TPTP Problem File: ITP130^2.p
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%------------------------------------------------------------------------------
% File : ITP130^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer NthRoot_Impl problem prob_360__3277026_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : NthRoot_Impl/prob_360__3277026_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 328 ( 120 unt; 36 typ; 0 def)
% Number of atoms : 768 ( 209 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 2926 ( 75 ~; 22 |; 38 &;2425 @)
% ( 0 <=>; 366 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 50 ( 50 >; 0 *; 0 +; 0 <<)
% Number of symbols : 34 ( 33 usr; 7 con; 0-3 aty)
% Number of variables : 648 ( 7 ^; 590 !; 27 ?; 648 :)
% ( 24 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:26:40.961
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
% Explicit typings (33)
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__bits,type,
semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
semiri134348788visors:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim1804426504_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1598680935umeral:
!>[A: $tType] : $o ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_NthRoot__Impl__Mirabelle__yrjxwmcnbt_Ofixed__root,type,
nthRoo220209705d_root: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_v_NY____,type,
ny: real ).
thf(sy_v_na____,type,
na: int ).
thf(sy_v_p,type,
p: nat ).
thf(sy_v_pm,type,
pm: nat ).
thf(sy_v_x,type,
x: int ).
thf(sy_v_y_H____,type,
y: int ).
thf(sy_v_ya____,type,
ya: int ).
% Relevant facts (253)
thf(fact_0_n00,axiom,
( na
!= ( zero_zero @ int ) ) ).
% n00
thf(fact_1_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B: int,W: nat,X: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( power_power @ int @ B @ W ) @ X ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_2_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B: int,W: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
= ( ord_less @ int @ X @ ( power_power @ int @ B @ W ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_3_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int,N: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N ) ) ) ).
% of_int_power
thf(fact_4_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B: int,W: nat,X: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W )
= ( ring_1_of_int @ A @ X ) )
= ( ( power_power @ int @ B @ W )
= X ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_5_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int @ A @ X )
= ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
= ( X
= ( power_power @ int @ B @ W ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ W @ Z ) ) ) ).
% of_int_less_iff
thf(fact_7_n0,axiom,
ord_less @ real @ ( zero_zero @ real ) @ ( ring_1_of_int @ real @ na ) ).
% n0
thf(fact_8_y0,axiom,
ord_less @ real @ ( zero_zero @ real ) @ ( ring_1_of_int @ real @ ya ) ).
% y0
thf(fact_9_of__int__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W: int,Z: int] :
( ( ( ring_1_of_int @ A @ W )
= ( ring_1_of_int @ A @ Z ) )
= ( W = Z ) ) ) ).
% of_int_eq_iff
thf(fact_10_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_11_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_12_y00,axiom,
( ya
!= ( zero_zero @ int ) ) ).
% y00
thf(fact_13__092_060open_0620_A_060_Ay_A_092_060or_062_Ay_A_061_A0_092_060close_062,axiom,
( ( ord_less @ int @ ( zero_zero @ int ) @ ya )
| ( ya
= ( zero_zero @ int ) ) ) ).
% \<open>0 < y \<or> y = 0\<close>
thf(fact_14_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_15_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_16_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_17_not_I1_J,axiom,
~ ( ( ord_less @ int @ ya @ ( zero_zero @ int ) )
| ( ord_less @ int @ na @ ( zero_zero @ int ) ) ) ).
% not(1)
thf(fact_18_not_I2_J,axiom,
~ ( ( ord_less @ int @ ya @ ( zero_zero @ int ) )
| ( ord_less @ int @ na @ ( zero_zero @ int ) ) ) ).
% not(2)
thf(fact_19__C1_Oprems_C_I2_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ya ).
% "1.prems"(2)
thf(fact_20__C1_Oprems_C_I3_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ na ).
% "1.prems"(3)
thf(fact_21_of__int__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
= ( zero_zero @ A ) ) ) ).
% of_int_0
thf(fact_22_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_23_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( ring_1_of_int @ A @ Z )
= ( zero_zero @ A ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_24_of__int__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
= ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% of_int_less_0_iff
thf(fact_25_of__int__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% of_int_0_less_iff
thf(fact_26_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_27_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_28_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_29_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K )
=> ( P @ I2 @ K ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_30_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ K2 )
=> ( ord_less @ nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_31_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_32_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_33_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M2: nat] :
( ( M
= ( suc @ M2 ) )
& ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% Suc_less_eq2
thf(fact_34_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_35_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_36_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_37_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_38_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_39_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_40_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_41_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_42_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_43_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less @ nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_44_of__int__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_pos
thf(fact_45_measure__induct__rule,axiom,
! [B2: $tType,A: $tType] :
( ( wellorder @ B2 )
=> ! [F: A > B2,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y: A] :
( ( ord_less @ B2 @ ( F @ Y ) @ ( F @ X3 ) )
=> ( P @ Y ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_46_measure__induct,axiom,
! [B2: $tType,A: $tType] :
( ( wellorder @ B2 )
=> ! [F: A > B2,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y: A] :
( ( ord_less @ B2 @ ( F @ Y ) @ ( F @ X3 ) )
=> ( P @ Y ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct
thf(fact_47_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_48_Suc__inject,axiom,
! [X: nat,Y3: nat] :
( ( ( suc @ X )
= ( suc @ Y3 ) )
=> ( X = Y3 ) ) ).
% Suc_inject
thf(fact_49_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_50_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N3 )
=> ( ord_less @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_51_power__0__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
= ( zero_zero @ A ) ) ) ).
% power_0_Suc
thf(fact_52_NY0,axiom,
ord_less @ real @ ( zero_zero @ real ) @ ny ).
% NY0
thf(fact_53_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri134348788visors @ A )
=> ! [A2: A,N: nat] :
( ( ( power_power @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_54_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_55_realpow__pos__nth2,axiom,
! [A2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
& ( ( power_power @ real @ R @ ( suc @ N ) )
= A2 ) ) ) ).
% realpow_pos_nth2
thf(fact_56_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_less_power
thf(fact_57_NY__def,axiom,
( ny
= ( divide_divide @ real @ ( ring_1_of_int @ real @ na ) @ ( power_power @ real @ ( ring_1_of_int @ real @ ya ) @ pm ) ) ) ).
% NY_def
thf(fact_58_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A )
=> ! [X: A] :
? [Z2: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ X ) ) ).
% ex_of_int_less
thf(fact_59_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A )
=> ! [X: A] :
? [Z2: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% ex_less_of_int
thf(fact_60_yyn,axiom,
ord_less @ int @ na @ ( power_power @ int @ ya @ p ) ).
% yyn
thf(fact_61__092_060open_062pm_A_061_A0_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
( pm
!= ( zero_zero @ nat ) ) ).
% \<open>pm = 0 \<Longrightarrow> False\<close>
thf(fact_62_p0,axiom,
( p
!= ( zero_zero @ nat ) ) ).
% p0
thf(fact_63_pm0,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ pm ).
% pm0
thf(fact_64__C1_Oprems_C_I1_J,axiom,
ord_less_eq @ int @ na @ ( power_power @ int @ ya @ p ) ).
% "1.prems"(1)
thf(fact_65_p,axiom,
( p
= ( suc @ pm ) ) ).
% p
thf(fact_66_fixed__root__axioms,axiom,
nthRoo220209705d_root @ p @ pm ).
% fixed_root_axioms
thf(fact_67_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_68_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power @ nat @ X @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_69_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_70_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_71_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_72_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_73_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_74_power__Suc0__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% power_Suc0_right
thf(fact_75_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_76_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_77_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ W @ Z ) ) ) ).
% of_int_le_iff
thf(fact_78_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_79_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_80_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% of_int_0_le_iff
thf(fact_81_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B: int,W: nat,X: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B @ W ) @ X ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_82_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B: int,W: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
= ( ord_less_eq @ int @ X @ ( power_power @ int @ B @ W ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_83_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_84_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_85_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_86_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_87_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_88_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_89_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_90_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_91_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_92_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_93_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_94_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_95_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_96_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less @ nat @ X @ Y3 )
=> ( ord_less @ nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_97_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X3 ) )
& ~ ( P @ Y ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_98_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_99_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X3: A] :
( ( ( V @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X3 ) )
& ~ ( P @ Y ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_100_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_101_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_102_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_103_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_104_power__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ A2 @ B ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ).
% power_divide
thf(fact_105_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A )
=> ! [X: A] :
? [Z2: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% ex_le_of_int
thf(fact_106_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_107_power__gt__expt,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ K2 @ ( power_power @ nat @ N @ K2 ) ) ) ).
% power_gt_expt
thf(fact_108_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_109_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_110_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_111_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_112_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_113_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_114_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P @ X3 @ Y4 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_115_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_116_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_117_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_118_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_119_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y3
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_120_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [Nat4: nat] :
( ( P @ Nat4 )
=> ( P @ ( suc @ Nat4 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_121_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_122_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( M
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_123_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_124_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_125_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_126_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_127_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B: A] :
( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_128_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A2: A,B: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B @ N ) )
= ( A2 = B ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_129_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_le_power
thf(fact_130_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B: A,N: nat] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ) ).
% power_mono
thf(fact_131_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_nonneg
thf(fact_132_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B: A] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less @ A @ A2 @ B ) ) ) ) ).
% power_less_imp_less_base
thf(fact_133_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B: A,N: nat] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_134_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B: A] :
( ( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( power_power @ A @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( A2 = B ) ) ) ) ) ).
% power_inject_base
thf(fact_135_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( power_power @ A @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% power_le_imp_le_base
thf(fact_136_realpow__pos__nth,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
& ( ( power_power @ real @ R @ N )
= A2 ) ) ) ) ).
% realpow_pos_nth
thf(fact_137_realpow__pos__nth__unique,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ( power_power @ real @ X3 @ N )
= A2 )
& ! [Y: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
& ( ( power_power @ real @ Y @ N )
= A2 ) )
=> ( Y = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_138_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_139_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_140_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_141_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_142_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_143_power__not__zero,axiom,
! [A: $tType] :
( ( semiri134348788visors @ A )
=> ! [A2: A,N: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_not_zero
thf(fact_144_zero__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ).
% zero_power
thf(fact_145_yyn_H,axiom,
ord_less @ int @ ( power_power @ int @ y @ p ) @ na ).
% yyn'
thf(fact_146_y_H0,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ y ).
% y'0
thf(fact_147_compare__pow__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: nat,X: A,Y3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ P2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less @ A @ ( power_power @ A @ X @ P2 ) @ ( power_power @ A @ Y3 @ P2 ) )
= ( ord_less @ A @ X @ Y3 ) ) ) ) ) ) ).
% compare_pow_less_iff
thf(fact_148_compare__pow__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: nat,X: A,Y3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ P2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X @ P2 ) @ ( power_power @ A @ Y3 @ P2 ) )
= ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ) ) ).
% compare_pow_le_iff
thf(fact_149_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_150_bits__div__by__0,axiom,
! [A: $tType] :
( ( semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_151_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,C: A,B: A] :
( ( ( divide_divide @ A @ A2 @ C )
= ( divide_divide @ A @ B @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B ) ) ) ) ).
% divide_cancel_right
thf(fact_152_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B: A] :
( ( ( divide_divide @ A @ A2 @ B )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_153_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C: A,A2: A,B: A] :
( ( ( divide_divide @ A @ C @ A2 )
= ( divide_divide @ A @ C @ B ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B ) ) ) ) ).
% divide_cancel_left
thf(fact_154_bits__div__0,axiom,
! [A: $tType] :
( ( semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_155_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_156_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_157_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq @ nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_158_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_159_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K2 )
=> ( ord_less_eq @ nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_160_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_161_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_162_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_163_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ Y @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_164_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_165_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_166_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_167_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_168_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z2: nat] :
( ( R2 @ X3 @ Y4 )
=> ( ( R2 @ Y4 @ Z2 )
=> ( R2 @ X3 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_169_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_170_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_171_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq @ nat @ M @ N ) )
= ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_172_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_173_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq @ nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_174_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_175_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_176_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_177_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_178_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_179_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( M != N )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_180_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less @ nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_181_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N4: nat] :
( ( ord_less @ nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_182_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_183_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq @ nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_184_fixed__root_Op0,axiom,
! [P2: nat,Pm: nat] :
( ( nthRoo220209705d_root @ P2 @ Pm )
=> ( P2
!= ( zero_zero @ nat ) ) ) ).
% fixed_root.p0
thf(fact_185_fixed__root_Ointro,axiom,
! [P2: nat,Pm: nat] :
( ( P2
= ( suc @ Pm ) )
=> ( nthRoo220209705d_root @ P2 @ Pm ) ) ).
% fixed_root.intro
thf(fact_186_fixed__root__def,axiom,
( nthRoo220209705d_root
= ( ^ [P3: nat,Pm2: nat] :
( P3
= ( suc @ Pm2 ) ) ) ) ).
% fixed_root_def
thf(fact_187_fixed__root_Op,axiom,
! [P2: nat,Pm: nat] :
( ( nthRoo220209705d_root @ P2 @ Pm )
=> ( P2
= ( suc @ Pm ) ) ) ).
% fixed_root.p
thf(fact_188_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_189_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less_eq @ nat @ K @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ~ ( P @ I4 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_190_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_191_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_192_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_193_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_194_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_195_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_196_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_197_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N4: nat] : ( ord_less_eq @ nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_198_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_199_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X4: A] :
? [X_12: A] : ( ord_less @ A @ X4 @ X_12 ) ) ).
% linordered_field_no_ub
thf(fact_200_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X4: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ).
% linordered_field_no_lb
thf(fact_201_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less @ nat @ K @ N )
& ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ K )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_202_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B @ C ) @ ( divide_divide @ A @ A2 @ C ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_203_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_204_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_205_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_206_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_207_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_208_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% divide_right_mono
thf(fact_209_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_210_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% divide_neg_neg
thf(fact_211_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_212_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_213_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% divide_pos_pos
thf(fact_214_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_215_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less @ A @ A2 @ B ) )
& ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B @ A2 ) )
& ( C
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_216_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_217_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_218_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_219_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% frac_le
thf(fact_220_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% frac_less
thf(fact_221_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A,W: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_222_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ A2 @ B ) )
& ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_223_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_224_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_225_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_226_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_227_div__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ K2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K2 )
=> ( ( divide_divide @ int @ K2 @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_228_div__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less @ int @ K2 @ L )
=> ( ( divide_divide @ int @ K2 @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_229_x,axiom,
x = y ).
% x
thf(fact_230_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_231_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_232_div__le__mono,axiom,
! [M: nat,N: nat,K2: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K2 ) @ ( divide_divide @ nat @ N @ K2 ) ) ) ).
% div_le_mono
thf(fact_233_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_234_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ M @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_235_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ ( divide_divide @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_236_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) )
= ( ( ord_less_eq @ nat @ N @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_237_div__le__mono2,axiom,
! [M: nat,N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K2 @ N ) @ ( divide_divide @ nat @ K2 @ M ) ) ) ) ).
% div_le_mono2
thf(fact_238_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_239_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_240_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( X != Y3 )
=> ( ~ ( ord_less @ A @ X @ Y3 )
=> ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_241_zdiv__mono1,axiom,
! [A2: int,A3: int,B: int] :
( ( ord_less_eq @ int @ A2 @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( divide_divide @ int @ A3 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_242_zdiv__mono2,axiom,
! [A2: int,B3: int,B: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( divide_divide @ int @ A2 @ B3 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_243_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_244_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less @ int @ B @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_245_div__neg__pos__less0,axiom,
! [A2: int,B: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
=> ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% div_neg_pos_less0
thf(fact_246_div__positive,axiom,
! [A: $tType] :
( ( unique1598680935umeral @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B )
=> ( ( ord_less_eq @ A @ B @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) ) ) ) ) ).
% div_positive
thf(fact_247_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1598680935umeral @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ B )
=> ( ( divide_divide @ A @ A2 @ B )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_248_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
= ( ( ord_less_eq @ int @ B @ A2 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_249_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_250_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less @ int @ B @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
= ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_251_pos__imp__zdiv__pos__iff,axiom,
! [K2: int,I: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K2 ) )
= ( ord_less_eq @ int @ K2 @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_252_div__nonpos__pos__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
% Type constructors (38)
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1598680935umeral @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri134348788visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__bits,axiom,
semiring_bits @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Orderings_Oorder,axiom,
order @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_1,axiom,
unique1598680935umeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_2,axiom,
semiri134348788visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_3,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_4,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__bits_5,axiom,
semiring_bits @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_6,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_7,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_8,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_9,axiom,
zero @ nat ).
thf(tcon_HOL_Obool___Orderings_Oorder_10,axiom,
order @ $o ).
thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
archim1804426504_field @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_11,axiom,
semiri134348788visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_12,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_13,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_14,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_15,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_16,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_17,axiom,
order @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_18,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_19,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Fields_Ofield,axiom,
field @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_20,axiom,
zero @ real ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less @ real @ ( ring_1_of_int @ real @ na ) @ ( power_power @ real @ ( ring_1_of_int @ real @ ya ) @ ( suc @ pm ) ) ).
%------------------------------------------------------------------------------