TPTP Problem File: SLH0683^1.p
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%------------------------------------------------------------------------------
% 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 : VYDRA_MDL/0011_Monitor/prob_00493_022981__17009960_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1421 ( 640 unt; 148 typ; 0 def)
% Number of atoms : 3456 (1213 equ; 0 cnn)
% Maximal formula atoms : 24 ( 2 avg)
% Number of connectives : 10663 ( 564 ~; 91 |; 181 &;8343 @)
% ( 0 <=>;1484 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 751 ( 751 >; 0 *; 0 +; 0 <<)
% Number of symbols : 126 ( 123 usr; 21 con; 0-3 aty)
% Number of variables : 4203 ( 150 ^;3920 !; 133 ?;4203 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:54:32.048
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Interval__O__092__060I__062_Itf__t_J,type,
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thf(ty_n_t__MDL__Oformula_Itf__a_Mtf__t_J,type,
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thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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thf(ty_n_t__MDL__Oregex_Itf__a_Mtf__t_J,type,
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thf(ty_n_t__List__Olist_Itf__t_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__String__Ochar,type,
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thf(ty_n_t__Real__Oreal,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_t__Int__Oint,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (123)
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thf(sy_c_GCD_Obezw,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
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thf(sy_c_If_001t__Int__Oint,type,
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thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
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thf(sy_c_Int_Onat,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_MDL_OAlways_001tf__t_001tf__a,type,
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thf(sy_c_MDL_OEventually_001tf__t_001tf__a,type,
eventually_t_a: i_t > formula_a_t > formula_a_t ).
thf(sy_c_MDL_OHistorically_001tf__t_001tf__a,type,
historically_t_a: i_t > formula_a_t > formula_a_t ).
thf(sy_c_MDL_OOnce_001tf__t_001tf__a,type,
once_t_a: i_t > formula_a_t > formula_a_t ).
thf(sy_c_MDL_Oformula_OAtom_001tf__a_001tf__t,type,
atom_a_t: a > formula_a_t ).
thf(sy_c_MDL_Oformula_OBin_001tf__a_001tf__t,type,
bin_a_t: ( $o > $o > $o ) > formula_a_t > formula_a_t > formula_a_t ).
thf(sy_c_MDL_Oformula_OBool_001tf__a_001tf__t,type,
bool_a_t: $o > formula_a_t ).
thf(sy_c_MDL_Oformula_OMatchF_001tf__t_001tf__a,type,
matchF_t_a: i_t > regex_a_t > formula_a_t ).
thf(sy_c_MDL_Oformula_OMatchP_001tf__t_001tf__a,type,
matchP_t_a: i_t > regex_a_t > formula_a_t ).
thf(sy_c_MDL_Oformula_ONeg_001tf__a_001tf__t,type,
neg_a_t: formula_a_t > formula_a_t ).
thf(sy_c_MDL_Oformula_ONext_001tf__t_001tf__a,type,
next_t_a: i_t > formula_a_t > formula_a_t ).
thf(sy_c_MDL_Oformula_OPrev_001tf__t_001tf__a,type,
prev_t_a: i_t > formula_a_t > formula_a_t ).
thf(sy_c_MDL_Oformula_OSince_001tf__a_001tf__t,type,
since_a_t: formula_a_t > i_t > formula_a_t > formula_a_t ).
thf(sy_c_MDL_Oformula_OUntil_001tf__a_001tf__t,type,
until_a_t: formula_a_t > i_t > formula_a_t > formula_a_t ).
thf(sy_c_MDL_Oformula_Orel__formula_001tf__a_001tf__a_001tf__t,type,
rel_formula_a_a_t: ( a > a > $o ) > formula_a_t > formula_a_t > $o ).
thf(sy_c_MDL_Oformula_Oset__formula_001tf__a_001tf__t,type,
set_formula_a_t: formula_a_t > set_a ).
thf(sy_c_MDL_Oregex_Orel__regex_001tf__a_001tf__a_001tf__t,type,
rel_regex_a_a_t: ( a > a > $o ) > regex_a_t > regex_a_t > $o ).
thf(sy_c_MDL_Oregex_Oset__regex_001tf__a_001tf__t,type,
set_regex_a_t: regex_a_t > set_a ).
thf(sy_c_Monitor_Omsize__fmla_001tf__a_001tf__t,type,
msize_fmla_a_t: formula_a_t > nat ).
thf(sy_c_Monitor_Omsize__regex_001tf__a_001tf__t,type,
msize_regex_a_t: regex_a_t > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat_Osize__class_Osize_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
size_s4016968051272393527la_a_t: formula_a_t > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__MDL__Oregex_Itf__a_Mtf__t_J,type,
size_size_regex_a_t: regex_a_t > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_Mt__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,type,
product_Pair_int_nat: int > nat > product_prod_int_nat ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Real__Oreal,type,
produc801115645435158769t_real: int > real > produc679980390762269497t_real ).
thf(sy_c_Product__Type_OPair_001t__MDL__Oformula_Itf__a_Mtf__t_J_001t__List__Olist_Itf__t_J,type,
produc8618728406956180955list_t: formula_a_t > list_t > produc3757808585008439209list_t ).
thf(sy_c_Product__Type_OPair_001t__MDL__Oformula_Itf__a_Mtf__t_J_001t__Nat__Onat,type,
produc373174216379635980_t_nat: formula_a_t > nat > produc6285428564250921684_t_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
product_Pair_nat_int: nat > int > product_prod_nat_int ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
produc2953812259323318284la_a_t: nat > formula_a_t > produc1078910373471009876la_a_t ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Real__Oreal,type,
produc7837566107596912789t_real: nat > real > produc7716430852924023517t_real ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
produc3646306378393792727nt_int: product_prod_int_int > product_prod_int_int > produc1219242969750017639nt_int ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Int__Oint,type,
produc3179012173361985393al_int: real > int > produc8786904178792722361al_int ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Timestamp_Oembed__nat__class_O_092_060iota_062_001t__Nat__Onat,type,
embed_nat_iota_nat: nat > nat ).
thf(sy_c_Timestamp_Oembed__nat__class_O_092_060iota_062_001t__Real__Oreal,type,
embed_nat_iota_real: nat > real ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_P,type,
p: nat > formula_a_t > $o ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_phi,type,
phi: formula_a_t ).
thf(sy_v_phia____,type,
phia: formula_a_t ).
% Relevant facts (1265)
thf(fact_0__C0_C,axiom,
na = zero_zero_nat ).
% "0"
thf(fact_1__C1_Oprems_C,axiom,
ord_less_eq_nat @ ( msize_fmla_a_t @ phia ) @ na ).
% "1.prems"
thf(fact_2_assms_I2_J,axiom,
! [N: nat,B: $o] : ( p @ N @ ( bool_a_t @ B ) ) ).
% assms(2)
thf(fact_3_assms_I3_J,axiom,
! [N: nat,A: a] : ( p @ N @ ( atom_a_t @ A ) ) ).
% assms(3)
thf(fact_4__C1_OIH_C,axiom,
! [M: nat] :
( ( ord_less_nat @ M @ na )
=> ! [X: formula_a_t] :
( ( ord_less_eq_nat @ ( msize_fmla_a_t @ X ) @ M )
=> ( p @ M @ X ) ) ) ).
% "1.IH"
thf(fact_5_assms_I7_J,axiom,
! [Phi: formula_a_t,N: nat,I: i_t] :
( ( ord_less_eq_nat @ ( msize_fmla_a_t @ Phi ) @ N )
=> ( ( p @ N @ Phi )
=> ( p @ ( suc @ N ) @ ( next_t_a @ I @ Phi ) ) ) ) ).
% assms(7)
thf(fact_6_assms_I6_J,axiom,
! [Phi: formula_a_t,N: nat,I: i_t] :
( ( ord_less_eq_nat @ ( msize_fmla_a_t @ Phi ) @ N )
=> ( ( p @ N @ Phi )
=> ( p @ ( suc @ N ) @ ( prev_t_a @ I @ Phi ) ) ) ) ).
% assms(6)
thf(fact_7_assms_I11_J,axiom,
! [I: i_t,R: regex_a_t,N: nat] :
( ( ord_less_eq_nat @ ( msize_fmla_a_t @ ( matchF_t_a @ I @ R ) ) @ ( suc @ N ) )
=> ( ! [X2: formula_a_t] :
( ( ord_less_eq_nat @ ( msize_fmla_a_t @ X2 ) @ N )
=> ( p @ N @ X2 ) )
=> ( p @ ( suc @ N ) @ ( matchF_t_a @ I @ R ) ) ) ) ).
% assms(11)
thf(fact_8_assms_I10_J,axiom,
! [I: i_t,R: regex_a_t,N: nat] :
( ( ord_less_eq_nat @ ( msize_fmla_a_t @ ( matchP_t_a @ I @ R ) ) @ ( suc @ N ) )
=> ( ! [X2: formula_a_t] :
( ( ord_less_eq_nat @ ( msize_fmla_a_t @ X2 ) @ N )
=> ( p @ N @ X2 ) )
=> ( p @ ( suc @ N ) @ ( matchP_t_a @ I @ R ) ) ) ) ).
% assms(10)
thf(fact_9_assms_I9_J,axiom,
! [Phi: formula_a_t,N: nat,Psi: formula_a_t,I: i_t] :
( ( ord_less_eq_nat @ ( msize_fmla_a_t @ Phi ) @ N )
=> ( ( ord_less_eq_nat @ ( msize_fmla_a_t @ Psi ) @ N )
=> ( ( p @ N @ Phi )
=> ( ( p @ N @ Psi )
=> ( p @ ( suc @ N ) @ ( until_a_t @ Phi @ I @ Psi ) ) ) ) ) ) ).
% assms(9)
thf(fact_10_assms_I8_J,axiom,
! [Phi: formula_a_t,N: nat,Psi: formula_a_t,I: i_t] :
( ( ord_less_eq_nat @ ( msize_fmla_a_t @ Phi ) @ N )
=> ( ( ord_less_eq_nat @ ( msize_fmla_a_t @ Psi ) @ N )
=> ( ( p @ N @ Phi )
=> ( ( p @ N @ Psi )
=> ( p @ ( suc @ N ) @ ( since_a_t @ Phi @ I @ Psi ) ) ) ) ) ) ).
% assms(8)
thf(fact_11_assms_I4_J,axiom,
! [Phi: formula_a_t,N: nat] :
( ( ord_less_eq_nat @ ( msize_fmla_a_t @ Phi ) @ N )
=> ( ( p @ N @ Phi )
=> ( p @ ( suc @ N ) @ ( neg_a_t @ Phi ) ) ) ) ).
% assms(4)
thf(fact_12_assms_I5_J,axiom,
! [F: $o > $o > $o,Phi: formula_a_t,Psi: formula_a_t,N: nat] :
( ( ord_less_eq_nat @ ( msize_fmla_a_t @ ( bin_a_t @ F @ Phi @ Psi ) ) @ ( suc @ N ) )
=> ( ( p @ N @ Phi )
=> ( ( p @ N @ Psi )
=> ( p @ ( suc @ N ) @ ( bin_a_t @ F @ Phi @ Psi ) ) ) ) ) ).
% assms(5)
thf(fact_13_assms_I1_J,axiom,
ord_less_eq_nat @ ( msize_fmla_a_t @ phi ) @ n ).
% assms(1)
thf(fact_14_msize__fmla_Osimps_I6_J,axiom,
! [I: i_t,Phi: formula_a_t] :
( ( msize_fmla_a_t @ ( next_t_a @ I @ Phi ) )
= ( suc @ ( msize_fmla_a_t @ Phi ) ) ) ).
% msize_fmla.simps(6)
thf(fact_15_msize__fmla_Osimps_I5_J,axiom,
! [I: i_t,Phi: formula_a_t] :
( ( msize_fmla_a_t @ ( prev_t_a @ I @ Phi ) )
= ( suc @ ( msize_fmla_a_t @ Phi ) ) ) ).
% msize_fmla.simps(5)
thf(fact_16_msize__fmla_Osimps_I3_J,axiom,
! [Phi: formula_a_t] :
( ( msize_fmla_a_t @ ( neg_a_t @ Phi ) )
= ( suc @ ( msize_fmla_a_t @ Phi ) ) ) ).
% msize_fmla.simps(3)
thf(fact_17_msize__fmla_Osimps_I2_J,axiom,
! [A: a] :
( ( msize_fmla_a_t @ ( atom_a_t @ A ) )
= zero_zero_nat ) ).
% msize_fmla.simps(2)
thf(fact_18_msize__fmla_Osimps_I1_J,axiom,
! [B: $o] :
( ( msize_fmla_a_t @ ( bool_a_t @ B ) )
= zero_zero_nat ) ).
% msize_fmla.simps(1)
thf(fact_19_msize__fmla_Ocases,axiom,
! [X3: formula_a_t] :
( ! [B2: $o] :
( X3
!= ( bool_a_t @ B2 ) )
=> ( ! [A2: a] :
( X3
!= ( atom_a_t @ A2 ) )
=> ( ! [Phi2: formula_a_t] :
( X3
!= ( neg_a_t @ Phi2 ) )
=> ( ! [F2: $o > $o > $o,Phi2: formula_a_t,Psi2: formula_a_t] :
( X3
!= ( bin_a_t @ F2 @ Phi2 @ Psi2 ) )
=> ( ! [I2: i_t,Phi2: formula_a_t] :
( X3
!= ( prev_t_a @ I2 @ Phi2 ) )
=> ( ! [I2: i_t,Phi2: formula_a_t] :
( X3
!= ( next_t_a @ I2 @ Phi2 ) )
=> ( ! [Phi2: formula_a_t,I2: i_t,Psi2: formula_a_t] :
( X3
!= ( since_a_t @ Phi2 @ I2 @ Psi2 ) )
=> ( ! [Phi2: formula_a_t,I2: i_t,Psi2: formula_a_t] :
( X3
!= ( until_a_t @ Phi2 @ I2 @ Psi2 ) )
=> ( ! [I2: i_t,R2: regex_a_t] :
( X3
!= ( matchP_t_a @ I2 @ R2 ) )
=> ~ ! [I2: i_t,R2: regex_a_t] :
( X3
!= ( matchF_t_a @ I2 @ R2 ) ) ) ) ) ) ) ) ) ) ) ).
% msize_fmla.cases
thf(fact_20_formula_Oexhaust,axiom,
! [Y: formula_a_t] :
( ! [X1: $o] :
( Y
!= ( bool_a_t @ X1 ) )
=> ( ! [X22: a] :
( Y
!= ( atom_a_t @ X22 ) )
=> ( ! [X32: formula_a_t] :
( Y
!= ( neg_a_t @ X32 ) )
=> ( ! [X41: $o > $o > $o,X42: formula_a_t,X43: formula_a_t] :
( Y
!= ( bin_a_t @ X41 @ X42 @ X43 ) )
=> ( ! [X51: i_t,X52: formula_a_t] :
( Y
!= ( prev_t_a @ X51 @ X52 ) )
=> ( ! [X61: i_t,X62: formula_a_t] :
( Y
!= ( next_t_a @ X61 @ X62 ) )
=> ( ! [X71: formula_a_t,X72: i_t,X73: formula_a_t] :
( Y
!= ( since_a_t @ X71 @ X72 @ X73 ) )
=> ( ! [X81: formula_a_t,X82: i_t,X83: formula_a_t] :
( Y
!= ( until_a_t @ X81 @ X82 @ X83 ) )
=> ( ! [X91: i_t,X92: regex_a_t] :
( Y
!= ( matchP_t_a @ X91 @ X92 ) )
=> ~ ! [X101: i_t,X102: regex_a_t] :
( Y
!= ( matchF_t_a @ X101 @ X102 ) ) ) ) ) ) ) ) ) ) ) ).
% formula.exhaust
thf(fact_21_bounded__future__fmla_Ocases,axiom,
! [X3: formula_a_t] :
( ! [B2: $o] :
( X3
!= ( bool_a_t @ B2 ) )
=> ( ! [A2: a] :
( X3
!= ( atom_a_t @ A2 ) )
=> ( ! [Phi2: formula_a_t] :
( X3
!= ( neg_a_t @ Phi2 ) )
=> ( ! [F2: $o > $o > $o,Phi2: formula_a_t,Psi2: formula_a_t] :
( X3
!= ( bin_a_t @ F2 @ Phi2 @ Psi2 ) )
=> ( ! [I2: i_t,Phi2: formula_a_t] :
( X3
!= ( prev_t_a @ I2 @ Phi2 ) )
=> ( ! [I2: i_t,Phi2: formula_a_t] :
( X3
!= ( next_t_a @ I2 @ Phi2 ) )
=> ( ! [Phi2: formula_a_t,I2: i_t,Psi2: formula_a_t] :
( X3
!= ( since_a_t @ Phi2 @ I2 @ Psi2 ) )
=> ( ! [Phi2: formula_a_t,I2: i_t,Psi2: formula_a_t] :
( X3
!= ( until_a_t @ Phi2 @ I2 @ Psi2 ) )
=> ( ! [I2: i_t,R2: regex_a_t] :
( X3
!= ( matchP_t_a @ I2 @ R2 ) )
=> ~ ! [I2: i_t,R2: regex_a_t] :
( X3
!= ( matchF_t_a @ I2 @ R2 ) ) ) ) ) ) ) ) ) ) ) ).
% bounded_future_fmla.cases
thf(fact_22_sat_Ocases,axiom,
! [X3: produc6285428564250921684_t_nat] :
( ! [B2: $o,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( bool_a_t @ B2 ) @ I3 ) )
=> ( ! [A2: a,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( atom_a_t @ A2 ) @ I3 ) )
=> ( ! [Phi3: formula_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( neg_a_t @ Phi3 ) @ I3 ) )
=> ( ! [F2: $o > $o > $o,Phi3: formula_a_t,Psi3: formula_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( bin_a_t @ F2 @ Phi3 @ Psi3 ) @ I3 ) )
=> ( ! [I2: i_t,Phi3: formula_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( prev_t_a @ I2 @ Phi3 ) @ I3 ) )
=> ( ! [I2: i_t,Phi3: formula_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( next_t_a @ I2 @ Phi3 ) @ I3 ) )
=> ( ! [Phi3: formula_a_t,I2: i_t,Psi3: formula_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( since_a_t @ Phi3 @ I2 @ Psi3 ) @ I3 ) )
=> ( ! [Phi3: formula_a_t,I2: i_t,Psi3: formula_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( until_a_t @ Phi3 @ I2 @ Psi3 ) @ I3 ) )
=> ( ! [I2: i_t,R2: regex_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( matchP_t_a @ I2 @ R2 ) @ I3 ) )
=> ~ ! [I2: i_t,R2: regex_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( matchF_t_a @ I2 @ R2 ) @ I3 ) ) ) ) ) ) ) ) ) ) ) ).
% sat.cases
thf(fact_23_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_24_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_25_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_26_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_27_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_28_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_29_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_30_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_31_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_32_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_33_nat_Oinject,axiom,
! [X23: nat,Y2: nat] :
( ( ( suc @ X23 )
= ( suc @ Y2 ) )
= ( X23 = Y2 ) ) ).
% nat.inject
thf(fact_34_formula_Oinject_I5_J,axiom,
! [X512: i_t,X522: formula_a_t,Y51: i_t,Y52: formula_a_t] :
( ( ( prev_t_a @ X512 @ X522 )
= ( prev_t_a @ Y51 @ Y52 ) )
= ( ( X512 = Y51 )
& ( X522 = Y52 ) ) ) ).
% formula.inject(5)
thf(fact_35_formula_Oinject_I6_J,axiom,
! [X612: i_t,X622: formula_a_t,Y61: i_t,Y62: formula_a_t] :
( ( ( next_t_a @ X612 @ X622 )
= ( next_t_a @ Y61 @ Y62 ) )
= ( ( X612 = Y61 )
& ( X622 = Y62 ) ) ) ).
% formula.inject(6)
thf(fact_36_formula_Oinject_I10_J,axiom,
! [X1012: i_t,X1022: regex_a_t,Y101: i_t,Y102: regex_a_t] :
( ( ( matchF_t_a @ X1012 @ X1022 )
= ( matchF_t_a @ Y101 @ Y102 ) )
= ( ( X1012 = Y101 )
& ( X1022 = Y102 ) ) ) ).
% formula.inject(10)
thf(fact_37_formula_Oinject_I9_J,axiom,
! [X912: i_t,X922: regex_a_t,Y91: i_t,Y92: regex_a_t] :
( ( ( matchP_t_a @ X912 @ X922 )
= ( matchP_t_a @ Y91 @ Y92 ) )
= ( ( X912 = Y91 )
& ( X922 = Y92 ) ) ) ).
% formula.inject(9)
thf(fact_38_formula_Oinject_I7_J,axiom,
! [X712: formula_a_t,X722: i_t,X732: formula_a_t,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t] :
( ( ( since_a_t @ X712 @ X722 @ X732 )
= ( since_a_t @ Y71 @ Y72 @ Y73 ) )
= ( ( X712 = Y71 )
& ( X722 = Y72 )
& ( X732 = Y73 ) ) ) ).
% formula.inject(7)
thf(fact_39_formula_Oinject_I8_J,axiom,
! [X812: formula_a_t,X822: i_t,X832: formula_a_t,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t] :
( ( ( until_a_t @ X812 @ X822 @ X832 )
= ( until_a_t @ Y81 @ Y82 @ Y83 ) )
= ( ( X812 = Y81 )
& ( X822 = Y82 )
& ( X832 = Y83 ) ) ) ).
% formula.inject(8)
thf(fact_40_formula_Oinject_I3_J,axiom,
! [X33: formula_a_t,Y3: formula_a_t] :
( ( ( neg_a_t @ X33 )
= ( neg_a_t @ Y3 ) )
= ( X33 = Y3 ) ) ).
% formula.inject(3)
thf(fact_41_formula_Oinject_I4_J,axiom,
! [X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,Y41: $o > $o > $o,Y42: formula_a_t,Y43: formula_a_t] :
( ( ( bin_a_t @ X412 @ X422 @ X432 )
= ( bin_a_t @ Y41 @ Y42 @ Y43 ) )
= ( ( X412 = Y41 )
& ( X422 = Y42 )
& ( X432 = Y43 ) ) ) ).
% formula.inject(4)
thf(fact_42_formula_Oinject_I1_J,axiom,
! [X12: $o,Y1: $o] :
( ( ( bool_a_t @ X12 )
= ( bool_a_t @ Y1 ) )
= ( X12 = Y1 ) ) ).
% formula.inject(1)
thf(fact_43_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A3: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_45_formula_Oinject_I2_J,axiom,
! [X23: a,Y2: a] :
( ( ( atom_a_t @ X23 )
= ( atom_a_t @ Y2 ) )
= ( X23 = Y2 ) ) ).
% formula.inject(2)
thf(fact_46_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_47_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_48_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_49_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_50_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_51_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_52_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_53_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_54_le__trans,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% le_trans
thf(fact_55_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_56_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_57_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P @ M ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_58_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_59_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_60_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_61_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_62_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_63_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_64_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_65_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_66_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_67_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_68_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_69_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X2: nat,Y4: nat] :
( ( P @ X2 @ Y4 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y4 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_70_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_71_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_72_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_73_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_74_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_75_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_76_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X2: nat] : ( R3 @ X2 @ X2 )
=> ( ! [X2: nat,Y4: nat,Z: nat] :
( ( R3 @ X2 @ Y4 )
=> ( ( R3 @ Y4 @ Z )
=> ( R3 @ X2 @ Z ) ) )
=> ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
=> ( R3 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_77_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_78_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_79_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_80_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_81_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_82_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M3: nat] :
( M4
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_83_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_84_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_85_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_86_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_87_strict__inc__induct,axiom,
! [I4: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I4 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I4 ) ) ) ) ).
% strict_inc_induct
thf(fact_88_less__Suc__induct,axiom,
! [I4: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I4 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I4 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_89_less__trans__Suc,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I4 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_90_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_91_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_92_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M5: nat] :
( ( M2
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_93_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ I5 ) ) ) ) ).
% All_less_Suc
thf(fact_94_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_95_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_96_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ N )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( P @ I5 ) ) ) ) ).
% Ex_less_Suc
thf(fact_97_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_98_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_99_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_100_Suc__lessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_101_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_102_Nat_OlessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( ( K
!= ( suc @ I4 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_103_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_104_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_105_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_106_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_107_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P @ M ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_108_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_109_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_110_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_111_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_112_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_113_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_114_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I4: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_115_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_116_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_117_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
| ( M6 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_118_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_119_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N3: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
& ( M6 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_120_formula_Odistinct_I61_J,axiom,
! [X512: i_t,X522: formula_a_t,X612: i_t,X622: formula_a_t] :
( ( prev_t_a @ X512 @ X522 )
!= ( next_t_a @ X612 @ X622 ) ) ).
% formula.distinct(61)
thf(fact_121_formula_Odistinct_I69_J,axiom,
! [X512: i_t,X522: formula_a_t,X1012: i_t,X1022: regex_a_t] :
( ( prev_t_a @ X512 @ X522 )
!= ( matchF_t_a @ X1012 @ X1022 ) ) ).
% formula.distinct(69)
thf(fact_122_formula_Odistinct_I77_J,axiom,
! [X612: i_t,X622: formula_a_t,X1012: i_t,X1022: regex_a_t] :
( ( next_t_a @ X612 @ X622 )
!= ( matchF_t_a @ X1012 @ X1022 ) ) ).
% formula.distinct(77)
thf(fact_123_formula_Odistinct_I67_J,axiom,
! [X512: i_t,X522: formula_a_t,X912: i_t,X922: regex_a_t] :
( ( prev_t_a @ X512 @ X522 )
!= ( matchP_t_a @ X912 @ X922 ) ) ).
% formula.distinct(67)
thf(fact_124_formula_Odistinct_I75_J,axiom,
! [X612: i_t,X622: formula_a_t,X912: i_t,X922: regex_a_t] :
( ( next_t_a @ X612 @ X622 )
!= ( matchP_t_a @ X912 @ X922 ) ) ).
% formula.distinct(75)
thf(fact_125_formula_Odistinct_I63_J,axiom,
! [X512: i_t,X522: formula_a_t,X712: formula_a_t,X722: i_t,X732: formula_a_t] :
( ( prev_t_a @ X512 @ X522 )
!= ( since_a_t @ X712 @ X722 @ X732 ) ) ).
% formula.distinct(63)
thf(fact_126_formula_Odistinct_I65_J,axiom,
! [X512: i_t,X522: formula_a_t,X812: formula_a_t,X822: i_t,X832: formula_a_t] :
( ( prev_t_a @ X512 @ X522 )
!= ( until_a_t @ X812 @ X822 @ X832 ) ) ).
% formula.distinct(65)
thf(fact_127_formula_Odistinct_I71_J,axiom,
! [X612: i_t,X622: formula_a_t,X712: formula_a_t,X722: i_t,X732: formula_a_t] :
( ( next_t_a @ X612 @ X622 )
!= ( since_a_t @ X712 @ X722 @ X732 ) ) ).
% formula.distinct(71)
thf(fact_128_formula_Odistinct_I73_J,axiom,
! [X612: i_t,X622: formula_a_t,X812: formula_a_t,X822: i_t,X832: formula_a_t] :
( ( next_t_a @ X612 @ X622 )
!= ( until_a_t @ X812 @ X822 @ X832 ) ) ).
% formula.distinct(73)
thf(fact_129_formula_Odistinct_I37_J,axiom,
! [X33: formula_a_t,X512: i_t,X522: formula_a_t] :
( ( neg_a_t @ X33 )
!= ( prev_t_a @ X512 @ X522 ) ) ).
% formula.distinct(37)
thf(fact_130_formula_Odistinct_I39_J,axiom,
! [X33: formula_a_t,X612: i_t,X622: formula_a_t] :
( ( neg_a_t @ X33 )
!= ( next_t_a @ X612 @ X622 ) ) ).
% formula.distinct(39)
thf(fact_131_formula_Odistinct_I89_J,axiom,
! [X912: i_t,X922: regex_a_t,X1012: i_t,X1022: regex_a_t] :
( ( matchP_t_a @ X912 @ X922 )
!= ( matchF_t_a @ X1012 @ X1022 ) ) ).
% formula.distinct(89)
thf(fact_132_formula_Odistinct_I49_J,axiom,
! [X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,X512: i_t,X522: formula_a_t] :
( ( bin_a_t @ X412 @ X422 @ X432 )
!= ( prev_t_a @ X512 @ X522 ) ) ).
% formula.distinct(49)
thf(fact_133_formula_Odistinct_I51_J,axiom,
! [X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,X612: i_t,X622: formula_a_t] :
( ( bin_a_t @ X412 @ X422 @ X432 )
!= ( next_t_a @ X612 @ X622 ) ) ).
% formula.distinct(51)
thf(fact_134_formula_Odistinct_I83_J,axiom,
! [X712: formula_a_t,X722: i_t,X732: formula_a_t,X1012: i_t,X1022: regex_a_t] :
( ( since_a_t @ X712 @ X722 @ X732 )
!= ( matchF_t_a @ X1012 @ X1022 ) ) ).
% formula.distinct(83)
thf(fact_135_formula_Odistinct_I87_J,axiom,
! [X812: formula_a_t,X822: i_t,X832: formula_a_t,X1012: i_t,X1022: regex_a_t] :
( ( until_a_t @ X812 @ X822 @ X832 )
!= ( matchF_t_a @ X1012 @ X1022 ) ) ).
% formula.distinct(87)
thf(fact_136_formula_Odistinct_I81_J,axiom,
! [X712: formula_a_t,X722: i_t,X732: formula_a_t,X912: i_t,X922: regex_a_t] :
( ( since_a_t @ X712 @ X722 @ X732 )
!= ( matchP_t_a @ X912 @ X922 ) ) ).
% formula.distinct(81)
thf(fact_137_formula_Odistinct_I85_J,axiom,
! [X812: formula_a_t,X822: i_t,X832: formula_a_t,X912: i_t,X922: regex_a_t] :
( ( until_a_t @ X812 @ X822 @ X832 )
!= ( matchP_t_a @ X912 @ X922 ) ) ).
% formula.distinct(85)
thf(fact_138_formula_Odistinct_I47_J,axiom,
! [X33: formula_a_t,X1012: i_t,X1022: regex_a_t] :
( ( neg_a_t @ X33 )
!= ( matchF_t_a @ X1012 @ X1022 ) ) ).
% formula.distinct(47)
thf(fact_139_formula_Odistinct_I79_J,axiom,
! [X712: formula_a_t,X722: i_t,X732: formula_a_t,X812: formula_a_t,X822: i_t,X832: formula_a_t] :
( ( since_a_t @ X712 @ X722 @ X732 )
!= ( until_a_t @ X812 @ X822 @ X832 ) ) ).
% formula.distinct(79)
thf(fact_140_formula_Odistinct_I45_J,axiom,
! [X33: formula_a_t,X912: i_t,X922: regex_a_t] :
( ( neg_a_t @ X33 )
!= ( matchP_t_a @ X912 @ X922 ) ) ).
% formula.distinct(45)
thf(fact_141_formula_Odistinct_I59_J,axiom,
! [X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,X1012: i_t,X1022: regex_a_t] :
( ( bin_a_t @ X412 @ X422 @ X432 )
!= ( matchF_t_a @ X1012 @ X1022 ) ) ).
% formula.distinct(59)
thf(fact_142_formula_Odistinct_I41_J,axiom,
! [X33: formula_a_t,X712: formula_a_t,X722: i_t,X732: formula_a_t] :
( ( neg_a_t @ X33 )
!= ( since_a_t @ X712 @ X722 @ X732 ) ) ).
% formula.distinct(41)
thf(fact_143_formula_Odistinct_I43_J,axiom,
! [X33: formula_a_t,X812: formula_a_t,X822: i_t,X832: formula_a_t] :
( ( neg_a_t @ X33 )
!= ( until_a_t @ X812 @ X822 @ X832 ) ) ).
% formula.distinct(43)
thf(fact_144_formula_Odistinct_I57_J,axiom,
! [X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,X912: i_t,X922: regex_a_t] :
( ( bin_a_t @ X412 @ X422 @ X432 )
!= ( matchP_t_a @ X912 @ X922 ) ) ).
% formula.distinct(57)
thf(fact_145_formula_Odistinct_I53_J,axiom,
! [X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,X712: formula_a_t,X722: i_t,X732: formula_a_t] :
( ( bin_a_t @ X412 @ X422 @ X432 )
!= ( since_a_t @ X712 @ X722 @ X732 ) ) ).
% formula.distinct(53)
thf(fact_146_formula_Odistinct_I55_J,axiom,
! [X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,X812: formula_a_t,X822: i_t,X832: formula_a_t] :
( ( bin_a_t @ X412 @ X422 @ X432 )
!= ( until_a_t @ X812 @ X822 @ X832 ) ) ).
% formula.distinct(55)
thf(fact_147_formula_Odistinct_I35_J,axiom,
! [X33: formula_a_t,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t] :
( ( neg_a_t @ X33 )
!= ( bin_a_t @ X412 @ X422 @ X432 ) ) ).
% formula.distinct(35)
thf(fact_148_formula_Odistinct_I7_J,axiom,
! [X12: $o,X512: i_t,X522: formula_a_t] :
( ( bool_a_t @ X12 )
!= ( prev_t_a @ X512 @ X522 ) ) ).
% formula.distinct(7)
thf(fact_149_formula_Odistinct_I9_J,axiom,
! [X12: $o,X612: i_t,X622: formula_a_t] :
( ( bool_a_t @ X12 )
!= ( next_t_a @ X612 @ X622 ) ) ).
% formula.distinct(9)
thf(fact_150_formula_Odistinct_I17_J,axiom,
! [X12: $o,X1012: i_t,X1022: regex_a_t] :
( ( bool_a_t @ X12 )
!= ( matchF_t_a @ X1012 @ X1022 ) ) ).
% formula.distinct(17)
thf(fact_151_formula_Odistinct_I15_J,axiom,
! [X12: $o,X912: i_t,X922: regex_a_t] :
( ( bool_a_t @ X12 )
!= ( matchP_t_a @ X912 @ X922 ) ) ).
% formula.distinct(15)
thf(fact_152_formula_Odistinct_I11_J,axiom,
! [X12: $o,X712: formula_a_t,X722: i_t,X732: formula_a_t] :
( ( bool_a_t @ X12 )
!= ( since_a_t @ X712 @ X722 @ X732 ) ) ).
% formula.distinct(11)
thf(fact_153_formula_Odistinct_I13_J,axiom,
! [X12: $o,X812: formula_a_t,X822: i_t,X832: formula_a_t] :
( ( bool_a_t @ X12 )
!= ( until_a_t @ X812 @ X822 @ X832 ) ) ).
% formula.distinct(13)
thf(fact_154_formula_Odistinct_I3_J,axiom,
! [X12: $o,X33: formula_a_t] :
( ( bool_a_t @ X12 )
!= ( neg_a_t @ X33 ) ) ).
% formula.distinct(3)
thf(fact_155_formula_Odistinct_I5_J,axiom,
! [X12: $o,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t] :
( ( bool_a_t @ X12 )
!= ( bin_a_t @ X412 @ X422 @ X432 ) ) ).
% formula.distinct(5)
thf(fact_156_formula_Odistinct_I23_J,axiom,
! [X23: a,X512: i_t,X522: formula_a_t] :
( ( atom_a_t @ X23 )
!= ( prev_t_a @ X512 @ X522 ) ) ).
% formula.distinct(23)
thf(fact_157_formula_Odistinct_I25_J,axiom,
! [X23: a,X612: i_t,X622: formula_a_t] :
( ( atom_a_t @ X23 )
!= ( next_t_a @ X612 @ X622 ) ) ).
% formula.distinct(25)
thf(fact_158_formula_Odistinct_I33_J,axiom,
! [X23: a,X1012: i_t,X1022: regex_a_t] :
( ( atom_a_t @ X23 )
!= ( matchF_t_a @ X1012 @ X1022 ) ) ).
% formula.distinct(33)
thf(fact_159_formula_Odistinct_I31_J,axiom,
! [X23: a,X912: i_t,X922: regex_a_t] :
( ( atom_a_t @ X23 )
!= ( matchP_t_a @ X912 @ X922 ) ) ).
% formula.distinct(31)
thf(fact_160_formula_Odistinct_I27_J,axiom,
! [X23: a,X712: formula_a_t,X722: i_t,X732: formula_a_t] :
( ( atom_a_t @ X23 )
!= ( since_a_t @ X712 @ X722 @ X732 ) ) ).
% formula.distinct(27)
thf(fact_161_formula_Odistinct_I29_J,axiom,
! [X23: a,X812: formula_a_t,X822: i_t,X832: formula_a_t] :
( ( atom_a_t @ X23 )
!= ( until_a_t @ X812 @ X822 @ X832 ) ) ).
% formula.distinct(29)
thf(fact_162_formula_Odistinct_I19_J,axiom,
! [X23: a,X33: formula_a_t] :
( ( atom_a_t @ X23 )
!= ( neg_a_t @ X33 ) ) ).
% formula.distinct(19)
thf(fact_163_formula_Odistinct_I21_J,axiom,
! [X23: a,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t] :
( ( atom_a_t @ X23 )
!= ( bin_a_t @ X412 @ X422 @ X432 ) ) ).
% formula.distinct(21)
thf(fact_164_formula_Odistinct_I1_J,axiom,
! [X12: $o,X23: a] :
( ( bool_a_t @ X12 )
!= ( atom_a_t @ X23 ) ) ).
% formula.distinct(1)
thf(fact_165_MDL_Osat_Ocases,axiom,
! [X3: produc6285428564250921684_t_nat] :
( ! [B2: $o,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( bool_a_t @ B2 ) @ I3 ) )
=> ( ! [A2: a,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( atom_a_t @ A2 ) @ I3 ) )
=> ( ! [Phi3: formula_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( neg_a_t @ Phi3 ) @ I3 ) )
=> ( ! [F2: $o > $o > $o,Phi3: formula_a_t,Psi3: formula_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( bin_a_t @ F2 @ Phi3 @ Psi3 ) @ I3 ) )
=> ( ! [I2: i_t,Phi3: formula_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( prev_t_a @ I2 @ Phi3 ) @ I3 ) )
=> ( ! [I2: i_t,Phi3: formula_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( next_t_a @ I2 @ Phi3 ) @ I3 ) )
=> ( ! [Phi3: formula_a_t,I2: i_t,Psi3: formula_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( since_a_t @ Phi3 @ I2 @ Psi3 ) @ I3 ) )
=> ( ! [Phi3: formula_a_t,I2: i_t,Psi3: formula_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( until_a_t @ Phi3 @ I2 @ Psi3 ) @ I3 ) )
=> ( ! [I2: i_t,R2: regex_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( matchP_t_a @ I2 @ R2 ) @ I3 ) )
=> ~ ! [I2: i_t,R2: regex_a_t,I3: nat] :
( X3
!= ( produc373174216379635980_t_nat @ ( matchF_t_a @ I2 @ R2 ) @ I3 ) ) ) ) ) ) ) ) ) ) ) ).
% MDL.sat.cases
thf(fact_166_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_167_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_168_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_169_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_170_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_171_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_172_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_173_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_174_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_175_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_176_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_177_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_178_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_179_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_180_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ ( suc @ I5 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_181_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_182_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( P @ ( suc @ I5 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_183_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_184_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_185_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_186_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_187_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_188_inc__induct,axiom,
! [I4: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% inc_induct
thf(fact_189_dec__induct,axiom,
! [I4: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( P @ I4 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_190_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_191_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_192_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ K2 )
=> ~ ( P @ I6 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_193_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I6: nat] :
( ( ord_less_eq_nat @ I6 @ K2 )
=> ~ ( P @ I6 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_194_less__eq__prod__simp,axiom,
! [X12: nat > $o,Y1: nat,X23: nat > $o,Y2: nat] :
( ( ord_le1595970101268698462_o_nat @ ( produc7277522915581678840_o_nat @ X12 @ Y1 ) @ ( produc7277522915581678840_o_nat @ X23 @ Y2 ) )
= ( ( ord_less_nat_o @ X12 @ X23 )
| ( ( ord_less_eq_nat_o @ X12 @ X23 )
& ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_195_less__eq__prod__simp,axiom,
! [X12: nat,Y1: nat,X23: nat,Y2: nat] :
( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X12 @ Y1 ) @ ( product_Pair_nat_nat @ X23 @ Y2 ) )
= ( ( ord_less_nat @ X12 @ X23 )
| ( ( ord_less_eq_nat @ X12 @ X23 )
& ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_196_less__eq__prod__simp,axiom,
! [X12: nat,Y1: int,X23: nat,Y2: int] :
( ( ord_le4282293441679094013at_int @ ( product_Pair_nat_int @ X12 @ Y1 ) @ ( product_Pair_nat_int @ X23 @ Y2 ) )
= ( ( ord_less_nat @ X12 @ X23 )
| ( ( ord_less_eq_nat @ X12 @ X23 )
& ( ord_less_eq_int @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_197_less__eq__prod__simp,axiom,
! [X12: nat,Y1: real,X23: nat,Y2: real] :
( ( ord_le8710666929947597437t_real @ ( produc7837566107596912789t_real @ X12 @ Y1 ) @ ( produc7837566107596912789t_real @ X23 @ Y2 ) )
= ( ( ord_less_nat @ X12 @ X23 )
| ( ( ord_less_eq_nat @ X12 @ X23 )
& ( ord_less_eq_real @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_198_less__eq__prod__simp,axiom,
! [X12: int,Y1: nat,X23: int,Y2: nat] :
( ( ord_le236126136234569469nt_nat @ ( product_Pair_int_nat @ X12 @ Y1 ) @ ( product_Pair_int_nat @ X23 @ Y2 ) )
= ( ( ord_less_int @ X12 @ X23 )
| ( ( ord_less_eq_int @ X12 @ X23 )
& ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_199_less__eq__prod__simp,axiom,
! [X12: int,Y1: int,X23: int,Y2: int] :
( ( ord_le5281647153580148569nt_int @ ( product_Pair_int_int @ X12 @ Y1 ) @ ( product_Pair_int_int @ X23 @ Y2 ) )
= ( ( ord_less_int @ X12 @ X23 )
| ( ( ord_less_eq_int @ X12 @ X23 )
& ( ord_less_eq_int @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_200_less__eq__prod__simp,axiom,
! [X12: int,Y1: real,X23: int,Y2: real] :
( ( ord_le1674216467785843417t_real @ ( produc801115645435158769t_real @ X12 @ Y1 ) @ ( produc801115645435158769t_real @ X23 @ Y2 ) )
= ( ( ord_less_int @ X12 @ X23 )
| ( ( ord_less_eq_int @ X12 @ X23 )
& ( ord_less_eq_real @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_201_less__eq__prod__simp,axiom,
! [X12: real,Y1: nat,X23: real,Y2: nat] :
( ( ord_le4735619238470717181al_nat @ ( produc3181502643871035669al_nat @ X12 @ Y1 ) @ ( produc3181502643871035669al_nat @ X23 @ Y2 ) )
= ( ( ord_less_real @ X12 @ X23 )
| ( ( ord_less_eq_real @ X12 @ X23 )
& ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_202_less__eq__prod__simp,axiom,
! [X12: real,Y1: int,X23: real,Y2: int] :
( ( ord_le557768218961520473al_int @ ( produc3179012173361985393al_int @ X12 @ Y1 ) @ ( produc3179012173361985393al_int @ X23 @ Y2 ) )
= ( ( ord_less_real @ X12 @ X23 )
| ( ( ord_less_eq_real @ X12 @ X23 )
& ( ord_less_eq_int @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_203_less__eq__prod__simp,axiom,
! [X12: real,Y1: real,X23: real,Y2: real] :
( ( ord_le1075799226346578649l_real @ ( produc4511245868158468465l_real @ X12 @ Y1 ) @ ( produc4511245868158468465l_real @ X23 @ Y2 ) )
= ( ( ord_less_real @ X12 @ X23 )
| ( ( ord_less_eq_real @ X12 @ X23 )
& ( ord_less_eq_real @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_204_less__prod__simp,axiom,
! [X12: nat > $o,Y1: nat,X23: nat > $o,Y2: nat] :
( ( ord_le1440161272000278354_o_nat @ ( produc7277522915581678840_o_nat @ X12 @ Y1 ) @ ( produc7277522915581678840_o_nat @ X23 @ Y2 ) )
= ( ( ord_less_nat_o @ X12 @ X23 )
| ( ( ord_less_eq_nat_o @ X12 @ X23 )
& ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_205_less__prod__simp,axiom,
! [X12: nat,Y1: nat,X23: nat,Y2: nat] :
( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X12 @ Y1 ) @ ( product_Pair_nat_nat @ X23 @ Y2 ) )
= ( ( ord_less_nat @ X12 @ X23 )
| ( ( ord_less_eq_nat @ X12 @ X23 )
& ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_206_less__prod__simp,axiom,
! [X12: nat,Y1: int,X23: nat,Y2: int] :
( ( ord_le6248945520114023945at_int @ ( product_Pair_nat_int @ X12 @ Y1 ) @ ( product_Pair_nat_int @ X23 @ Y2 ) )
= ( ( ord_less_nat @ X12 @ X23 )
| ( ( ord_less_eq_nat @ X12 @ X23 )
& ( ord_less_int @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_207_less__prod__simp,axiom,
! [X12: nat,Y1: real,X23: nat,Y2: real] :
( ( ord_le5560589655369113993t_real @ ( produc7837566107596912789t_real @ X12 @ Y1 ) @ ( produc7837566107596912789t_real @ X23 @ Y2 ) )
= ( ( ord_less_nat @ X12 @ X23 )
| ( ( ord_less_eq_nat @ X12 @ X23 )
& ( ord_less_real @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_208_less__prod__simp,axiom,
! [X12: int,Y1: nat,X23: int,Y2: nat] :
( ( ord_le2202778214669499401nt_nat @ ( product_Pair_int_nat @ X12 @ Y1 ) @ ( product_Pair_int_nat @ X23 @ Y2 ) )
= ( ( ord_less_int @ X12 @ X23 )
| ( ( ord_less_eq_int @ X12 @ X23 )
& ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_209_less__prod__simp,axiom,
! [X12: int,Y1: int,X23: int,Y2: int] :
( ( ord_le7248299232015078501nt_int @ ( product_Pair_int_int @ X12 @ Y1 ) @ ( product_Pair_int_int @ X23 @ Y2 ) )
= ( ( ord_less_int @ X12 @ X23 )
| ( ( ord_less_eq_int @ X12 @ X23 )
& ( ord_less_int @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_210_less__prod__simp,axiom,
! [X12: int,Y1: real,X23: int,Y2: real] :
( ( ord_le7747511230062135781t_real @ ( produc801115645435158769t_real @ X12 @ Y1 ) @ ( produc801115645435158769t_real @ X23 @ Y2 ) )
= ( ( ord_less_int @ X12 @ X23 )
| ( ( ord_less_eq_int @ X12 @ X23 )
& ( ord_less_real @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_211_less__prod__simp,axiom,
! [X12: real,Y1: nat,X23: real,Y2: nat] :
( ( ord_le1585541963892233737al_nat @ ( produc3181502643871035669al_nat @ X12 @ Y1 ) @ ( produc3181502643871035669al_nat @ X23 @ Y2 ) )
= ( ( ord_less_real @ X12 @ X23 )
| ( ( ord_less_eq_real @ X12 @ X23 )
& ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_212_less__prod__simp,axiom,
! [X12: real,Y1: int,X23: real,Y2: int] :
( ( ord_le6631062981237812837al_int @ ( produc3179012173361985393al_int @ X12 @ Y1 ) @ ( produc3179012173361985393al_int @ X23 @ Y2 ) )
= ( ( ord_less_real @ X12 @ X23 )
| ( ( ord_less_eq_real @ X12 @ X23 )
& ( ord_less_int @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_213_less__prod__simp,axiom,
! [X12: real,Y1: real,X23: real,Y2: real] :
( ( ord_le4389378000187319269l_real @ ( produc4511245868158468465l_real @ X12 @ Y1 ) @ ( produc4511245868158468465l_real @ X23 @ Y2 ) )
= ( ( ord_less_real @ X12 @ X23 )
| ( ( ord_less_eq_real @ X12 @ X23 )
& ( ord_less_real @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_214_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_215_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_216_sub_Ocases,axiom,
! [X3: produc1078910373471009876la_a_t] :
( ! [N2: nat,B2: $o] :
( X3
!= ( produc2953812259323318284la_a_t @ N2 @ ( bool_a_t @ B2 ) ) )
=> ( ! [N2: nat,A2: a] :
( X3
!= ( produc2953812259323318284la_a_t @ N2 @ ( atom_a_t @ A2 ) ) )
=> ( ! [N2: nat,Phi2: formula_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ ( suc @ N2 ) @ ( neg_a_t @ Phi2 ) ) )
=> ( ! [N2: nat,F2: $o > $o > $o,Phi2: formula_a_t,Psi2: formula_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ ( suc @ N2 ) @ ( bin_a_t @ F2 @ Phi2 @ Psi2 ) ) )
=> ( ! [N2: nat,I2: i_t,Phi2: formula_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ ( suc @ N2 ) @ ( prev_t_a @ I2 @ Phi2 ) ) )
=> ( ! [N2: nat,I2: i_t,Phi2: formula_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ ( suc @ N2 ) @ ( next_t_a @ I2 @ Phi2 ) ) )
=> ( ! [N2: nat,Phi2: formula_a_t,I2: i_t,Psi2: formula_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ ( suc @ N2 ) @ ( since_a_t @ Phi2 @ I2 @ Psi2 ) ) )
=> ( ! [N2: nat,Phi2: formula_a_t,I2: i_t,Psi2: formula_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ ( suc @ N2 ) @ ( until_a_t @ Phi2 @ I2 @ Psi2 ) ) )
=> ( ! [N2: nat,I2: i_t,R2: regex_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ ( suc @ N2 ) @ ( matchP_t_a @ I2 @ R2 ) ) )
=> ( ! [N2: nat,I2: i_t,R2: regex_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ ( suc @ N2 ) @ ( matchF_t_a @ I2 @ R2 ) ) )
=> ( ! [V: formula_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ zero_zero_nat @ ( neg_a_t @ V ) ) )
=> ( ! [V: $o > $o > $o,Va: formula_a_t,Vb: formula_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ zero_zero_nat @ ( bin_a_t @ V @ Va @ Vb ) ) )
=> ( ! [V: i_t,Va: formula_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ zero_zero_nat @ ( prev_t_a @ V @ Va ) ) )
=> ( ! [V: i_t,Va: formula_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ zero_zero_nat @ ( next_t_a @ V @ Va ) ) )
=> ( ! [V: formula_a_t,Va: i_t,Vb: formula_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ zero_zero_nat @ ( since_a_t @ V @ Va @ Vb ) ) )
=> ( ! [V: formula_a_t,Va: i_t,Vb: formula_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ zero_zero_nat @ ( until_a_t @ V @ Va @ Vb ) ) )
=> ( ! [V: i_t,Va: regex_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ zero_zero_nat @ ( matchP_t_a @ V @ Va ) ) )
=> ~ ! [V: i_t,Va: regex_a_t] :
( X3
!= ( produc2953812259323318284la_a_t @ zero_zero_nat @ ( matchF_t_a @ V @ Va ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sub.cases
thf(fact_217_progress_Ocases,axiom,
! [X3: produc3757808585008439209list_t] :
( ! [B2: $o,Ts: list_t] :
( X3
!= ( produc8618728406956180955list_t @ ( bool_a_t @ B2 ) @ Ts ) )
=> ( ! [A2: a,Ts: list_t] :
( X3
!= ( produc8618728406956180955list_t @ ( atom_a_t @ A2 ) @ Ts ) )
=> ( ! [Phi2: formula_a_t,Ts: list_t] :
( X3
!= ( produc8618728406956180955list_t @ ( neg_a_t @ Phi2 ) @ Ts ) )
=> ( ! [F2: $o > $o > $o,Phi2: formula_a_t,Psi2: formula_a_t,Ts: list_t] :
( X3
!= ( produc8618728406956180955list_t @ ( bin_a_t @ F2 @ Phi2 @ Psi2 ) @ Ts ) )
=> ( ! [I2: i_t,Phi2: formula_a_t,Ts: list_t] :
( X3
!= ( produc8618728406956180955list_t @ ( prev_t_a @ I2 @ Phi2 ) @ Ts ) )
=> ( ! [I2: i_t,Phi2: formula_a_t,Ts: list_t] :
( X3
!= ( produc8618728406956180955list_t @ ( next_t_a @ I2 @ Phi2 ) @ Ts ) )
=> ( ! [Phi2: formula_a_t,I2: i_t,Psi2: formula_a_t,Ts: list_t] :
( X3
!= ( produc8618728406956180955list_t @ ( since_a_t @ Phi2 @ I2 @ Psi2 ) @ Ts ) )
=> ( ! [Phi2: formula_a_t,I2: i_t,Psi2: formula_a_t,Ts: list_t] :
( X3
!= ( produc8618728406956180955list_t @ ( until_a_t @ Phi2 @ I2 @ Psi2 ) @ Ts ) )
=> ( ! [I2: i_t,R2: regex_a_t,Ts: list_t] :
( X3
!= ( produc8618728406956180955list_t @ ( matchP_t_a @ I2 @ R2 ) @ Ts ) )
=> ~ ! [I2: i_t,R2: regex_a_t,Ts: list_t] :
( X3
!= ( produc8618728406956180955list_t @ ( matchF_t_a @ I2 @ R2 ) @ Ts ) ) ) ) ) ) ) ) ) ) ) ).
% progress.cases
thf(fact_218_Eventually__def,axiom,
( eventually_t_a
= ( until_a_t @ ( bool_a_t @ $true ) ) ) ).
% Eventually_def
thf(fact_219_Once__def,axiom,
( once_t_a
= ( since_a_t @ ( bool_a_t @ $true ) ) ) ).
% Once_def
thf(fact_220_old_Oprod_Oinject,axiom,
! [A: formula_a_t,B: nat,A4: formula_a_t,B3: nat] :
( ( ( produc373174216379635980_t_nat @ A @ B )
= ( produc373174216379635980_t_nat @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_221_old_Oprod_Oinject,axiom,
! [A: nat > $o,B: nat,A4: nat > $o,B3: nat] :
( ( ( produc7277522915581678840_o_nat @ A @ B )
= ( produc7277522915581678840_o_nat @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_222_old_Oprod_Oinject,axiom,
! [A: int,B: int,A4: int,B3: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_223_prod_Oinject,axiom,
! [X12: formula_a_t,X23: nat,Y1: formula_a_t,Y2: nat] :
( ( ( produc373174216379635980_t_nat @ X12 @ X23 )
= ( produc373174216379635980_t_nat @ Y1 @ Y2 ) )
= ( ( X12 = Y1 )
& ( X23 = Y2 ) ) ) ).
% prod.inject
thf(fact_224_prod_Oinject,axiom,
! [X12: nat > $o,X23: nat,Y1: nat > $o,Y2: nat] :
( ( ( produc7277522915581678840_o_nat @ X12 @ X23 )
= ( produc7277522915581678840_o_nat @ Y1 @ Y2 ) )
= ( ( X12 = Y1 )
& ( X23 = Y2 ) ) ) ).
% prod.inject
thf(fact_225_prod_Oinject,axiom,
! [X12: int,X23: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X12 @ X23 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X12 = Y1 )
& ( X23 = Y2 ) ) ) ).
% prod.inject
thf(fact_226_last__before_Ocases,axiom,
! [X3: produc3074792404157404414_o_nat] :
( ! [P2: nat > $o] :
( X3
!= ( produc7277522915581678840_o_nat @ P2 @ zero_zero_nat ) )
=> ~ ! [P2: nat > $o,N2: nat] :
( X3
!= ( produc7277522915581678840_o_nat @ P2 @ ( suc @ N2 ) ) ) ) ).
% last_before.cases
thf(fact_227_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_228_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_229_zero__reorient,axiom,
! [X3: real] :
( ( zero_zero_real = X3 )
= ( X3 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_230_old_Oprod_Oexhaust,axiom,
! [Y: produc6285428564250921684_t_nat] :
~ ! [A2: formula_a_t,B2: nat] :
( Y
!= ( produc373174216379635980_t_nat @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_231_old_Oprod_Oexhaust,axiom,
! [Y: produc3074792404157404414_o_nat] :
~ ! [A2: nat > $o,B2: nat] :
( Y
!= ( produc7277522915581678840_o_nat @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_232_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A2: int,B2: int] :
( Y
!= ( product_Pair_int_int @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_233_surj__pair,axiom,
! [P3: produc6285428564250921684_t_nat] :
? [X2: formula_a_t,Y4: nat] :
( P3
= ( produc373174216379635980_t_nat @ X2 @ Y4 ) ) ).
% surj_pair
thf(fact_234_surj__pair,axiom,
! [P3: produc3074792404157404414_o_nat] :
? [X2: nat > $o,Y4: nat] :
( P3
= ( produc7277522915581678840_o_nat @ X2 @ Y4 ) ) ).
% surj_pair
thf(fact_235_surj__pair,axiom,
! [P3: product_prod_int_int] :
? [X2: int,Y4: int] :
( P3
= ( product_Pair_int_int @ X2 @ Y4 ) ) ).
% surj_pair
thf(fact_236_prod__cases,axiom,
! [P: produc6285428564250921684_t_nat > $o,P3: produc6285428564250921684_t_nat] :
( ! [A2: formula_a_t,B2: nat] : ( P @ ( produc373174216379635980_t_nat @ A2 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_237_prod__cases,axiom,
! [P: produc3074792404157404414_o_nat > $o,P3: produc3074792404157404414_o_nat] :
( ! [A2: nat > $o,B2: nat] : ( P @ ( produc7277522915581678840_o_nat @ A2 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_238_prod__cases,axiom,
! [P: product_prod_int_int > $o,P3: product_prod_int_int] :
( ! [A2: int,B2: int] : ( P @ ( product_Pair_int_int @ A2 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_239_Pair__inject,axiom,
! [A: formula_a_t,B: nat,A4: formula_a_t,B3: nat] :
( ( ( produc373174216379635980_t_nat @ A @ B )
= ( produc373174216379635980_t_nat @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_240_Pair__inject,axiom,
! [A: nat > $o,B: nat,A4: nat > $o,B3: nat] :
( ( ( produc7277522915581678840_o_nat @ A @ B )
= ( produc7277522915581678840_o_nat @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_241_Pair__inject,axiom,
! [A: int,B: int,A4: int,B3: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_242_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_243_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_244_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_245_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_246_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_247_Always__def,axiom,
( always_t_a
= ( ^ [I7: i_t,Phi4: formula_a_t] : ( neg_a_t @ ( eventually_t_a @ I7 @ ( neg_a_t @ Phi4 ) ) ) ) ) ).
% Always_def
thf(fact_248_Historically__def,axiom,
( historically_t_a
= ( ^ [I7: i_t,Phi4: formula_a_t] : ( neg_a_t @ ( once_t_a @ I7 @ ( neg_a_t @ Phi4 ) ) ) ) ) ).
% Historically_def
thf(fact_249_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_250_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_251_order__refl,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% order_refl
thf(fact_252_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_253_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_254_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_255_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I6: nat] :
( ( ord_less_nat @ K2 @ I6 )
=> ( P @ I6 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_256_list__decode_Ocases,axiom,
! [X3: nat] :
( ( X3 != zero_zero_nat )
=> ~ ! [N2: nat] :
( X3
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_257_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_258_order__antisym__conv,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_259_order__antisym__conv,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_260_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_261_linorder__le__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_262_linorder__le__cases,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_263_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_264_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_265_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_266_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_267_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_268_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_269_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_270_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_271_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_272_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_273_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_274_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_275_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_276_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_277_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_278_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_279_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_280_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_281_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_282_linorder__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_283_linorder__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
| ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_284_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_285_order__eq__refl,axiom,
! [X3: int,Y: int] :
( ( X3 = Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_286_order__eq__refl,axiom,
! [X3: real,Y: real] :
( ( X3 = Y )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_287_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_288_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_289_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_290_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_291_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_292_order__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_293_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_294_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_295_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_296_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_297_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_298_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_299_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_300_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_301_order__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_302_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_303_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_304_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_305_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_306_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z2: int] : ( Y6 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_307_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z2: real] : ( Y6 = Z2 ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_308_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_309_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_310_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_311_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_312_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_313_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_314_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_315_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_316_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_317_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_318_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: int,Z2: int] : ( Y6 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_319_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: real,Z2: real] : ( Y6 = Z2 ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_320_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_321_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: int,B2: int] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_322_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: real,B2: real] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_323_order__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_324_order__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_325_order__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_326_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_327_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_328_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_329_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_330_order__antisym,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_331_order__antisym,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_332_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_333_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_334_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_335_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_336_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_337_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_338_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [X4: nat,Y7: nat] :
( ( ord_less_eq_nat @ X4 @ Y7 )
& ( ord_less_eq_nat @ Y7 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_339_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z2: int] : ( Y6 = Z2 ) )
= ( ^ [X4: int,Y7: int] :
( ( ord_less_eq_int @ X4 @ Y7 )
& ( ord_less_eq_int @ Y7 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_340_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z2: real] : ( Y6 = Z2 ) )
= ( ^ [X4: real,Y7: real] :
( ( ord_less_eq_real @ X4 @ Y7 )
& ( ord_less_eq_real @ Y7 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_341_le__cases3,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_342_le__cases3,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_343_le__cases3,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X3 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X3 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_344_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_345_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_346_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_347_order__less__imp__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_348_order__less__imp__not__less,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_349_order__less__imp__not__less,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ~ ( ord_less_real @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_350_order__less__imp__not__eq2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_351_order__less__imp__not__eq2,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_352_order__less__imp__not__eq2,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_353_order__less__imp__not__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_354_order__less__imp__not__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_355_order__less__imp__not__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_356_linorder__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_357_linorder__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_358_linorder__less__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_real @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_359_order__less__imp__triv,axiom,
! [X3: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_360_order__less__imp__triv,axiom,
! [X3: int,Y: int,P: $o] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_361_order__less__imp__triv,axiom,
! [X3: real,Y: real,P: $o] :
( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_362_order__less__not__sym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_363_order__less__not__sym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_364_order__less__not__sym,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ~ ( ord_less_real @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_365_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_366_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_367_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_368_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_369_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_370_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_371_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_372_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_373_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_374_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_375_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_376_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_377_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_378_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_379_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_380_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_381_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_382_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_383_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_384_order__less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_385_order__less__irrefl,axiom,
! [X3: real] :
~ ( ord_less_real @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_386_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_387_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_388_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_389_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_390_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_391_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_392_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_393_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_394_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_395_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_396_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_397_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_398_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_399_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_400_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_401_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_402_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_403_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_404_order__less__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_405_order__less__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_406_order__less__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_407_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_408_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_409_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_410_linorder__neq__iff,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
= ( ( ord_less_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_411_linorder__neq__iff,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
= ( ( ord_less_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_412_linorder__neq__iff,axiom,
! [X3: real,Y: real] :
( ( X3 != Y )
= ( ( ord_less_real @ X3 @ Y )
| ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_413_order__less__asym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_414_order__less__asym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_415_order__less__asym,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ~ ( ord_less_real @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_416_linorder__neqE,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_417_linorder__neqE,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_418_linorder__neqE,axiom,
! [X3: real,Y: real] :
( ( X3 != Y )
=> ( ~ ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_419_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_420_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_421_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_422_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_423_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_424_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_425_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_426_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_427_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_428_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ( ord_less_nat @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_429_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ( ord_less_int @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_430_not__less__iff__gr__or__eq,axiom,
! [X3: real,Y: real] :
( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( ( ord_less_real @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_431_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_432_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_433_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_434_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ A2 )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_435_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: int] : ( P @ A2 @ A2 )
=> ( ! [A2: int,B2: int] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_436_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: real] : ( P @ A2 @ A2 )
=> ( ! [A2: real,B2: real] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_437_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [N3: nat] :
( ( P5 @ N3 )
& ! [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ~ ( P5 @ M6 ) ) ) ) ) ).
% exists_least_iff
thf(fact_438_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_439_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_440_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_441_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_442_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_443_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_444_linorder__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_445_linorder__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_446_linorder__cases,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_real @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_447_antisym__conv3,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_nat @ Y @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_448_antisym__conv3,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_int @ Y @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_449_antisym__conv3,axiom,
! [Y: real,X3: real] :
( ~ ( ord_less_real @ Y @ X3 )
=> ( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_450_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_451_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_452_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_453_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_454_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_455_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_456_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_457_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_458_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_459_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_460_less__imp__neq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_461_less__imp__neq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_462_less__imp__neq,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_463_dense,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ? [Z: real] :
( ( ord_less_real @ X3 @ Z )
& ( ord_less_real @ Z @ Y ) ) ) ).
% dense
thf(fact_464_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_465_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_466_gt__ex,axiom,
! [X3: real] :
? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% gt_ex
thf(fact_467_lt__ex,axiom,
! [X3: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X3 ) ).
% lt_ex
thf(fact_468_lt__ex,axiom,
! [X3: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X3 ) ).
% lt_ex
thf(fact_469_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_470_order__le__imp__less__or__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_int @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_471_order__le__imp__less__or__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_472_linorder__le__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_473_linorder__le__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_474_linorder__le__less__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
| ( ord_less_real @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_475_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_476_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_477_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_478_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_479_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_480_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_481_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_482_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_483_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_484_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_485_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_486_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_487_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_488_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_489_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_490_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_491_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_492_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_493_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_494_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_495_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_496_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_497_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_498_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_499_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_500_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_501_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_502_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_503_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_504_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_505_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_506_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_507_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_508_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_509_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_510_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_511_order__less__le__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_512_order__less__le__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_513_order__less__le__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_514_order__le__less__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_515_order__le__less__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_516_order__le__less__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_517_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_518_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_519_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_520_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_521_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_522_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_523_order__less__imp__le,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_524_order__less__imp__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_525_order__less__imp__le,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_526_linorder__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_527_linorder__not__less,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_528_linorder__not__less,axiom,
! [X3: real,Y: real] :
( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_529_linorder__not__le,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
= ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_530_linorder__not__le,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X3 @ Y ) )
= ( ord_less_int @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_531_linorder__not__le,axiom,
! [X3: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X3 @ Y ) )
= ( ord_less_real @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_532_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y7: nat] :
( ( ord_less_eq_nat @ X4 @ Y7 )
& ( X4 != Y7 ) ) ) ) ).
% order_less_le
thf(fact_533_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y7: int] :
( ( ord_less_eq_int @ X4 @ Y7 )
& ( X4 != Y7 ) ) ) ) ).
% order_less_le
thf(fact_534_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y7: real] :
( ( ord_less_eq_real @ X4 @ Y7 )
& ( X4 != Y7 ) ) ) ) ).
% order_less_le
thf(fact_535_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y7: nat] :
( ( ord_less_nat @ X4 @ Y7 )
| ( X4 = Y7 ) ) ) ) ).
% order_le_less
thf(fact_536_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y7: int] :
( ( ord_less_int @ X4 @ Y7 )
| ( X4 = Y7 ) ) ) ) ).
% order_le_less
thf(fact_537_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y7: real] :
( ( ord_less_real @ X4 @ Y7 )
| ( X4 = Y7 ) ) ) ) ).
% order_le_less
thf(fact_538_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_539_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_540_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_541_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_542_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_543_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_544_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_545_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_546_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_547_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_548_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_549_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_550_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_551_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_552_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_553_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_554_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_555_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_556_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_557_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_558_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_real @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_559_dense__le__bounded,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X3 @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_560_dense__ge__bounded,axiom,
! [Z3: real,X3: real,Y: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X3 )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_561_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_562_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_563_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_564_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_565_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_566_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_567_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_568_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_569_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_570_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_571_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_572_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_573_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_574_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_575_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_real @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_576_not__le__imp__less,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y @ X3 )
=> ( ord_less_nat @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_577_not__le__imp__less,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_eq_int @ Y @ X3 )
=> ( ord_less_int @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_578_not__le__imp__less,axiom,
! [Y: real,X3: real] :
( ~ ( ord_less_eq_real @ Y @ X3 )
=> ( ord_less_real @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_579_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y7: nat] :
( ( ord_less_eq_nat @ X4 @ Y7 )
& ~ ( ord_less_eq_nat @ Y7 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_580_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y7: int] :
( ( ord_less_eq_int @ X4 @ Y7 )
& ~ ( ord_less_eq_int @ Y7 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_581_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y7: real] :
( ( ord_less_eq_real @ X4 @ Y7 )
& ~ ( ord_less_eq_real @ Y7 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_582_dense__le,axiom,
! [Y: real,Z3: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_le
thf(fact_583_dense__ge,axiom,
! [Z3: real,Y: real] :
( ! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_584_antisym__conv2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_585_antisym__conv2,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_586_antisym__conv2,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_587_antisym__conv1,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_588_antisym__conv1,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_589_antisym__conv1,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_590_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_591_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_592_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_593_leI,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% leI
thf(fact_594_leI,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% leI
thf(fact_595_leI,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ Y @ X3 ) ) ).
% leI
thf(fact_596_leD,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y ) ) ).
% leD
thf(fact_597_leD,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_int @ X3 @ Y ) ) ).
% leD
thf(fact_598_leD,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ~ ( ord_less_real @ X3 @ Y ) ) ).
% leD
thf(fact_599_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_600_zero__prod__def,axiom,
( zero_z3979849011205770936at_nat
= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_601_zero__prod__def,axiom,
( zero_z9025370028551350036at_int
= ( product_Pair_nat_int @ zero_zero_nat @ zero_zero_int ) ) ).
% zero_prod_def
thf(fact_602_zero__prod__def,axiom,
( zero_z738777567634093332t_real
= ( produc7837566107596912789t_real @ zero_zero_nat @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_603_zero__prod__def,axiom,
( zero_z4979202723106825492nt_nat
= ( product_Pair_int_nat @ zero_zero_int @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_604_zero__prod__def,axiom,
( zero_z801351703597628784nt_int
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) ).
% zero_prod_def
thf(fact_605_zero__prod__def,axiom,
( zero_z2925699142327115120t_real
= ( produc801115645435158769t_real @ zero_zero_int @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_606_zero__prod__def,axiom,
( zero_z5987101913011988884al_nat
= ( produc3181502643871035669al_nat @ zero_zero_real @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_607_zero__prod__def,axiom,
( zero_z1809250893502792176al_int
= ( produc3179012173361985393al_int @ zero_zero_real @ zero_zero_int ) ) ).
% zero_prod_def
thf(fact_608_zero__prod__def,axiom,
( zero_z1365759597461889520l_real
= ( produc4511245868158468465l_real @ zero_zero_real @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_609_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_610_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_611_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_612_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_613_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ C2 ) )
=> ( P @ X ) )
& ! [D: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_614_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X: int] :
( ( ( ord_less_eq_int @ A @ X )
& ( ord_less_int @ X @ C2 ) )
=> ( P @ X ) )
& ! [D: int] :
( ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_615_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X: real] :
( ( ( ord_less_eq_real @ A @ X )
& ( ord_less_real @ X @ C2 ) )
=> ( P @ X ) )
& ! [D: real] :
( ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_real @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_real @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_616_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A4 ) )
= ( ord_less_nat @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_617_verit__comp__simplify1_I3_J,axiom,
! [B3: int,A4: int] :
( ( ~ ( ord_less_eq_int @ B3 @ A4 ) )
= ( ord_less_int @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_618_verit__comp__simplify1_I3_J,axiom,
! [B3: real,A4: real] :
( ( ~ ( ord_less_eq_real @ B3 @ A4 ) )
= ( ord_less_real @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_619_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_620_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_621_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_622_timestamp__total,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
| ( ord_less_eq_nat @ B @ A ) ) ).
% timestamp_total
thf(fact_623_timestamp__total,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
| ( ord_less_eq_real @ B @ A ) ) ).
% timestamp_total
thf(fact_624_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_625_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_626_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_627_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_628_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_629_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_630_ex__gt__or__lt,axiom,
! [A: real] :
? [B2: real] :
( ( ord_less_real @ A @ B2 )
| ( ord_less_real @ B2 @ A ) ) ).
% ex_gt_or_lt
thf(fact_631_plus__prod_Ocases,axiom,
! [X3: produc1219242969750017639nt_int] :
~ ! [A2: int,B2: int,C2: int,D3: int] :
( X3
!= ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ ( product_Pair_int_int @ C2 @ D3 ) ) ) ).
% plus_prod.cases
thf(fact_632_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_633_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_634_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_635_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ T @ X ) ) ).
% minf(8)
thf(fact_636_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ~ ( ord_less_eq_int @ T @ X ) ) ).
% minf(8)
thf(fact_637_minf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ~ ( ord_less_eq_real @ T @ X ) ) ).
% minf(8)
thf(fact_638_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ T ) ) ).
% minf(6)
thf(fact_639_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ord_less_eq_int @ X @ T ) ) ).
% minf(6)
thf(fact_640_minf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( ord_less_eq_real @ X @ T ) ) ).
% minf(6)
thf(fact_641_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ord_less_eq_nat @ T @ X ) ) ).
% pinf(8)
thf(fact_642_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ord_less_eq_int @ T @ X ) ) ).
% pinf(8)
thf(fact_643_pinf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ord_less_eq_real @ T @ X ) ) ).
% pinf(8)
thf(fact_644_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ T ) ) ).
% pinf(6)
thf(fact_645_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ T ) ) ).
% pinf(6)
thf(fact_646_pinf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ T ) ) ).
% pinf(6)
thf(fact_647_formula_Oset__cases,axiom,
! [E2: a,A: formula_a_t] :
( ( member_a @ E2 @ ( set_formula_a_t @ A ) )
=> ( ( A
!= ( atom_a_t @ E2 ) )
=> ( ! [Z: formula_a_t] :
( ( A
= ( neg_a_t @ Z ) )
=> ~ ( member_a @ E2 @ ( set_formula_a_t @ Z ) ) )
=> ( ! [Z1: $o > $o > $o,Z22: formula_a_t] :
( ? [Z32: formula_a_t] :
( A
= ( bin_a_t @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member_a @ E2 @ ( set_formula_a_t @ Z22 ) ) )
=> ( ! [Z1: $o > $o > $o,Z22: formula_a_t,Z32: formula_a_t] :
( ( A
= ( bin_a_t @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member_a @ E2 @ ( set_formula_a_t @ Z32 ) ) )
=> ( ! [Z1: i_t,Z22: formula_a_t] :
( ( A
= ( prev_t_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E2 @ ( set_formula_a_t @ Z22 ) ) )
=> ( ! [Z1: i_t,Z22: formula_a_t] :
( ( A
= ( next_t_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E2 @ ( set_formula_a_t @ Z22 ) ) )
=> ( ! [Z1: formula_a_t] :
( ? [Z22: i_t,Z32: formula_a_t] :
( A
= ( since_a_t @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member_a @ E2 @ ( set_formula_a_t @ Z1 ) ) )
=> ( ! [Z1: formula_a_t,Z22: i_t,Z32: formula_a_t] :
( ( A
= ( since_a_t @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member_a @ E2 @ ( set_formula_a_t @ Z32 ) ) )
=> ( ! [Z1: formula_a_t] :
( ? [Z22: i_t,Z32: formula_a_t] :
( A
= ( until_a_t @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member_a @ E2 @ ( set_formula_a_t @ Z1 ) ) )
=> ( ! [Z1: formula_a_t,Z22: i_t,Z32: formula_a_t] :
( ( A
= ( until_a_t @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member_a @ E2 @ ( set_formula_a_t @ Z32 ) ) )
=> ( ! [Z1: i_t,Z22: regex_a_t] :
( ( A
= ( matchP_t_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E2 @ ( set_regex_a_t @ Z22 ) ) )
=> ~ ! [Z1: i_t,Z22: regex_a_t] :
( ( A
= ( matchF_t_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E2 @ ( set_regex_a_t @ Z22 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% formula.set_cases
thf(fact_648_size__neq__size__imp__neq,axiom,
! [X3: char,Y: char] :
( ( ( size_size_char @ X3 )
!= ( size_size_char @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_649_formula_Oset__intros_I12_J,axiom,
! [Yl: a,X1022: regex_a_t,X1012: i_t] :
( ( member_a @ Yl @ ( set_regex_a_t @ X1022 ) )
=> ( member_a @ Yl @ ( set_formula_a_t @ ( matchF_t_a @ X1012 @ X1022 ) ) ) ) ).
% formula.set_intros(12)
thf(fact_650_formula_Osimps_I240_J,axiom,
! [X1012: i_t,X1022: regex_a_t] :
( ( set_formula_a_t @ ( matchF_t_a @ X1012 @ X1022 ) )
= ( set_regex_a_t @ X1022 ) ) ).
% formula.simps(240)
thf(fact_651_formula_Oset__intros_I11_J,axiom,
! [Yk: a,X922: regex_a_t,X912: i_t] :
( ( member_a @ Yk @ ( set_regex_a_t @ X922 ) )
=> ( member_a @ Yk @ ( set_formula_a_t @ ( matchP_t_a @ X912 @ X922 ) ) ) ) ).
% formula.set_intros(11)
thf(fact_652_formula_Osimps_I239_J,axiom,
! [X912: i_t,X922: regex_a_t] :
( ( set_formula_a_t @ ( matchP_t_a @ X912 @ X922 ) )
= ( set_regex_a_t @ X922 ) ) ).
% formula.simps(239)
thf(fact_653_formula_Oset__intros_I5_J,axiom,
! [Ye: a,X522: formula_a_t,X512: i_t] :
( ( member_a @ Ye @ ( set_formula_a_t @ X522 ) )
=> ( member_a @ Ye @ ( set_formula_a_t @ ( prev_t_a @ X512 @ X522 ) ) ) ) ).
% formula.set_intros(5)
thf(fact_654_formula_Oset__intros_I6_J,axiom,
! [Yf: a,X622: formula_a_t,X612: i_t] :
( ( member_a @ Yf @ ( set_formula_a_t @ X622 ) )
=> ( member_a @ Yf @ ( set_formula_a_t @ ( next_t_a @ X612 @ X622 ) ) ) ) ).
% formula.set_intros(6)
thf(fact_655_formula_Osimps_I235_J,axiom,
! [X512: i_t,X522: formula_a_t] :
( ( set_formula_a_t @ ( prev_t_a @ X512 @ X522 ) )
= ( set_formula_a_t @ X522 ) ) ).
% formula.simps(235)
thf(fact_656_formula_Osimps_I236_J,axiom,
! [X612: i_t,X622: formula_a_t] :
( ( set_formula_a_t @ ( next_t_a @ X612 @ X622 ) )
= ( set_formula_a_t @ X622 ) ) ).
% formula.simps(236)
thf(fact_657_formula_Oset__intros_I7_J,axiom,
! [Yg: a,X712: formula_a_t,X722: i_t,X732: formula_a_t] :
( ( member_a @ Yg @ ( set_formula_a_t @ X712 ) )
=> ( member_a @ Yg @ ( set_formula_a_t @ ( since_a_t @ X712 @ X722 @ X732 ) ) ) ) ).
% formula.set_intros(7)
thf(fact_658_formula_Oset__intros_I8_J,axiom,
! [Yh: a,X732: formula_a_t,X712: formula_a_t,X722: i_t] :
( ( member_a @ Yh @ ( set_formula_a_t @ X732 ) )
=> ( member_a @ Yh @ ( set_formula_a_t @ ( since_a_t @ X712 @ X722 @ X732 ) ) ) ) ).
% formula.set_intros(8)
thf(fact_659_formula_Oset__intros_I9_J,axiom,
! [Yi: a,X812: formula_a_t,X822: i_t,X832: formula_a_t] :
( ( member_a @ Yi @ ( set_formula_a_t @ X812 ) )
=> ( member_a @ Yi @ ( set_formula_a_t @ ( until_a_t @ X812 @ X822 @ X832 ) ) ) ) ).
% formula.set_intros(9)
thf(fact_660_formula_Oset__intros_I10_J,axiom,
! [Yj: a,X832: formula_a_t,X812: formula_a_t,X822: i_t] :
( ( member_a @ Yj @ ( set_formula_a_t @ X832 ) )
=> ( member_a @ Yj @ ( set_formula_a_t @ ( until_a_t @ X812 @ X822 @ X832 ) ) ) ) ).
% formula.set_intros(10)
thf(fact_661_formula_Oset__intros_I2_J,axiom,
! [Y: a,X33: formula_a_t] :
( ( member_a @ Y @ ( set_formula_a_t @ X33 ) )
=> ( member_a @ Y @ ( set_formula_a_t @ ( neg_a_t @ X33 ) ) ) ) ).
% formula.set_intros(2)
thf(fact_662_formula_Osimps_I233_J,axiom,
! [X33: formula_a_t] :
( ( set_formula_a_t @ ( neg_a_t @ X33 ) )
= ( set_formula_a_t @ X33 ) ) ).
% formula.simps(233)
thf(fact_663_formula_Oset__intros_I3_J,axiom,
! [Yc: a,X422: formula_a_t,X412: $o > $o > $o,X432: formula_a_t] :
( ( member_a @ Yc @ ( set_formula_a_t @ X422 ) )
=> ( member_a @ Yc @ ( set_formula_a_t @ ( bin_a_t @ X412 @ X422 @ X432 ) ) ) ) ).
% formula.set_intros(3)
thf(fact_664_formula_Oset__intros_I4_J,axiom,
! [Yd: a,X432: formula_a_t,X412: $o > $o > $o,X422: formula_a_t] :
( ( member_a @ Yd @ ( set_formula_a_t @ X432 ) )
=> ( member_a @ Yd @ ( set_formula_a_t @ ( bin_a_t @ X412 @ X422 @ X432 ) ) ) ) ).
% formula.set_intros(4)
thf(fact_665_formula_Oset__intros_I1_J,axiom,
! [X23: a] : ( member_a @ X23 @ ( set_formula_a_t @ ( atom_a_t @ X23 ) ) ) ).
% formula.set_intros(1)
thf(fact_666_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P6 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_667_pinf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P6 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_668_pinf_I1_J,axiom,
! [P: real > $o,P6: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P6 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_669_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P6 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_670_pinf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P6 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_671_pinf_I2_J,axiom,
! [P: real > $o,P6: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P6 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_672_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_673_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_674_pinf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_675_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_676_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_677_pinf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_678_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ~ ( ord_less_nat @ X @ T ) ) ).
% pinf(5)
thf(fact_679_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ~ ( ord_less_int @ X @ T ) ) ).
% pinf(5)
thf(fact_680_pinf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ~ ( ord_less_real @ X @ T ) ) ).
% pinf(5)
thf(fact_681_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ord_less_nat @ T @ X ) ) ).
% pinf(7)
thf(fact_682_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ord_less_int @ T @ X ) ) ).
% pinf(7)
thf(fact_683_pinf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ord_less_real @ T @ X ) ) ).
% pinf(7)
thf(fact_684_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P6 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_685_minf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P6 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_686_minf_I1_J,axiom,
! [P: real > $o,P6: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P6 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_687_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P6 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_688_minf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P6 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_689_minf_I2_J,axiom,
! [P: real > $o,P6: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P6 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_690_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_691_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_692_minf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_693_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_694_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_695_minf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_696_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ord_less_nat @ X @ T ) ) ).
% minf(5)
thf(fact_697_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ord_less_int @ X @ T ) ) ).
% minf(5)
thf(fact_698_minf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( ord_less_real @ X @ T ) ) ).
% minf(5)
thf(fact_699_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ~ ( ord_less_nat @ T @ X ) ) ).
% minf(7)
thf(fact_700_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ~ ( ord_less_int @ T @ X ) ) ).
% minf(7)
thf(fact_701_minf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ~ ( ord_less_real @ T @ X ) ) ).
% minf(7)
thf(fact_702__092_060iota_062__mono,axiom,
! [I4: nat,J: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( embed_nat_iota_nat @ I4 ) @ ( embed_nat_iota_nat @ J ) ) ) ).
% \<iota>_mono
thf(fact_703__092_060iota_062__mono,axiom,
! [I4: nat,J: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_real @ ( embed_nat_iota_real @ I4 ) @ ( embed_nat_iota_real @ J ) ) ) ).
% \<iota>_mono
thf(fact_704_msize__fmla_Oelims,axiom,
! [X3: formula_a_t,Y: nat] :
( ( ( msize_fmla_a_t @ X3 )
= Y )
=> ( ( ? [B2: $o] :
( X3
= ( bool_a_t @ B2 ) )
=> ( Y != zero_zero_nat ) )
=> ( ( ? [A2: a] :
( X3
= ( atom_a_t @ A2 ) )
=> ( Y != zero_zero_nat ) )
=> ( ! [Phi2: formula_a_t] :
( ( X3
= ( neg_a_t @ Phi2 ) )
=> ( Y
!= ( suc @ ( msize_fmla_a_t @ Phi2 ) ) ) )
=> ( ! [F2: $o > $o > $o,Phi2: formula_a_t,Psi2: formula_a_t] :
( ( X3
= ( bin_a_t @ F2 @ Phi2 @ Psi2 ) )
=> ( Y
!= ( suc @ ( plus_plus_nat @ ( msize_fmla_a_t @ Phi2 ) @ ( msize_fmla_a_t @ Psi2 ) ) ) ) )
=> ( ! [I2: i_t,Phi2: formula_a_t] :
( ( X3
= ( prev_t_a @ I2 @ Phi2 ) )
=> ( Y
!= ( suc @ ( msize_fmla_a_t @ Phi2 ) ) ) )
=> ( ! [I2: i_t,Phi2: formula_a_t] :
( ( X3
= ( next_t_a @ I2 @ Phi2 ) )
=> ( Y
!= ( suc @ ( msize_fmla_a_t @ Phi2 ) ) ) )
=> ( ! [Phi2: formula_a_t,I2: i_t,Psi2: formula_a_t] :
( ( X3
= ( since_a_t @ Phi2 @ I2 @ Psi2 ) )
=> ( Y
!= ( suc @ ( ord_max_nat @ ( msize_fmla_a_t @ Phi2 ) @ ( msize_fmla_a_t @ Psi2 ) ) ) ) )
=> ( ! [Phi2: formula_a_t,I2: i_t,Psi2: formula_a_t] :
( ( X3
= ( until_a_t @ Phi2 @ I2 @ Psi2 ) )
=> ( Y
!= ( suc @ ( ord_max_nat @ ( msize_fmla_a_t @ Phi2 ) @ ( msize_fmla_a_t @ Psi2 ) ) ) ) )
=> ( ! [I2: i_t,R2: regex_a_t] :
( ( X3
= ( matchP_t_a @ I2 @ R2 ) )
=> ( Y
!= ( suc @ ( msize_regex_a_t @ R2 ) ) ) )
=> ~ ! [I2: i_t,R2: regex_a_t] :
( ( X3
= ( matchF_t_a @ I2 @ R2 ) )
=> ( Y
!= ( suc @ ( msize_regex_a_t @ R2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% msize_fmla.elims
thf(fact_705_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_706_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_707_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_708_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_709_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_710_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_711_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_712_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_713_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_714_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_715_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_716_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_717_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_718_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_719_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_720_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_721_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_722_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_723_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_724_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_725_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_726_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_727_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_728_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_729_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_730_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_731_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_732_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_733_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_734_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_735_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_736_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_737_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_738_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_739_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_740_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y ) )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_741_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_742_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_743_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_744_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_745_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_746_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_747_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_748_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_749_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_750_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_751_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_752_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_753_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_754_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_755_max__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M2 @ N ) ) ) ).
% max_Suc_Suc
thf(fact_756_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_757_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_758_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_759_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_760_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_761_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_762_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_763_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_764_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_765_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_766_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_767_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_768_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_769_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_770_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_771_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_772_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_773_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_774_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_775_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_776_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_777_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_778_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_779_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_780_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_781_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_782_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_783_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_784_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_785_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_786_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_787_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_788_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_789_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_790_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_791_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_792_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_793_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_794_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_795_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_796_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_797_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_798_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_799_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_800_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_801_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_802_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_803_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_804_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_805_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_806_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_807_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_808_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_809_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_810_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_811_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_812_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_813_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_814_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_815_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_816_plus__prod_Osimps,axiom,
! [A: nat,B: nat,C: nat,D4: nat] :
( ( plus_p9057090461656269880at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D4 ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D4 ) ) ) ).
% plus_prod.simps
thf(fact_817_plus__prod_Osimps,axiom,
! [A: nat,B: int,C: nat,D4: int] :
( ( plus_p4879239442147073172at_int @ ( product_Pair_nat_int @ A @ B ) @ ( product_Pair_nat_int @ C @ D4 ) )
= ( product_Pair_nat_int @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_int @ B @ D4 ) ) ) ).
% plus_prod.simps
thf(fact_818_plus__prod_Osimps,axiom,
! [A: nat,B: real,C: nat,D4: real] :
( ( plus_p8900843186509212308t_real @ ( produc7837566107596912789t_real @ A @ B ) @ ( produc7837566107596912789t_real @ C @ D4 ) )
= ( produc7837566107596912789t_real @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_real @ B @ D4 ) ) ) ).
% plus_prod.simps
thf(fact_819_plus__prod_Osimps,axiom,
! [A: int,B: nat,C: int,D4: nat] :
( ( plus_p833072136702548628nt_nat @ ( product_Pair_int_nat @ A @ B ) @ ( product_Pair_int_nat @ C @ D4 ) )
= ( product_Pair_int_nat @ ( plus_plus_int @ A @ C ) @ ( plus_plus_nat @ B @ D4 ) ) ) ).
% plus_prod.simps
thf(fact_820_plus__prod_Osimps,axiom,
! [A: int,B: int,C: int,D4: int] :
( ( plus_p5878593154048127728nt_int @ ( product_Pair_int_int @ A @ B ) @ ( product_Pair_int_int @ C @ D4 ) )
= ( product_Pair_int_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D4 ) ) ) ).
% plus_prod.simps
thf(fact_821_plus__prod_Osimps,axiom,
! [A: int,B: real,C: int,D4: real] :
( ( plus_p1864392724347458288t_real @ ( produc801115645435158769t_real @ A @ B ) @ ( produc801115645435158769t_real @ C @ D4 ) )
= ( produc801115645435158769t_real @ ( plus_plus_int @ A @ C ) @ ( plus_plus_real @ B @ D4 ) ) ) ).
% plus_prod.simps
thf(fact_822_plus__prod_Osimps,axiom,
! [A: real,B: nat,C: real,D4: nat] :
( ( plus_p4925795495032332052al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc3181502643871035669al_nat @ C @ D4 ) )
= ( produc3181502643871035669al_nat @ ( plus_plus_real @ A @ C ) @ ( plus_plus_nat @ B @ D4 ) ) ) ).
% plus_prod.simps
thf(fact_823_plus__prod_Osimps,axiom,
! [A: real,B: int,C: real,D4: int] :
( ( plus_p747944475523135344al_int @ ( produc3179012173361985393al_int @ A @ B ) @ ( produc3179012173361985393al_int @ C @ D4 ) )
= ( produc3179012173361985393al_int @ ( plus_plus_real @ A @ C ) @ ( plus_plus_int @ B @ D4 ) ) ) ).
% plus_prod.simps
thf(fact_824_plus__prod_Osimps,axiom,
! [A: real,B: real,C: real,D4: real] :
( ( plus_p1196244663705802608l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( produc4511245868158468465l_real @ C @ D4 ) )
= ( produc4511245868158468465l_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D4 ) ) ) ).
% plus_prod.simps
thf(fact_825_plus__prod_Oelims,axiom,
! [X3: product_prod_nat_nat,Xa: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ( plus_p9057090461656269880at_nat @ X3 @ Xa )
= Y )
=> ~ ! [A2: nat,B2: nat] :
( ( X3
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ! [C2: nat,D3: nat] :
( ( Xa
= ( product_Pair_nat_nat @ C2 @ D3 ) )
=> ( Y
!= ( product_Pair_nat_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D3 ) ) ) ) ) ) ).
% plus_prod.elims
thf(fact_826_plus__prod_Oelims,axiom,
! [X3: product_prod_nat_int,Xa: product_prod_nat_int,Y: product_prod_nat_int] :
( ( ( plus_p4879239442147073172at_int @ X3 @ Xa )
= Y )
=> ~ ! [A2: nat,B2: int] :
( ( X3
= ( product_Pair_nat_int @ A2 @ B2 ) )
=> ! [C2: nat,D3: int] :
( ( Xa
= ( product_Pair_nat_int @ C2 @ D3 ) )
=> ( Y
!= ( product_Pair_nat_int @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D3 ) ) ) ) ) ) ).
% plus_prod.elims
thf(fact_827_plus__prod_Oelims,axiom,
! [X3: produc7716430852924023517t_real,Xa: produc7716430852924023517t_real,Y: produc7716430852924023517t_real] :
( ( ( plus_p8900843186509212308t_real @ X3 @ Xa )
= Y )
=> ~ ! [A2: nat,B2: real] :
( ( X3
= ( produc7837566107596912789t_real @ A2 @ B2 ) )
=> ! [C2: nat,D3: real] :
( ( Xa
= ( produc7837566107596912789t_real @ C2 @ D3 ) )
=> ( Y
!= ( produc7837566107596912789t_real @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ D3 ) ) ) ) ) ) ).
% plus_prod.elims
thf(fact_828_plus__prod_Oelims,axiom,
! [X3: product_prod_int_nat,Xa: product_prod_int_nat,Y: product_prod_int_nat] :
( ( ( plus_p833072136702548628nt_nat @ X3 @ Xa )
= Y )
=> ~ ! [A2: int,B2: nat] :
( ( X3
= ( product_Pair_int_nat @ A2 @ B2 ) )
=> ! [C2: int,D3: nat] :
( ( Xa
= ( product_Pair_int_nat @ C2 @ D3 ) )
=> ( Y
!= ( product_Pair_int_nat @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D3 ) ) ) ) ) ) ).
% plus_prod.elims
thf(fact_829_plus__prod_Oelims,axiom,
! [X3: product_prod_int_int,Xa: product_prod_int_int,Y: product_prod_int_int] :
( ( ( plus_p5878593154048127728nt_int @ X3 @ Xa )
= Y )
=> ~ ! [A2: int,B2: int] :
( ( X3
= ( product_Pair_int_int @ A2 @ B2 ) )
=> ! [C2: int,D3: int] :
( ( Xa
= ( product_Pair_int_int @ C2 @ D3 ) )
=> ( Y
!= ( product_Pair_int_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D3 ) ) ) ) ) ) ).
% plus_prod.elims
thf(fact_830_plus__prod_Oelims,axiom,
! [X3: produc679980390762269497t_real,Xa: produc679980390762269497t_real,Y: produc679980390762269497t_real] :
( ( ( plus_p1864392724347458288t_real @ X3 @ Xa )
= Y )
=> ~ ! [A2: int,B2: real] :
( ( X3
= ( produc801115645435158769t_real @ A2 @ B2 ) )
=> ! [C2: int,D3: real] :
( ( Xa
= ( produc801115645435158769t_real @ C2 @ D3 ) )
=> ( Y
!= ( produc801115645435158769t_real @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ D3 ) ) ) ) ) ) ).
% plus_prod.elims
thf(fact_831_plus__prod_Oelims,axiom,
! [X3: produc3741383161447143261al_nat,Xa: produc3741383161447143261al_nat,Y: produc3741383161447143261al_nat] :
( ( ( plus_p4925795495032332052al_nat @ X3 @ Xa )
= Y )
=> ~ ! [A2: real,B2: nat] :
( ( X3
= ( produc3181502643871035669al_nat @ A2 @ B2 ) )
=> ! [C2: real,D3: nat] :
( ( Xa
= ( produc3181502643871035669al_nat @ C2 @ D3 ) )
=> ( Y
!= ( produc3181502643871035669al_nat @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D3 ) ) ) ) ) ) ).
% plus_prod.elims
thf(fact_832_plus__prod_Oelims,axiom,
! [X3: produc8786904178792722361al_int,Xa: produc8786904178792722361al_int,Y: produc8786904178792722361al_int] :
( ( ( plus_p747944475523135344al_int @ X3 @ Xa )
= Y )
=> ~ ! [A2: real,B2: int] :
( ( X3
= ( produc3179012173361985393al_int @ A2 @ B2 ) )
=> ! [C2: real,D3: int] :
( ( Xa
= ( produc3179012173361985393al_int @ C2 @ D3 ) )
=> ( Y
!= ( produc3179012173361985393al_int @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D3 ) ) ) ) ) ) ).
% plus_prod.elims
thf(fact_833_plus__prod_Oelims,axiom,
! [X3: produc2422161461964618553l_real,Xa: produc2422161461964618553l_real,Y: produc2422161461964618553l_real] :
( ( ( plus_p1196244663705802608l_real @ X3 @ Xa )
= Y )
=> ~ ! [A2: real,B2: real] :
( ( X3
= ( produc4511245868158468465l_real @ A2 @ B2 ) )
=> ! [C2: real,D3: real] :
( ( Xa
= ( produc4511245868158468465l_real @ C2 @ D3 ) )
=> ( Y
!= ( produc4511245868158468465l_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ D3 ) ) ) ) ) ) ).
% plus_prod.elims
thf(fact_834_nat__add__max__right,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N @ Q3 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ N ) @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_835_nat__add__max__left,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N ) @ Q3 )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_836_of__nat__max,axiom,
! [X3: nat,Y: nat] :
( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X3 @ Y ) )
= ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% of_nat_max
thf(fact_837_of__nat__max,axiom,
! [X3: nat,Y: nat] :
( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X3 @ Y ) )
= ( ord_max_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% of_nat_max
thf(fact_838_of__nat__max,axiom,
! [X3: nat,Y: nat] :
( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X3 @ Y ) )
= ( ord_max_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).
% of_nat_max
thf(fact_839_formula_Orel__inject_I9_J,axiom,
! [R3: a > a > $o,X912: i_t,X922: regex_a_t,Y91: i_t,Y92: regex_a_t] :
( ( rel_formula_a_a_t @ R3 @ ( matchP_t_a @ X912 @ X922 ) @ ( matchP_t_a @ Y91 @ Y92 ) )
= ( ( X912 = Y91 )
& ( rel_regex_a_a_t @ R3 @ X922 @ Y92 ) ) ) ).
% formula.rel_inject(9)
thf(fact_840_formula_Orel__intros_I9_J,axiom,
! [X912: i_t,Y91: i_t,R3: a > a > $o,X922: regex_a_t,Y92: regex_a_t] :
( ( X912 = Y91 )
=> ( ( rel_regex_a_a_t @ R3 @ X922 @ Y92 )
=> ( rel_formula_a_a_t @ R3 @ ( matchP_t_a @ X912 @ X922 ) @ ( matchP_t_a @ Y91 @ Y92 ) ) ) ) ).
% formula.rel_intros(9)
thf(fact_841_formula_Orel__inject_I10_J,axiom,
! [R3: a > a > $o,X1012: i_t,X1022: regex_a_t,Y101: i_t,Y102: regex_a_t] :
( ( rel_formula_a_a_t @ R3 @ ( matchF_t_a @ X1012 @ X1022 ) @ ( matchF_t_a @ Y101 @ Y102 ) )
= ( ( X1012 = Y101 )
& ( rel_regex_a_a_t @ R3 @ X1022 @ Y102 ) ) ) ).
% formula.rel_inject(10)
thf(fact_842_formula_Orel__intros_I10_J,axiom,
! [X1012: i_t,Y101: i_t,R3: a > a > $o,X1022: regex_a_t,Y102: regex_a_t] :
( ( X1012 = Y101 )
=> ( ( rel_regex_a_a_t @ R3 @ X1022 @ Y102 )
=> ( rel_formula_a_a_t @ R3 @ ( matchF_t_a @ X1012 @ X1022 ) @ ( matchF_t_a @ Y101 @ Y102 ) ) ) ) ).
% formula.rel_intros(10)
thf(fact_843_max__add__distrib__right,axiom,
! [X3: int,Y: int,Z3: int] :
( ( plus_plus_int @ X3 @ ( ord_max_int @ Y @ Z3 ) )
= ( ord_max_int @ ( plus_plus_int @ X3 @ Y ) @ ( plus_plus_int @ X3 @ Z3 ) ) ) ).
% max_add_distrib_right
thf(fact_844_max__add__distrib__right,axiom,
! [X3: real,Y: real,Z3: real] :
( ( plus_plus_real @ X3 @ ( ord_max_real @ Y @ Z3 ) )
= ( ord_max_real @ ( plus_plus_real @ X3 @ Y ) @ ( plus_plus_real @ X3 @ Z3 ) ) ) ).
% max_add_distrib_right
thf(fact_845_max__add__distrib__right,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( plus_plus_nat @ X3 @ ( ord_max_nat @ Y @ Z3 ) )
= ( ord_max_nat @ ( plus_plus_nat @ X3 @ Y ) @ ( plus_plus_nat @ X3 @ Z3 ) ) ) ).
% max_add_distrib_right
thf(fact_846_max__add__distrib__left,axiom,
! [X3: int,Y: int,Z3: int] :
( ( plus_plus_int @ ( ord_max_int @ X3 @ Y ) @ Z3 )
= ( ord_max_int @ ( plus_plus_int @ X3 @ Z3 ) @ ( plus_plus_int @ Y @ Z3 ) ) ) ).
% max_add_distrib_left
thf(fact_847_max__add__distrib__left,axiom,
! [X3: real,Y: real,Z3: real] :
( ( plus_plus_real @ ( ord_max_real @ X3 @ Y ) @ Z3 )
= ( ord_max_real @ ( plus_plus_real @ X3 @ Z3 ) @ ( plus_plus_real @ Y @ Z3 ) ) ) ).
% max_add_distrib_left
thf(fact_848_max__add__distrib__left,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X3 @ Y ) @ Z3 )
= ( ord_max_nat @ ( plus_plus_nat @ X3 @ Z3 ) @ ( plus_plus_nat @ Y @ Z3 ) ) ) ).
% max_add_distrib_left
thf(fact_849_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_850_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_851_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_852_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_853_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_854_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_855_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_856_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_857_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_858_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_859_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_860_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A5: real,B4: real] : ( plus_plus_real @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_861_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_862_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_863_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_864_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_865_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_866_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_867_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_868_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_869_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_870_group__cancel_Oadd2,axiom,
! [B5: real,K: real,B: real,A: real] :
( ( B5
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B5 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_871_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_872_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_873_group__cancel_Oadd1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_874_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( I4 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I4 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_875_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( I4 = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I4 @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_876_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( I4 = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I4 @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_877_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_878_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_879_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_880_max__def,axiom,
( ord_max_nat
= ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_881_max__def,axiom,
( ord_max_int
= ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_882_max__def,axiom,
( ord_max_real
= ( ^ [A5: real,B4: real] : ( if_real @ ( ord_less_eq_real @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_883_max__absorb1,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_max_nat @ X3 @ Y )
= X3 ) ) ).
% max_absorb1
thf(fact_884_max__absorb1,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ( ( ord_max_int @ X3 @ Y )
= X3 ) ) ).
% max_absorb1
thf(fact_885_max__absorb1,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ( ( ord_max_real @ X3 @ Y )
= X3 ) ) ).
% max_absorb1
thf(fact_886_max__absorb2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_max_nat @ X3 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_887_max__absorb2,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_max_int @ X3 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_888_max__absorb2,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_max_real @ X3 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_889_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_890_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_891_real__arch__simple,axiom,
! [X3: real] :
? [N2: nat] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% real_arch_simple
thf(fact_892_reals__Archimedean2,axiom,
! [X3: real] :
? [N2: nat] : ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% reals_Archimedean2
thf(fact_893_formula_Orel__inject_I6_J,axiom,
! [R3: a > a > $o,X612: i_t,X622: formula_a_t,Y61: i_t,Y62: formula_a_t] :
( ( rel_formula_a_a_t @ R3 @ ( next_t_a @ X612 @ X622 ) @ ( next_t_a @ Y61 @ Y62 ) )
= ( ( X612 = Y61 )
& ( rel_formula_a_a_t @ R3 @ X622 @ Y62 ) ) ) ).
% formula.rel_inject(6)
thf(fact_894_formula_Orel__inject_I5_J,axiom,
! [R3: a > a > $o,X512: i_t,X522: formula_a_t,Y51: i_t,Y52: formula_a_t] :
( ( rel_formula_a_a_t @ R3 @ ( prev_t_a @ X512 @ X522 ) @ ( prev_t_a @ Y51 @ Y52 ) )
= ( ( X512 = Y51 )
& ( rel_formula_a_a_t @ R3 @ X522 @ Y52 ) ) ) ).
% formula.rel_inject(5)
thf(fact_895_formula_Orel__intros_I6_J,axiom,
! [X612: i_t,Y61: i_t,R3: a > a > $o,X622: formula_a_t,Y62: formula_a_t] :
( ( X612 = Y61 )
=> ( ( rel_formula_a_a_t @ R3 @ X622 @ Y62 )
=> ( rel_formula_a_a_t @ R3 @ ( next_t_a @ X612 @ X622 ) @ ( next_t_a @ Y61 @ Y62 ) ) ) ) ).
% formula.rel_intros(6)
thf(fact_896_formula_Orel__intros_I5_J,axiom,
! [X512: i_t,Y51: i_t,R3: a > a > $o,X522: formula_a_t,Y52: formula_a_t] :
( ( X512 = Y51 )
=> ( ( rel_formula_a_a_t @ R3 @ X522 @ Y52 )
=> ( rel_formula_a_a_t @ R3 @ ( prev_t_a @ X512 @ X522 ) @ ( prev_t_a @ Y51 @ Y52 ) ) ) ) ).
% formula.rel_intros(5)
thf(fact_897_formula_Orel__inject_I8_J,axiom,
! [R3: a > a > $o,X812: formula_a_t,X822: i_t,X832: formula_a_t,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t] :
( ( rel_formula_a_a_t @ R3 @ ( until_a_t @ X812 @ X822 @ X832 ) @ ( until_a_t @ Y81 @ Y82 @ Y83 ) )
= ( ( rel_formula_a_a_t @ R3 @ X812 @ Y81 )
& ( X822 = Y82 )
& ( rel_formula_a_a_t @ R3 @ X832 @ Y83 ) ) ) ).
% formula.rel_inject(8)
thf(fact_898_formula_Orel__inject_I7_J,axiom,
! [R3: a > a > $o,X712: formula_a_t,X722: i_t,X732: formula_a_t,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t] :
( ( rel_formula_a_a_t @ R3 @ ( since_a_t @ X712 @ X722 @ X732 ) @ ( since_a_t @ Y71 @ Y72 @ Y73 ) )
= ( ( rel_formula_a_a_t @ R3 @ X712 @ Y71 )
& ( X722 = Y72 )
& ( rel_formula_a_a_t @ R3 @ X732 @ Y73 ) ) ) ).
% formula.rel_inject(7)
thf(fact_899_formula_Orel__intros_I8_J,axiom,
! [R3: a > a > $o,X812: formula_a_t,Y81: formula_a_t,X822: i_t,Y82: i_t,X832: formula_a_t,Y83: formula_a_t] :
( ( rel_formula_a_a_t @ R3 @ X812 @ Y81 )
=> ( ( X822 = Y82 )
=> ( ( rel_formula_a_a_t @ R3 @ X832 @ Y83 )
=> ( rel_formula_a_a_t @ R3 @ ( until_a_t @ X812 @ X822 @ X832 ) @ ( until_a_t @ Y81 @ Y82 @ Y83 ) ) ) ) ) ).
% formula.rel_intros(8)
thf(fact_900_formula_Orel__intros_I7_J,axiom,
! [R3: a > a > $o,X712: formula_a_t,Y71: formula_a_t,X722: i_t,Y72: i_t,X732: formula_a_t,Y73: formula_a_t] :
( ( rel_formula_a_a_t @ R3 @ X712 @ Y71 )
=> ( ( X722 = Y72 )
=> ( ( rel_formula_a_a_t @ R3 @ X732 @ Y73 )
=> ( rel_formula_a_a_t @ R3 @ ( since_a_t @ X712 @ X722 @ X732 ) @ ( since_a_t @ Y71 @ Y72 @ Y73 ) ) ) ) ) ).
% formula.rel_intros(7)
thf(fact_901_formula_Orel__inject_I3_J,axiom,
! [R3: a > a > $o,X33: formula_a_t,Y3: formula_a_t] :
( ( rel_formula_a_a_t @ R3 @ ( neg_a_t @ X33 ) @ ( neg_a_t @ Y3 ) )
= ( rel_formula_a_a_t @ R3 @ X33 @ Y3 ) ) ).
% formula.rel_inject(3)
thf(fact_902_formula_Orel__intros_I3_J,axiom,
! [R3: a > a > $o,X33: formula_a_t,Y3: formula_a_t] :
( ( rel_formula_a_a_t @ R3 @ X33 @ Y3 )
=> ( rel_formula_a_a_t @ R3 @ ( neg_a_t @ X33 ) @ ( neg_a_t @ Y3 ) ) ) ).
% formula.rel_intros(3)
thf(fact_903_formula_Orel__inject_I4_J,axiom,
! [R3: a > a > $o,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,Y41: $o > $o > $o,Y42: formula_a_t,Y43: formula_a_t] :
( ( rel_formula_a_a_t @ R3 @ ( bin_a_t @ X412 @ X422 @ X432 ) @ ( bin_a_t @ Y41 @ Y42 @ Y43 ) )
= ( ( X412 = Y41 )
& ( rel_formula_a_a_t @ R3 @ X422 @ Y42 )
& ( rel_formula_a_a_t @ R3 @ X432 @ Y43 ) ) ) ).
% formula.rel_inject(4)
thf(fact_904_formula_Orel__intros_I4_J,axiom,
! [X412: $o > $o > $o,Y41: $o > $o > $o,R3: a > a > $o,X422: formula_a_t,Y42: formula_a_t,X432: formula_a_t,Y43: formula_a_t] :
( ( X412 = Y41 )
=> ( ( rel_formula_a_a_t @ R3 @ X422 @ Y42 )
=> ( ( rel_formula_a_a_t @ R3 @ X432 @ Y43 )
=> ( rel_formula_a_a_t @ R3 @ ( bin_a_t @ X412 @ X422 @ X432 ) @ ( bin_a_t @ Y41 @ Y42 @ Y43 ) ) ) ) ) ).
% formula.rel_intros(4)
thf(fact_905_formula_Orel__inject_I1_J,axiom,
! [R3: a > a > $o,X12: $o,Y1: $o] :
( ( rel_formula_a_a_t @ R3 @ ( bool_a_t @ X12 ) @ ( bool_a_t @ Y1 ) )
= ( X12 = Y1 ) ) ).
% formula.rel_inject(1)
thf(fact_906_formula_Orel__intros_I1_J,axiom,
! [X12: $o,Y1: $o,R3: a > a > $o] :
( ( X12 = Y1 )
=> ( rel_formula_a_a_t @ R3 @ ( bool_a_t @ X12 ) @ ( bool_a_t @ Y1 ) ) ) ).
% formula.rel_intros(1)
thf(fact_907_formula_Orel__inject_I2_J,axiom,
! [R3: a > a > $o,X23: a,Y2: a] :
( ( rel_formula_a_a_t @ R3 @ ( atom_a_t @ X23 ) @ ( atom_a_t @ Y2 ) )
= ( R3 @ X23 @ Y2 ) ) ).
% formula.rel_inject(2)
thf(fact_908_formula_Orel__intros_I2_J,axiom,
! [R3: a > a > $o,X23: a,Y2: a] :
( ( R3 @ X23 @ Y2 )
=> ( rel_formula_a_a_t @ R3 @ ( atom_a_t @ X23 ) @ ( atom_a_t @ Y2 ) ) ) ).
% formula.rel_intros(2)
thf(fact_909_add__mono__comm,axiom,
! [C: nat,D4: nat,A: nat] :
( ( ord_less_eq_nat @ C @ D4 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ D4 @ A ) ) ) ).
% add_mono_comm
thf(fact_910_add__mono__comm,axiom,
! [C: real,D4: real,A: real] :
( ( ord_less_eq_real @ C @ D4 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ D4 @ A ) ) ) ).
% add_mono_comm
thf(fact_911_timestamp__class_Oadd__mono,axiom,
! [C: nat,D4: nat,A: nat] :
( ( ord_less_eq_nat @ C @ D4 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ A @ D4 ) ) ) ).
% timestamp_class.add_mono
thf(fact_912_timestamp__class_Oadd__mono,axiom,
! [C: real,D4: real,A: real] :
( ( ord_less_eq_real @ C @ D4 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ A @ D4 ) ) ) ).
% timestamp_class.add_mono
thf(fact_913_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_914_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_915_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I4 @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_916_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( I4 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_917_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( I4 = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_918_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( I4 = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_919_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_920_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_921_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I4 @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_922_ordered__ab__semigroup__add__class_Oadd__mono,axiom,
! [A: nat,B: nat,C: nat,D4: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D4 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).
% ordered_ab_semigroup_add_class.add_mono
thf(fact_923_ordered__ab__semigroup__add__class_Oadd__mono,axiom,
! [A: int,B: int,C: int,D4: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D4 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D4 ) ) ) ) ).
% ordered_ab_semigroup_add_class.add_mono
thf(fact_924_ordered__ab__semigroup__add__class_Oadd__mono,axiom,
! [A: real,B: real,C: real,D4: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D4 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D4 ) ) ) ) ).
% ordered_ab_semigroup_add_class.add_mono
thf(fact_925_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_926_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_927_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_928_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_929_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_930_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_931_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_932_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_933_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_934_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_935_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_936_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_937_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_938_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_939_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_940_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_941_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_942_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_943_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_944_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_945_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_946_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_947_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_948_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_949_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_950_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_951_add__mono__strict,axiom,
! [C: nat,D4: nat,A: nat] :
( ( ord_less_nat @ C @ D4 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ A @ D4 ) ) ) ).
% add_mono_strict
thf(fact_952_add__mono__strict,axiom,
! [C: real,D4: real,A: real] :
( ( ord_less_real @ C @ D4 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ A @ D4 ) ) ) ).
% add_mono_strict
thf(fact_953_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_954_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I4 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_955_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I4 @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_956_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( I4 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_957_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( I4 = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_958_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( I4 = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_959_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_960_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I4 @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_961_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I4 @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_962_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D4: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D4 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).
% add_strict_mono
thf(fact_963_add__strict__mono,axiom,
! [A: int,B: int,C: int,D4: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D4 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D4 ) ) ) ) ).
% add_strict_mono
thf(fact_964_add__strict__mono,axiom,
! [A: real,B: real,C: real,D4: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D4 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D4 ) ) ) ) ).
% add_strict_mono
thf(fact_965_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_966_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_967_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_968_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_969_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_970_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_971_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_972_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_973_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_974_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_975_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_976_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_977_formula_Osize_I20_J,axiom,
! [X1101: i_t,X1102: regex_a_t] :
( ( size_s4016968051272393527la_a_t @ ( matchF_t_a @ X1101 @ X1102 ) )
= ( plus_plus_nat @ ( size_size_regex_a_t @ X1102 ) @ ( suc @ zero_zero_nat ) ) ) ).
% formula.size(20)
thf(fact_978_formula_Osize_I19_J,axiom,
! [X191: i_t,X192: regex_a_t] :
( ( size_s4016968051272393527la_a_t @ ( matchP_t_a @ X191 @ X192 ) )
= ( plus_plus_nat @ ( size_size_regex_a_t @ X192 ) @ ( suc @ zero_zero_nat ) ) ) ).
% formula.size(19)
thf(fact_979_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_980_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_981_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_982_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_983_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_984_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_985_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_986_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_987_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_988_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_989_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_990_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_991_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_992_add__le__mono,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_993_add__le__mono1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_994_trans__le__add1,axiom,
! [I4: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_995_trans__le__add2,axiom,
! [I4: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_996_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_997_add__lessD1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J ) @ K )
=> ( ord_less_nat @ I4 @ K ) ) ).
% add_lessD1
thf(fact_998_add__less__mono,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_999_not__add__less1,axiom,
! [I4: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J ) @ I4 ) ).
% not_add_less1
thf(fact_1000_not__add__less2,axiom,
! [J: nat,I4: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I4 ) @ I4 ) ).
% not_add_less2
thf(fact_1001_add__less__mono1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1002_trans__less__add1,axiom,
! [I4: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1003_trans__less__add2,axiom,
! [I4: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1004_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1005_formula_Osize_I16_J,axiom,
! [X161: i_t,X162: formula_a_t] :
( ( size_s4016968051272393527la_a_t @ ( next_t_a @ X161 @ X162 ) )
= ( plus_plus_nat @ ( size_s4016968051272393527la_a_t @ X162 ) @ ( suc @ zero_zero_nat ) ) ) ).
% formula.size(16)
thf(fact_1006_formula_Osize_I15_J,axiom,
! [X151: i_t,X152: formula_a_t] :
( ( size_s4016968051272393527la_a_t @ ( prev_t_a @ X151 @ X152 ) )
= ( plus_plus_nat @ ( size_s4016968051272393527la_a_t @ X152 ) @ ( suc @ zero_zero_nat ) ) ) ).
% formula.size(15)
thf(fact_1007_formula_Osize_I18_J,axiom,
! [X181: formula_a_t,X182: i_t,X183: formula_a_t] :
( ( size_s4016968051272393527la_a_t @ ( until_a_t @ X181 @ X182 @ X183 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_s4016968051272393527la_a_t @ X181 ) @ ( size_s4016968051272393527la_a_t @ X183 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% formula.size(18)
thf(fact_1008_formula_Osize_I17_J,axiom,
! [X171: formula_a_t,X172: i_t,X173: formula_a_t] :
( ( size_s4016968051272393527la_a_t @ ( since_a_t @ X171 @ X172 @ X173 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_s4016968051272393527la_a_t @ X171 ) @ ( size_s4016968051272393527la_a_t @ X173 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% formula.size(17)
thf(fact_1009_formula_Osize_I13_J,axiom,
! [X13: formula_a_t] :
( ( size_s4016968051272393527la_a_t @ ( neg_a_t @ X13 ) )
= ( plus_plus_nat @ ( size_s4016968051272393527la_a_t @ X13 ) @ ( suc @ zero_zero_nat ) ) ) ).
% formula.size(13)
thf(fact_1010_formula_Osize_I14_J,axiom,
! [X141: $o > $o > $o,X142: formula_a_t,X143: formula_a_t] :
( ( size_s4016968051272393527la_a_t @ ( bin_a_t @ X141 @ X142 @ X143 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_s4016968051272393527la_a_t @ X142 ) @ ( size_s4016968051272393527la_a_t @ X143 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% formula.size(14)
thf(fact_1011_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_1012_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_1013_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1014_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_1015_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_1016_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_1017_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_1018_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_1019_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_1020_of__nat__mono,axiom,
! [I4: nat,J: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_1021_of__nat__mono,axiom,
! [I4: nat,J: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I4 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_1022_of__nat__mono,axiom,
! [I4: nat,J: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I4 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_1023_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_1024_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_1025_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_1026_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_1027_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_1028_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_1029_formula_Orel__distinct_I62_J,axiom,
! [R3: a > a > $o,Y61: i_t,Y62: formula_a_t,X512: i_t,X522: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( next_t_a @ Y61 @ Y62 ) @ ( prev_t_a @ X512 @ X522 ) ) ).
% formula.rel_distinct(62)
thf(fact_1030_formula_Orel__distinct_I61_J,axiom,
! [R3: a > a > $o,X512: i_t,X522: formula_a_t,Y61: i_t,Y62: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( prev_t_a @ X512 @ X522 ) @ ( next_t_a @ Y61 @ Y62 ) ) ).
% formula.rel_distinct(61)
thf(fact_1031_formula_Orel__distinct_I78_J,axiom,
! [R3: a > a > $o,Y101: i_t,Y102: regex_a_t,X612: i_t,X622: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchF_t_a @ Y101 @ Y102 ) @ ( next_t_a @ X612 @ X622 ) ) ).
% formula.rel_distinct(78)
thf(fact_1032_formula_Orel__distinct_I77_J,axiom,
! [R3: a > a > $o,X612: i_t,X622: formula_a_t,Y101: i_t,Y102: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( next_t_a @ X612 @ X622 ) @ ( matchF_t_a @ Y101 @ Y102 ) ) ).
% formula.rel_distinct(77)
thf(fact_1033_formula_Orel__distinct_I70_J,axiom,
! [R3: a > a > $o,Y101: i_t,Y102: regex_a_t,X512: i_t,X522: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchF_t_a @ Y101 @ Y102 ) @ ( prev_t_a @ X512 @ X522 ) ) ).
% formula.rel_distinct(70)
thf(fact_1034_formula_Orel__distinct_I69_J,axiom,
! [R3: a > a > $o,X512: i_t,X522: formula_a_t,Y101: i_t,Y102: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( prev_t_a @ X512 @ X522 ) @ ( matchF_t_a @ Y101 @ Y102 ) ) ).
% formula.rel_distinct(69)
thf(fact_1035_formula_Orel__distinct_I76_J,axiom,
! [R3: a > a > $o,Y91: i_t,Y92: regex_a_t,X612: i_t,X622: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchP_t_a @ Y91 @ Y92 ) @ ( next_t_a @ X612 @ X622 ) ) ).
% formula.rel_distinct(76)
thf(fact_1036_formula_Orel__distinct_I75_J,axiom,
! [R3: a > a > $o,X612: i_t,X622: formula_a_t,Y91: i_t,Y92: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( next_t_a @ X612 @ X622 ) @ ( matchP_t_a @ Y91 @ Y92 ) ) ).
% formula.rel_distinct(75)
thf(fact_1037_formula_Orel__distinct_I68_J,axiom,
! [R3: a > a > $o,Y91: i_t,Y92: regex_a_t,X512: i_t,X522: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchP_t_a @ Y91 @ Y92 ) @ ( prev_t_a @ X512 @ X522 ) ) ).
% formula.rel_distinct(68)
thf(fact_1038_formula_Orel__distinct_I67_J,axiom,
! [R3: a > a > $o,X512: i_t,X522: formula_a_t,Y91: i_t,Y92: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( prev_t_a @ X512 @ X522 ) @ ( matchP_t_a @ Y91 @ Y92 ) ) ).
% formula.rel_distinct(67)
thf(fact_1039_formula_Orel__distinct_I74_J,axiom,
! [R3: a > a > $o,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t,X612: i_t,X622: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( until_a_t @ Y81 @ Y82 @ Y83 ) @ ( next_t_a @ X612 @ X622 ) ) ).
% formula.rel_distinct(74)
thf(fact_1040_formula_Orel__distinct_I73_J,axiom,
! [R3: a > a > $o,X612: i_t,X622: formula_a_t,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( next_t_a @ X612 @ X622 ) @ ( until_a_t @ Y81 @ Y82 @ Y83 ) ) ).
% formula.rel_distinct(73)
thf(fact_1041_formula_Orel__distinct_I72_J,axiom,
! [R3: a > a > $o,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t,X612: i_t,X622: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( since_a_t @ Y71 @ Y72 @ Y73 ) @ ( next_t_a @ X612 @ X622 ) ) ).
% formula.rel_distinct(72)
thf(fact_1042_formula_Orel__distinct_I71_J,axiom,
! [R3: a > a > $o,X612: i_t,X622: formula_a_t,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( next_t_a @ X612 @ X622 ) @ ( since_a_t @ Y71 @ Y72 @ Y73 ) ) ).
% formula.rel_distinct(71)
thf(fact_1043_formula_Orel__distinct_I66_J,axiom,
! [R3: a > a > $o,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t,X512: i_t,X522: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( until_a_t @ Y81 @ Y82 @ Y83 ) @ ( prev_t_a @ X512 @ X522 ) ) ).
% formula.rel_distinct(66)
thf(fact_1044_formula_Orel__distinct_I65_J,axiom,
! [R3: a > a > $o,X512: i_t,X522: formula_a_t,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( prev_t_a @ X512 @ X522 ) @ ( until_a_t @ Y81 @ Y82 @ Y83 ) ) ).
% formula.rel_distinct(65)
thf(fact_1045_formula_Orel__distinct_I64_J,axiom,
! [R3: a > a > $o,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t,X512: i_t,X522: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( since_a_t @ Y71 @ Y72 @ Y73 ) @ ( prev_t_a @ X512 @ X522 ) ) ).
% formula.rel_distinct(64)
thf(fact_1046_formula_Orel__distinct_I63_J,axiom,
! [R3: a > a > $o,X512: i_t,X522: formula_a_t,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( prev_t_a @ X512 @ X522 ) @ ( since_a_t @ Y71 @ Y72 @ Y73 ) ) ).
% formula.rel_distinct(63)
thf(fact_1047_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1048_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1049_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1050_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1051_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1052_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1053_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1054_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1055_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1056_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1057_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1058_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1059_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1060_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1061_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1062_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1063_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1064_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_1065_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1066_add__nonneg__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1067_add__nonneg__eq__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X3 @ Y )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1068_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1069_add__nonpos__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1070_add__nonpos__eq__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X3 @ Y )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1071_formula_Orel__distinct_I40_J,axiom,
! [R3: a > a > $o,Y61: i_t,Y62: formula_a_t,X33: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( next_t_a @ Y61 @ Y62 ) @ ( neg_a_t @ X33 ) ) ).
% formula.rel_distinct(40)
thf(fact_1072_formula_Orel__distinct_I39_J,axiom,
! [R3: a > a > $o,X33: formula_a_t,Y61: i_t,Y62: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( neg_a_t @ X33 ) @ ( next_t_a @ Y61 @ Y62 ) ) ).
% formula.rel_distinct(39)
thf(fact_1073_formula_Orel__distinct_I38_J,axiom,
! [R3: a > a > $o,Y51: i_t,Y52: formula_a_t,X33: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( prev_t_a @ Y51 @ Y52 ) @ ( neg_a_t @ X33 ) ) ).
% formula.rel_distinct(38)
thf(fact_1074_formula_Orel__distinct_I37_J,axiom,
! [R3: a > a > $o,X33: formula_a_t,Y51: i_t,Y52: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( neg_a_t @ X33 ) @ ( prev_t_a @ Y51 @ Y52 ) ) ).
% formula.rel_distinct(37)
thf(fact_1075_formula_Orel__distinct_I90_J,axiom,
! [R3: a > a > $o,Y101: i_t,Y102: regex_a_t,X912: i_t,X922: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchF_t_a @ Y101 @ Y102 ) @ ( matchP_t_a @ X912 @ X922 ) ) ).
% formula.rel_distinct(90)
thf(fact_1076_formula_Orel__distinct_I89_J,axiom,
! [R3: a > a > $o,X912: i_t,X922: regex_a_t,Y101: i_t,Y102: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchP_t_a @ X912 @ X922 ) @ ( matchF_t_a @ Y101 @ Y102 ) ) ).
% formula.rel_distinct(89)
thf(fact_1077_formula_Orel__distinct_I52_J,axiom,
! [R3: a > a > $o,Y61: i_t,Y62: formula_a_t,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( next_t_a @ Y61 @ Y62 ) @ ( bin_a_t @ X412 @ X422 @ X432 ) ) ).
% formula.rel_distinct(52)
thf(fact_1078_formula_Orel__distinct_I51_J,axiom,
! [R3: a > a > $o,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,Y61: i_t,Y62: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bin_a_t @ X412 @ X422 @ X432 ) @ ( next_t_a @ Y61 @ Y62 ) ) ).
% formula.rel_distinct(51)
thf(fact_1079_formula_Orel__distinct_I50_J,axiom,
! [R3: a > a > $o,Y51: i_t,Y52: formula_a_t,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( prev_t_a @ Y51 @ Y52 ) @ ( bin_a_t @ X412 @ X422 @ X432 ) ) ).
% formula.rel_distinct(50)
thf(fact_1080_formula_Orel__distinct_I49_J,axiom,
! [R3: a > a > $o,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,Y51: i_t,Y52: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bin_a_t @ X412 @ X422 @ X432 ) @ ( prev_t_a @ Y51 @ Y52 ) ) ).
% formula.rel_distinct(49)
thf(fact_1081_formula_Orel__distinct_I88_J,axiom,
! [R3: a > a > $o,Y101: i_t,Y102: regex_a_t,X812: formula_a_t,X822: i_t,X832: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchF_t_a @ Y101 @ Y102 ) @ ( until_a_t @ X812 @ X822 @ X832 ) ) ).
% formula.rel_distinct(88)
thf(fact_1082_formula_Orel__distinct_I87_J,axiom,
! [R3: a > a > $o,X812: formula_a_t,X822: i_t,X832: formula_a_t,Y101: i_t,Y102: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( until_a_t @ X812 @ X822 @ X832 ) @ ( matchF_t_a @ Y101 @ Y102 ) ) ).
% formula.rel_distinct(87)
thf(fact_1083_formula_Orel__distinct_I84_J,axiom,
! [R3: a > a > $o,Y101: i_t,Y102: regex_a_t,X712: formula_a_t,X722: i_t,X732: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchF_t_a @ Y101 @ Y102 ) @ ( since_a_t @ X712 @ X722 @ X732 ) ) ).
% formula.rel_distinct(84)
thf(fact_1084_formula_Orel__distinct_I83_J,axiom,
! [R3: a > a > $o,X712: formula_a_t,X722: i_t,X732: formula_a_t,Y101: i_t,Y102: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( since_a_t @ X712 @ X722 @ X732 ) @ ( matchF_t_a @ Y101 @ Y102 ) ) ).
% formula.rel_distinct(83)
thf(fact_1085_formula_Orel__distinct_I86_J,axiom,
! [R3: a > a > $o,Y91: i_t,Y92: regex_a_t,X812: formula_a_t,X822: i_t,X832: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchP_t_a @ Y91 @ Y92 ) @ ( until_a_t @ X812 @ X822 @ X832 ) ) ).
% formula.rel_distinct(86)
thf(fact_1086_formula_Orel__distinct_I85_J,axiom,
! [R3: a > a > $o,X812: formula_a_t,X822: i_t,X832: formula_a_t,Y91: i_t,Y92: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( until_a_t @ X812 @ X822 @ X832 ) @ ( matchP_t_a @ Y91 @ Y92 ) ) ).
% formula.rel_distinct(85)
thf(fact_1087_formula_Orel__distinct_I82_J,axiom,
! [R3: a > a > $o,Y91: i_t,Y92: regex_a_t,X712: formula_a_t,X722: i_t,X732: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchP_t_a @ Y91 @ Y92 ) @ ( since_a_t @ X712 @ X722 @ X732 ) ) ).
% formula.rel_distinct(82)
thf(fact_1088_formula_Orel__distinct_I81_J,axiom,
! [R3: a > a > $o,X712: formula_a_t,X722: i_t,X732: formula_a_t,Y91: i_t,Y92: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( since_a_t @ X712 @ X722 @ X732 ) @ ( matchP_t_a @ Y91 @ Y92 ) ) ).
% formula.rel_distinct(81)
thf(fact_1089_formula_Orel__distinct_I48_J,axiom,
! [R3: a > a > $o,Y101: i_t,Y102: regex_a_t,X33: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchF_t_a @ Y101 @ Y102 ) @ ( neg_a_t @ X33 ) ) ).
% formula.rel_distinct(48)
thf(fact_1090_formula_Orel__distinct_I47_J,axiom,
! [R3: a > a > $o,X33: formula_a_t,Y101: i_t,Y102: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( neg_a_t @ X33 ) @ ( matchF_t_a @ Y101 @ Y102 ) ) ).
% formula.rel_distinct(47)
thf(fact_1091_formula_Orel__distinct_I80_J,axiom,
! [R3: a > a > $o,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t,X712: formula_a_t,X722: i_t,X732: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( until_a_t @ Y81 @ Y82 @ Y83 ) @ ( since_a_t @ X712 @ X722 @ X732 ) ) ).
% formula.rel_distinct(80)
thf(fact_1092_formula_Orel__distinct_I79_J,axiom,
! [R3: a > a > $o,X712: formula_a_t,X722: i_t,X732: formula_a_t,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( since_a_t @ X712 @ X722 @ X732 ) @ ( until_a_t @ Y81 @ Y82 @ Y83 ) ) ).
% formula.rel_distinct(79)
thf(fact_1093_formula_Orel__distinct_I46_J,axiom,
! [R3: a > a > $o,Y91: i_t,Y92: regex_a_t,X33: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchP_t_a @ Y91 @ Y92 ) @ ( neg_a_t @ X33 ) ) ).
% formula.rel_distinct(46)
thf(fact_1094_formula_Orel__distinct_I45_J,axiom,
! [R3: a > a > $o,X33: formula_a_t,Y91: i_t,Y92: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( neg_a_t @ X33 ) @ ( matchP_t_a @ Y91 @ Y92 ) ) ).
% formula.rel_distinct(45)
thf(fact_1095_formula_Orel__distinct_I60_J,axiom,
! [R3: a > a > $o,Y101: i_t,Y102: regex_a_t,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchF_t_a @ Y101 @ Y102 ) @ ( bin_a_t @ X412 @ X422 @ X432 ) ) ).
% formula.rel_distinct(60)
thf(fact_1096_formula_Orel__distinct_I59_J,axiom,
! [R3: a > a > $o,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,Y101: i_t,Y102: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bin_a_t @ X412 @ X422 @ X432 ) @ ( matchF_t_a @ Y101 @ Y102 ) ) ).
% formula.rel_distinct(59)
thf(fact_1097_formula_Orel__distinct_I44_J,axiom,
! [R3: a > a > $o,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t,X33: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( until_a_t @ Y81 @ Y82 @ Y83 ) @ ( neg_a_t @ X33 ) ) ).
% formula.rel_distinct(44)
thf(fact_1098_formula_Orel__distinct_I43_J,axiom,
! [R3: a > a > $o,X33: formula_a_t,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( neg_a_t @ X33 ) @ ( until_a_t @ Y81 @ Y82 @ Y83 ) ) ).
% formula.rel_distinct(43)
thf(fact_1099_formula_Orel__distinct_I42_J,axiom,
! [R3: a > a > $o,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t,X33: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( since_a_t @ Y71 @ Y72 @ Y73 ) @ ( neg_a_t @ X33 ) ) ).
% formula.rel_distinct(42)
thf(fact_1100_formula_Orel__distinct_I41_J,axiom,
! [R3: a > a > $o,X33: formula_a_t,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( neg_a_t @ X33 ) @ ( since_a_t @ Y71 @ Y72 @ Y73 ) ) ).
% formula.rel_distinct(41)
thf(fact_1101_formula_Orel__distinct_I58_J,axiom,
! [R3: a > a > $o,Y91: i_t,Y92: regex_a_t,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( matchP_t_a @ Y91 @ Y92 ) @ ( bin_a_t @ X412 @ X422 @ X432 ) ) ).
% formula.rel_distinct(58)
thf(fact_1102_formula_Orel__distinct_I57_J,axiom,
! [R3: a > a > $o,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,Y91: i_t,Y92: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bin_a_t @ X412 @ X422 @ X432 ) @ ( matchP_t_a @ Y91 @ Y92 ) ) ).
% formula.rel_distinct(57)
thf(fact_1103_formula_Orel__distinct_I56_J,axiom,
! [R3: a > a > $o,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( until_a_t @ Y81 @ Y82 @ Y83 ) @ ( bin_a_t @ X412 @ X422 @ X432 ) ) ).
% formula.rel_distinct(56)
thf(fact_1104_formula_Orel__distinct_I55_J,axiom,
! [R3: a > a > $o,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bin_a_t @ X412 @ X422 @ X432 ) @ ( until_a_t @ Y81 @ Y82 @ Y83 ) ) ).
% formula.rel_distinct(55)
thf(fact_1105_formula_Orel__distinct_I54_J,axiom,
! [R3: a > a > $o,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( since_a_t @ Y71 @ Y72 @ Y73 ) @ ( bin_a_t @ X412 @ X422 @ X432 ) ) ).
% formula.rel_distinct(54)
thf(fact_1106_formula_Orel__distinct_I53_J,axiom,
! [R3: a > a > $o,X412: $o > $o > $o,X422: formula_a_t,X432: formula_a_t,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bin_a_t @ X412 @ X422 @ X432 ) @ ( since_a_t @ Y71 @ Y72 @ Y73 ) ) ).
% formula.rel_distinct(53)
thf(fact_1107_formula_Orel__distinct_I36_J,axiom,
! [R3: a > a > $o,Y41: $o > $o > $o,Y42: formula_a_t,Y43: formula_a_t,X33: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bin_a_t @ Y41 @ Y42 @ Y43 ) @ ( neg_a_t @ X33 ) ) ).
% formula.rel_distinct(36)
thf(fact_1108_formula_Orel__distinct_I35_J,axiom,
! [R3: a > a > $o,X33: formula_a_t,Y41: $o > $o > $o,Y42: formula_a_t,Y43: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( neg_a_t @ X33 ) @ ( bin_a_t @ Y41 @ Y42 @ Y43 ) ) ).
% formula.rel_distinct(35)
thf(fact_1109_formula_Orel__distinct_I10_J,axiom,
! [R3: a > a > $o,Y61: i_t,Y62: formula_a_t,X12: $o] :
~ ( rel_formula_a_a_t @ R3 @ ( next_t_a @ Y61 @ Y62 ) @ ( bool_a_t @ X12 ) ) ).
% formula.rel_distinct(10)
thf(fact_1110_formula_Orel__distinct_I9_J,axiom,
! [R3: a > a > $o,X12: $o,Y61: i_t,Y62: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bool_a_t @ X12 ) @ ( next_t_a @ Y61 @ Y62 ) ) ).
% formula.rel_distinct(9)
thf(fact_1111_formula_Orel__distinct_I8_J,axiom,
! [R3: a > a > $o,Y51: i_t,Y52: formula_a_t,X12: $o] :
~ ( rel_formula_a_a_t @ R3 @ ( prev_t_a @ Y51 @ Y52 ) @ ( bool_a_t @ X12 ) ) ).
% formula.rel_distinct(8)
thf(fact_1112_formula_Orel__distinct_I7_J,axiom,
! [R3: a > a > $o,X12: $o,Y51: i_t,Y52: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bool_a_t @ X12 ) @ ( prev_t_a @ Y51 @ Y52 ) ) ).
% formula.rel_distinct(7)
thf(fact_1113_formula_Orel__distinct_I18_J,axiom,
! [R3: a > a > $o,Y101: i_t,Y102: regex_a_t,X12: $o] :
~ ( rel_formula_a_a_t @ R3 @ ( matchF_t_a @ Y101 @ Y102 ) @ ( bool_a_t @ X12 ) ) ).
% formula.rel_distinct(18)
thf(fact_1114_formula_Orel__distinct_I17_J,axiom,
! [R3: a > a > $o,X12: $o,Y101: i_t,Y102: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bool_a_t @ X12 ) @ ( matchF_t_a @ Y101 @ Y102 ) ) ).
% formula.rel_distinct(17)
thf(fact_1115_formula_Orel__distinct_I16_J,axiom,
! [R3: a > a > $o,Y91: i_t,Y92: regex_a_t,X12: $o] :
~ ( rel_formula_a_a_t @ R3 @ ( matchP_t_a @ Y91 @ Y92 ) @ ( bool_a_t @ X12 ) ) ).
% formula.rel_distinct(16)
thf(fact_1116_formula_Orel__distinct_I15_J,axiom,
! [R3: a > a > $o,X12: $o,Y91: i_t,Y92: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bool_a_t @ X12 ) @ ( matchP_t_a @ Y91 @ Y92 ) ) ).
% formula.rel_distinct(15)
thf(fact_1117_formula_Orel__distinct_I14_J,axiom,
! [R3: a > a > $o,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t,X12: $o] :
~ ( rel_formula_a_a_t @ R3 @ ( until_a_t @ Y81 @ Y82 @ Y83 ) @ ( bool_a_t @ X12 ) ) ).
% formula.rel_distinct(14)
thf(fact_1118_formula_Orel__distinct_I13_J,axiom,
! [R3: a > a > $o,X12: $o,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bool_a_t @ X12 ) @ ( until_a_t @ Y81 @ Y82 @ Y83 ) ) ).
% formula.rel_distinct(13)
thf(fact_1119_formula_Orel__distinct_I12_J,axiom,
! [R3: a > a > $o,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t,X12: $o] :
~ ( rel_formula_a_a_t @ R3 @ ( since_a_t @ Y71 @ Y72 @ Y73 ) @ ( bool_a_t @ X12 ) ) ).
% formula.rel_distinct(12)
thf(fact_1120_formula_Orel__distinct_I11_J,axiom,
! [R3: a > a > $o,X12: $o,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bool_a_t @ X12 ) @ ( since_a_t @ Y71 @ Y72 @ Y73 ) ) ).
% formula.rel_distinct(11)
thf(fact_1121_formula_Orel__distinct_I4_J,axiom,
! [R3: a > a > $o,Y3: formula_a_t,X12: $o] :
~ ( rel_formula_a_a_t @ R3 @ ( neg_a_t @ Y3 ) @ ( bool_a_t @ X12 ) ) ).
% formula.rel_distinct(4)
thf(fact_1122_formula_Orel__distinct_I3_J,axiom,
! [R3: a > a > $o,X12: $o,Y3: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bool_a_t @ X12 ) @ ( neg_a_t @ Y3 ) ) ).
% formula.rel_distinct(3)
thf(fact_1123_formula_Orel__distinct_I6_J,axiom,
! [R3: a > a > $o,Y41: $o > $o > $o,Y42: formula_a_t,Y43: formula_a_t,X12: $o] :
~ ( rel_formula_a_a_t @ R3 @ ( bin_a_t @ Y41 @ Y42 @ Y43 ) @ ( bool_a_t @ X12 ) ) ).
% formula.rel_distinct(6)
thf(fact_1124_formula_Orel__distinct_I5_J,axiom,
! [R3: a > a > $o,X12: $o,Y41: $o > $o > $o,Y42: formula_a_t,Y43: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( bool_a_t @ X12 ) @ ( bin_a_t @ Y41 @ Y42 @ Y43 ) ) ).
% formula.rel_distinct(5)
thf(fact_1125_formula_Orel__distinct_I26_J,axiom,
! [R3: a > a > $o,Y61: i_t,Y62: formula_a_t,X23: a] :
~ ( rel_formula_a_a_t @ R3 @ ( next_t_a @ Y61 @ Y62 ) @ ( atom_a_t @ X23 ) ) ).
% formula.rel_distinct(26)
thf(fact_1126_formula_Orel__distinct_I25_J,axiom,
! [R3: a > a > $o,X23: a,Y61: i_t,Y62: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( atom_a_t @ X23 ) @ ( next_t_a @ Y61 @ Y62 ) ) ).
% formula.rel_distinct(25)
thf(fact_1127_formula_Orel__distinct_I24_J,axiom,
! [R3: a > a > $o,Y51: i_t,Y52: formula_a_t,X23: a] :
~ ( rel_formula_a_a_t @ R3 @ ( prev_t_a @ Y51 @ Y52 ) @ ( atom_a_t @ X23 ) ) ).
% formula.rel_distinct(24)
thf(fact_1128_formula_Orel__distinct_I23_J,axiom,
! [R3: a > a > $o,X23: a,Y51: i_t,Y52: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( atom_a_t @ X23 ) @ ( prev_t_a @ Y51 @ Y52 ) ) ).
% formula.rel_distinct(23)
thf(fact_1129_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1130_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1131_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I4 @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1132_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1133_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I4 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1134_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I4 @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1135_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D4: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D4 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).
% add_le_less_mono
thf(fact_1136_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D4: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D4 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D4 ) ) ) ) ).
% add_le_less_mono
thf(fact_1137_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D4: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D4 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D4 ) ) ) ) ).
% add_le_less_mono
thf(fact_1138_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D4: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D4 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).
% add_less_le_mono
thf(fact_1139_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D4: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D4 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D4 ) ) ) ) ).
% add_less_le_mono
thf(fact_1140_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D4: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D4 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D4 ) ) ) ) ).
% add_less_le_mono
thf(fact_1141_formula_Orel__distinct_I34_J,axiom,
! [R3: a > a > $o,Y101: i_t,Y102: regex_a_t,X23: a] :
~ ( rel_formula_a_a_t @ R3 @ ( matchF_t_a @ Y101 @ Y102 ) @ ( atom_a_t @ X23 ) ) ).
% formula.rel_distinct(34)
thf(fact_1142_formula_Orel__distinct_I33_J,axiom,
! [R3: a > a > $o,X23: a,Y101: i_t,Y102: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( atom_a_t @ X23 ) @ ( matchF_t_a @ Y101 @ Y102 ) ) ).
% formula.rel_distinct(33)
thf(fact_1143_formula_Orel__distinct_I32_J,axiom,
! [R3: a > a > $o,Y91: i_t,Y92: regex_a_t,X23: a] :
~ ( rel_formula_a_a_t @ R3 @ ( matchP_t_a @ Y91 @ Y92 ) @ ( atom_a_t @ X23 ) ) ).
% formula.rel_distinct(32)
thf(fact_1144_formula_Orel__distinct_I31_J,axiom,
! [R3: a > a > $o,X23: a,Y91: i_t,Y92: regex_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( atom_a_t @ X23 ) @ ( matchP_t_a @ Y91 @ Y92 ) ) ).
% formula.rel_distinct(31)
thf(fact_1145_formula_Orel__distinct_I30_J,axiom,
! [R3: a > a > $o,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t,X23: a] :
~ ( rel_formula_a_a_t @ R3 @ ( until_a_t @ Y81 @ Y82 @ Y83 ) @ ( atom_a_t @ X23 ) ) ).
% formula.rel_distinct(30)
thf(fact_1146_formula_Orel__distinct_I29_J,axiom,
! [R3: a > a > $o,X23: a,Y81: formula_a_t,Y82: i_t,Y83: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( atom_a_t @ X23 ) @ ( until_a_t @ Y81 @ Y82 @ Y83 ) ) ).
% formula.rel_distinct(29)
thf(fact_1147_formula_Orel__distinct_I28_J,axiom,
! [R3: a > a > $o,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t,X23: a] :
~ ( rel_formula_a_a_t @ R3 @ ( since_a_t @ Y71 @ Y72 @ Y73 ) @ ( atom_a_t @ X23 ) ) ).
% formula.rel_distinct(28)
thf(fact_1148_formula_Orel__distinct_I27_J,axiom,
! [R3: a > a > $o,X23: a,Y71: formula_a_t,Y72: i_t,Y73: formula_a_t] :
~ ( rel_formula_a_a_t @ R3 @ ( atom_a_t @ X23 ) @ ( since_a_t @ Y71 @ Y72 @ Y73 ) ) ).
% formula.rel_distinct(27)
thf(fact_1149_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1150_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_1151_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_1152_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1153_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1154_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1155_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1156_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1157_less__add__Suc1,axiom,
! [I4: nat,M2: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ I4 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1158_less__add__Suc2,axiom,
! [I4: nat,M2: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ M2 @ I4 ) ) ) ).
% less_add_Suc2
thf(fact_1159_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1160_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1161_less__imp__add__positive,axiom,
! [I4: nat,J: nat] :
( ( ord_less_nat @ I4 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I4 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1162_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1163_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1164_nat__int__comparison_I1_J,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1165_verit__la__generic,axiom,
! [A: int,X3: int] :
( ( ord_less_eq_int @ A @ X3 )
| ( A = X3 )
| ( ord_less_eq_int @ X3 @ A ) ) ).
% verit_la_generic
thf(fact_1166_conj__le__cong,axiom,
! [X3: int,X6: int,P: $o,P6: $o] :
( ( X3 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P6 ) ) ) ) ).
% conj_le_cong
thf(fact_1167_imp__le__cong,axiom,
! [X3: int,X6: int,P: $o,P6: $o] :
( ( X3 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P6 ) ) ) ) ).
% imp_le_cong
thf(fact_1168_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1169_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1170_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_1171_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1172_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1173_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1174_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1175_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1176_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z5: int] :
? [N3: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1177_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z5: int] :
? [N3: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1178_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_1179_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1180_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1181_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_1182_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_1183_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( P @ A2 @ B2 )
= ( P @ B2 @ A2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ zero_zero_nat )
=> ( ! [A2: nat,B2: nat] :
( ( P @ A2 @ B2 )
=> ( P @ A2 @ ( plus_plus_nat @ A2 @ B2 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1184_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1185_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_1186_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_1187_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1188_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_1189_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_1190_int__of__nat__induct,axiom,
! [P: int > $o,Z3: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z3 ) ) ) ).
% int_of_nat_induct
thf(fact_1191_int__cases,axiom,
! [Z3: int] :
( ! [N2: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_1192_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1193_int__cases4,axiom,
! [M2: int] :
( ! [N2: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_1194_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1195_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1196_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1197_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_1198_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1199_negD,axiom,
! [X3: int] :
( ( ord_less_int @ X3 @ zero_zero_int )
=> ? [N2: nat] :
( X3
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_1200_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1201_nat__less__iff,axiom,
! [W3: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( ( ord_less_nat @ ( nat2 @ W3 ) @ M2 )
= ( ord_less_int @ W3 @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_1202_nat__0__iff,axiom,
! [I4: int] :
( ( ( nat2 @ I4 )
= zero_zero_nat )
= ( ord_less_eq_int @ I4 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1203_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ Z3 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1204_zless__nat__conj,axiom,
! [W3: int,Z3: int] :
( ( ord_less_nat @ ( nat2 @ W3 ) @ ( nat2 @ Z3 ) )
= ( ( ord_less_int @ zero_zero_int @ Z3 )
& ( ord_less_int @ W3 @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_1205_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1206_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1207_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_1208_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1209_eq__nat__nat__iff,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z6 ) )
= ( Z3 = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1210_nat__mono,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1211_all__nat,axiom,
( ( ^ [P4: nat > $o] :
! [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
! [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P5 @ ( nat2 @ X4 ) ) ) ) ) ).
% all_nat
thf(fact_1212_ex__nat,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
& ( P5 @ ( nat2 @ X4 ) ) ) ) ) ).
% ex_nat
thf(fact_1213_nat__mono__iff,axiom,
! [Z3: int,W3: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_nat @ ( nat2 @ W3 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W3 @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_1214_nat__le__iff,axiom,
! [X3: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X3 ) @ N )
= ( ord_less_eq_int @ X3 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1215_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z3: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1216_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_1217_int__eq__iff,axiom,
! [M2: nat,Z3: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z3 )
= ( ( M2
= ( nat2 @ Z3 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_1218_nat__eq__iff2,axiom,
! [M2: nat,W3: int] :
( ( M2
= ( nat2 @ W3 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( W3
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1219_nat__eq__iff,axiom,
! [W3: int,M2: nat] :
( ( ( nat2 @ W3 )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( W3
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1220_nat__le__eq__zle,axiom,
! [W3: int,Z3: int] :
( ( ( ord_less_int @ zero_zero_int @ W3 )
| ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W3 ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq_int @ W3 @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_1221_nat__less__eq__zless,axiom,
! [W3: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( ( ord_less_nat @ ( nat2 @ W3 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W3 @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_1222_split__nat,axiom,
! [P: nat > $o,I4: int] :
( ( P @ ( nat2 @ I4 ) )
= ( ! [N3: nat] :
( ( I4
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less_int @ I4 @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1223_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_1224_nat__add__distrib,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( nat2 @ ( plus_plus_int @ Z3 @ Z6 ) )
= ( plus_plus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1225_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_1226_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1227_zle__add1__eq__le,axiom,
! [W3: int,Z3: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W3 @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1228_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1229_int__ge__induct,axiom,
! [K: int,I4: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% int_ge_induct
thf(fact_1230_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1231_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1232_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1233_odd__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1234_add1__zle__eq,axiom,
! [W3: int,Z3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z3 )
= ( ord_less_int @ W3 @ Z3 ) ) ).
% add1_zle_eq
thf(fact_1235_zless__imp__add1__zle,axiom,
! [W3: int,Z3: int] :
( ( ord_less_int @ W3 @ Z3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_1236_le__imp__0__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ).
% le_imp_0_less
thf(fact_1237_Suc__as__int,axiom,
( suc
= ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1238_Suc__nat__eq__nat__zadd1,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( suc @ ( nat2 @ Z3 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1239_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [I3: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
=> ( ( ( ord_less_eq_int @ I3 @ J2 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
=> ( P @ I3 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_1240_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1241_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1242_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N3: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_1243_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N3: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).
% int_less_real_le
thf(fact_1244_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N3: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1245_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N3: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).
% nat_less_real_le
thf(fact_1246_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1247_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1248_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1249_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1250_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1251_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1252_bezw__0,axiom,
! [X3: nat] :
( ( bezw @ X3 @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_1253_real__add__minus__iff,axiom,
! [X3: real,A: real] :
( ( ( plus_plus_real @ X3 @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X3 = A ) ) ).
% real_add_minus_iff
thf(fact_1254_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y7: real] :
( ( ord_less_real @ X4 @ Y7 )
| ( X4 = Y7 ) ) ) ) ).
% less_eq_real_def
thf(fact_1255_complete__real,axiom,
! [S2: set_real] :
( ? [X: real] : ( member_real @ X @ S2 )
=> ( ? [Z4: real] :
! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ? [Y4: real] :
( ! [X: real] :
( ( member_real @ X @ S2 )
=> ( ord_less_eq_real @ X @ Y4 ) )
& ! [Z4: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ( ord_less_eq_real @ Y4 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1256__092_060iota_062__real__def,axiom,
embed_nat_iota_real = semiri5074537144036343181t_real ).
% \<iota>_real_def
thf(fact_1257_real__add__le__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X3 @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X3 ) ) ) ).
% real_add_le_0_iff
thf(fact_1258_real__0__le__add__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X3 ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1259_real__add__less__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X3 @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X3 ) ) ) ).
% real_add_less_0_iff
thf(fact_1260_real__0__less__add__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X3 ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1261_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M7: nat] :
( ( P @ X3 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M7 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1262_nat__ceiling__le__eq,axiom,
! [X3: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) @ A )
= ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1263_real__nat__ceiling__ge,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1264_real__archimedian__rdiv__eq__0,axiom,
! [X3: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X3 ) @ C ) )
=> ( X3 = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y: int] :
( ( if_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y: int] :
( ( if_int @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X3: real,Y: real] :
( ( if_real @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X3: real,Y: real] :
( ( if_real @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
p @ na @ phia ).
%------------------------------------------------------------------------------