minus_269717477326218608t_char @ A3 @ B4 ) )
=> ~ ( member_list_char @ C @ B4 ) ) ).
% DiffD2
thf(fact_823_DiffD1,axiom,
! [C: list_char,A3: set_list_char,B4: set_list_char] :
( ( member_list_char @ C @ ( minus_269717477326218608t_char @ A3 @ B4 ) )
=> ( member_list_char @ C @ A3 ) ) ).
% DiffD1
thf(fact_824_DiffE,axiom,
! [C: list_char,A3: set_list_char,B4: set_list_char] :
( ( member_list_char @ C @ ( minus_269717477326218608t_char @ A3 @ B4 ) )
=> ~ ( ( member_list_char @ C @ A3 )
=> ( member_list_char @ C @ B4 ) ) ) ).
% DiffE
thf(fact_825_less__set__def,axiom,
( ord_le5286566838938544235t_char
= ( ^ [A6: set_list_char,B6: set_list_char] :
( ord_less_list_char_o
@ ^ [X2: list_char] : ( member_list_char @ X2 @ A6 )
@ ^ [X2: list_char] : ( member_list_char @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_826_accp__subset,axiom,
! [R1: produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o,R22: produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o] :
( ( ord_le1819137518486948652_int_o @ R1 @ R22 )
=> ( ord_le3703957343819488690_int_o @ ( accp_P4562858270263085236nt_int @ R22 ) @ ( accp_P4562858270263085236nt_int @ R1 ) ) ) ).
% accp_subset
thf(fact_827_accp__subset,axiom,
! [R1: list_instr > list_instr > $o,R22: list_instr > list_instr > $o] :
( ( ord_le7616546472596271438nstr_o @ R1 @ R22 )
=> ( ord_le4570788554581971896nstr_o @ ( accp_list_instr @ R22 ) @ ( accp_list_instr @ R1 ) ) ) ).
% accp_subset
thf(fact_828_accp__subset,axiom,
! [R1: produc6385869745582161021t_char > produc6385869745582161021t_char > $o,R22: produc6385869745582161021t_char > produc6385869745582161021t_char > $o] :
( ( ord_le1046979081363672876char_o @ R1 @ R22 )
=> ( ord_le1172707689240788914char_o @ ( accp_P6759666029860677300t_char @ R22 ) @ ( accp_P6759666029860677300t_char @ R1 ) ) ) ).
% accp_subset
thf(fact_829_accp__subset__induct,axiom,
! [D3: produc4189061985984568957nt_int > $o,R3: produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o,X: produc4189061985984568957nt_int,P: produc4189061985984568957nt_int > $o] :
( ( ord_le3703957343819488690_int_o @ D3 @ ( accp_P4562858270263085236nt_int @ R3 ) )
=> ( ! [X3: produc4189061985984568957nt_int,Z4: produc4189061985984568957nt_int] :
( ( D3 @ X3 )
=> ( ( R3 @ Z4 @ X3 )
=> ( D3 @ Z4 ) ) )
=> ( ( D3 @ X )
=> ( ! [X3: produc4189061985984568957nt_int] :
( ( D3 @ X3 )
=> ( ! [Z5: produc4189061985984568957nt_int] :
( ( R3 @ Z5 @ X3 )
=> ( P @ Z5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_830_accp__subset__induct,axiom,
! [D3: list_instr > $o,R3: list_instr > list_instr > $o,X: list_instr,P: list_instr > $o] :
( ( ord_le4570788554581971896nstr_o @ D3 @ ( accp_list_instr @ R3 ) )
=> ( ! [X3: list_instr,Z4: list_instr] :
( ( D3 @ X3 )
=> ( ( R3 @ Z4 @ X3 )
=> ( D3 @ Z4 ) ) )
=> ( ( D3 @ X )
=> ( ! [X3: list_instr] :
( ( D3 @ X3 )
=> ( ! [Z5: list_instr] :
( ( R3 @ Z5 @ X3 )
=> ( P @ Z5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_831_accp__subset__induct,axiom,
! [D3: produc6385869745582161021t_char > $o,R3: produc6385869745582161021t_char > produc6385869745582161021t_char > $o,X: produc6385869745582161021t_char,P: produc6385869745582161021t_char > $o] :
( ( ord_le1172707689240788914char_o @ D3 @ ( accp_P6759666029860677300t_char @ R3 ) )
=> ( ! [X3: produc6385869745582161021t_char,Z4: produc6385869745582161021t_char] :
( ( D3 @ X3 )
=> ( ( R3 @ Z4 @ X3 )
=> ( D3 @ Z4 ) ) )
=> ( ( D3 @ X )
=> ( ! [X3: produc6385869745582161021t_char] :
( ( D3 @ X3 )
=> ( ! [Z5: produc6385869745582161021t_char] :
( ( R3 @ Z5 @ X3 )
=> ( P @ Z5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_832_inth__map,axiom,
! [I: int,Xs: list_m_instr,F: m_instr > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s5914166952900237872_instr @ Xs ) ) )
=> ( ( inth_instr @ ( map_m_instr_instr @ F @ Xs ) @ I )
= ( F @ ( inth_m_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_833_inth__map,axiom,
! [I: int,Xs: list_m_instr,F: m_instr > m_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s5914166952900237872_instr @ Xs ) ) )
=> ( ( inth_m_instr @ ( map_m_instr_m_instr @ F @ Xs ) @ I )
= ( F @ ( inth_m_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_834_inth__map,axiom,
! [I: int,Xs: list_instr,F: instr > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( map_instr_instr @ F @ Xs ) @ I )
= ( F @ ( inth_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_835_inth__map,axiom,
! [I: int,Xs: list_instr,F: instr > m_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_m_instr @ ( map_instr_m_instr @ F @ Xs ) @ I )
= ( F @ ( inth_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_836_inth__map,axiom,
! [I: int,Xs: list_int,F: int > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_instr @ ( map_int_instr @ F @ Xs ) @ I )
= ( F @ ( inth_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_837_inth__map,axiom,
! [I: int,Xs: list_int,F: int > m_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_m_instr @ ( map_int_m_instr @ F @ Xs ) @ I )
= ( F @ ( inth_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_838_inth__map,axiom,
! [I: int,Xs: list_m_instr,F: m_instr > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s5914166952900237872_instr @ Xs ) ) )
=> ( ( inth_list_char @ ( map_m_3283148507038589619t_char @ F @ Xs ) @ I )
= ( F @ ( inth_m_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_839_inth__map,axiom,
! [I: int,Xs: list_instr,F: instr > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_list_char @ ( map_instr_list_char @ F @ Xs ) @ I )
= ( F @ ( inth_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_840_inth__map,axiom,
! [I: int,Xs: list_list_char,F: list_char > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_instr @ ( map_list_char_instr @ F @ Xs ) @ I )
= ( F @ ( inth_list_char @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_841_inth__map,axiom,
! [I: int,Xs: list_list_char,F: list_char > m_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_m_instr @ ( map_li1986238532096774195_instr @ F @ Xs ) @ I )
= ( F @ ( inth_list_char @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_842_complete__interval,axiom,
! [A2: int,B2: int,P: int > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C5: int] :
( ( ord_less_eq_int @ A2 @ C5 )
& ( ord_less_eq_int @ C5 @ B2 )
& ! [X5: int] :
( ( ( ord_less_eq_int @ A2 @ X5 )
& ( ord_less_int @ X5 @ C5 ) )
=> ( P @ X5 ) )
& ! [D4: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A2 @ X3 )
& ( ord_less_int @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D4 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_843_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A2 @ C5 )
& ( ord_less_eq_nat @ C5 @ B2 )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A2 @ X5 )
& ( ord_less_nat @ X5 @ C5 ) )
=> ( P @ X5 ) )
& ! [D4: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D4 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_844_length__map,axiom,
! [F: m_instr > m_instr,Xs: list_m_instr] :
( ( size_s5914166952900237872_instr @ ( map_m_instr_m_instr @ F @ Xs ) )
= ( size_s5914166952900237872_instr @ Xs ) ) ).
% length_map
thf(fact_845_length__map,axiom,
! [F: instr > m_instr,Xs: list_instr] :
( ( size_s5914166952900237872_instr @ ( map_instr_m_instr @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_846_length__map,axiom,
! [F: int > m_instr,Xs: list_int] :
( ( size_s5914166952900237872_instr @ ( map_int_m_instr @ F @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_map
thf(fact_847_length__map,axiom,
! [F: m_instr > instr,Xs: list_m_instr] :
( ( size_size_list_instr @ ( map_m_instr_instr @ F @ Xs ) )
= ( size_s5914166952900237872_instr @ Xs ) ) ).
% length_map
thf(fact_848_length__map,axiom,
! [F: instr > instr,Xs: list_instr] :
( ( size_size_list_instr @ ( map_instr_instr @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_849_length__map,axiom,
! [F: int > instr,Xs: list_int] :
( ( size_size_list_instr @ ( map_int_instr @ F @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_map
thf(fact_850_length__map,axiom,
! [F: m_instr > int,Xs: list_m_instr] :
( ( size_size_list_int @ ( map_m_instr_int @ F @ Xs ) )
= ( size_s5914166952900237872_instr @ Xs ) ) ).
% length_map
thf(fact_851_length__map,axiom,
! [F: instr > int,Xs: list_instr] :
( ( size_size_list_int @ ( map_instr_int @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_852_length__map,axiom,
! [F: int > int,Xs: list_int] :
( ( size_size_list_int @ ( map_int_int @ F @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_map
thf(fact_853_length__map,axiom,
! [F: list_char > m_instr,Xs: list_list_char] :
( ( size_s5914166952900237872_instr @ ( map_li1986238532096774195_instr @ F @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_map
thf(fact_854_map__append,axiom,
! [F: instr > instr,Xs: list_instr,Ys: list_instr] :
( ( map_instr_instr @ F @ ( append_instr @ Xs @ Ys ) )
= ( append_instr @ ( map_instr_instr @ F @ Xs ) @ ( map_instr_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_855_map__append,axiom,
! [F: instr > int,Xs: list_instr,Ys: list_instr] :
( ( map_instr_int @ F @ ( append_instr @ Xs @ Ys ) )
= ( append_int @ ( map_instr_int @ F @ Xs ) @ ( map_instr_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_856_map__append,axiom,
! [F: int > instr,Xs: list_int,Ys: list_int] :
( ( map_int_instr @ F @ ( append_int @ Xs @ Ys ) )
= ( append_instr @ ( map_int_instr @ F @ Xs ) @ ( map_int_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_857_map__append,axiom,
! [F: int > int,Xs: list_int,Ys: list_int] :
( ( map_int_int @ F @ ( append_int @ Xs @ Ys ) )
= ( append_int @ ( map_int_int @ F @ Xs ) @ ( map_int_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_858_map__append,axiom,
! [F: int > m_instr,Xs: list_int,Ys: list_int] :
( ( map_int_m_instr @ F @ ( append_int @ Xs @ Ys ) )
= ( append_m_instr @ ( map_int_m_instr @ F @ Xs ) @ ( map_int_m_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_859_map__append,axiom,
! [F: m_instr > instr,Xs: list_m_instr,Ys: list_m_instr] :
( ( map_m_instr_instr @ F @ ( append_m_instr @ Xs @ Ys ) )
= ( append_instr @ ( map_m_instr_instr @ F @ Xs ) @ ( map_m_instr_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_860_map__append,axiom,
! [F: m_instr > int,Xs: list_m_instr,Ys: list_m_instr] :
( ( map_m_instr_int @ F @ ( append_m_instr @ Xs @ Ys ) )
= ( append_int @ ( map_m_instr_int @ F @ Xs ) @ ( map_m_instr_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_861_map__append,axiom,
! [F: m_instr > m_instr,Xs: list_m_instr,Ys: list_m_instr] :
( ( map_m_instr_m_instr @ F @ ( append_m_instr @ Xs @ Ys ) )
= ( append_m_instr @ ( map_m_instr_m_instr @ F @ Xs ) @ ( map_m_instr_m_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_862_map__append,axiom,
! [F: instr > m_instr,Xs: list_instr,Ys: list_instr] :
( ( map_instr_m_instr @ F @ ( append_instr @ Xs @ Ys ) )
= ( append_m_instr @ ( map_instr_m_instr @ F @ Xs ) @ ( map_instr_m_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_863_map__append,axiom,
! [F: instr > list_char,Xs: list_instr,Ys: list_instr] :
( ( map_instr_list_char @ F @ ( append_instr @ Xs @ Ys ) )
= ( append_list_char @ ( map_instr_list_char @ F @ Xs ) @ ( map_instr_list_char @ F @ Ys ) ) ) ).
% map_append
thf(fact_864_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: instr > instr,Xs: list_instr] :
( ( ( append_instr @ Ys @ Zs )
= ( map_instr_instr @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_865_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: int > instr,Xs: list_int] :
( ( ( append_instr @ Ys @ Zs )
= ( map_int_instr @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_instr @ F @ Us2 ) )
& ( Zs
= ( map_int_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_866_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: m_instr > instr,Xs: list_m_instr] :
( ( ( append_instr @ Ys @ Zs )
= ( map_m_instr_instr @ F @ Xs ) )
= ( ? [Us2: list_m_instr,Vs2: list_m_instr] :
( ( Xs
= ( append_m_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_m_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_m_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_867_append__eq__map__conv,axiom,
! [Ys: list_int,Zs: list_int,F: instr > int,Xs: list_instr] :
( ( ( append_int @ Ys @ Zs )
= ( map_instr_int @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_int @ F @ Us2 ) )
& ( Zs
= ( map_instr_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_868_append__eq__map__conv,axiom,
! [Ys: list_int,Zs: list_int,F: int > int,Xs: list_int] :
( ( ( append_int @ Ys @ Zs )
= ( map_int_int @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_int @ F @ Us2 ) )
& ( Zs
= ( map_int_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_869_append__eq__map__conv,axiom,
! [Ys: list_int,Zs: list_int,F: m_instr > int,Xs: list_m_instr] :
( ( ( append_int @ Ys @ Zs )
= ( map_m_instr_int @ F @ Xs ) )
= ( ? [Us2: list_m_instr,Vs2: list_m_instr] :
( ( Xs
= ( append_m_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_m_instr_int @ F @ Us2 ) )
& ( Zs
= ( map_m_instr_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_870_append__eq__map__conv,axiom,
! [Ys: list_m_instr,Zs: list_m_instr,F: int > m_instr,Xs: list_int] :
( ( ( append_m_instr @ Ys @ Zs )
= ( map_int_m_instr @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_m_instr @ F @ Us2 ) )
& ( Zs
= ( map_int_m_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_871_append__eq__map__conv,axiom,
! [Ys: list_m_instr,Zs: list_m_instr,F: m_instr > m_instr,Xs: list_m_instr] :
( ( ( append_m_instr @ Ys @ Zs )
= ( map_m_instr_m_instr @ F @ Xs ) )
= ( ? [Us2: list_m_instr,Vs2: list_m_instr] :
( ( Xs
= ( append_m_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_m_instr_m_instr @ F @ Us2 ) )
& ( Zs
= ( map_m_instr_m_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_872_append__eq__map__conv,axiom,
! [Ys: list_m_instr,Zs: list_m_instr,F: instr > m_instr,Xs: list_instr] :
( ( ( append_m_instr @ Ys @ Zs )
= ( map_instr_m_instr @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_m_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_m_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_873_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: list_char > instr,Xs: list_list_char] :
( ( ( append_instr @ Ys @ Zs )
= ( map_list_char_instr @ F @ Xs ) )
= ( ? [Us2: list_list_char,Vs2: list_list_char] :
( ( Xs
= ( append_list_char @ Us2 @ Vs2 ) )
& ( Ys
= ( map_list_char_instr @ F @ Us2 ) )
& ( Zs
= ( map_list_char_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_874_map__eq__append__conv,axiom,
! [F: instr > instr,Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_875_map__eq__append__conv,axiom,
! [F: int > instr,Xs: list_int,Ys: list_instr,Zs: list_instr] :
( ( ( map_int_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_instr @ F @ Us2 ) )
& ( Zs
= ( map_int_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_876_map__eq__append__conv,axiom,
! [F: m_instr > instr,Xs: list_m_instr,Ys: list_instr,Zs: list_instr] :
( ( ( map_m_instr_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_m_instr,Vs2: list_m_instr] :
( ( Xs
= ( append_m_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_m_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_m_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_877_map__eq__append__conv,axiom,
! [F: instr > int,Xs: list_instr,Ys: list_int,Zs: list_int] :
( ( ( map_instr_int @ F @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_int @ F @ Us2 ) )
& ( Zs
= ( map_instr_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_878_map__eq__append__conv,axiom,
! [F: int > int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_int @ F @ Us2 ) )
& ( Zs
= ( map_int_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_879_map__eq__append__conv,axiom,
! [F: m_instr > int,Xs: list_m_instr,Ys: list_int,Zs: list_int] :
( ( ( map_m_instr_int @ F @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ? [Us2: list_m_instr,Vs2: list_m_instr] :
( ( Xs
= ( append_m_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_m_instr_int @ F @ Us2 ) )
& ( Zs
= ( map_m_instr_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_880_map__eq__append__conv,axiom,
! [F: int > m_instr,Xs: list_int,Ys: list_m_instr,Zs: list_m_instr] :
( ( ( map_int_m_instr @ F @ Xs )
= ( append_m_instr @ Ys @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_m_instr @ F @ Us2 ) )
& ( Zs
= ( map_int_m_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_881_map__eq__append__conv,axiom,
! [F: m_instr > m_instr,Xs: list_m_instr,Ys: list_m_instr,Zs: list_m_instr] :
( ( ( map_m_instr_m_instr @ F @ Xs )
= ( append_m_instr @ Ys @ Zs ) )
= ( ? [Us2: list_m_instr,Vs2: list_m_instr] :
( ( Xs
= ( append_m_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_m_instr_m_instr @ F @ Us2 ) )
& ( Zs
= ( map_m_instr_m_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_882_map__eq__append__conv,axiom,
! [F: instr > m_instr,Xs: list_instr,Ys: list_m_instr,Zs: list_m_instr] :
( ( ( map_instr_m_instr @ F @ Xs )
= ( append_m_instr @ Ys @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_m_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_m_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_883_map__eq__append__conv,axiom,
! [F: list_char > instr,Xs: list_list_char,Ys: list_instr,Zs: list_instr] :
( ( ( map_list_char_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_list_char,Vs2: list_list_char] :
( ( Xs
= ( append_list_char @ Us2 @ Vs2 ) )
& ( Ys
= ( map_list_char_instr @ F @ Us2 ) )
& ( Zs
= ( map_list_char_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_884_map__eq__imp__length__eq,axiom,
! [F: m_instr > m_instr,Xs: list_m_instr,G: instr > m_instr,Ys: list_instr] :
( ( ( map_m_instr_m_instr @ F @ Xs )
= ( map_instr_m_instr @ G @ Ys ) )
=> ( ( size_s5914166952900237872_instr @ Xs )
= ( size_size_list_instr @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_885_map__eq__imp__length__eq,axiom,
! [F: instr > m_instr,Xs: list_instr,G: m_instr > m_instr,Ys: list_m_instr] :
( ( ( map_instr_m_instr @ F @ Xs )
= ( map_m_instr_m_instr @ G @ Ys ) )
=> ( ( size_size_list_instr @ Xs )
= ( size_s5914166952900237872_instr @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_886_map__eq__imp__length__eq,axiom,
! [F: instr > m_instr,Xs: list_instr,G: instr > m_instr,Ys: list_instr] :
( ( ( map_instr_m_instr @ F @ Xs )
= ( map_instr_m_instr @ G @ Ys ) )
=> ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_887_map__eq__imp__length__eq,axiom,
! [F: instr > m_instr,Xs: list_instr,G: list_char > m_instr,Ys: list_list_char] :
( ( ( map_instr_m_instr @ F @ Xs )
= ( map_li1986238532096774195_instr @ G @ Ys ) )
=> ( ( size_size_list_instr @ Xs )
= ( size_s356637359517785349t_char @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_888_map__eq__imp__length__eq,axiom,
! [F: instr > m_instr,Xs: list_instr,G: int > m_instr,Ys: list_int] :
( ( ( map_instr_m_instr @ F @ Xs )
= ( map_int_m_instr @ G @ Ys ) )
=> ( ( size_size_list_instr @ Xs )
= ( size_size_list_int @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_889_map__eq__imp__length__eq,axiom,
! [F: list_char > m_instr,Xs: list_list_char,G: instr > m_instr,Ys: list_instr] :
( ( ( map_li1986238532096774195_instr @ F @ Xs )
= ( map_instr_m_instr @ G @ Ys ) )
=> ( ( size_s356637359517785349t_char @ Xs )
= ( size_size_list_instr @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_890_map__eq__imp__length__eq,axiom,
! [F: int > m_instr,Xs: list_int,G: instr > m_instr,Ys: list_instr] :
( ( ( map_int_m_instr @ F @ Xs )
= ( map_instr_m_instr @ G @ Ys ) )
=> ( ( size_size_list_int @ Xs )
= ( size_size_list_instr @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_891_inth__set,axiom,
! [I: int,Xs: list_m_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s5914166952900237872_instr @ Xs ) ) )
=> ( member_m_instr @ ( inth_m_instr @ Xs @ I ) @ ( set_m_instr2 @ Xs ) ) ) ) ).
% inth_set
thf(fact_892_inth__set,axiom,
! [I: int,Xs: list_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( member_instr @ ( inth_instr @ Xs @ I ) @ ( set_instr2 @ Xs ) ) ) ) ).
% inth_set
thf(fact_893_inth__set,axiom,
! [I: int,Xs: list_list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( member_list_char @ ( inth_list_char @ Xs @ I ) @ ( set_list_char2 @ Xs ) ) ) ) ).
% inth_set
thf(fact_894_inth__set,axiom,
! [I: int,Xs: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( member_int @ ( inth_int @ Xs @ I ) @ ( set_int2 @ Xs ) ) ) ) ).
% inth_set
thf(fact_895_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X2 )
@ ^ [X2: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X2 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_896_exec1__def,axiom,
( exec1
= ( ^ [P5: list_instr,Cf: produc6425607678544837394st_int,Cf2: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P5 @ ( produc8941309049508465880st_int @ Cf ) ) @ Cf ) @ Cf2 )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ Cf ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P5 ) ) ) ) ) ) ).
% exec1_def
thf(fact_897_map__eq__conv,axiom,
! [F: instr > m_instr,Xs: list_instr,G: instr > m_instr] :
( ( ( map_instr_m_instr @ F @ Xs )
= ( map_instr_m_instr @ G @ Xs ) )
= ( ! [X2: instr] :
( ( member_instr @ X2 @ ( set_instr2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_898_ex__map__conv,axiom,
! [Ys: list_m_instr,F: instr > m_instr] :
( ( ? [Xs3: list_instr] :
( Ys
= ( map_instr_m_instr @ F @ Xs3 ) ) )
= ( ! [X2: m_instr] :
( ( member_m_instr @ X2 @ ( set_m_instr2 @ Ys ) )
=> ? [Y2: instr] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_899_map__cong,axiom,
! [Xs: list_instr,Ys: list_instr,F: instr > m_instr,G: instr > m_instr] :
( ( Xs = Ys )
=> ( ! [X3: instr] :
( ( member_instr @ X3 @ ( set_instr2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_instr_m_instr @ F @ Xs )
= ( map_instr_m_instr @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_900_map__idI,axiom,
! [Xs: list_list_char,F: list_char > list_char] :
( ! [X3: list_char] :
( ( member_list_char @ X3 @ ( set_list_char2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li116305933131242120t_char @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_901_map__idI,axiom,
! [Xs: list_int,F: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_int_int @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_902_map__ext,axiom,
! [Xs: list_instr,F: instr > m_instr,G: instr > m_instr] :
( ! [X3: instr] :
( ( member_instr @ X3 @ ( set_instr2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_instr_m_instr @ F @ Xs )
= ( map_instr_m_instr @ G @ Xs ) ) ) ).
% map_ext
thf(fact_903_list_Omap__ident__strong,axiom,
! [T: list_list_char,F: list_char > list_char] :
( ! [Z4: list_char] :
( ( member_list_char @ Z4 @ ( set_list_char2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_li116305933131242120t_char @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_904_list_Omap__ident__strong,axiom,
! [T: list_int,F: int > int] :
( ! [Z4: int] :
( ( member_int @ Z4 @ ( set_int2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_int_int @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_905_list_Oinj__map__strong,axiom,
! [X: list_instr,Xa: list_instr,F: instr > m_instr,Fa: instr > m_instr] :
( ! [Z4: instr,Za: instr] :
( ( member_instr @ Z4 @ ( set_instr2 @ X ) )
=> ( ( member_instr @ Za @ ( set_instr2 @ Xa ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_instr_m_instr @ F @ X )
= ( map_instr_m_instr @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_906_list_Omap__cong0,axiom,
! [X: list_instr,F: instr > m_instr,G: instr > m_instr] :
( ! [Z4: instr] :
( ( member_instr @ Z4 @ ( set_instr2 @ X ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_instr_m_instr @ F @ X )
= ( map_instr_m_instr @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_907_list_Omap__cong,axiom,
! [X: list_instr,Ya: list_instr,F: instr > m_instr,G: instr > m_instr] :
( ( X = Ya )
=> ( ! [Z4: instr] :
( ( member_instr @ Z4 @ ( set_instr2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_instr_m_instr @ F @ X )
= ( map_instr_m_instr @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_908_subset__code_I1_J,axiom,
! [Xs: list_list_char,B4: set_list_char] :
( ( ord_le1207710660481778807t_char @ ( set_list_char2 @ Xs ) @ B4 )
= ( ! [X2: list_char] :
( ( member_list_char @ X2 @ ( set_list_char2 @ Xs ) )
=> ( member_list_char @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_909_subset__code_I1_J,axiom,
! [Xs: list_int,B4: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B4 )
= ( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ( member_int @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_910_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A2 @ Top ) ) ).
% ordering_top.extremum
thf(fact_911_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A2 ) ) ).
% ordering_top.extremum_strict
thf(fact_912_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A2 )
= ( A2 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_913_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A2 != Top )
= ( Less @ A2 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_914_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A2 )
=> ( A2 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_915_fst__eqD,axiom,
! [X: com,Y: list_char > int,A2: com] :
( ( ( produc8829524412605799873ar_int @ ( produc5595214716300948949ar_int @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_916_fst__eqD,axiom,
! [X: list_char > int,Y: list_int,A2: list_char > int] :
( ( ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_917_fst__eqD,axiom,
! [X: list_char > int,Y: int,A2: list_char > int] :
( ( ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_918_fst__eqD,axiom,
! [X: int > int,Y: list_char,A2: int > int] :
( ( ( produc7904525314154352605t_char @ ( produc4670215617849501681t_char @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_919_fst__eqD,axiom,
! [X: int > int,Y: nat,A2: int > int] :
( ( ( produc8715251273072885836nt_nat @ ( produc2630796075472406112nt_nat @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_920_fst__eqD,axiom,
! [X: int,Y: produc369741286924889929st_int,A2: int] :
( ( ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_921_fst__eqD,axiom,
! [X: int,Y: produc1534919314292393896nt_nat,A2: int] :
( ( ( produc2827224793399753527nt_nat @ ( produc8342123264692915235nt_nat @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_922_fst__conv,axiom,
! [X1: com,X22: list_char > int] :
( ( produc8829524412605799873ar_int @ ( produc5595214716300948949ar_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_923_fst__conv,axiom,
! [X1: list_char > int,X22: list_int] :
( ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_924_fst__conv,axiom,
! [X1: list_char > int,X22: int] :
( ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_925_fst__conv,axiom,
! [X1: int > int,X22: list_char] :
( ( produc7904525314154352605t_char @ ( produc4670215617849501681t_char @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_926_fst__conv,axiom,
! [X1: int > int,X22: nat] :
( ( produc8715251273072885836nt_nat @ ( produc2630796075472406112nt_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_927_fst__conv,axiom,
! [X1: int,X22: produc369741286924889929st_int] :
( ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_928_fst__conv,axiom,
! [X1: int,X22: produc1534919314292393896nt_nat] :
( ( produc2827224793399753527nt_nat @ ( produc8342123264692915235nt_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_929_length__pos__if__in__set,axiom,
! [X: m_instr,Xs: list_m_instr] :
( ( member_m_instr @ X @ ( set_m_instr2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s5914166952900237872_instr @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_930_length__pos__if__in__set,axiom,
! [X: instr,Xs: list_instr] :
( ( member_instr @ X @ ( set_instr2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_instr @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_931_length__pos__if__in__set,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s356637359517785349t_char @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_932_length__pos__if__in__set,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_933_iexec__simp,axiom,
! [Ins2: instr,Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ Cf3 ) @ Cf4 )
= ( Cf4
= ( iexec @ Ins2 @ Cf3 ) ) ) ).
% iexec_simp
thf(fact_934_iexec__pred_Ointros,axiom,
! [Ins2: instr,Cf3: produc6425607678544837394st_int] : ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ Cf3 ) @ ( iexec @ Ins2 @ Cf3 ) ) ).
% iexec_pred.intros
thf(fact_935_iexec__pred_Osimps,axiom,
( iexec_pred
= ( ^ [A1: produc5995290525303592096st_int,A22: produc6425607678544837394st_int] :
? [Ins3: instr,Cf: produc6425607678544837394st_int] :
( ( A1
= ( produc1484136438817787418st_int @ Ins3 @ Cf ) )
& ( A22
= ( iexec @ Ins3 @ Cf ) ) ) ) ) ).
% iexec_pred.simps
thf(fact_936_iexec__pred_Ocases,axiom,
! [A12: produc5995290525303592096st_int,A23: produc6425607678544837394st_int] :
( ( iexec_pred @ A12 @ A23 )
=> ~ ! [Ins: instr,Cf5: produc6425607678544837394st_int] :
( ( A12
= ( produc1484136438817787418st_int @ Ins @ Cf5 ) )
=> ( A23
!= ( iexec @ Ins @ Cf5 ) ) ) ) ).
% iexec_pred.cases
thf(fact_937_eq__fst__iff,axiom,
! [A2: com,P3: produc2931317944591925149ar_int] :
( ( A2
= ( produc8829524412605799873ar_int @ P3 ) )
= ( ? [B3: list_char > int] :
( P3
= ( produc5595214716300948949ar_int @ A2 @ B3 ) ) ) ) ).
% eq_fst_iff
thf(fact_938_eq__fst__iff,axiom,
! [A2: list_char > int,P3: produc369741286924889929st_int] :
( ( A2
= ( produc2155683374303800813st_int @ P3 ) )
= ( ? [B3: list_int] :
( P3
= ( produc8650753666468850689st_int @ A2 @ B3 ) ) ) ) ).
% eq_fst_iff
thf(fact_939_eq__fst__iff,axiom,
! [A2: list_char > int,P3: produc4435102495419491129nt_int] :
( ( A2
= ( produc9025023058967219549nt_int @ P3 ) )
= ( ? [B3: int] :
( P3
= ( produc5790713362662368625nt_int @ A2 @ B3 ) ) ) ) ).
% eq_fst_iff
thf(fact_940_eq__fst__iff,axiom,
! [A2: int > int,P3: produc699483346519933241t_char] :
( ( A2
= ( produc7904525314154352605t_char @ P3 ) )
= ( ? [B3: list_char] :
( P3
= ( produc4670215617849501681t_char @ A2 @ B3 ) ) ) ) ).
% eq_fst_iff
thf(fact_941_eq__fst__iff,axiom,
! [A2: int > int,P3: produc1534919314292393896nt_nat] :
( ( A2
= ( produc8715251273072885836nt_nat @ P3 ) )
= ( ? [B3: nat] :
( P3
= ( produc2630796075472406112nt_nat @ A2 @ B3 ) ) ) ) ).
% eq_fst_iff
thf(fact_942_eq__fst__iff,axiom,
! [A2: int,P3: produc6425607678544837394st_int] :
( ( A2
= ( produc8941309049508465880st_int @ P3 ) )
= ( ? [B3: produc369741286924889929st_int] :
( P3
= ( produc5086643055186798020st_int @ A2 @ B3 ) ) ) ) ).
% eq_fst_iff
thf(fact_943_eq__fst__iff,axiom,
! [A2: int,P3: produc1019184382078353393nt_nat] :
( ( A2
= ( produc2827224793399753527nt_nat @ P3 ) )
= ( ? [B3: produc1534919314292393896nt_nat] :
( P3
= ( produc8342123264692915235nt_nat @ A2 @ B3 ) ) ) ) ).
% eq_fst_iff
thf(fact_944_fstI,axiom,
! [X: produc2931317944591925149ar_int,Y: com,Z: list_char > int] :
( ( X
= ( produc5595214716300948949ar_int @ Y @ Z ) )
=> ( ( produc8829524412605799873ar_int @ X )
= Y ) ) ).
% fstI
thf(fact_945_fstI,axiom,
! [X: produc369741286924889929st_int,Y: list_char > int,Z: list_int] :
( ( X
= ( produc8650753666468850689st_int @ Y @ Z ) )
=> ( ( produc2155683374303800813st_int @ X )
= Y ) ) ).
% fstI
thf(fact_946_fstI,axiom,
! [X: produc4435102495419491129nt_int,Y: list_char > int,Z: int] :
( ( X
= ( produc5790713362662368625nt_int @ Y @ Z ) )
=> ( ( produc9025023058967219549nt_int @ X )
= Y ) ) ).
% fstI
thf(fact_947_fstI,axiom,
! [X: produc699483346519933241t_char,Y: int > int,Z: list_char] :
( ( X
= ( produc4670215617849501681t_char @ Y @ Z ) )
=> ( ( produc7904525314154352605t_char @ X )
= Y ) ) ).
% fstI
thf(fact_948_fstI,axiom,
! [X: produc1534919314292393896nt_nat,Y: int > int,Z: nat] :
( ( X
= ( produc2630796075472406112nt_nat @ Y @ Z ) )
=> ( ( produc8715251273072885836nt_nat @ X )
= Y ) ) ).
% fstI
thf(fact_949_fstI,axiom,
! [X: produc6425607678544837394st_int,Y: int,Z: produc369741286924889929st_int] :
( ( X
= ( produc5086643055186798020st_int @ Y @ Z ) )
=> ( ( produc8941309049508465880st_int @ X )
= Y ) ) ).
% fstI
thf(fact_950_fstI,axiom,
! [X: produc1019184382078353393nt_nat,Y: int,Z: produc1534919314292393896nt_nat] :
( ( X
= ( produc8342123264692915235nt_nat @ Y @ Z ) )
=> ( ( produc2827224793399753527nt_nat @ X )
= Y ) ) ).
% fstI
thf(fact_951_addr__of__correct,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ( ( inth_list_char @ Xs @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) @ ( addr_of @ Xs @ X ) ) )
= X ) ) ).
% addr_of_correct
thf(fact_952_set__n__lists,axiom,
! [N: nat,Xs: list_char] :
( ( set_list_char2 @ ( n_lists_char @ N @ Xs ) )
= ( collect_list_char
@ ^ [Ys3: list_char] :
( ( ( size_size_list_char @ Ys3 )
= N )
& ( ord_less_eq_set_char @ ( set_char2 @ Ys3 ) @ ( set_char2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_953_set__n__lists,axiom,
! [N: nat,Xs: list_m_instr] :
( ( set_list_m_instr2 @ ( n_lists_m_instr @ N @ Xs ) )
= ( collect_list_m_instr
@ ^ [Ys3: list_m_instr] :
( ( ( size_s5914166952900237872_instr @ Ys3 )
= N )
& ( ord_le9000437750415933090_instr @ ( set_m_instr2 @ Ys3 ) @ ( set_m_instr2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_954_set__n__lists,axiom,
! [N: nat,Xs: list_instr] :
( ( set_list_instr2 @ ( n_lists_instr @ N @ Xs ) )
= ( collect_list_instr
@ ^ [Ys3: list_instr] :
( ( ( size_size_list_instr @ Ys3 )
= N )
& ( ord_le4956778034350794079_instr @ ( set_instr2 @ Ys3 ) @ ( set_instr2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_955_set__n__lists,axiom,
! [N: nat,Xs: list_list_char] :
( ( set_list_list_char2 @ ( n_lists_list_char @ N @ Xs ) )
= ( collec7404787413733948998t_char
@ ^ [Ys3: list_list_char] :
( ( ( size_s356637359517785349t_char @ Ys3 )
= N )
& ( ord_le1207710660481778807t_char @ ( set_list_char2 @ Ys3 ) @ ( set_list_char2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_956_set__n__lists,axiom,
! [N: nat,Xs: list_int] :
( ( set_list_int2 @ ( n_lists_int @ N @ Xs ) )
= ( collect_list_int
@ ^ [Ys3: list_int] :
( ( ( size_size_list_int @ Ys3 )
= N )
& ( ord_less_eq_set_int @ ( set_int2 @ Ys3 ) @ ( set_int2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_957_addr__of__nset,axiom,
! [X: list_char,Xs: list_list_char] :
( ~ ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ( ( addr_of @ Xs @ X )
= zero_zero_int ) ) ).
% addr_of_nset
thf(fact_958_addr__of__neq2,axiom,
! [X: list_char,Xs: list_list_char,X4: list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ( ( X4 != X )
=> ( ( addr_of @ Xs @ X4 )
!= ( addr_of @ Xs @ X ) ) ) ) ).
% addr_of_neq2
thf(fact_959_length__n__lists__elem,axiom,
! [Ys: list_char,N: nat,Xs: list_char] :
( ( member_list_char @ Ys @ ( set_list_char2 @ ( n_lists_char @ N @ Xs ) ) )
=> ( ( size_size_list_char @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_960_length__n__lists__elem,axiom,
! [Ys: list_m_instr,N: nat,Xs: list_m_instr] :
( ( member_list_m_instr @ Ys @ ( set_list_m_instr2 @ ( n_lists_m_instr @ N @ Xs ) ) )
=> ( ( size_s5914166952900237872_instr @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_961_length__n__lists__elem,axiom,
! [Ys: list_instr,N: nat,Xs: list_instr] :
( ( member_list_instr @ Ys @ ( set_list_instr2 @ ( n_lists_instr @ N @ Xs ) ) )
=> ( ( size_size_list_instr @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_962_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_963_length__n__lists__elem,axiom,
! [Ys: list_int,N: nat,Xs: list_int] :
( ( member_list_int @ Ys @ ( set_list_int2 @ ( n_lists_int @ N @ Xs ) ) )
=> ( ( size_size_list_int @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_964_addr__of__nneg,axiom,
! [Xs: list_list_char,X: list_char] : ( ord_less_eq_int @ zero_zero_int @ ( addr_of @ Xs @ X ) ) ).
% addr_of_nneg
thf(fact_965_addr__of__set,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ( ord_less_int @ zero_zero_int @ ( addr_of @ Xs @ X ) ) ) ).
% addr_of_set
thf(fact_966_addr__of__max,axiom,
! [Xs: list_list_char,X: list_char] : ( ord_less_eq_int @ ( addr_of @ Xs @ X ) @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) ) ).
% addr_of_max
thf(fact_967_addr__of__unique,axiom,
! [Xs: list_list_char,A2: int] :
( ( distinct_list_char @ Xs )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( addr_of @ Xs @ ( inth_list_char @ Xs @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) @ A2 ) ) )
= A2 ) ) ) ) ).
% addr_of_unique
thf(fact_968_in__set__product__lists__length,axiom,
! [Xs: list_char,Xss: list_list_char] :
( ( member_list_char @ Xs @ ( set_list_char2 @ ( product_lists_char @ Xss ) ) )
=> ( ( size_size_list_char @ Xs )
= ( size_s356637359517785349t_char @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_969_in__set__product__lists__length,axiom,
! [Xs: list_m_instr,Xss: list_list_m_instr] :
( ( member_list_m_instr @ Xs @ ( set_list_m_instr2 @ ( produc5735103596895602909_instr @ Xss ) ) )
=> ( ( size_s5914166952900237872_instr @ Xs )
= ( size_s5217651917374353344_instr @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_970_in__set__product__lists__length,axiom,
! [Xs: list_instr,Xss: list_list_instr] :
( ( member_list_instr @ Xs @ ( set_list_instr2 @ ( product_lists_instr @ Xss ) ) )
=> ( ( size_size_list_instr @ Xs )
= ( size_s1289461246499836759_instr @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_971_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_972_in__set__product__lists__length,axiom,
! [Xs: list_int,Xss: list_list_int] :
( ( member_list_int @ Xs @ ( set_list_int2 @ ( product_lists_int @ Xss ) ) )
=> ( ( size_size_list_int @ Xs )
= ( size_s533118279054570080st_int @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_973_vars__store,axiom,
! [I: int,P: list_instr,X: list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) )
=> ( ( ( inth_instr @ P @ I )
= ( store @ X ) )
=> ( member_list_char @ X @ ( set_list_char2 @ ( vars @ P ) ) ) ) ) ) ).
% vars_store
thf(fact_974_instr_Oinject_I3_J,axiom,
! [X42: list_char,Y42: list_char] :
( ( ( store @ X42 )
= ( store @ Y42 ) )
= ( X42 = Y42 ) ) ).
% instr.inject(3)
thf(fact_975_distinct__product__lists,axiom,
! [Xss: list_list_list_char] :
( ! [X3: list_list_char] :
( ( member4762036990631661320t_char @ X3 @ ( set_list_list_char2 @ Xss ) )
=> ( distinct_list_char @ X3 ) )
=> ( distin92326681159624120t_char @ ( produc4928866681682397746t_char @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_976_distinct__product__lists,axiom,
! [Xss: list_list_int] :
( ! [X3: list_int] :
( ( member_list_int @ X3 @ ( set_list_int2 @ Xss ) )
=> ( distinct_int @ X3 ) )
=> ( distinct_list_int @ ( product_lists_int @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_977_distinct__product__lists,axiom,
! [Xss: list_list_char] :
( ! [X3: list_char] :
( ( member_list_char @ X3 @ ( set_list_char2 @ Xss ) )
=> ( distinct_char @ X3 ) )
=> ( distinct_list_char @ ( product_lists_char @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_978_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs: list_int] :
( ( distinct_int @ Xs )
=> ( distinct_int @ Xs ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_979_vars__dist,axiom,
! [P: list_instr] : ( distinct_list_char @ ( vars @ P ) ) ).
% vars_dist
thf(fact_980_instr_Osize_I11_J,axiom,
! [X42: list_char] :
( ( size_size_instr @ ( store @ X42 ) )
= zero_zero_nat ) ).
% instr.size(11)
thf(fact_981_vars__load,axiom,
! [I: int,P: list_instr,X: list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) )
=> ( ( ( inth_instr @ P @ I )
= ( load @ X ) )
=> ( member_list_char @ X @ ( set_list_char2 @ ( vars @ P ) ) ) ) ) ) ).
% vars_load
thf(fact_982_eq__key__imp__eq__value,axiom,
! [Xs: list_P1039738593281864227ar_int,K: com,V1: list_char > int,V2: list_char > int] :
( ( distinct_com @ ( map_Pr7622010126730545513nt_com @ produc8829524412605799873ar_int @ Xs ) )
=> ( ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ K @ V1 ) @ ( set_Pr7490742738769445810ar_int @ Xs ) )
=> ( ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ K @ V2 ) @ ( set_Pr7490742738769445810ar_int @ Xs ) )
=> ( V1 = V2 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_983_eq__key__imp__eq__value,axiom,
! [Xs: list_P8654944379044127311st_int,K: list_char > int,V1: list_int,V2: list_int] :
( ( distin6806876563479763699ar_int @ ( map_Pr4838045029083743153ar_int @ produc2155683374303800813st_int @ Xs ) )
=> ( ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ K @ V1 ) @ ( set_Pr5507350050178589918st_int @ Xs ) )
=> ( ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ K @ V2 ) @ ( set_Pr5507350050178589918st_int @ Xs ) )
=> ( V1 = V2 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_984_eq__key__imp__eq__value,axiom,
! [Xs: list_P98921565119350719nt_int,K: list_char > int,V1: int,V2: int] :
( ( distin6806876563479763699ar_int @ ( map_Pr6267657629804605729ar_int @ produc9025023058967219549nt_int @ Xs ) )
=> ( ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ K @ V1 ) @ ( set_Pr8994527289597011790nt_int @ Xs ) )
=> ( ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ K @ V2 ) @ ( set_Pr8994527289597011790nt_int @ Xs ) )
=> ( V1 = V2 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_985_eq__key__imp__eq__value,axiom,
! [Xs: list_P2295729324716942783t_char,K: int > int,V1: list_char,V2: list_char] :
( ( distinct_int_int @ ( map_Pr6680895987495608812nt_int @ produc7904525314154352605t_char @ Xs ) )
=> ( ( member9086947928420754658t_char @ ( produc4670215617849501681t_char @ K @ V1 ) @ ( set_Pr5258908140697453902t_char @ Xs ) )
=> ( ( member9086947928420754658t_char @ ( produc4670215617849501681t_char @ K @ V2 ) @ ( set_Pr5258908140697453902t_char @ Xs ) )
=> ( V1 = V2 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_986_eq__key__imp__eq__value,axiom,
! [Xs: list_P5370272772617341230nt_nat,K: int > int,V1: nat,V2: nat] :
( ( distinct_int_int @ ( map_Pr1664154291802054235nt_int @ produc8715251273072885836nt_nat @ Xs ) )
=> ( ( member561943782142090449nt_nat @ ( produc2630796075472406112nt_nat @ K @ V1 ) @ ( set_Pr4499741033739358013nt_nat @ Xs ) )
=> ( ( member561943782142090449nt_nat @ ( produc2630796075472406112nt_nat @ K @ V2 ) @ ( set_Pr4499741033739358013nt_nat @ Xs ) )
=> ( V1 = V2 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_987_eq__key__imp__eq__value,axiom,
! [Xs: list_P419547167332244514st_int,K: int,V1: produc369741286924889929st_int,V2: produc369741286924889929st_int] :
( ( distinct_int @ ( map_Pr3224476576807795204nt_int @ produc8941309049508465880st_int @ Xs ) )
=> ( ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ K @ V1 ) @ ( set_Pr3303432077563008829st_int @ Xs ) )
=> ( ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ K @ V2 ) @ ( set_Pr3303432077563008829st_int @ Xs ) )
=> ( V1 = V2 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_988_eq__key__imp__eq__value,axiom,
! [Xs: list_P586826539370926977nt_nat,K: int,V1: produc1534919314292393896nt_nat,V2: produc1534919314292393896nt_nat] :
( ( distinct_int @ ( map_Pr1329617156475778275at_int @ produc2827224793399753527nt_nat @ Xs ) )
=> ( ( member7326949433154712456nt_nat @ ( produc8342123264692915235nt_nat @ K @ V1 ) @ ( set_Pr7406552540789836316nt_nat @ Xs ) )
=> ( ( member7326949433154712456nt_nat @ ( produc8342123264692915235nt_nat @ K @ V2 ) @ ( set_Pr7406552540789836316nt_nat @ Xs ) )
=> ( V1 = V2 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_989_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_990_distinct__union,axiom,
! [Xs: list_int,Ys: list_int] :
( ( distinct_int @ ( union_int @ Xs @ Ys ) )
= ( distinct_int @ Ys ) ) ).
% distinct_union
thf(fact_991_instr_Oinject_I2_J,axiom,
! [X22: list_char,Y22: list_char] :
( ( ( load @ X22 )
= ( load @ Y22 ) )
= ( X22 = Y22 ) ) ).
% instr.inject(2)
thf(fact_992_instr_Osize_I9_J,axiom,
! [X22: list_char] :
( ( size_size_instr @ ( load @ X22 ) )
= zero_zero_nat ) ).
% instr.size(9)
thf(fact_993_instr_Odistinct_I15_J,axiom,
! [X22: list_char,X42: list_char] :
( ( load @ X22 )
!= ( store @ X42 ) ) ).
% instr.distinct(15)
thf(fact_994_exec1__m__exec1,axiom,
! [P: list_instr,Pc: int,S: list_char > int,Stk: list_int,Pc2: int,S2: list_char > int,Stk2: list_int,Sp: nat] :
( ( exec1 @ P @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ Pc2 @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) )
=> ( ( ord_less_eq_nat @ ( msp @ P @ Pc ) @ Sp )
=> ( ( ord_less_eq_nat @ Sp @ ( size_size_list_int @ Stk ) )
=> ( m_exec1 @ ( to_m_prog @ P ) @ ( produc8342123264692915235nt_nat @ Pc @ ( produc2630796075472406112nt_nat @ ( add_m_stack @ Sp @ Stk @ ( to_m_state @ ( vars @ P ) @ S ) ) @ Sp ) ) @ ( produc8342123264692915235nt_nat @ Pc2 @ ( produc2630796075472406112nt_nat @ ( add_m_stack @ ( minus_minus_nat @ ( plus_plus_nat @ Sp @ ( size_size_list_int @ Stk2 ) ) @ ( size_size_list_int @ Stk ) ) @ Stk2 @ ( to_m_state @ ( vars @ P ) @ S2 ) ) @ ( minus_minus_nat @ ( plus_plus_nat @ Sp @ ( size_size_list_int @ Stk2 ) ) @ ( size_size_list_int @ Stk ) ) ) ) ) ) ) ) ).
% exec1_m_exec1
thf(fact_995_diff__nat__eq__if,axiom,
! [Z6: int,Z: int] :
( ( ( ord_less_int @ Z6 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_996_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_997_verit__minus__simplify_I4_J,axiom,
! [B2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
= B2 ) ).
% verit_minus_simplify(4)
thf(fact_998_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_999_neg__equal__iff__equal,axiom,
! [A2: int,B2: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B2 ) )
= ( A2 = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_1000_neg__le__iff__le,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% neg_le_iff_le
thf(fact_1001_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_1002_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_1003_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_1004_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_1005_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_1006_neg__less__iff__less,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% neg_less_iff_less
thf(fact_1007_add__minus__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B2 ) )
= B2 ) ).
% add_minus_cancel
thf(fact_1008_minus__add__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B2 ) )
= B2 ) ).
% minus_add_cancel
thf(fact_1009_minus__add__distrib,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) ) ) ).
% minus_add_distrib
thf(fact_1010_minus__diff__eq,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( minus_minus_int @ B2 @ A2 ) ) ).
% minus_diff_eq
thf(fact_1011_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_1012_neg__0__le__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_1013_neg__le__0__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_1014_less__eq__neg__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_1015_neg__less__eq__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_1016_neg__less__0__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_1017_neg__0__less__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_1018_neg__less__pos,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_pos
thf(fact_1019_less__neg__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_1020_ab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_1021_add_Oright__inverse,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1022_verit__minus__simplify_I3_J,axiom,
! [B2: int] :
( ( minus_minus_int @ zero_zero_int @ B2 )
= ( uminus_uminus_int @ B2 ) ) ).
% verit_minus_simplify(3)
thf(fact_1023_diff__0,axiom,
! [A2: int] :
( ( minus_minus_int @ zero_zero_int @ A2 )
= ( uminus_uminus_int @ A2 ) ) ).
% diff_0
thf(fact_1024_diff__minus__eq__add,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( plus_plus_int @ A2 @ B2 ) ) ).
% diff_minus_eq_add
thf(fact_1025_uminus__add__conv__diff,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( minus_minus_int @ B2 @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_1026_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1027_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1028_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1029_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1030_zless__nat__conj,axiom,
! [W2: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W2 @ Z ) ) ) ).
% zless_nat_conj
thf(fact_1031_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1032_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1033_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_1034_minus__less__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).
% minus_less_iff
thf(fact_1035_less__minus__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_1036_verit__negate__coefficient_I2_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1037_minus__diff__commute,axiom,
! [B2: int,A2: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A2 )
= ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ).
% minus_diff_commute
thf(fact_1038_le__imp__neg__le,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_1039_minus__le__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).
% minus_le_iff
thf(fact_1040_le__minus__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_minus_iff
thf(fact_1041_group__cancel_Oneg1,axiom,
! [A3: int,K: int,A2: int] :
( ( A3
= ( plus_plus_int @ K @ A2 ) )
=> ( ( uminus_uminus_int @ A3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_1042_add_Oinverse__distrib__swap,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1043_is__num__normalize_I8_J,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_1044_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1045_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_1046_verit__negate__coefficient_I3_J,axiom,
! [A2: int,B2: int] :
( ( A2 = B2 )
=> ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B2 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1047_equation__minus__iff,axiom,
! [A2: int,B2: int] :
( ( A2
= ( uminus_uminus_int @ B2 ) )
= ( B2
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_1048_minus__equation__iff,axiom,
! [A2: int,B2: int] :
( ( ( uminus_uminus_int @ A2 )
= B2 )
= ( ( uminus_uminus_int @ B2 )
= A2 ) ) ).
% minus_equation_iff
thf(fact_1049_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1050_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1051_eq__nat__nat__iff,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z6 ) )
= ( Z = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1052_all__nat,axiom,
( ( ^ [P4: nat > $o] :
! [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P5: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P5 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_1053_ex__nat,axiom,
( ( ^ [P4: nat > $o] :
? [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P5: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P5 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_1054_neg__eq__iff__add__eq__0,axiom,
! [A2: int,B2: int] :
( ( ( uminus_uminus_int @ A2 )
= B2 )
= ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1055_eq__neg__iff__add__eq__0,axiom,
! [A2: int,B2: int] :
( ( A2
= ( uminus_uminus_int @ B2 ) )
= ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1056_add_Oinverse__unique,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A2 )
= B2 ) ) ).
% add.inverse_unique
thf(fact_1057_ab__group__add__class_Oab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1058_add__eq__0__iff,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int )
= ( B2
= ( uminus_uminus_int @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_1059_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1060_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1061_group__cancel_Osub2,axiom,
! [B4: int,K: int,B2: int,A2: int] :
( ( B4
= ( plus_plus_int @ K @ B2 ) )
=> ( ( minus_minus_int @ A2 @ B4 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.sub2
thf(fact_1062_add__m__stack_Osimps_I1_J,axiom,
! [Uu: list_int,Ms: int > int] :
( ( add_m_stack @ zero_zero_nat @ Uu @ Ms )
= Ms ) ).
% add_m_stack.simps(1)
thf(fact_1063_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1064_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1065_nat__mono__iff,axiom,
! [Z: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_mono_iff
thf(fact_1066_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_1067_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1068_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_1069_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z )
= ( ( M
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_1070_nat__int__add,axiom,
! [A2: nat,B2: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
= ( plus_plus_nat @ A2 @ B2 ) ) ).
% nat_int_add
thf(fact_1071_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% int_minus
thf(fact_1072_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1073_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1074_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1075_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1076_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1077_nat__minus__as__int,axiom,
( minus_minus_nat
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_1078_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1079_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1080_add__m__stack__nneg,axiom,
! [A2: int,N: nat,Stk: list_int,Ms: int > int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( add_m_stack @ N @ Stk @ Ms @ A2 )
= ( Ms @ A2 ) ) ) ).
% add_m_stack_nneg
thf(fact_1081_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N2: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N2 ) )
=> ( P @ N2 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1082_nat__less__eq__zless,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_1083_nat__le__eq__zle,axiom,
! [W2: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_1084_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1085_nat__add__distrib,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
= ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1086_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1087_nat__diff__distrib,axiom,
! [Z6: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ord_less_eq_int @ Z6 @ Z )
=> ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
= ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1088_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1089_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1090_m__config__equiv_Osimps,axiom,
! [I: int,Ms: int > int,Sp: nat,I2: int,Ms2: int > int,Sp2: nat] :
( ( m_config_equiv @ ( produc8342123264692915235nt_nat @ I @ ( produc2630796075472406112nt_nat @ Ms @ Sp ) ) @ ( produc8342123264692915235nt_nat @ I2 @ ( produc2630796075472406112nt_nat @ Ms2 @ Sp2 ) ) )
= ( ( I = I2 )
& ( Sp = Sp2 )
& ! [A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ Sp ) ) @ A4 )
=> ( ( Ms @ A4 )
= ( Ms2 @ A4 ) ) ) ) ) ).
% m_config_equiv.simps
thf(fact_1091_add__m__stack__nth,axiom,
! [N: nat,A2: int,Stk: list_int,Ms: int > int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ A2 )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Stk ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( add_m_stack @ N @ Stk @ Ms @ A2 )
= ( Ms @ A2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( add_m_stack @ N @ Stk @ Ms @ A2 )
= ( nth_int @ Stk @ ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ A2 ) ) ) ) ) ) ) ) ).
% add_m_stack_nth
thf(fact_1092_m__exec__exec,axiom,
! [P: list_instr,Pc: int,Ms: int > int,Sp: nat,Pc2: int,Ms2: int > int,Sp2: nat,Stk: list_int] :
( ( star_P5461765526710342210nt_nat @ ( m_exec1 @ ( to_m_prog @ P ) ) @ ( produc8342123264692915235nt_nat @ Pc @ ( produc2630796075472406112nt_nat @ Ms @ Sp ) ) @ ( produc8342123264692915235nt_nat @ Pc2 @ ( produc2630796075472406112nt_nat @ Ms2 @ Sp2 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ ( to_state @ ( vars @ P ) @ Ms ) @ ( append_int @ ( add_stack @ Sp @ Ms ) @ Stk ) ) ) @ ( produc5086643055186798020st_int @ Pc2 @ ( produc8650753666468850689st_int @ ( to_state @ ( vars @ P ) @ Ms2 ) @ ( append_int @ ( add_stack @ Sp2 @ Ms2 ) @ Stk ) ) ) ) ) ).
% m_exec_exec
thf(fact_1093_to__m__prog_Osimps,axiom,
( to_m_prog
= ( ^ [P5: list_instr] : ( map_instr_m_instr @ ( to_m_instr @ ( vars @ P5 ) ) @ P5 ) ) ) ).
% to_m_prog.simps
thf(fact_1094_nth__map,axiom,
! [N: nat,Xs: list_m_instr,F: m_instr > int] :
( ( ord_less_nat @ N @ ( size_s5914166952900237872_instr @ Xs ) )
=> ( ( nth_int @ ( map_m_instr_int @ F @ Xs ) @ N )
= ( F @ ( nth_m_instr @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_1095_nth__map,axiom,
! [N: nat,Xs: list_instr,F: instr > int] :
( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_int @ ( map_instr_int @ F @ Xs ) @ N )
= ( F @ ( nth_instr @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_1096_nth__map,axiom,
! [N: nat,Xs: list_instr,F: instr > m_instr] :
( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_m_instr @ ( map_instr_m_instr @ F @ Xs ) @ N )
= ( F @ ( nth_instr @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_1097_nth__map,axiom,
! [N: nat,Xs: list_list_char,F: list_char > int] :
( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_int @ ( map_list_char_int @ F @ Xs ) @ N )
= ( F @ ( nth_list_char @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_1098_nth__map,axiom,
! [N: nat,Xs: list_int,F: int > int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( map_int_int @ F @ Xs ) @ N )
= ( F @ ( nth_int @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_1099_nth__append__length__plus,axiom,
! [Xs: list_m_instr,Ys: list_m_instr,N: nat] :
( ( nth_m_instr @ ( append_m_instr @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s5914166952900237872_instr @ Xs ) @ N ) )
= ( nth_m_instr @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1100_nth__append__length__plus,axiom,
! [Xs: list_instr,Ys: list_instr,N: nat] :
( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_instr @ Xs ) @ N ) )
= ( nth_instr @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1101_nth__append__length__plus,axiom,
! [Xs: list_list_char,Ys: list_list_char,N: nat] :
( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s356637359517785349t_char @ Xs ) @ N ) )
= ( nth_list_char @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1102_nth__append__length__plus,axiom,
! [Xs: list_int,Ys: list_int,N: nat] :
( ( nth_int @ ( append_int @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ N ) )
= ( nth_int @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1103_uminus__set__def,axiom,
( uminus5517088710498760928t_char
= ( ^ [A6: set_list_char] :
( collect_list_char
@ ( uminus7154032370948614053char_o
@ ^ [X2: list_char] : ( member_list_char @ X2 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1104_Compl__eq,axiom,
( uminus5517088710498760928t_char
= ( ^ [A6: set_list_char] :
( collect_list_char
@ ^ [X2: list_char] :
~ ( member_list_char @ X2 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_1105_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_m_instr,N: nat] :
( ( ord_less_nat @ M @ ( size_s5914166952900237872_instr @ Xs ) )
=> ( ( nth_Pr8384913817476697284_instr @ ( enumerate_m_instr @ N @ Xs ) @ M )
= ( produc2580942203496701173_instr @ ( plus_plus_nat @ N @ M ) @ ( nth_m_instr @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1106_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_instr,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_Pr4329546499383151805_instr @ ( enumerate_instr @ N @ Xs ) @ M )
= ( produc3129998205563682742_instr @ ( plus_plus_nat @ N @ M ) @ ( nth_instr @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1107_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_list_char,N: nat] :
( ( ord_less_nat @ M @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_Pr7433773183981329817t_char @ ( enumerate_list_char @ N @ Xs ) @ M )
= ( produc120197956887798346t_char @ ( plus_plus_nat @ N @ M ) @ ( nth_list_char @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1108_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_int,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
=> ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs ) @ M )
= ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1109_nth__equalityI,axiom,
! [Xs: list_m_instr,Ys: list_m_instr] :
( ( ( size_s5914166952900237872_instr @ Xs )
= ( size_s5914166952900237872_instr @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5914166952900237872_instr @ Xs ) )
=> ( ( nth_m_instr @ Xs @ I3 )
= ( nth_m_instr @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1110_nth__equalityI,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ Xs @ I3 )
= ( nth_instr @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1111_nth__equalityI,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ Xs @ I3 )
= ( nth_list_char @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1112_nth__equalityI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I3 )
= ( nth_int @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1113_Skolem__list__nth,axiom,
! [K: nat,P: nat > m_instr > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ K )
=> ? [X7: m_instr] : ( P @ I6 @ X7 ) ) )
= ( ? [Xs3: list_m_instr] :
( ( ( size_s5914166952900237872_instr @ Xs3 )
= K )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ K )
=> ( P @ I6 @ ( nth_m_instr @ Xs3 @ I6 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1114_Skolem__list__nth,axiom,
! [K: nat,P: nat > instr > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ K )
=> ? [X7: instr] : ( P @ I6 @ X7 ) ) )
= ( ? [Xs3: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= K )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ K )
=> ( P @ I6 @ ( nth_instr @ Xs3 @ I6 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1115_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_char > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ K )
=> ? [X7: list_char] : ( P @ I6 @ X7 ) ) )
= ( ? [Xs3: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs3 )
= K )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ K )
=> ( P @ I6 @ ( nth_list_char @ Xs3 @ I6 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1116_Skolem__list__nth,axiom,
! [K: nat,P: nat > int > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ K )
=> ? [X7: int] : ( P @ I6 @ X7 ) ) )
= ( ? [Xs3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= K )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ K )
=> ( P @ I6 @ ( nth_int @ Xs3 @ I6 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1117_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_m_instr,Z3: list_m_instr] : ( Y4 = Z3 ) )
= ( ^ [Xs3: list_m_instr,Ys3: list_m_instr] :
( ( ( size_s5914166952900237872_instr @ Xs3 )
= ( size_s5914166952900237872_instr @ Ys3 ) )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_s5914166952900237872_instr @ Xs3 ) )
=> ( ( nth_m_instr @ Xs3 @ I6 )
= ( nth_m_instr @ Ys3 @ I6 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1118_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_instr,Z3: list_instr] : ( Y4 = Z3 ) )
= ( ^ [Xs3: list_instr,Ys3: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_instr @ Ys3 ) )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_instr @ Xs3 ) )
=> ( ( nth_instr @ Xs3 @ I6 )
= ( nth_instr @ Ys3 @ I6 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1119_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_list_char,Z3: list_list_char] : ( Y4 = Z3 ) )
= ( ^ [Xs3: list_list_char,Ys3: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs3 )
= ( size_s356637359517785349t_char @ Ys3 ) )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_s356637359517785349t_char @ Xs3 ) )
=> ( ( nth_list_char @ Xs3 @ I6 )
= ( nth_list_char @ Ys3 @ I6 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1120_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_int,Z3: list_int] : ( Y4 = Z3 ) )
= ( ^ [Xs3: list_int,Ys3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_int @ Xs3 ) )
=> ( ( nth_int @ Xs3 @ I6 )
= ( nth_int @ Ys3 @ I6 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1121_to__state_Osimps,axiom,
( to_state
= ( ^ [Xs3: list_list_char,Ms3: int > int,X2: list_char] : ( Ms3 @ ( addr_of @ Xs3 @ X2 ) ) ) ) ).
% to_state.simps
thf(fact_1122_to__state_Oelims,axiom,
! [X: list_list_char,Xa: int > int,Xb: list_char,Y: int] :
( ( ( to_state @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( Xa @ ( addr_of @ X @ Xb ) ) ) ) ).
% to_state.elims
thf(fact_1123_nth__mem,axiom,
! [N: nat,Xs: list_m_instr] :
( ( ord_less_nat @ N @ ( size_s5914166952900237872_instr @ Xs ) )
=> ( member_m_instr @ ( nth_m_instr @ Xs @ N ) @ ( set_m_instr2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1124_nth__mem,axiom,
! [N: nat,Xs: list_instr] :
( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( member_instr @ ( nth_instr @ Xs @ N ) @ ( set_instr2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1125_nth__mem,axiom,
! [N: nat,Xs: list_list_char] :
( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( member_list_char @ ( nth_list_char @ Xs @ N ) @ ( set_list_char2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1126_nth__mem,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1127_list__ball__nth,axiom,
! [N: nat,Xs: list_m_instr,P: m_instr > $o] :
( ( ord_less_nat @ N @ ( size_s5914166952900237872_instr @ Xs ) )
=> ( ! [X3: m_instr] :
( ( member_m_instr @ X3 @ ( set_m_instr2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_m_instr @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1128_list__ball__nth,axiom,
! [N: nat,Xs: list_instr,P: instr > $o] :
( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ! [X3: instr] :
( ( member_instr @ X3 @ ( set_instr2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_instr @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1129_list__ball__nth,axiom,
! [N: nat,Xs: list_list_char,P: list_char > $o] :
( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ! [X3: list_char] :
( ( member_list_char @ X3 @ ( set_list_char2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_list_char @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1130_list__ball__nth,axiom,
! [N: nat,Xs: list_int,P: int > $o] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1131_in__set__conv__nth,axiom,
! [X: m_instr,Xs: list_m_instr] :
( ( member_m_instr @ X @ ( set_m_instr2 @ Xs ) )
= ( ? [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_s5914166952900237872_instr @ Xs ) )
& ( ( nth_m_instr @ Xs @ I6 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1132_in__set__conv__nth,axiom,
! [X: instr,Xs: list_instr] :
( ( member_instr @ X @ ( set_instr2 @ Xs ) )
= ( ? [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_instr @ Xs ) )
& ( ( nth_instr @ Xs @ I6 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1133_in__set__conv__nth,axiom,
! [X: list_char,Xs: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
= ( ? [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_s356637359517785349t_char @ Xs ) )
& ( ( nth_list_char @ Xs @ I6 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1134_in__set__conv__nth,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ I6 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1135_all__nth__imp__all__set,axiom,
! [Xs: list_m_instr,P: m_instr > $o,X: m_instr] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5914166952900237872_instr @ Xs ) )
=> ( P @ ( nth_m_instr @ Xs @ I3 ) ) )
=> ( ( member_m_instr @ X @ ( set_m_instr2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1136_all__nth__imp__all__set,axiom,
! [Xs: list_instr,P: instr > $o,X: instr] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_instr @ Xs ) )
=> ( P @ ( nth_instr @ Xs @ I3 ) ) )
=> ( ( member_instr @ X @ ( set_instr2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1137_all__nth__imp__all__set,axiom,
! [Xs: list_list_char,P: list_char > $o,X: list_char] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s356637359517785349t_char @ Xs ) )
=> ( P @ ( nth_list_char @ Xs @ I3 ) ) )
=> ( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1138_all__nth__imp__all__set,axiom,
! [Xs: list_int,P: int > $o,X: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( P @ ( nth_int @ Xs @ I3 ) ) )
=> ( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1139_all__set__conv__all__nth,axiom,
! [Xs: list_m_instr,P: m_instr > $o] :
( ( ! [X2: m_instr] :
( ( member_m_instr @ X2 @ ( set_m_instr2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_s5914166952900237872_instr @ Xs ) )
=> ( P @ ( nth_m_instr @ Xs @ I6 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1140_all__set__conv__all__nth,axiom,
! [Xs: list_instr,P: instr > $o] :
( ( ! [X2: instr] :
( ( member_instr @ X2 @ ( set_instr2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_instr @ Xs ) )
=> ( P @ ( nth_instr @ Xs @ I6 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1141_all__set__conv__all__nth,axiom,
! [Xs: list_list_char,P: list_char > $o] :
( ( ! [X2: list_char] :
( ( member_list_char @ X2 @ ( set_list_char2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_s356637359517785349t_char @ Xs ) )
=> ( P @ ( nth_list_char @ Xs @ I6 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1142_all__set__conv__all__nth,axiom,
! [Xs: list_int,P: int > $o] :
( ( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_int @ Xs ) )
=> ( P @ ( nth_int @ Xs @ I6 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1143_map__equality__iff,axiom,
! [F: m_instr > m_instr,Xs: list_m_instr,G: instr > m_instr,Ys: list_instr] :
( ( ( map_m_instr_m_instr @ F @ Xs )
= ( map_instr_m_instr @ G @ Ys ) )
= ( ( ( size_s5914166952900237872_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_instr @ Ys ) )
=> ( ( F @ ( nth_m_instr @ Xs @ I6 ) )
= ( G @ ( nth_instr @ Ys @ I6 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1144_map__equality__iff,axiom,
! [F: instr > m_instr,Xs: list_instr,G: m_instr > m_instr,Ys: list_m_instr] :
( ( ( map_instr_m_instr @ F @ Xs )
= ( map_m_instr_m_instr @ G @ Ys ) )
= ( ( ( size_size_list_instr @ Xs )
= ( size_s5914166952900237872_instr @ Ys ) )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_s5914166952900237872_instr @ Ys ) )
=> ( ( F @ ( nth_instr @ Xs @ I6 ) )
= ( G @ ( nth_m_instr @ Ys @ I6 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1145_map__equality__iff,axiom,
! [F: instr > m_instr,Xs: list_instr,G: instr > m_instr,Ys: list_instr] :
( ( ( map_instr_m_instr @ F @ Xs )
= ( map_instr_m_instr @ G @ Ys ) )
= ( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_instr @ Ys ) )
=> ( ( F @ ( nth_instr @ Xs @ I6 ) )
= ( G @ ( nth_instr @ Ys @ I6 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1146_map__equality__iff,axiom,
! [F: instr > m_instr,Xs: list_instr,G: list_char > m_instr,Ys: list_list_char] :
( ( ( map_instr_m_instr @ F @ Xs )
= ( map_li1986238532096774195_instr @ G @ Ys ) )
= ( ( ( size_size_list_instr @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_s356637359517785349t_char @ Ys ) )
=> ( ( F @ ( nth_instr @ Xs @ I6 ) )
= ( G @ ( nth_list_char @ Ys @ I6 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1147_map__equality__iff,axiom,
! [F: instr > m_instr,Xs: list_instr,G: int > m_instr,Ys: list_int] :
( ( ( map_instr_m_instr @ F @ Xs )
= ( map_int_m_instr @ G @ Ys ) )
= ( ( ( size_size_list_instr @ Xs )
= ( size_size_list_int @ Ys ) )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_int @ Ys ) )
=> ( ( F @ ( nth_instr @ Xs @ I6 ) )
= ( G @ ( nth_int @ Ys @ I6 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1148_map__equality__iff,axiom,
! [F: list_char > m_instr,Xs: list_list_char,G: instr > m_instr,Ys: list_instr] :
( ( ( map_li1986238532096774195_instr @ F @ Xs )
= ( map_instr_m_instr @ G @ Ys ) )
= ( ( ( size_s356637359517785349t_char @ Xs )
= ( size_size_list_instr @ Ys ) )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_instr @ Ys ) )
=> ( ( F @ ( nth_list_char @ Xs @ I6 ) )
= ( G @ ( nth_instr @ Ys @ I6 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1149_map__equality__iff,axiom,
! [F: int > m_instr,Xs: list_int,G: instr > m_instr,Ys: list_instr] :
( ( ( map_int_m_instr @ F @ Xs )
= ( map_instr_m_instr @ G @ Ys ) )
= ( ( ( size_size_list_int @ Xs )
= ( size_size_list_instr @ Ys ) )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_instr @ Ys ) )
=> ( ( F @ ( nth_int @ Xs @ I6 ) )
= ( G @ ( nth_instr @ Ys @ I6 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1150_distinct__conv__nth,axiom,
( distinct_m_instr
= ( ^ [Xs3: list_m_instr] :
! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_s5914166952900237872_instr @ Xs3 ) )
=> ! [J4: nat] :
( ( ord_less_nat @ J4 @ ( size_s5914166952900237872_instr @ Xs3 ) )
=> ( ( I6 != J4 )
=> ( ( nth_m_instr @ Xs3 @ I6 )
!= ( nth_m_instr @ Xs3 @ J4 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_1151_distinct__conv__nth,axiom,
( distinct_instr
= ( ^ [Xs3: list_instr] :
! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_instr @ Xs3 ) )
=> ! [J4: nat] :
( ( ord_less_nat @ J4 @ ( size_size_list_instr @ Xs3 ) )
=> ( ( I6 != J4 )
=> ( ( nth_instr @ Xs3 @ I6 )
!= ( nth_instr @ Xs3 @ J4 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_1152_distinct__conv__nth,axiom,
( distinct_list_char
= ( ^ [Xs3: list_list_char] :
! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_s356637359517785349t_char @ Xs3 ) )
=> ! [J4: nat] :
( ( ord_less_nat @ J4 @ ( size_s356637359517785349t_char @ Xs3 ) )
=> ( ( I6 != J4 )
=> ( ( nth_list_char @ Xs3 @ I6 )
!= ( nth_list_char @ Xs3 @ J4 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_1153_distinct__conv__nth,axiom,
( distinct_int
= ( ^ [Xs3: list_int] :
! [I6: nat] :
( ( ord_less_nat @ I6 @ ( size_size_list_int @ Xs3 ) )
=> ! [J4: nat] :
( ( ord_less_nat @ J4 @ ( size_size_list_int @ Xs3 ) )
=> ( ( I6 != J4 )
=> ( ( nth_int @ Xs3 @ I6 )
!= ( nth_int @ Xs3 @ J4 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_1154_nth__eq__iff__index__eq,axiom,
! [Xs: list_m_instr,I: nat,J: nat] :
( ( distinct_m_instr @ Xs )
=> ( ( ord_less_nat @ I @ ( size_s5914166952900237872_instr @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s5914166952900237872_instr @ Xs ) )
=> ( ( ( nth_m_instr @ Xs @ I )
= ( nth_m_instr @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_1155_nth__eq__iff__index__eq,axiom,
! [Xs: list_instr,I: nat,J: nat] :
( ( distinct_instr @ Xs )
=> ( ( ord_less_nat @ I @ ( size_size_list_instr @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_instr @ Xs ) )
=> ( ( ( nth_instr @ Xs @ I )
= ( nth_instr @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_1156_nth__eq__iff__index__eq,axiom,
! [Xs: list_list_char,I: nat,J: nat] :
( ( distinct_list_char @ Xs )
=> ( ( ord_less_nat @ I @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( ( nth_list_char @ Xs @ I )
= ( nth_list_char @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_1157_nth__eq__iff__index__eq,axiom,
! [Xs: list_int,I: nat,J: nat] :
( ( distinct_int @ Xs )
=> ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ( ( nth_int @ Xs @ I )
= ( nth_int @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_1158_to__state__eq,axiom,
! [Ms2: int > int,Ms: int > int,Xs: list_list_char] :
( ! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( Ms2 @ A )
= ( Ms @ A ) ) )
=> ( ( to_state @ Xs @ Ms2 )
= ( to_state @ Xs @ Ms ) ) ) ).
% to_state_eq
thf(fact_1159_nth__append,axiom,
! [N: nat,Xs: list_m_instr,Ys: list_m_instr] :
( ( ( ord_less_nat @ N @ ( size_s5914166952900237872_instr @ Xs ) )
=> ( ( nth_m_instr @ ( append_m_instr @ Xs @ Ys ) @ N )
= ( nth_m_instr @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s5914166952900237872_instr @ Xs ) )
=> ( ( nth_m_instr @ ( append_m_instr @ Xs @ Ys ) @ N )
= ( nth_m_instr @ Ys @ ( minus_minus_nat @ N @ ( size_s5914166952900237872_instr @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1160_nth__append,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ N )
= ( nth_instr @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ N )
= ( nth_instr @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1161_nth__append,axiom,
! [N: nat,Xs: list_list_char,Ys: list_list_char] :
( ( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ N )
= ( nth_list_char @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ N )
= ( nth_list_char @ Ys @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1162_nth__append,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1163_distinct__Ex1,axiom,
! [Xs: list_m_instr,X: m_instr] :
( ( distinct_m_instr @ Xs )
=> ( ( member_m_instr @ X @ ( set_m_instr2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s5914166952900237872_instr @ Xs ) )
& ( ( nth_m_instr @ Xs @ X3 )
= X )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_s5914166952900237872_instr @ Xs ) )
& ( ( nth_m_instr @ Xs @ Y5 )
= X ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1164_distinct__Ex1,axiom,
! [Xs: list_instr,X: instr] :
( ( distinct_instr @ Xs )
=> ( ( member_instr @ X @ ( set_instr2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_instr @ Xs ) )
& ( ( nth_instr @ Xs @ X3 )
= X )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_instr @ Xs ) )
& ( ( nth_instr @ Xs @ Y5 )
= X ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1165_distinct__Ex1,axiom,
! [Xs: list_list_char,X: list_char] :
( ( distinct_list_char @ Xs )
=> ( ( member_list_char @ X @ ( set_list_char2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s356637359517785349t_char @ Xs ) )
& ( ( nth_list_char @ Xs @ X3 )
= X )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_s356637359517785349t_char @ Xs ) )
& ( ( nth_list_char @ Xs @ Y5 )
= X ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1166_distinct__Ex1,axiom,
! [Xs: list_int,X: int] :
( ( distinct_int @ Xs )
=> ( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ X3 )
= X )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ Y5 )
= X ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1167_m__exec1__exec1,axiom,
! [P: list_instr,Pc: int,Ms: int > int,Sp: nat,Pc2: int,Ms2: int > int,Sp2: nat,Stk: list_int] :
( ( m_exec1 @ ( map_instr_m_instr @ ( to_m_instr @ ( vars @ P ) ) @ P ) @ ( produc8342123264692915235nt_nat @ Pc @ ( produc2630796075472406112nt_nat @ Ms @ Sp ) ) @ ( produc8342123264692915235nt_nat @ Pc2 @ ( produc2630796075472406112nt_nat @ Ms2 @ Sp2 ) ) )
=> ( exec1 @ P @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ ( to_state @ ( vars @ P ) @ Ms ) @ ( append_int @ ( add_stack @ Sp @ Ms ) @ Stk ) ) ) @ ( produc5086643055186798020st_int @ Pc2 @ ( produc8650753666468850689st_int @ ( to_state @ ( vars @ P ) @ Ms2 ) @ ( append_int @ ( add_stack @ Sp2 @ Ms2 ) @ Stk ) ) ) ) ) ).
% m_exec1_exec1
thf(fact_1168_to__m__prog_Oelims,axiom,
! [X: list_instr,Y: list_m_instr] :
( ( ( to_m_prog @ X )
= Y )
=> ( Y
= ( map_instr_m_instr @ ( to_m_instr @ ( vars @ X ) ) @ X ) ) ) ).
% to_m_prog.elims
thf(fact_1169_to__m__prog_Opelims,axiom,
! [X: list_instr,Y: list_m_instr] :
( ( ( to_m_prog @ X )
= Y )
=> ( ( accp_list_instr @ to_m_prog_rel @ X )
=> ~ ( ( Y
= ( map_instr_m_instr @ ( to_m_instr @ ( vars @ X ) ) @ X ) )
=> ~ ( accp_list_instr @ to_m_prog_rel @ X ) ) ) ) ).
% to_m_prog.pelims
thf(fact_1170_to__state_Opelims,axiom,
! [X: list_list_char,Xa: int > int,Xb: list_char,Y: int] :
( ( ( to_state @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P6759666029860677300t_char @ to_state_rel @ ( produc229435045275838383t_char @ X @ ( produc4670215617849501681t_char @ Xa @ Xb ) ) )
=> ~ ( ( Y
= ( Xa @ ( addr_of @ X @ Xb ) ) )
=> ~ ( accp_P6759666029860677300t_char @ to_state_rel @ ( produc229435045275838383t_char @ X @ ( produc4670215617849501681t_char @ Xa @ Xb ) ) ) ) ) ) ).
% to_state.pelims
thf(fact_1171_in__set__enumerate__eq,axiom,
! [P3: produc3073582426511957693_instr,N: nat,Xs: list_m_instr] :
( ( member1049328020672055910_instr @ P3 @ ( set_Pr4664272659350407890_instr @ ( enumerate_m_instr @ N @ Xs ) ) )
= ( ( ord_less_eq_nat @ N @ ( produc961630787621463265_instr @ P3 ) )
& ( ord_less_nat @ ( produc961630787621463265_instr @ P3 ) @ ( plus_plus_nat @ ( size_s5914166952900237872_instr @ Xs ) @ N ) )
& ( ( nth_m_instr @ Xs @ ( minus_minus_nat @ ( produc961630787621463265_instr @ P3 ) @ N ) )
= ( produc2690975792901314339_instr @ P3 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_1172_in__set__enumerate__eq,axiom,
! [P3: produc2078578717123253124_instr,N: nat,Xs: list_instr] :
( ( member6409878718942995611_instr @ P3 @ ( set_Pr2644756094545354031_instr @ ( enumerate_instr @ N @ Xs ) ) )
= ( ( ord_less_eq_nat @ N @ ( produc7244106053685809354_instr @ P3 ) )
& ( ord_less_nat @ ( produc7244106053685809354_instr @ P3 ) @ ( plus_plus_nat @ ( size_size_list_instr @ Xs ) @ N ) )
& ( ( nth_instr @ Xs @ ( minus_minus_nat @ ( produc7244106053685809354_instr @ P3 ) @ N ) )
= ( produc4771465314307615752_instr @ P3 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_1173_in__set__enumerate__eq,axiom,
! [P3: produc1457211279475724562t_char,N: nat,Xs: list_list_char] :
( ( member484235747325421115t_char @ P3 @ ( set_Pr4422032998922688679t_char @ ( enumerate_list_char @ N @ Xs ) ) )
= ( ( ord_less_eq_nat @ N @ ( produc6204653154488278070t_char @ P3 ) )
& ( ord_less_nat @ ( produc6204653154488278070t_char @ P3 ) @ ( plus_plus_nat @ ( size_s356637359517785349t_char @ Xs ) @ N ) )
& ( ( nth_list_char @ Xs @ ( minus_minus_nat @ ( produc6204653154488278070t_char @ P3 ) @ N ) )
= ( produc1900778367047502200t_char @ P3 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_1174_in__set__enumerate__eq,axiom,
! [P3: product_prod_nat_int,N: nat,Xs: list_int] :
( ( member4262671552274231302at_int @ P3 @ ( set_Pr1470767568048878706at_int @ ( enumerate_int @ N @ Xs ) ) )
= ( ( ord_less_eq_nat @ N @ ( product_fst_nat_int @ P3 ) )
& ( ord_less_nat @ ( product_fst_nat_int @ P3 ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ N ) )
& ( ( nth_int @ Xs @ ( minus_minus_nat @ ( product_fst_nat_int @ P3 ) @ N ) )
= ( product_snd_nat_int @ P3 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_1175_prod_Ocollapse,axiom,
! [Prod: produc2931317944591925149ar_int] :
( ( produc5595214716300948949ar_int @ ( produc8829524412605799873ar_int @ Prod ) @ ( produc9207319146083860483ar_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1176_prod_Ocollapse,axiom,
! [Prod: produc369741286924889929st_int] :
( ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ Prod ) @ ( produc6244060863719409455st_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1177_prod_Ocollapse,axiom,
! [Prod: produc4435102495419491129nt_int] :
( ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ Prod ) @ ( produc179445755590504351nt_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1178_prod_Ocollapse,axiom,
! [Prod: produc699483346519933241t_char] :
( ( produc4670215617849501681t_char @ ( produc7904525314154352605t_char @ Prod ) @ ( produc8282320047632413215t_char @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1179_prod_Ocollapse,axiom,
! [Prod: produc6425607678544837394st_int] :
( ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ Prod ) @ ( produc1296517004818028822st_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1180_prod_Ocollapse,axiom,
! [Prod: produc1534919314292393896nt_nat] :
( ( produc2630796075472406112nt_nat @ ( produc8715251273072885836nt_nat @ Prod ) @ ( produc4411376485632109966nt_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1181_prod_Ocollapse,axiom,
! [Prod: produc1019184382078353393nt_nat] :
( ( produc8342123264692915235nt_nat @ ( produc2827224793399753527nt_nat @ Prod ) @ ( produc2259646285430092661nt_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1182_pair__list__eqI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_Pr3224476576807795204nt_int @ produc8941309049508465880st_int @ Xs )
= ( map_Pr3224476576807795204nt_int @ produc8941309049508465880st_int @ Ys ) )
=> ( ( ( map_Pr4101632105909876081st_int @ produc1296517004818028822st_int @ Xs )
= ( map_Pr4101632105909876081st_int @ produc1296517004818028822st_int @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_1183_pair__list__eqI,axiom,
! [Xs: list_P5370272772617341230nt_nat,Ys: list_P5370272772617341230nt_nat] :
( ( ( map_Pr1664154291802054235nt_int @ produc8715251273072885836nt_nat @ Xs )
= ( map_Pr1664154291802054235nt_int @ produc8715251273072885836nt_nat @ Ys ) )
=> ( ( ( map_Pr4188089722216089652at_nat @ produc4411376485632109966nt_nat @ Xs )
= ( map_Pr4188089722216089652at_nat @ produc4411376485632109966nt_nat @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_1184_pair__list__eqI,axiom,
! [Xs: list_P586826539370926977nt_nat,Ys: list_P586826539370926977nt_nat] :
( ( ( map_Pr1329617156475778275at_int @ produc2827224793399753527nt_nat @ Xs )
= ( map_Pr1329617156475778275at_int @ produc2827224793399753527nt_nat @ Ys ) )
=> ( ( ( map_Pr3887368998649301681nt_nat @ produc2259646285430092661nt_nat @ Xs )
= ( map_Pr3887368998649301681nt_nat @ produc2259646285430092661nt_nat @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_1185_sndI,axiom,
! [X: produc2931317944591925149ar_int,Y: com,Z: list_char > int] :
( ( X
= ( produc5595214716300948949ar_int @ Y @ Z ) )
=> ( ( produc9207319146083860483ar_int @ X )
= Z ) ) ).
% sndI
thf(fact_1186_sndI,axiom,
! [X: produc369741286924889929st_int,Y: list_char > int,Z: list_int] :
( ( X
= ( produc8650753666468850689st_int @ Y @ Z ) )
=> ( ( produc6244060863719409455st_int @ X )
= Z ) ) ).
% sndI
thf(fact_1187_sndI,axiom,
! [X: produc4435102495419491129nt_int,Y: list_char > int,Z: int] :
( ( X
= ( produc5790713362662368625nt_int @ Y @ Z ) )
=> ( ( produc179445755590504351nt_int @ X )
= Z ) ) ).
% sndI
thf(fact_1188_sndI,axiom,
! [X: produc699483346519933241t_char,Y: int > int,Z: list_char] :
( ( X
= ( produc4670215617849501681t_char @ Y @ Z ) )
=> ( ( produc8282320047632413215t_char @ X )
= Z ) ) ).
% sndI
thf(fact_1189_sndI,axiom,
! [X: produc1534919314292393896nt_nat,Y: int > int,Z: nat] :
( ( X
= ( produc2630796075472406112nt_nat @ Y @ Z ) )
=> ( ( produc4411376485632109966nt_nat @ X )
= Z ) ) ).
% sndI
thf(fact_1190_sndI,axiom,
! [X: produc1019184382078353393nt_nat,Y: int,Z: produc1534919314292393896nt_nat] :
( ( X
= ( produc8342123264692915235nt_nat @ Y @ Z ) )
=> ( ( produc2259646285430092661nt_nat @ X )
= Z ) ) ).
% sndI
thf(fact_1191_eq__snd__iff,axiom,
! [B2: list_int,P3: produc369741286924889929st_int] :
( ( B2
= ( produc6244060863719409455st_int @ P3 ) )
= ( ? [A4: list_char > int] :
( P3
= ( produc8650753666468850689st_int @ A4 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_1192_eq__snd__iff,axiom,
! [B2: int,P3: produc4435102495419491129nt_int] :
( ( B2
= ( produc179445755590504351nt_int @ P3 ) )
= ( ? [A4: list_char > int] :
( P3
= ( produc5790713362662368625nt_int @ A4 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_1193_eq__snd__iff,axiom,
! [B2: list_char,P3: produc699483346519933241t_char] :
( ( B2
= ( produc8282320047632413215t_char @ P3 ) )
= ( ? [A4: int > int] :
( P3
= ( produc4670215617849501681t_char @ A4 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_1194_eq__snd__iff,axiom,
! [B2: nat,P3: produc1534919314292393896nt_nat] :
( ( B2
= ( produc4411376485632109966nt_nat @ P3 ) )
= ( ? [A4: int > int] :
( P3
= ( produc2630796075472406112nt_nat @ A4 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_1195_eq__snd__iff,axiom,
! [B2: produc1534919314292393896nt_nat,P3: produc1019184382078353393nt_nat] :
( ( B2
= ( produc2259646285430092661nt_nat @ P3 ) )
= ( ? [A4: int] :
( P3
= ( produc8342123264692915235nt_nat @ A4 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_1196_m__exec1__def,axiom,
( m_exec1
= ( ^ [MP: list_m_instr,Mcf: produc1019184382078353393nt_nat,Mcf2: produc1019184382078353393nt_nat] :
( ( m_config_equiv @ Mcf2 @ ( m_iexec @ ( inth_m_instr @ MP @ ( produc2827224793399753527nt_nat @ Mcf ) ) @ Mcf ) )
& ( ord_less_eq_int @ zero_zero_int @ ( produc2827224793399753527nt_nat @ Mcf ) )
& ( ord_less_int @ ( produc2827224793399753527nt_nat @ Mcf ) @ ( semiri1314217659103216013at_int @ ( size_s5914166952900237872_instr @ MP ) ) )
& ( ord_less_eq_nat @ ( m_msp @ ( inth_m_instr @ MP @ ( produc2827224793399753527nt_nat @ Mcf ) ) ) @ ( produc4411376485632109966nt_nat @ ( produc2259646285430092661nt_nat @ Mcf ) ) ) ) ) ) ).
% m_exec1_def
thf(fact_1197_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
=> ( ( nth_int @ ( upto @ I @ J ) @ K )
= ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% nth_upto
thf(fact_1198_distinct__upto,axiom,
! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).
% distinct_upto
thf(fact_1199_length__upto,axiom,
! [I: int,J: int] :
( ( size_size_list_int @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% length_upto
thf(fact_1200_upto__aux__def,axiom,
( upto_aux
= ( ^ [I6: int,J4: int] : ( append_int @ ( upto @ I6 @ J4 ) ) ) ) ).
% upto_aux_def
thf(fact_1201_zle__add1__eq__le,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1202_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_1203_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1204_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1205_zless__add1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z )
| ( W2 = Z ) ) ) ).
% zless_add1_eq
thf(fact_1206_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1207_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1208_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1209_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_1210_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1211_zless__imp__add1__zle,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1212_add1__zle__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
= ( ord_less_int @ W2 @ Z ) ) ).
% add1_zle_eq
thf(fact_1213_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1214_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1215_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_1216_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_1217_addr__of__neq,axiom,
! [Xs: list_list_char,X: list_char] :
( ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
!= ( addr_of @ Xs @ X ) ) ).
% addr_of_neq
thf(fact_1218_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1219_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1220_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1221_JmpE,axiom,
! [I: int,Pc: int,S: list_char > int,Stk: list_int,Cf3: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( jmp @ I ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk ) ) ) @ Cf3 )
=> ( Cf3
= ( produc5086643055186798020st_int @ ( plus_plus_int @ ( plus_plus_int @ Pc @ one_one_int ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk ) ) ) ) ).
% JmpE
thf(fact_1222_instr_Oinject_I4_J,axiom,
! [X52: int,Y52: int] :
( ( ( jmp @ X52 )
= ( jmp @ Y52 ) )
= ( X52 = Y52 ) ) ).
% instr.inject(4)
thf(fact_1223_instr_Odistinct_I31_J,axiom,
! [X42: list_char,X52: int] :
( ( store @ X42 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(31)
thf(fact_1224_instr_Odistinct_I17_J,axiom,
! [X22: list_char,X52: int] :
( ( load @ X22 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(17)
thf(fact_1225_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1226_instr_Osize_I12_J,axiom,
! [X52: int] :
( ( size_size_instr @ ( jmp @ X52 ) )
= zero_zero_nat ) ).
% instr.size(12)
thf(fact_1227_exec__Cons__1,axiom,
! [P: list_instr,S: list_char > int,Stk: list_int,J: int,T: list_char > int,Stk2: list_int,Ins2: instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ T @ Stk2 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( cons_instr @ Ins2 @ P ) ) @ ( produc5086643055186798020st_int @ one_one_int @ ( produc8650753666468850689st_int @ S @ Stk ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ one_one_int @ J ) @ ( produc8650753666468850689st_int @ T @ Stk2 ) ) ) ) ).
% exec_Cons_1
thf(fact_1228_ccomp_Osimps_I4_J,axiom,
! [B2: bexp,C_1: com,C_2: com] :
( ( ccomp @ ( if @ B2 @ C_1 @ C_2 ) )
= ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B2 @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C_1 ) ) ) @ one_one_int ) ) ) ) @ ( append_instr @ ( ccomp @ C_1 ) @ ( cons_instr @ ( jmp @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C_2 ) ) ) ) @ ( ccomp @ C_2 ) ) ) ) ) ).
% ccomp.simps(4)
thf(fact_1229_triv__if,axiom,
! [B2: bexp,C: com,S4: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B2 @ C @ C ) @ S4 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S4 ) @ T3 ) ) ).
% triv_if
thf(fact_1230_commute__if,axiom,
! [B1: bexp,B22: bexp,C11: com,C12: com,C22: com,S4: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B1 @ ( if @ B22 @ C11 @ C12 ) @ C22 ) @ S4 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B22 @ ( if @ B1 @ C11 @ C22 ) @ ( if @ B1 @ C12 @ C22 ) ) @ S4 ) @ T3 ) ) ).
% commute_if
thf(fact_1231_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_1232_IfTrue,axiom,
! [B2: bexp,S: list_char > int,C_1: com,T: list_char > int,C_2: com] :
( ( bval @ B2 @ S )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ C_1 @ S ) @ T )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B2 @ C_1 @ C_2 ) @ S ) @ T ) ) ) ).
% IfTrue
thf(fact_1233_IfFalse,axiom,
! [B2: bexp,S: list_char > int,C_2: com,T: list_char > int,C_1: com] :
( ~ ( bval @ B2 @ S )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ C_2 @ S ) @ T )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B2 @ C_1 @ C_2 ) @ S ) @ T ) ) ) ).
% IfFalse
thf(fact_1234_IfE,axiom,
! [B2: bexp,C1: com,C22: com,S: list_char > int,T: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B2 @ C1 @ C22 ) @ S ) @ T )
=> ( ( ( bval @ B2 @ S )
=> ~ ( big_big_step @ ( produc5595214716300948949ar_int @ C1 @ S ) @ T ) )
=> ~ ( ~ ( bval @ B2 @ S )
=> ~ ( big_big_step @ ( produc5595214716300948949ar_int @ C22 @ S ) @ T ) ) ) ) ).
% IfE
thf(fact_1235_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I6: int,J4: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J4 @ I6 ) @ Js @ ( upto_aux @ I6 @ ( minus_minus_int @ J4 @ one_one_int ) @ ( cons_int @ J4 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1236_addr__of_Osimps_I2_J,axiom,
! [X: list_char,Y: list_char,Xs: list_list_char] :
( ( ( X = Y )
=> ( ( addr_of @ ( cons_list_char @ X @ Xs ) @ Y )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) @ one_one_int ) ) )
& ( ( X != Y )
=> ( ( addr_of @ ( cons_list_char @ X @ Xs ) @ Y )
= ( addr_of @ Xs @ Y ) ) ) ) ).
% addr_of.simps(2)
thf(fact_1237_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_1238_LoadE,axiom,
! [X: list_char,Pc: int,S: list_char > int,Stk: list_int,Cf3: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( load @ X ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk ) ) ) @ Cf3 )
=> ( Cf3
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ ( S @ X ) @ Stk ) ) ) ) ) ).
% LoadE
thf(fact_1239_LoadIE,axiom,
! [I: int,Pc: int,S: list_char > int,Stk: list_int,Cf3: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( loadi @ I ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk ) ) ) @ Cf3 )
=> ( Cf3
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ I @ Stk ) ) ) ) ) ).
% LoadIE
thf(fact_1240_instr_Oinject_I1_J,axiom,
! [X1: int,Y1: int] :
( ( ( loadi @ X1 )
= ( loadi @ Y1 ) )
= ( X1 = Y1 ) ) ).
% instr.inject(1)
thf(fact_1241_instr_Odistinct_I7_J,axiom,
! [X1: int,X52: int] :
( ( loadi @ X1 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(7)
thf(fact_1242_instr_Odistinct_I1_J,axiom,
! [X1: int,X22: list_char] :
( ( loadi @ X1 )
!= ( load @ X22 ) ) ).
% instr.distinct(1)
thf(fact_1243_instr_Odistinct_I5_J,axiom,
! [X1: int,X42: list_char] :
( ( loadi @ X1 )
!= ( store @ X42 ) ) ).
% instr.distinct(5)
thf(fact_1244_instr_Osize_I8_J,axiom,
! [X1: int] :
( ( size_size_instr @ ( loadi @ X1 ) )
= zero_zero_nat ) ).
% instr.size(8)
thf(fact_1245_add__stack__eq2,axiom,
! [N: nat,Ms2: int > int,I: int,Ms: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_or4662586982721622107an_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) )
=> ( ( ( X3
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
=> ( ( Ms2 @ X3 )
= I ) )
& ( ( X3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
=> ( ( Ms2 @ X3 )
= ( Ms @ X3 ) ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( add_stack @ N @ Ms2 )
= ( cons_int @ I @ ( add_stack @ ( minus_minus_nat @ N @ one_one_nat ) @ Ms ) ) ) ) ) ).
% add_stack_eq2
thf(fact_1246_ccomp_Osimps_I5_J,axiom,
! [B2: bexp,C: com] :
( ( ccomp @ ( while @ B2 @ C ) )
= ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B2 @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C ) ) ) @ one_one_int ) ) ) ) @ ( append_instr @ ( ccomp @ C ) @ ( cons_instr @ ( jmp @ ( uminus_uminus_int @ ( plus_plus_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B2 @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C ) ) ) @ one_one_int ) ) ) ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C ) ) ) ) @ one_one_int ) ) ) @ nil_instr ) ) ) ) ).
% ccomp.simps(5)
thf(fact_1247_atLeastLessThan__upto,axiom,
( set_or4662586982721622107an_int
= ( ^ [I6: int,J4: int] : ( set_int2 @ ( upto @ I6 @ ( minus_minus_int @ J4 @ one_one_int ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_1248_sim__while__cong,axiom,
! [C: com,C2: com,B2: bexp] :
( ! [S3: list_char > int,T2: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S3 ) @ T2 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C2 @ S3 ) @ T2 ) )
=> ! [S4: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B2 @ C ) @ S4 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B2 @ C2 ) @ S4 ) @ T3 ) ) ) ).
% sim_while_cong
thf(fact_1249_sim__while__cong__aux,axiom,
! [B2: bexp,C: com,S: list_char > int,T: list_char > int,C2: com] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B2 @ C ) @ S ) @ T )
=> ( ! [S3: list_char > int,T2: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S3 ) @ T2 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C2 @ S3 ) @ T2 ) )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B2 @ C2 ) @ S ) @ T ) ) ) ).
% sim_while_cong_aux
thf(fact_1250_add__stack__eq,axiom,
! [M: nat,Ms2: int > int,Ms: int > int,N: nat] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_or4662586982721622107an_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) @ zero_zero_int ) )
=> ( ( Ms2 @ X3 )
= ( Ms @ X3 ) ) )
=> ( ( M = N )
=> ( ( add_stack @ M @ Ms2 )
= ( add_stack @ N @ Ms ) ) ) ) ).
% add_stack_eq
thf(fact_1251_WhileTrue,axiom,
! [B2: bexp,S_1: list_char > int,C: com,S_2: list_char > int,S_3: list_char > int] :
( ( bval @ B2 @ S_1 )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S_1 ) @ S_2 )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B2 @ C ) @ S_2 ) @ S_3 )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B2 @ C ) @ S_1 ) @ S_3 ) ) ) ) ).
% WhileTrue
thf(fact_1252_WhileFalse,axiom,
! [B2: bexp,S: list_char > int,C: com] :
( ~ ( bval @ B2 @ S )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B2 @ C ) @ S ) @ S ) ) ).
% WhileFalse
thf(fact_1253_WhileE,axiom,
! [B2: bexp,C: com,S: list_char > int,T: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B2 @ C ) @ S ) @ T )
=> ( ( ( T = S )
=> ( bval @ B2 @ S ) )
=> ~ ( ( bval @ B2 @ S )
=> ! [S_22: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S ) @ S_22 )
=> ~ ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B2 @ C ) @ S_22 ) @ T ) ) ) ) ) ).
% WhileE
thf(fact_1254_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1255_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= nil_int )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil
thf(fact_1256_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil_int
= ( upto @ I @ J ) )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil2
thf(fact_1257_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less_int @ J @ I )
=> ( ( upto @ I @ J )
= nil_int ) ) ).
% upto_empty
thf(fact_1258_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons_int @ I @ nil_int ) ) ).
% upto_single
thf(fact_1259_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1260_vars_Osimps_I1_J,axiom,
( ( vars @ nil_instr )
= nil_list_char ) ).
% vars.simps(1)
thf(fact_1261_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1262_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1263_add__stack_Osimps_I1_J,axiom,
! [Uu: int > int] :
( ( add_stack @ zero_zero_nat @ Uu )
= nil_int ) ).
% add_stack.simps(1)
thf(fact_1264_addr__of_Osimps_I1_J,axiom,
! [Uu: list_char] :
( ( addr_of @ nil_list_char @ Uu )
= zero_zero_int ) ).
% addr_of.simps(1)
thf(fact_1265_zabs__def,axiom,
( abs_abs_int
= ( ^ [I6: int] : ( if_int @ ( ord_less_int @ I6 @ zero_zero_int ) @ ( uminus_uminus_int @ I6 ) @ I6 ) ) ) ).
% zabs_def
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ ( ccomp @ c_1 ) @ ( ccomp @ c_2 ) ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ s_1 @ stka ) ) @ ( produc5086643055186798020st_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c_1 ) ) ) @ ( produc8650753666468850689st_int @ s_2 @ stka ) ) ).
%------------------------------------------------------------------------------