TPTP Problem File: SLH0623^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : LP_Duality/0001_LP_Duality/prob_00251_011664__28850146_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1375 ( 656 unt; 98 typ; 0 def)
% Number of atoms : 3656 (1359 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10380 ( 291 ~; 201 |; 142 &;8411 @)
% ( 0 <=>;1335 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 215 ( 215 >; 0 *; 0 +; 0 <<)
% Number of symbols : 87 ( 84 usr; 18 con; 0-3 aty)
% Number of variables : 3090 ( 164 ^;2902 !; 24 ?;3090 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:59:28.488
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J,type,
poly_poly_a: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Polynomial__Opoly_Itf__a_J_J,type,
vec_poly_a: $tType ).
thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_Itf__a_J_J,type,
set_poly_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Num__Onum_J_J,type,
set_set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Polynomial__Opoly_Itf__a_J,type,
poly_a: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_Itf__a_J,type,
vec_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (84)
thf(sy_c_Groups_Ogroup_001tf__a,type,
group_a: ( a > a > a ) > a > ( a > a ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Ovec_Itf__a_J,type,
minus_minus_vec_a: vec_a > vec_a > vec_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_Itf__a_J,type,
minus_minus_poly_a: poly_a > poly_a > poly_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001tf__a,type,
minus_minus_a: a > a > a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001tf__a,type,
one_one_a: a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_Itf__a_J,type,
plus_plus_poly_a: poly_a > poly_a > poly_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Num__Onum_J,type,
plus_plus_set_num: set_num > set_num > set_num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Polynomial__Opoly_Itf__a_J_J,type,
plus_plus_set_poly_a: set_poly_a > set_poly_a > set_poly_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
plus_p4817606893110106565et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Set__Oset_It__Num__Onum_J_J,type,
plus_p532826482549453327et_num: set_set_num > set_set_num > set_set_num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
plus_plus_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_Itf__a_J,type,
plus_plus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a,type,
plus_plus_a: a > a > a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Num__Onum_J,type,
times_times_set_num: set_num > set_num > set_num ).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a,type,
times_times_a: a > a > a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Ovec_Itf__a_J,type,
uminus_uminus_vec_a: vec_a > vec_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_Itf__a_J,type,
uminus_uminus_poly_a: poly_a > poly_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001tf__a,type,
uminus_uminus_a: a > a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_Itf__a_J,type,
zero_zero_poly_a: poly_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_Itf__a_J,type,
zero_zero_set_a: set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Matrix_Oscalar__prod_001t__Polynomial__Opoly_Itf__a_J,type,
scalar_prod_poly_a: vec_poly_a > vec_poly_a > poly_a ).
thf(sy_c_Matrix_Oscalar__prod_001tf__a,type,
scalar_prod_a: vec_a > vec_a > a ).
thf(sy_c_Norms_Olinf__norm__poly_001tf__a,type,
linf_norm_poly_a: poly_a > a ).
thf(sy_c_Norms_Olinf__norm__vec_001tf__a,type,
linf_norm_vec_a: vec_a > a ).
thf(sy_c_Norms_Onorm1_001t__Polynomial__Opoly_Itf__a_J,type,
norm1_poly_a: poly_poly_a > poly_a ).
thf(sy_c_Norms_Onorm1_001tf__a,type,
norm1_a: poly_a > a ).
thf(sy_c_Norms_Osq__norm__poly_001tf__a,type,
sq_norm_poly_a: poly_a > a ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001tf__a,type,
neg_numeral_dbl_a: a > a ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001tf__a,type,
neg_nu181380926503873385_dec_a: a > a ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001tf__a,type,
neg_nu6917059380386235053_inc_a: a > a ).
thf(sy_c_Num_Oneg__numeral__class_Ois__num_001tf__a,type,
neg_numeral_is_num_a: a > $o ).
thf(sy_c_Num_Oneg__numeral__class_Osub_001tf__a,type,
neg_numeral_sub_a: num > num > a ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001tf__a,type,
numeral_numeral_a: num > a ).
thf(sy_c_Num_Oring__1__class_Oiszero_001tf__a,type,
ring_1_iszero_a: a > $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__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_Itf__a_J,type,
ord_less_eq_poly_a: poly_a > poly_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Polynomial__Opoly_Itf__a_J_J,type,
ord_le1582614204914470028poly_a: set_poly_a > set_poly_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_RBT__Impl_Oplog2,type,
rBT_plog2: nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001tf__a,type,
divide_divide_a: a > a > a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Num__Onum,type,
collect_num: ( num > $o ) > set_num ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Polynomial__Opoly_Itf__a_J,type,
member_poly_a: poly_a > set_poly_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Num__Onum_J,type,
member_set_num: set_num > set_set_num > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_b,type,
b: vec_a ).
thf(sy_v_bc____,type,
bc: vec_a ).
thf(sy_v_c,type,
c: vec_a ).
thf(sy_v_lam____,type,
lam: a ).
thf(sy_v_u____,type,
u: vec_a ).
thf(sy_v_ulv____,type,
ulv: vec_a ).
thf(sy_v_v____,type,
v: vec_a ).
thf(sy_v_w____,type,
w: vec_a ).
% Relevant facts (1273)
thf(fact_0_lam0,axiom,
ord_less_eq_a @ zero_zero_a @ lam ).
% lam0
thf(fact_1_le__add__diff__inverse,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( plus_plus_a @ B @ ( minus_minus_a @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_2_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_3_le__add__diff__inverse2,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_4_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_5_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_6_diff__ge__0__iff__ge,axiom,
! [A: poly_a,B: poly_a] :
( ( ord_less_eq_poly_a @ zero_zero_poly_a @ ( minus_minus_poly_a @ A @ B ) )
= ( ord_less_eq_poly_a @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_7_diff__ge__0__iff__ge,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( minus_minus_a @ A @ B ) )
= ( ord_less_eq_a @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_8_add__le__same__cancel1,axiom,
! [B: poly_a,A: poly_a] :
( ( ord_less_eq_poly_a @ ( plus_plus_poly_a @ B @ A ) @ B )
= ( ord_less_eq_poly_a @ A @ zero_zero_poly_a ) ) ).
% add_le_same_cancel1
thf(fact_9_add__le__same__cancel1,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ B @ A ) @ B )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% add_le_same_cancel1
thf(fact_10_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_11_add__le__same__cancel2,axiom,
! [A: poly_a,B: poly_a] :
( ( ord_less_eq_poly_a @ ( plus_plus_poly_a @ A @ B ) @ B )
= ( ord_less_eq_poly_a @ A @ zero_zero_poly_a ) ) ).
% add_le_same_cancel2
thf(fact_12_add__le__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ B )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% add_le_same_cancel2
thf(fact_13_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_14_le__add__same__cancel1,axiom,
! [A: poly_a,B: poly_a] :
( ( ord_less_eq_poly_a @ A @ ( plus_plus_poly_a @ A @ B ) )
= ( ord_less_eq_poly_a @ zero_zero_poly_a @ B ) ) ).
% le_add_same_cancel1
thf(fact_15_le__add__same__cancel1,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ ( plus_plus_a @ A @ B ) )
= ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% le_add_same_cancel1
thf(fact_16_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_17_le__add__same__cancel2,axiom,
! [A: poly_a,B: poly_a] :
( ( ord_less_eq_poly_a @ A @ ( plus_plus_poly_a @ B @ A ) )
= ( ord_less_eq_poly_a @ zero_zero_poly_a @ B ) ) ).
% le_add_same_cancel2
thf(fact_18_le__add__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ ( plus_plus_a @ B @ A ) )
= ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% le_add_same_cancel2
thf(fact_19_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_20_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: poly_a] :
( ( ord_less_eq_poly_a @ ( plus_plus_poly_a @ A @ A ) @ zero_zero_poly_a )
= ( ord_less_eq_poly_a @ A @ zero_zero_poly_a ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_21_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ A ) @ zero_zero_a )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_22_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: poly_a] :
( ( ord_less_eq_poly_a @ zero_zero_poly_a @ ( plus_plus_poly_a @ A @ A ) )
= ( ord_less_eq_poly_a @ zero_zero_poly_a @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_23_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ A @ A ) )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_24__092_060open_062v_A_092_060bullet_062_Ac_A_L_Aw_A_092_060bullet_062_A_N_Ac_A_061_Ac_A_092_060bullet_062_Av_A_L_A_N_Ac_A_092_060bullet_062_Aw_092_060close_062,axiom,
( ( plus_plus_a @ ( scalar_prod_a @ v @ c ) @ ( scalar_prod_a @ w @ ( uminus_uminus_vec_a @ c ) ) )
= ( plus_plus_a @ ( scalar_prod_a @ c @ v ) @ ( scalar_prod_a @ ( uminus_uminus_vec_a @ c ) @ w ) ) ) ).
% \<open>v \<bullet> c + w \<bullet> - c = c \<bullet> v + - c \<bullet> w\<close>
thf(fact_25_add__right__cancel,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_26_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_27_add__left__cancel,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_28_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_29_add_Oinverse__inverse,axiom,
! [A: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_30_neg__equal__iff__equal,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= ( uminus_uminus_a @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_31_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_32_add__0,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ zero_zero_set_nat @ A )
= A ) ).
% add_0
thf(fact_33_add__0,axiom,
! [A: set_a] :
( ( plus_plus_set_a @ zero_zero_set_a @ A )
= A ) ).
% add_0
thf(fact_34_add__0,axiom,
! [A: poly_a] :
( ( plus_plus_poly_a @ zero_zero_poly_a @ A )
= A ) ).
% add_0
thf(fact_35_add__0,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add_0
thf(fact_36_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_37_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_38_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_39_add__cancel__right__right,axiom,
! [A: poly_a,B: poly_a] :
( ( A
= ( plus_plus_poly_a @ A @ B ) )
= ( B = zero_zero_poly_a ) ) ).
% add_cancel_right_right
thf(fact_40_add__cancel__right__right,axiom,
! [A: a,B: a] :
( ( A
= ( plus_plus_a @ A @ B ) )
= ( B = zero_zero_a ) ) ).
% add_cancel_right_right
thf(fact_41_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_42_add__cancel__right__left,axiom,
! [A: poly_a,B: poly_a] :
( ( A
= ( plus_plus_poly_a @ B @ A ) )
= ( B = zero_zero_poly_a ) ) ).
% add_cancel_right_left
thf(fact_43_add__cancel__right__left,axiom,
! [A: a,B: a] :
( ( A
= ( plus_plus_a @ B @ A ) )
= ( B = zero_zero_a ) ) ).
% add_cancel_right_left
thf(fact_44_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_45_add__cancel__left__right,axiom,
! [A: poly_a,B: poly_a] :
( ( ( plus_plus_poly_a @ A @ B )
= A )
= ( B = zero_zero_poly_a ) ) ).
% add_cancel_left_right
thf(fact_46_add__cancel__left__right,axiom,
! [A: a,B: a] :
( ( ( plus_plus_a @ A @ B )
= A )
= ( B = zero_zero_a ) ) ).
% add_cancel_left_right
thf(fact_47_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_48_add__cancel__left__left,axiom,
! [B: poly_a,A: poly_a] :
( ( ( plus_plus_poly_a @ B @ A )
= A )
= ( B = zero_zero_poly_a ) ) ).
% add_cancel_left_left
thf(fact_49_add__cancel__left__left,axiom,
! [B: a,A: a] :
( ( ( plus_plus_a @ B @ A )
= A )
= ( B = zero_zero_a ) ) ).
% add_cancel_left_left
thf(fact_50_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_51_double__zero__sym,axiom,
! [A: poly_a] :
( ( zero_zero_poly_a
= ( plus_plus_poly_a @ A @ A ) )
= ( A = zero_zero_poly_a ) ) ).
% double_zero_sym
thf(fact_52_double__zero__sym,axiom,
! [A: a] :
( ( zero_zero_a
= ( plus_plus_a @ A @ A ) )
= ( A = zero_zero_a ) ) ).
% double_zero_sym
thf(fact_53_add_Oright__neutral,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ A @ zero_zero_set_nat )
= A ) ).
% add.right_neutral
thf(fact_54_add_Oright__neutral,axiom,
! [A: set_a] :
( ( plus_plus_set_a @ A @ zero_zero_set_a )
= A ) ).
% add.right_neutral
thf(fact_55_add_Oright__neutral,axiom,
! [A: poly_a] :
( ( plus_plus_poly_a @ A @ zero_zero_poly_a )
= A ) ).
% add.right_neutral
thf(fact_56_add_Oright__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.right_neutral
thf(fact_57_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_58_add__le__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_59_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_60_add__le__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_61_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_62_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: poly_a] :
( ( minus_minus_poly_a @ A @ A )
= zero_zero_poly_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_63_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: a] :
( ( minus_minus_a @ A @ A )
= zero_zero_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_64_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_65_diff__zero,axiom,
! [A: poly_a] :
( ( minus_minus_poly_a @ A @ zero_zero_poly_a )
= A ) ).
% diff_zero
thf(fact_66_diff__zero,axiom,
! [A: a] :
( ( minus_minus_a @ A @ zero_zero_a )
= A ) ).
% diff_zero
thf(fact_67_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_68_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_69_diff__0__right,axiom,
! [A: poly_a] :
( ( minus_minus_poly_a @ A @ zero_zero_poly_a )
= A ) ).
% diff_0_right
thf(fact_70_diff__0__right,axiom,
! [A: a] :
( ( minus_minus_a @ A @ zero_zero_a )
= A ) ).
% diff_0_right
thf(fact_71_diff__self,axiom,
! [A: poly_a] :
( ( minus_minus_poly_a @ A @ A )
= zero_zero_poly_a ) ).
% diff_self
thf(fact_72_diff__self,axiom,
! [A: a] :
( ( minus_minus_a @ A @ A )
= zero_zero_a ) ).
% diff_self
thf(fact_73_neg__equal__zero,axiom,
! [A: poly_a] :
( ( ( uminus_uminus_poly_a @ A )
= A )
= ( A = zero_zero_poly_a ) ) ).
% neg_equal_zero
thf(fact_74_neg__equal__zero,axiom,
! [A: a] :
( ( ( uminus_uminus_a @ A )
= A )
= ( A = zero_zero_a ) ) ).
% neg_equal_zero
thf(fact_75_equal__neg__zero,axiom,
! [A: poly_a] :
( ( A
= ( uminus_uminus_poly_a @ A ) )
= ( A = zero_zero_poly_a ) ) ).
% equal_neg_zero
thf(fact_76_equal__neg__zero,axiom,
! [A: a] :
( ( A
= ( uminus_uminus_a @ A ) )
= ( A = zero_zero_a ) ) ).
% equal_neg_zero
thf(fact_77_neg__equal__0__iff__equal,axiom,
! [A: poly_a] :
( ( ( uminus_uminus_poly_a @ A )
= zero_zero_poly_a )
= ( A = zero_zero_poly_a ) ) ).
% neg_equal_0_iff_equal
thf(fact_78_neg__equal__0__iff__equal,axiom,
! [A: a] :
( ( ( uminus_uminus_a @ A )
= zero_zero_a )
= ( A = zero_zero_a ) ) ).
% neg_equal_0_iff_equal
thf(fact_79_neg__0__equal__iff__equal,axiom,
! [A: poly_a] :
( ( zero_zero_poly_a
= ( uminus_uminus_poly_a @ A ) )
= ( zero_zero_poly_a = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_80_neg__0__equal__iff__equal,axiom,
! [A: a] :
( ( zero_zero_a
= ( uminus_uminus_a @ A ) )
= ( zero_zero_a = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_81_add_Oinverse__neutral,axiom,
( ( uminus_uminus_poly_a @ zero_zero_poly_a )
= zero_zero_poly_a ) ).
% add.inverse_neutral
thf(fact_82_add_Oinverse__neutral,axiom,
( ( uminus_uminus_a @ zero_zero_a )
= zero_zero_a ) ).
% add.inverse_neutral
thf(fact_83_add__diff__cancel__right_H,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_84_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_85_add__diff__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( minus_minus_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_86_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_87_add__diff__cancel__left_H,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_88_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_89_add__diff__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( minus_minus_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_90_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_91_diff__add__cancel,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_92_add__diff__cancel,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_93_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
! [A: num,P: num > $o] :
( ( member_num @ A @ ( collect_num @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_96_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_97_Collect__mem__eq,axiom,
! [A2: set_num] :
( ( collect_num
@ ^ [X2: num] : ( member_num @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_98_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_99_neg__le__iff__le,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_100_minus__add__distrib,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
= ( plus_plus_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_a @ B ) ) ) ).
% minus_add_distrib
thf(fact_101_minus__add__cancel,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ ( plus_plus_a @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_102_add__minus__cancel,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ A @ ( plus_plus_a @ ( uminus_uminus_a @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_103_minus__diff__eq,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( minus_minus_a @ A @ B ) )
= ( minus_minus_a @ B @ A ) ) ).
% minus_diff_eq
thf(fact_104__092_060open_062c_A_092_060bullet_062_Av_A_N_Ac_A_092_060bullet_062_Aw_A_061_Ac_A_092_060bullet_062_A_Iv_A_N_Aw_J_092_060close_062,axiom,
( ( minus_minus_a @ ( scalar_prod_a @ c @ v ) @ ( scalar_prod_a @ c @ w ) )
= ( scalar_prod_a @ c @ ( minus_minus_vec_a @ v @ w ) ) ) ).
% \<open>c \<bullet> v - c \<bullet> w = c \<bullet> (v - w)\<close>
thf(fact_105__092_060open_062c_A_092_060bullet_062_Av_A_L_A_N_Ac_A_092_060bullet_062_Aw_A_061_Ac_A_092_060bullet_062_Av_A_N_Ac_A_092_060bullet_062_Aw_092_060close_062,axiom,
( ( plus_plus_a @ ( scalar_prod_a @ c @ v ) @ ( scalar_prod_a @ ( uminus_uminus_vec_a @ c ) @ w ) )
= ( minus_minus_a @ ( scalar_prod_a @ c @ v ) @ ( scalar_prod_a @ c @ w ) ) ) ).
% \<open>c \<bullet> v + - c \<bullet> w = c \<bullet> v - c \<bullet> w\<close>
thf(fact_106_neg__less__eq__nonneg,axiom,
! [A: poly_a] :
( ( ord_less_eq_poly_a @ ( uminus_uminus_poly_a @ A ) @ A )
= ( ord_less_eq_poly_a @ zero_zero_poly_a @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_107_neg__less__eq__nonneg,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ A )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_108_less__eq__neg__nonpos,axiom,
! [A: poly_a] :
( ( ord_less_eq_poly_a @ A @ ( uminus_uminus_poly_a @ A ) )
= ( ord_less_eq_poly_a @ A @ zero_zero_poly_a ) ) ).
% less_eq_neg_nonpos
thf(fact_109_less__eq__neg__nonpos,axiom,
! [A: a] :
( ( ord_less_eq_a @ A @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% less_eq_neg_nonpos
thf(fact_110_neg__le__0__iff__le,axiom,
! [A: poly_a] :
( ( ord_less_eq_poly_a @ ( uminus_uminus_poly_a @ A ) @ zero_zero_poly_a )
= ( ord_less_eq_poly_a @ zero_zero_poly_a @ A ) ) ).
% neg_le_0_iff_le
thf(fact_111_neg__le__0__iff__le,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% neg_le_0_iff_le
thf(fact_112_neg__0__le__iff__le,axiom,
! [A: poly_a] :
( ( ord_less_eq_poly_a @ zero_zero_poly_a @ ( uminus_uminus_poly_a @ A ) )
= ( ord_less_eq_poly_a @ A @ zero_zero_poly_a ) ) ).
% neg_0_le_iff_le
thf(fact_113_neg__0__le__iff__le,axiom,
! [A: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% neg_0_le_iff_le
thf(fact_114_ab__left__minus,axiom,
! [A: poly_a] :
( ( plus_plus_poly_a @ ( uminus_uminus_poly_a @ A ) @ A )
= zero_zero_poly_a ) ).
% ab_left_minus
thf(fact_115_ab__left__minus,axiom,
! [A: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ A )
= zero_zero_a ) ).
% ab_left_minus
thf(fact_116_add_Oright__inverse,axiom,
! [A: poly_a] :
( ( plus_plus_poly_a @ A @ ( uminus_uminus_poly_a @ A ) )
= zero_zero_poly_a ) ).
% add.right_inverse
thf(fact_117_add_Oright__inverse,axiom,
! [A: a] :
( ( plus_plus_a @ A @ ( uminus_uminus_a @ A ) )
= zero_zero_a ) ).
% add.right_inverse
thf(fact_118_diff__0,axiom,
! [A: poly_a] :
( ( minus_minus_poly_a @ zero_zero_poly_a @ A )
= ( uminus_uminus_poly_a @ A ) ) ).
% diff_0
thf(fact_119_diff__0,axiom,
! [A: a] :
( ( minus_minus_a @ zero_zero_a @ A )
= ( uminus_uminus_a @ A ) ) ).
% diff_0
thf(fact_120_uminus__add__conv__diff,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ B )
= ( minus_minus_a @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_121_diff__minus__eq__add,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ A @ ( uminus_uminus_a @ B ) )
= ( plus_plus_a @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_122_equation__minus__iff,axiom,
! [A: a,B: a] :
( ( A
= ( uminus_uminus_a @ B ) )
= ( B
= ( uminus_uminus_a @ A ) ) ) ).
% equation_minus_iff
thf(fact_123_minus__equation__iff,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= B )
= ( ( uminus_uminus_a @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_124_le__imp__neg__le,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% le_imp_neg_le
thf(fact_125_minus__le__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ B )
= ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_126_le__minus__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ ( uminus_uminus_a @ B ) )
= ( ord_less_eq_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% le_minus_iff
thf(fact_127_add_Oinverse__distrib__swap,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
= ( plus_plus_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_128_group__cancel_Oneg1,axiom,
! [A2: a,K: a,A: a] :
( ( A2
= ( plus_plus_a @ K @ A ) )
=> ( ( uminus_uminus_a @ A2 )
= ( plus_plus_a @ ( uminus_uminus_a @ K ) @ ( uminus_uminus_a @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_129_minus__diff__commute,axiom,
! [B: a,A: a] :
( ( minus_minus_a @ ( uminus_uminus_a @ B ) @ A )
= ( minus_minus_a @ ( uminus_uminus_a @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_130_neg__eq__iff__add__eq__0,axiom,
! [A: poly_a,B: poly_a] :
( ( ( uminus_uminus_poly_a @ A )
= B )
= ( ( plus_plus_poly_a @ A @ B )
= zero_zero_poly_a ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_131_neg__eq__iff__add__eq__0,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= B )
= ( ( plus_plus_a @ A @ B )
= zero_zero_a ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_132_eq__neg__iff__add__eq__0,axiom,
! [A: poly_a,B: poly_a] :
( ( A
= ( uminus_uminus_poly_a @ B ) )
= ( ( plus_plus_poly_a @ A @ B )
= zero_zero_poly_a ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_133_eq__neg__iff__add__eq__0,axiom,
! [A: a,B: a] :
( ( A
= ( uminus_uminus_a @ B ) )
= ( ( plus_plus_a @ A @ B )
= zero_zero_a ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_134_add_Oinverse__unique,axiom,
! [A: poly_a,B: poly_a] :
( ( ( plus_plus_poly_a @ A @ B )
= zero_zero_poly_a )
=> ( ( uminus_uminus_poly_a @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_135_add_Oinverse__unique,axiom,
! [A: a,B: a] :
( ( ( plus_plus_a @ A @ B )
= zero_zero_a )
=> ( ( uminus_uminus_a @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_136_ab__group__add__class_Oab__left__minus,axiom,
! [A: poly_a] :
( ( plus_plus_poly_a @ ( uminus_uminus_poly_a @ A ) @ A )
= zero_zero_poly_a ) ).
% ab_group_add_class.ab_left_minus
thf(fact_137_ab__group__add__class_Oab__left__minus,axiom,
! [A: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ A )
= zero_zero_a ) ).
% ab_group_add_class.ab_left_minus
thf(fact_138_add__eq__0__iff,axiom,
! [A: poly_a,B: poly_a] :
( ( ( plus_plus_poly_a @ A @ B )
= zero_zero_poly_a )
= ( B
= ( uminus_uminus_poly_a @ A ) ) ) ).
% add_eq_0_iff
thf(fact_139_add__eq__0__iff,axiom,
! [A: a,B: a] :
( ( ( plus_plus_a @ A @ B )
= zero_zero_a )
= ( B
= ( uminus_uminus_a @ A ) ) ) ).
% add_eq_0_iff
thf(fact_140_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_a
= ( ^ [A3: a,B2: a] : ( plus_plus_a @ A3 @ ( uminus_uminus_a @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_141_diff__conv__add__uminus,axiom,
( minus_minus_a
= ( ^ [A3: a,B2: a] : ( plus_plus_a @ A3 @ ( uminus_uminus_a @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_142_group__cancel_Osub2,axiom,
! [B3: a,K: a,B: a,A: a] :
( ( B3
= ( plus_plus_a @ K @ B ) )
=> ( ( minus_minus_a @ A @ B3 )
= ( plus_plus_a @ ( uminus_uminus_a @ K ) @ ( minus_minus_a @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_143_zero__reorient,axiom,
! [X: poly_a] :
( ( zero_zero_poly_a = X )
= ( X = zero_zero_poly_a ) ) ).
% zero_reorient
thf(fact_144_zero__reorient,axiom,
! [X: a] :
( ( zero_zero_a = X )
= ( X = zero_zero_a ) ) ).
% zero_reorient
thf(fact_145_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_146_add__right__imp__eq,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_147_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_148_add__left__imp__eq,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_149_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_150_add_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ B @ ( plus_plus_set_nat @ A @ C ) )
= ( plus_plus_set_nat @ A @ ( plus_plus_set_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_151_add_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( plus_plus_set_a @ B @ ( plus_plus_set_a @ A @ C ) )
= ( plus_plus_set_a @ A @ ( plus_plus_set_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_152_add_Oleft__commute,axiom,
! [B: a,A: a,C: a] :
( ( plus_plus_a @ B @ ( plus_plus_a @ A @ C ) )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_153_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_154_add_Ocommute,axiom,
( plus_plus_set_nat
= ( ^ [A3: set_nat,B2: set_nat] : ( plus_plus_set_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_155_add_Ocommute,axiom,
( plus_plus_set_a
= ( ^ [A3: set_a,B2: set_a] : ( plus_plus_set_a @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_156_add_Ocommute,axiom,
( plus_plus_a
= ( ^ [A3: a,B2: a] : ( plus_plus_a @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_157_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_158_add_Oright__cancel,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_159_add_Oleft__cancel,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_160_add_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ ( plus_plus_set_nat @ A @ B ) @ C )
= ( plus_plus_set_nat @ A @ ( plus_plus_set_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_161_add_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( plus_plus_set_a @ ( plus_plus_set_a @ A @ B ) @ C )
= ( plus_plus_set_a @ A @ ( plus_plus_set_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_162_add_Oassoc,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_163_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_164_group__cancel_Oadd2,axiom,
! [B3: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B3
= ( plus_plus_set_nat @ K @ B ) )
=> ( ( plus_plus_set_nat @ A @ B3 )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_165_group__cancel_Oadd2,axiom,
! [B3: set_a,K: set_a,B: set_a,A: set_a] :
( ( B3
= ( plus_plus_set_a @ K @ B ) )
=> ( ( plus_plus_set_a @ A @ B3 )
= ( plus_plus_set_a @ K @ ( plus_plus_set_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_166_group__cancel_Oadd2,axiom,
! [B3: a,K: a,B: a,A: a] :
( ( B3
= ( plus_plus_a @ K @ B ) )
=> ( ( plus_plus_a @ A @ B3 )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_167_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_168_group__cancel_Oadd1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A2
= ( plus_plus_set_nat @ K @ A ) )
=> ( ( plus_plus_set_nat @ A2 @ B )
= ( plus_plus_set_nat @ K @ ( plus_plus_set_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_169_group__cancel_Oadd1,axiom,
! [A2: set_a,K: set_a,A: set_a,B: set_a] :
( ( A2
= ( plus_plus_set_a @ K @ A ) )
=> ( ( plus_plus_set_a @ A2 @ B )
= ( plus_plus_set_a @ K @ ( plus_plus_set_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_170_group__cancel_Oadd1,axiom,
! [A2: a,K: a,A: a,B: a] :
( ( A2
= ( plus_plus_a @ K @ A ) )
=> ( ( plus_plus_a @ A2 @ B )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_171_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_172_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_a @ I @ K )
= ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_173_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_174_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( plus_plus_set_nat @ ( plus_plus_set_nat @ A @ B ) @ C )
= ( plus_plus_set_nat @ A @ ( plus_plus_set_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( plus_plus_set_a @ ( plus_plus_set_a @ A @ B ) @ C )
= ( plus_plus_set_a @ A @ ( plus_plus_set_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_176_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_177_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_178_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: a,C: a,B: a] :
( ( minus_minus_a @ ( minus_minus_a @ A @ C ) @ B )
= ( minus_minus_a @ ( minus_minus_a @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_179_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_180_diff__eq__diff__eq,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_181_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_182_add_Ogroup__left__neutral,axiom,
! [A: poly_a] :
( ( plus_plus_poly_a @ zero_zero_poly_a @ A )
= A ) ).
% add.group_left_neutral
thf(fact_183_add_Ogroup__left__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add.group_left_neutral
thf(fact_184_add_Ocomm__neutral,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ A @ zero_zero_set_nat )
= A ) ).
% add.comm_neutral
thf(fact_185_add_Ocomm__neutral,axiom,
! [A: set_a] :
( ( plus_plus_set_a @ A @ zero_zero_set_a )
= A ) ).
% add.comm_neutral
thf(fact_186_add_Ocomm__neutral,axiom,
! [A: poly_a] :
( ( plus_plus_poly_a @ A @ zero_zero_poly_a )
= A ) ).
% add.comm_neutral
thf(fact_187_add_Ocomm__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.comm_neutral
thf(fact_188_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_189_comm__monoid__add__class_Oadd__0,axiom,
! [A: set_nat] :
( ( plus_plus_set_nat @ zero_zero_set_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_190_comm__monoid__add__class_Oadd__0,axiom,
! [A: set_a] :
( ( plus_plus_set_a @ zero_zero_set_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_191_comm__monoid__add__class_Oadd__0,axiom,
! [A: poly_a] :
( ( plus_plus_poly_a @ zero_zero_poly_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_192_comm__monoid__add__class_Oadd__0,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_193_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_194_add__le__imp__le__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
=> ( ord_less_eq_a @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_195_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_196_add__le__imp__le__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
=> ( ord_less_eq_a @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_197_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_198_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_199_add__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) ) ) ).
% add_right_mono
thf(fact_200_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_201_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_202_add__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) ) ) ).
% add_left_mono
thf(fact_203_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_204_add__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% add_mono
thf(fact_205_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_206_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_eq_a @ I @ J )
& ( ord_less_eq_a @ K @ L ) )
=> ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_207_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_208_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( I = J )
& ( ord_less_eq_a @ K @ L ) )
=> ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_209_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_210_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_eq_a @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_211_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_212_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: poly_a,Z: poly_a] : ( Y2 = Z ) )
= ( ^ [A3: poly_a,B2: poly_a] :
( ( minus_minus_poly_a @ A3 @ B2 )
= zero_zero_poly_a ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_213_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
= ( ^ [A3: a,B2: a] :
( ( minus_minus_a @ A3 @ B2 )
= zero_zero_a ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_214_diff__eq__diff__less__eq,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D ) )
=> ( ( ord_less_eq_a @ A @ B )
= ( ord_less_eq_a @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_215_diff__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_216_diff__left__mono,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ord_less_eq_a @ ( minus_minus_a @ C @ A ) @ ( minus_minus_a @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_217_diff__mono,axiom,
! [A: a,B: a,D: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ D @ C )
=> ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_218_diff__diff__eq,axiom,
! [A: a,B: a,C: a] :
( ( minus_minus_a @ ( minus_minus_a @ A @ B ) @ C )
= ( minus_minus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_219_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_220_add__implies__diff,axiom,
! [C: a,B: a,A: a] :
( ( ( plus_plus_a @ C @ B )
= A )
=> ( C
= ( minus_minus_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_221_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_222_diff__add__eq__diff__diff__swap,axiom,
! [A: a,B: a,C: a] :
( ( minus_minus_a @ A @ ( plus_plus_a @ B @ C ) )
= ( minus_minus_a @ ( minus_minus_a @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_223_diff__add__eq,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ C )
= ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_224_diff__diff__eq2,axiom,
! [A: a,B: a,C: a] :
( ( minus_minus_a @ A @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_225_add__diff__eq,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ A @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_226_eq__diff__eq,axiom,
! [A: a,C: a,B: a] :
( ( A
= ( minus_minus_a @ C @ B ) )
= ( ( plus_plus_a @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_227_diff__eq__eq,axiom,
! [A: a,B: a,C: a] :
( ( ( minus_minus_a @ A @ B )
= C )
= ( A
= ( plus_plus_a @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_228_group__cancel_Osub1,axiom,
! [A2: a,K: a,A: a,B: a] :
( ( A2
= ( plus_plus_a @ K @ A ) )
=> ( ( minus_minus_a @ A2 @ B )
= ( plus_plus_a @ K @ ( minus_minus_a @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_229_add__nonpos__eq__0__iff,axiom,
! [X: poly_a,Y: poly_a] :
( ( ord_less_eq_poly_a @ X @ zero_zero_poly_a )
=> ( ( ord_less_eq_poly_a @ Y @ zero_zero_poly_a )
=> ( ( ( plus_plus_poly_a @ X @ Y )
= zero_zero_poly_a )
= ( ( X = zero_zero_poly_a )
& ( Y = zero_zero_poly_a ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_230_add__nonpos__eq__0__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ zero_zero_a )
=> ( ( ord_less_eq_a @ Y @ zero_zero_a )
=> ( ( ( plus_plus_a @ X @ Y )
= zero_zero_a )
= ( ( X = zero_zero_a )
& ( Y = zero_zero_a ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_231_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_232_add__nonneg__eq__0__iff,axiom,
! [X: poly_a,Y: poly_a] :
( ( ord_less_eq_poly_a @ zero_zero_poly_a @ X )
=> ( ( ord_less_eq_poly_a @ zero_zero_poly_a @ Y )
=> ( ( ( plus_plus_poly_a @ X @ Y )
= zero_zero_poly_a )
= ( ( X = zero_zero_poly_a )
& ( Y = zero_zero_poly_a ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_233_add__nonneg__eq__0__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ( plus_plus_a @ X @ Y )
= zero_zero_a )
= ( ( X = zero_zero_a )
& ( Y = zero_zero_a ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_234_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_235_add__nonpos__nonpos,axiom,
! [A: poly_a,B: poly_a] :
( ( ord_less_eq_poly_a @ A @ zero_zero_poly_a )
=> ( ( ord_less_eq_poly_a @ B @ zero_zero_poly_a )
=> ( ord_less_eq_poly_a @ ( plus_plus_poly_a @ A @ B ) @ zero_zero_poly_a ) ) ) ).
% add_nonpos_nonpos
thf(fact_236_add__nonpos__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_nonpos_nonpos
thf(fact_237_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_238_add__nonneg__nonneg,axiom,
! [A: poly_a,B: poly_a] :
( ( ord_less_eq_poly_a @ zero_zero_poly_a @ A )
=> ( ( ord_less_eq_poly_a @ zero_zero_poly_a @ B )
=> ( ord_less_eq_poly_a @ zero_zero_poly_a @ ( plus_plus_poly_a @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_239_add__nonneg__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_240_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_241_add__increasing2,axiom,
! [C: poly_a,B: poly_a,A: poly_a] :
( ( ord_less_eq_poly_a @ zero_zero_poly_a @ C )
=> ( ( ord_less_eq_poly_a @ B @ A )
=> ( ord_less_eq_poly_a @ B @ ( plus_plus_poly_a @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_242_add__increasing2,axiom,
! [C: a,B: a,A: a] :
( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ( ord_less_eq_a @ B @ A )
=> ( ord_less_eq_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_243_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_244_add__decreasing2,axiom,
! [C: poly_a,A: poly_a,B: poly_a] :
( ( ord_less_eq_poly_a @ C @ zero_zero_poly_a )
=> ( ( ord_less_eq_poly_a @ A @ B )
=> ( ord_less_eq_poly_a @ ( plus_plus_poly_a @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_245_add__decreasing2,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_246_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_247_add__increasing,axiom,
! [A: poly_a,B: poly_a,C: poly_a] :
( ( ord_less_eq_poly_a @ zero_zero_poly_a @ A )
=> ( ( ord_less_eq_poly_a @ B @ C )
=> ( ord_less_eq_poly_a @ B @ ( plus_plus_poly_a @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_248_add__increasing,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_249_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_250_add__decreasing,axiom,
! [A: poly_a,C: poly_a,B: poly_a] :
( ( ord_less_eq_poly_a @ A @ zero_zero_poly_a )
=> ( ( ord_less_eq_poly_a @ C @ B )
=> ( ord_less_eq_poly_a @ ( plus_plus_poly_a @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_251_add__decreasing,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_252_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_253_le__iff__diff__le__0,axiom,
( ord_less_eq_poly_a
= ( ^ [A3: poly_a,B2: poly_a] : ( ord_less_eq_poly_a @ ( minus_minus_poly_a @ A3 @ B2 ) @ zero_zero_poly_a ) ) ) ).
% le_iff_diff_le_0
thf(fact_254_le__iff__diff__le__0,axiom,
( ord_less_eq_a
= ( ^ [A3: a,B2: a] : ( ord_less_eq_a @ ( minus_minus_a @ A3 @ B2 ) @ zero_zero_a ) ) ) ).
% le_iff_diff_le_0
thf(fact_255_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_256_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_257_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_258_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_259_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_260_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_261_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_262_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_263_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_264_add__le__add__imp__diff__le,axiom,
! [I: a,K: a,N: a,J: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ N )
=> ( ( ord_less_eq_a @ N @ ( plus_plus_a @ J @ K ) )
=> ( ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ N )
=> ( ( ord_less_eq_a @ N @ ( plus_plus_a @ J @ K ) )
=> ( ord_less_eq_a @ ( minus_minus_a @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_265_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_266_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_267_add__le__imp__le__diff,axiom,
! [I: a,K: a,N: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ N )
=> ( ord_less_eq_a @ I @ ( minus_minus_a @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_268_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_269_le__diff__eq,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ A @ ( minus_minus_a @ C @ B ) )
= ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_270_diff__le__eq,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ ( minus_minus_a @ A @ B ) @ C )
= ( ord_less_eq_a @ A @ ( plus_plus_a @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_271_verit__minus__simplify_I3_J,axiom,
! [B: poly_a] :
( ( minus_minus_poly_a @ zero_zero_poly_a @ B )
= ( uminus_uminus_poly_a @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_272_verit__minus__simplify_I3_J,axiom,
! [B: a] :
( ( minus_minus_a @ zero_zero_a @ B )
= ( uminus_uminus_a @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_273_ulvbc,axiom,
( ( scalar_prod_a @ ulv @ bc )
= ( plus_plus_a @ ( scalar_prod_a @ u @ b ) @ ( scalar_prod_a @ c @ ( minus_minus_vec_a @ v @ w ) ) ) ) ).
% ulvbc
thf(fact_274__092_060open_062ulv_A_092_060bullet_062_Abc_A_061_Au_A_092_060bullet_062_Ab_A_L_A_Iv_A_092_060bullet_062_Ac_A_L_Aw_A_092_060bullet_062_A_N_Ac_J_092_060close_062,axiom,
( ( scalar_prod_a @ ulv @ bc )
= ( plus_plus_a @ ( scalar_prod_a @ u @ b ) @ ( plus_plus_a @ ( scalar_prod_a @ v @ c ) @ ( scalar_prod_a @ w @ ( uminus_uminus_vec_a @ c ) ) ) ) ) ).
% \<open>ulv \<bullet> bc = u \<bullet> b + (v \<bullet> c + w \<bullet> - c)\<close>
thf(fact_275_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_276_compl__le__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_277_class__ring_Ominus__zero,axiom,
( ( uminus_uminus_poly_a @ zero_zero_poly_a )
= zero_zero_poly_a ) ).
% class_ring.minus_zero
thf(fact_278_class__ring_Ominus__zero,axiom,
( ( uminus_uminus_a @ zero_zero_a )
= zero_zero_a ) ).
% class_ring.minus_zero
thf(fact_279_double__eq__0__iff,axiom,
! [A: poly_a] :
( ( ( plus_plus_poly_a @ A @ A )
= zero_zero_poly_a )
= ( A = zero_zero_poly_a ) ) ).
% double_eq_0_iff
thf(fact_280_double__eq__0__iff,axiom,
! [A: a] :
( ( ( plus_plus_a @ A @ A )
= zero_zero_a )
= ( A = zero_zero_a ) ) ).
% double_eq_0_iff
thf(fact_281_scalar__prod__ge__0,axiom,
! [X: vec_poly_a] : ( ord_less_eq_poly_a @ zero_zero_poly_a @ ( scalar_prod_poly_a @ X @ X ) ) ).
% scalar_prod_ge_0
thf(fact_282_scalar__prod__ge__0,axiom,
! [X: vec_a] : ( ord_less_eq_a @ zero_zero_a @ ( scalar_prod_a @ X @ X ) ) ).
% scalar_prod_ge_0
thf(fact_283_uminus__eq__vec,axiom,
! [V: vec_a,W: vec_a] :
( ( ( uminus_uminus_vec_a @ V )
= ( uminus_uminus_vec_a @ W ) )
= ( V = W ) ) ).
% uminus_eq_vec
thf(fact_284_uminus__uminus__vec,axiom,
! [V: vec_a] :
( ( uminus_uminus_vec_a @ ( uminus_uminus_vec_a @ V ) )
= V ) ).
% uminus_uminus_vec
thf(fact_285_class__ring_Ominus__eq,axiom,
( minus_minus_a
= ( ^ [X2: a,Y3: a] : ( plus_plus_a @ X2 @ ( uminus_uminus_a @ Y3 ) ) ) ) ).
% class_ring.minus_eq
thf(fact_286_verit__minus__simplify_I4_J,axiom,
! [B: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_287_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_288_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_289_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_290_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_291_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_292_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_293_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_294_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_295_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: a,B: a,C: a,D: a] :
( ( A = B )
=> ( ( C = D )
=> ( ( plus_plus_a @ A @ C )
= ( plus_plus_a @ B @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_296_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( A = B )
=> ( ( C = D )
=> ( ( plus_plus_nat @ A @ C )
= ( plus_plus_nat @ B @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_297_verit__negate__coefficient_I3_J,axiom,
! [A: a,B: a] :
( ( A = B )
=> ( ( uminus_uminus_a @ A )
= ( uminus_uminus_a @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_298_verit__sum__simplify,axiom,
! [A: poly_a] :
( ( plus_plus_poly_a @ A @ zero_zero_poly_a )
= A ) ).
% verit_sum_simplify
thf(fact_299_verit__sum__simplify,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% verit_sum_simplify
thf(fact_300_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_301_class__cring_Ocring__simprules_I22_J,axiom,
( ( uminus_uminus_poly_a @ zero_zero_poly_a )
= zero_zero_poly_a ) ).
% class_cring.cring_simprules(22)
thf(fact_302_class__cring_Ocring__simprules_I22_J,axiom,
( ( uminus_uminus_a @ zero_zero_a )
= zero_zero_a ) ).
% class_cring.cring_simprules(22)
thf(fact_303_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% compl_mono
thf(fact_304_compl__mono,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X ) ) ) ).
% compl_mono
thf(fact_305_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
=> ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_306_compl__le__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_307_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_308_compl__le__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_309_set__plus__intro,axiom,
! [A: set_nat,C4: set_set_nat,B: set_nat,D2: set_set_nat] :
( ( member_set_nat @ A @ C4 )
=> ( ( member_set_nat @ B @ D2 )
=> ( member_set_nat @ ( plus_plus_set_nat @ A @ B ) @ ( plus_p4817606893110106565et_nat @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_310_set__plus__intro,axiom,
! [A: set_num,C4: set_set_num,B: set_num,D2: set_set_num] :
( ( member_set_num @ A @ C4 )
=> ( ( member_set_num @ B @ D2 )
=> ( member_set_num @ ( plus_plus_set_num @ A @ B ) @ ( plus_p532826482549453327et_num @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_311_set__plus__intro,axiom,
! [A: set_a,C4: set_set_a,B: set_a,D2: set_set_a] :
( ( member_set_a @ A @ C4 )
=> ( ( member_set_a @ B @ D2 )
=> ( member_set_a @ ( plus_plus_set_a @ A @ B ) @ ( plus_plus_set_set_a @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_312_set__plus__intro,axiom,
! [A: a,C4: set_a,B: a,D2: set_a] :
( ( member_a @ A @ C4 )
=> ( ( member_a @ B @ D2 )
=> ( member_a @ ( plus_plus_a @ A @ B ) @ ( plus_plus_set_a @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_313_set__plus__intro,axiom,
! [A: num,C4: set_num,B: num,D2: set_num] :
( ( member_num @ A @ C4 )
=> ( ( member_num @ B @ D2 )
=> ( member_num @ ( plus_plus_num @ A @ B ) @ ( plus_plus_set_num @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_314_set__plus__intro,axiom,
! [A: nat,C4: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A @ C4 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_315_norm1__ge__0,axiom,
! [F: poly_poly_a] : ( ord_less_eq_poly_a @ zero_zero_poly_a @ ( norm1_poly_a @ F ) ) ).
% norm1_ge_0
thf(fact_316_norm1__ge__0,axiom,
! [F: poly_a] : ( ord_less_eq_a @ zero_zero_a @ ( norm1_a @ F ) ) ).
% norm1_ge_0
thf(fact_317_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_318_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_319_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_320_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_321_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_322_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_323_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_324_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_325_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_326_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_327_minus__diff__minus,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_a @ B ) )
= ( uminus_uminus_a @ ( minus_minus_a @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_328_is__num__normalize_I8_J,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
= ( plus_plus_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_329_add__diff__add,axiom,
! [A: a,C: a,B: a,D: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) )
= ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ ( minus_minus_a @ C @ D ) ) ) ).
% add_diff_add
thf(fact_330_set__plus__mono2,axiom,
! [C4: set_num,D2: set_num,E: set_num,F2: set_num] :
( ( ord_less_eq_set_num @ C4 @ D2 )
=> ( ( ord_less_eq_set_num @ E @ F2 )
=> ( ord_less_eq_set_num @ ( plus_plus_set_num @ C4 @ E ) @ ( plus_plus_set_num @ D2 @ F2 ) ) ) ) ).
% set_plus_mono2
thf(fact_331_set__plus__mono2,axiom,
! [C4: set_nat,D2: set_nat,E: set_nat,F2: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ D2 )
=> ( ( ord_less_eq_set_nat @ E @ F2 )
=> ( ord_less_eq_set_nat @ ( plus_plus_set_nat @ C4 @ E ) @ ( plus_plus_set_nat @ D2 @ F2 ) ) ) ) ).
% set_plus_mono2
thf(fact_332_set__plus__mono2,axiom,
! [C4: set_a,D2: set_a,E: set_a,F2: set_a] :
( ( ord_less_eq_set_a @ C4 @ D2 )
=> ( ( ord_less_eq_set_a @ E @ F2 )
=> ( ord_less_eq_set_a @ ( plus_plus_set_a @ C4 @ E ) @ ( plus_plus_set_a @ D2 @ F2 ) ) ) ) ).
% set_plus_mono2
thf(fact_333_set__zero__plus2,axiom,
! [A2: set_poly_a,B3: set_poly_a] :
( ( member_poly_a @ zero_zero_poly_a @ A2 )
=> ( ord_le1582614204914470028poly_a @ B3 @ ( plus_plus_set_poly_a @ A2 @ B3 ) ) ) ).
% set_zero_plus2
thf(fact_334_set__zero__plus2,axiom,
! [A2: set_a,B3: set_a] :
( ( member_a @ zero_zero_a @ A2 )
=> ( ord_less_eq_set_a @ B3 @ ( plus_plus_set_a @ A2 @ B3 ) ) ) ).
% set_zero_plus2
thf(fact_335_set__zero__plus2,axiom,
! [A2: set_nat,B3: set_nat] :
( ( member_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_set_nat @ B3 @ ( plus_plus_set_nat @ A2 @ B3 ) ) ) ).
% set_zero_plus2
thf(fact_336_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_337_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_338_order__antisym__conv,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_339_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_340_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_341_linorder__le__cases,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_342_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_343_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_344_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_345_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > num,C: num] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_346_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_347_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > a,C: a] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_348_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_349_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_350_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_351_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_352_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_353_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_354_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_355_ord__eq__le__subst,axiom,
! [A: num,F: a > num,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_356_ord__eq__le__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_357_ord__eq__le__subst,axiom,
! [A: a,F: num > a,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_358_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_359_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_360_ord__eq__le__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_361_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_362_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_363_ord__eq__le__subst,axiom,
! [A: set_nat,F: a > set_nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_364_linorder__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_linear
thf(fact_365_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_366_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_367_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_368_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_369_order__eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_370_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_371_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_372_order__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_373_order__subst2,axiom,
! [A: a,B: a,F: a > num,C: num] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_374_order__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_375_order__subst2,axiom,
! [A: num,B: num,F: num > a,C: a] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_376_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_377_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_378_order__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_379_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_380_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_381_order__subst2,axiom,
! [A: a,B: a,F: a > set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_382_order__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_383_order__subst1,axiom,
! [A: a,F: num > a,B: num,C: num] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_384_order__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_385_order__subst1,axiom,
! [A: num,F: a > num,B: a,C: a] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_386_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_387_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_388_order__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_389_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_390_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_391_order__subst1,axiom,
! [A: a,F: set_nat > a,B: set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_392_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_393_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_394_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
= ( ^ [A3: a,B2: a] :
( ( ord_less_eq_a @ A3 @ B2 )
& ( ord_less_eq_a @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_395_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: num,Z: num] : ( Y2 = Z ) )
= ( ^ [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
& ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_396_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_397_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_398_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_399_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_400_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_401_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_402_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_403_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_404_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_405_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_406_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_407_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_408_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_409_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_410_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_411_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_412_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_413_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_414_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
= ( ^ [A3: a,B2: a] :
( ( ord_less_eq_a @ B2 @ A3 )
& ( ord_less_eq_a @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_415_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: num,Z: num] : ( Y2 = Z ) )
= ( ^ [A3: num,B2: num] :
( ( ord_less_eq_num @ B2 @ A3 )
& ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_416_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_417_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: a,B4: a] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_418_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B4: num] :
( ( ord_less_eq_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_419_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_420_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_421_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_422_order__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z2 )
=> ( ord_less_eq_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_423_order__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_eq_num @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_424_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_425_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_426_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_427_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_428_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_429_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_430_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_431_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_432_order__antisym,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_433_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_434_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_435_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_436_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_437_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_438_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_439_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_440_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_441_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_442_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_443_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_444_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_445_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_446_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_447_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
= ( ^ [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
& ( ord_less_eq_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_448_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: num,Z: num] : ( Y2 = Z ) )
= ( ^ [X2: num,Y3: num] :
( ( ord_less_eq_num @ X2 @ Y3 )
& ( ord_less_eq_num @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_449_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_450_le__cases3,axiom,
! [X: a,Y: a,Z2: a] :
( ( ( ord_less_eq_a @ X @ Y )
=> ~ ( ord_less_eq_a @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_eq_a @ X @ Z2 ) )
=> ( ( ( ord_less_eq_a @ X @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_a @ Z2 @ Y )
=> ~ ( ord_less_eq_a @ Y @ X ) )
=> ( ( ( ord_less_eq_a @ Y @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_a @ Z2 @ X )
=> ~ ( ord_less_eq_a @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_451_le__cases3,axiom,
! [X: num,Y: num,Z2: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_452_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_453_nle__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_eq_a @ A @ B ) )
= ( ( ord_less_eq_a @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_454_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_455_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_456_set__plus__elim,axiom,
! [X: set_num,A2: set_set_num,B3: set_set_num] :
( ( member_set_num @ X @ ( plus_p532826482549453327et_num @ A2 @ B3 ) )
=> ~ ! [A4: set_num,B4: set_num] :
( ( X
= ( plus_plus_set_num @ A4 @ B4 ) )
=> ( ( member_set_num @ A4 @ A2 )
=> ~ ( member_set_num @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_457_set__plus__elim,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ X @ ( plus_plus_set_set_a @ A2 @ B3 ) )
=> ~ ! [A4: set_a,B4: set_a] :
( ( X
= ( plus_plus_set_a @ A4 @ B4 ) )
=> ( ( member_set_a @ A4 @ A2 )
=> ~ ( member_set_a @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_458_set__plus__elim,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ( member_a @ X @ ( plus_plus_set_a @ A2 @ B3 ) )
=> ~ ! [A4: a,B4: a] :
( ( X
= ( plus_plus_a @ A4 @ B4 ) )
=> ( ( member_a @ A4 @ A2 )
=> ~ ( member_a @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_459_set__plus__elim,axiom,
! [X: num,A2: set_num,B3: set_num] :
( ( member_num @ X @ ( plus_plus_set_num @ A2 @ B3 ) )
=> ~ ! [A4: num,B4: num] :
( ( X
= ( plus_plus_num @ A4 @ B4 ) )
=> ( ( member_num @ A4 @ A2 )
=> ~ ( member_num @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_460_set__plus__elim,axiom,
! [X: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A2 @ B3 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X
= ( plus_plus_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A2 )
=> ~ ( member_nat @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_461_is__num__normalize_I1_J,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_462_le__numeral__extra_I3_J,axiom,
ord_less_eq_a @ zero_zero_a @ zero_zero_a ).
% le_numeral_extra(3)
thf(fact_463_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_464_add__0__iff,axiom,
! [B: a,A: a] :
( ( B
= ( plus_plus_a @ B @ A ) )
= ( A = zero_zero_a ) ) ).
% add_0_iff
thf(fact_465_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_466_sq__norm__poly__ge__0,axiom,
! [P2: poly_a] : ( ord_less_eq_a @ zero_zero_a @ ( sq_norm_poly_a @ P2 ) ) ).
% sq_norm_poly_ge_0
thf(fact_467_linf__norm__poly__ge__0,axiom,
! [F: poly_a] : ( ord_less_eq_a @ zero_zero_a @ ( linf_norm_poly_a @ F ) ) ).
% linf_norm_poly_ge_0
thf(fact_468_add_Ogroup__axioms,axiom,
group_a @ plus_plus_a @ zero_zero_a @ uminus_uminus_a ).
% add.group_axioms
thf(fact_469_linf__norm__vec__ge__0,axiom,
! [V: vec_a] : ( ord_less_eq_a @ zero_zero_a @ ( linf_norm_vec_a @ V ) ) ).
% linf_norm_vec_ge_0
thf(fact_470_diff__numeral__special_I12_J,axiom,
( ( minus_minus_a @ ( uminus_uminus_a @ one_one_a ) @ ( uminus_uminus_a @ one_one_a ) )
= zero_zero_a ) ).
% diff_numeral_special(12)
thf(fact_471_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_a @ one_one_a @ ( uminus_uminus_a @ one_one_a ) )
= zero_zero_a ) ).
% add_neg_numeral_special(7)
thf(fact_472_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_a @ ( uminus_uminus_a @ one_one_a ) @ one_one_a )
= zero_zero_a ) ).
% add_neg_numeral_special(8)
thf(fact_473_linf__norm__poly__0,axiom,
( ( linf_norm_poly_a @ zero_zero_poly_a )
= zero_zero_a ) ).
% linf_norm_poly_0
thf(fact_474_linf__norm__poly__eq__0,axiom,
! [F: poly_a] :
( ( ( linf_norm_poly_a @ F )
= zero_zero_a )
= ( F = zero_zero_poly_a ) ) ).
% linf_norm_poly_eq_0
thf(fact_475_sq__norm__poly__0,axiom,
( ( sq_norm_poly_a @ zero_zero_poly_a )
= zero_zero_a ) ).
% sq_norm_poly_0
thf(fact_476_sq__norm__poly__eq__0,axiom,
! [P2: poly_a] :
( ( ( sq_norm_poly_a @ P2 )
= zero_zero_a )
= ( P2 = zero_zero_poly_a ) ) ).
% sq_norm_poly_eq_0
thf(fact_477_diff__numeral__special_I9_J,axiom,
( ( minus_minus_a @ one_one_a @ one_one_a )
= zero_zero_a ) ).
% diff_numeral_special(9)
thf(fact_478_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_479_class__field_Ozero__not__one,axiom,
zero_zero_a != one_one_a ).
% class_field.zero_not_one
thf(fact_480_zero__neq__one,axiom,
zero_zero_a != one_one_a ).
% zero_neq_one
thf(fact_481_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_482_le__numeral__extra_I4_J,axiom,
ord_less_eq_a @ one_one_a @ one_one_a ).
% le_numeral_extra(4)
thf(fact_483_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_484_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_a @ zero_zero_a @ one_one_a ).
% zero_less_one_class.zero_le_one
thf(fact_485_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_486_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_a @ zero_zero_a @ one_one_a ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_487_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_488_not__one__le__zero,axiom,
~ ( ord_less_eq_a @ one_one_a @ zero_zero_a ) ).
% not_one_le_zero
thf(fact_489_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_490_class__field_Oneg__1__not__0,axiom,
( ( uminus_uminus_a @ one_one_a )
!= zero_zero_a ) ).
% class_field.neg_1_not_0
thf(fact_491_zero__neq__neg__one,axiom,
( zero_zero_a
!= ( uminus_uminus_a @ one_one_a ) ) ).
% zero_neq_neg_one
thf(fact_492_le__minus__one__simps_I2_J,axiom,
ord_less_eq_a @ ( uminus_uminus_a @ one_one_a ) @ one_one_a ).
% le_minus_one_simps(2)
thf(fact_493_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_a @ one_one_a @ ( uminus_uminus_a @ one_one_a ) ) ).
% le_minus_one_simps(4)
thf(fact_494_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ one_one_a ) ) ).
% le_minus_one_simps(3)
thf(fact_495_le__minus__one__simps_I1_J,axiom,
ord_less_eq_a @ ( uminus_uminus_a @ one_one_a ) @ zero_zero_a ).
% le_minus_one_simps(1)
thf(fact_496_dbl__dec__simps_I2_J,axiom,
( ( neg_nu181380926503873385_dec_a @ zero_zero_a )
= ( uminus_uminus_a @ one_one_a ) ) ).
% dbl_dec_simps(2)
thf(fact_497_dbl__inc__simps_I2_J,axiom,
( ( neg_nu6917059380386235053_inc_a @ zero_zero_a )
= one_one_a ) ).
% dbl_inc_simps(2)
thf(fact_498_dbl__inc__def,axiom,
( neg_nu6917059380386235053_inc_a
= ( ^ [X2: a] : ( plus_plus_a @ ( plus_plus_a @ X2 @ X2 ) @ one_one_a ) ) ) ).
% dbl_inc_def
thf(fact_499_dbl__dec__def,axiom,
( neg_nu181380926503873385_dec_a
= ( ^ [X2: a] : ( minus_minus_a @ ( plus_plus_a @ X2 @ X2 ) @ one_one_a ) ) ) ).
% dbl_dec_def
thf(fact_500_sq__norm__poly__pos,axiom,
! [P2: poly_a] :
( ( ord_less_a @ zero_zero_a @ ( sq_norm_poly_a @ P2 ) )
= ( P2 != zero_zero_poly_a ) ) ).
% sq_norm_poly_pos
thf(fact_501_linf__norm__poly__greater__0,axiom,
! [F: poly_a] :
( ( ord_less_a @ zero_zero_a @ ( linf_norm_poly_a @ F ) )
= ( F != zero_zero_poly_a ) ) ).
% linf_norm_poly_greater_0
thf(fact_502_convex__bound__le,axiom,
! [X: a,A: a,Y: a,U: a,V: a] :
( ( ord_less_eq_a @ X @ A )
=> ( ( ord_less_eq_a @ Y @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ U )
=> ( ( ord_less_eq_a @ zero_zero_a @ V )
=> ( ( ( plus_plus_a @ U @ V )
= one_one_a )
=> ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ U @ X ) @ ( times_times_a @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_503_is__num_Ocases,axiom,
! [A: a] :
( ( neg_numeral_is_num_a @ A )
=> ( ( A != one_one_a )
=> ( ! [X3: a] :
( ( A
= ( uminus_uminus_a @ X3 ) )
=> ~ ( neg_numeral_is_num_a @ X3 ) )
=> ~ ! [X3: a,Y4: a] :
( ( A
= ( plus_plus_a @ X3 @ Y4 ) )
=> ( ( neg_numeral_is_num_a @ X3 )
=> ~ ( neg_numeral_is_num_a @ Y4 ) ) ) ) ) ) ).
% is_num.cases
thf(fact_504_set__times__intro,axiom,
! [A: nat,C4: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A @ C4 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C4 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_505_set__times__intro,axiom,
! [A: num,C4: set_num,B: num,D2: set_num] :
( ( member_num @ A @ C4 )
=> ( ( member_num @ B @ D2 )
=> ( member_num @ ( times_times_num @ A @ B ) @ ( times_times_set_num @ C4 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_506_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_507_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_508_mult__zero__left,axiom,
! [A: a] :
( ( times_times_a @ zero_zero_a @ A )
= zero_zero_a ) ).
% mult_zero_left
thf(fact_509_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_510_mult__zero__right,axiom,
! [A: a] :
( ( times_times_a @ A @ zero_zero_a )
= zero_zero_a ) ).
% mult_zero_right
thf(fact_511_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_512_mult__eq__0__iff,axiom,
! [A: a,B: a] :
( ( ( times_times_a @ A @ B )
= zero_zero_a )
= ( ( A = zero_zero_a )
| ( B = zero_zero_a ) ) ) ).
% mult_eq_0_iff
thf(fact_513_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_514_mult__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ( times_times_a @ C @ A )
= ( times_times_a @ C @ B ) )
= ( ( C = zero_zero_a )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_515_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_516_mult__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ( times_times_a @ A @ C )
= ( times_times_a @ B @ C ) )
= ( ( C = zero_zero_a )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_517_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_518_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_a @ ( numeral_numeral_a @ M ) @ ( numeral_numeral_a @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_519_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_520_add__less__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( ord_less_a @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_521_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_522_add__less__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( ord_less_a @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_523_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_524_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_525_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_526_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_527_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_528_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_529_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_a @ ( numeral_numeral_a @ M ) @ ( numeral_numeral_a @ N ) )
= ( numeral_numeral_a @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_530_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_531_add__numeral__left,axiom,
! [V: num,W: num,Z2: a] :
( ( plus_plus_a @ ( numeral_numeral_a @ V ) @ ( plus_plus_a @ ( numeral_numeral_a @ W ) @ Z2 ) )
= ( plus_plus_a @ ( numeral_numeral_a @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_532_add__numeral__left,axiom,
! [V: num,W: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_533_add__less__same__cancel1,axiom,
! [B: a,A: a] :
( ( ord_less_a @ ( plus_plus_a @ B @ A ) @ B )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% add_less_same_cancel1
thf(fact_534_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_535_add__less__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_a @ ( plus_plus_a @ A @ B ) @ B )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% add_less_same_cancel2
thf(fact_536_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_537_less__add__same__cancel1,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ ( plus_plus_a @ A @ B ) )
= ( ord_less_a @ zero_zero_a @ B ) ) ).
% less_add_same_cancel1
thf(fact_538_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_539_less__add__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ ( plus_plus_a @ B @ A ) )
= ( ord_less_a @ zero_zero_a @ B ) ) ).
% less_add_same_cancel2
thf(fact_540_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_541_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: a] :
( ( ord_less_a @ ( plus_plus_a @ A @ A ) @ zero_zero_a )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_542_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ A ) )
= ( ord_less_a @ zero_zero_a @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_543_mult__cancel__left1,axiom,
! [C: a,B: a] :
( ( C
= ( times_times_a @ C @ B ) )
= ( ( C = zero_zero_a )
| ( B = one_one_a ) ) ) ).
% mult_cancel_left1
thf(fact_544_mult__cancel__left2,axiom,
! [C: a,A: a] :
( ( ( times_times_a @ C @ A )
= C )
= ( ( C = zero_zero_a )
| ( A = one_one_a ) ) ) ).
% mult_cancel_left2
thf(fact_545_mult__cancel__right1,axiom,
! [C: a,B: a] :
( ( C
= ( times_times_a @ B @ C ) )
= ( ( C = zero_zero_a )
| ( B = one_one_a ) ) ) ).
% mult_cancel_right1
thf(fact_546_mult__cancel__right2,axiom,
! [A: a,C: a] :
( ( ( times_times_a @ A @ C )
= C )
= ( ( C = zero_zero_a )
| ( A = one_one_a ) ) ) ).
% mult_cancel_right2
thf(fact_547_diff__gt__0__iff__gt,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ ( minus_minus_a @ A @ B ) )
= ( ord_less_a @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_548_neg__less__0__iff__less,axiom,
! [A: a] :
( ( ord_less_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
= ( ord_less_a @ zero_zero_a @ A ) ) ).
% neg_less_0_iff_less
thf(fact_549_neg__0__less__iff__less,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% neg_0_less_iff_less
thf(fact_550_neg__less__pos,axiom,
! [A: a] :
( ( ord_less_a @ ( uminus_uminus_a @ A ) @ A )
= ( ord_less_a @ zero_zero_a @ A ) ) ).
% neg_less_pos
thf(fact_551_less__neg__neg,axiom,
! [A: a] :
( ( ord_less_a @ A @ ( uminus_uminus_a @ A ) )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% less_neg_neg
thf(fact_552_distrib__right__numeral,axiom,
! [A: a,B: a,V: num] :
( ( times_times_a @ ( plus_plus_a @ A @ B ) @ ( numeral_numeral_a @ V ) )
= ( plus_plus_a @ ( times_times_a @ A @ ( numeral_numeral_a @ V ) ) @ ( times_times_a @ B @ ( numeral_numeral_a @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_553_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_554_distrib__left__numeral,axiom,
! [V: num,B: a,C: a] :
( ( times_times_a @ ( numeral_numeral_a @ V ) @ ( plus_plus_a @ B @ C ) )
= ( plus_plus_a @ ( times_times_a @ ( numeral_numeral_a @ V ) @ B ) @ ( times_times_a @ ( numeral_numeral_a @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_555_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_556_left__diff__distrib__numeral,axiom,
! [A: a,B: a,V: num] :
( ( times_times_a @ ( minus_minus_a @ A @ B ) @ ( numeral_numeral_a @ V ) )
= ( minus_minus_a @ ( times_times_a @ A @ ( numeral_numeral_a @ V ) ) @ ( times_times_a @ B @ ( numeral_numeral_a @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_557_right__diff__distrib__numeral,axiom,
! [V: num,B: a,C: a] :
( ( times_times_a @ ( numeral_numeral_a @ V ) @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( times_times_a @ ( numeral_numeral_a @ V ) @ B ) @ ( times_times_a @ ( numeral_numeral_a @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_558_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) @ ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_559_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) @ ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) )
= ( uminus_uminus_a @ ( plus_plus_a @ ( numeral_numeral_a @ M ) @ ( numeral_numeral_a @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_560_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_a @ ( numeral_numeral_a @ M ) @ ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) )
= ( numeral_numeral_a @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_561_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) @ ( numeral_numeral_a @ N ) )
= ( uminus_uminus_a @ ( numeral_numeral_a @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_562_semiring__real__line__class_Omult__neg__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% semiring_real_line_class.mult_neg_neg
thf(fact_563_semiring__real__line__class_Omult__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% semiring_real_line_class.mult_neg_neg
thf(fact_564_real__mult__eq__0__iff,axiom,
! [A: a,B: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ( times_times_a @ A @ B )
= zero_zero_a )
= ( ( A = zero_zero_a )
| ( B = zero_zero_a ) ) ) ) ) ).
% real_mult_eq_0_iff
thf(fact_565_real__mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ) ) ).
% real_mult_eq_0_iff
thf(fact_566_real__mult__less__cancel__left__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_a @ A @ B ) ) ) ) ) ).
% real_mult_less_cancel_left_pos
thf(fact_567_real__mult__less__cancel__left__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ) ) ) ).
% real_mult_less_cancel_left_pos
thf(fact_568_real__mult__less__cancel__right__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( ord_less_a @ A @ B ) ) ) ) ) ).
% real_mult_less_cancel_right_pos
thf(fact_569_real__mult__less__cancel__right__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ) ) ) ).
% real_mult_less_cancel_right_pos
thf(fact_570_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_571_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_572_mult__less__cancel__right__disj,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
& ( ord_less_a @ A @ B ) )
| ( ( ord_less_a @ C @ zero_zero_a )
& ( ord_less_a @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_573_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_574_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_575_mult__strict__right__mono__neg,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_576_mult__less__cancel__left__disj,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
& ( ord_less_a @ A @ B ) )
| ( ( ord_less_a @ C @ zero_zero_a )
& ( ord_less_a @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_577_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_578_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_579_mult__strict__left__mono__neg,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_580_mult__less__cancel__left__pos,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_a @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_581_mult__less__cancel__left__neg,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ C @ zero_zero_a )
=> ( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_a @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_582_zero__less__mult__pos2,axiom,
! [B: a,A: a] :
( ( ord_less_a @ zero_zero_a @ ( times_times_a @ B @ A ) )
=> ( ( ord_less_a @ zero_zero_a @ A )
=> ( ord_less_a @ zero_zero_a @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_583_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_584_zero__less__mult__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ A )
=> ( ord_less_a @ zero_zero_a @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_585_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_586_zero__less__mult__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) )
= ( ( ( ord_less_a @ zero_zero_a @ A )
& ( ord_less_a @ zero_zero_a @ B ) )
| ( ( ord_less_a @ A @ zero_zero_a )
& ( ord_less_a @ B @ zero_zero_a ) ) ) ) ).
% zero_less_mult_iff
thf(fact_587_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( times_times_a @ B @ A ) @ zero_zero_a ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_588_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_589_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_590_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_591_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_592_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_593_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_594_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_595_mult__less__0__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ ( times_times_a @ A @ B ) @ zero_zero_a )
= ( ( ( ord_less_a @ zero_zero_a @ A )
& ( ord_less_a @ B @ zero_zero_a ) )
| ( ( ord_less_a @ A @ zero_zero_a )
& ( ord_less_a @ zero_zero_a @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_596_not__square__less__zero,axiom,
! [A: a] :
~ ( ord_less_a @ ( times_times_a @ A @ A ) @ zero_zero_a ) ).
% not_square_less_zero
thf(fact_597_linordered__ring__strict__class_Omult__neg__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% linordered_ring_strict_class.mult_neg_neg
thf(fact_598_zero__less__numeral,axiom,
! [N: num] : ( ord_less_a @ zero_zero_a @ ( numeral_numeral_a @ N ) ) ).
% zero_less_numeral
thf(fact_599_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_600_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_a @ ( numeral_numeral_a @ N ) @ zero_zero_a ) ).
% not_numeral_less_zero
thf(fact_601_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_602_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_603_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_604_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_605_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_606_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_607_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_608_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_609_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_610_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_611_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_612_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X4: nat] : ( P3 @ X4 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_613_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_614_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_615_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_616_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_617_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_618_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_619_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_620_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_621_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_622_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_623_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_624_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_625_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_626_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_627_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_628_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_629_set__times__elim,axiom,
! [X: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ X @ ( times_times_set_nat @ A2 @ B3 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X
= ( times_times_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A2 )
=> ~ ( member_nat @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_630_set__times__elim,axiom,
! [X: num,A2: set_num,B3: set_num] :
( ( member_num @ X @ ( times_times_set_num @ A2 @ B3 ) )
=> ~ ! [A4: num,B4: num] :
( ( X
= ( times_times_num @ A4 @ B4 ) )
=> ( ( member_num @ A4 @ A2 )
=> ~ ( member_num @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_631_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_632_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_633_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_634_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_635_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_636_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_637_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_638_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_639_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_640_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_641_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_642_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_643_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_644_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_a @ zero_zero_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_645_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) @ zero_zero_a ) ).
% neg_numeral_less_zero
thf(fact_646_real__mult__le__cancel__left__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ) ).
% real_mult_le_cancel_left_pos
thf(fact_647_real__mult__le__cancel__left__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% real_mult_le_cancel_left_pos
thf(fact_648_real__mult__le__cancel__right__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ) ).
% real_mult_le_cancel_right_pos
thf(fact_649_real__mult__le__cancel__right__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% real_mult_le_cancel_right_pos
thf(fact_650_mult__le__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ B ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_651_mult__le__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ B ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_652_mult__left__less__imp__less,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_653_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_654_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_655_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_656_mult__less__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ B ) )
& ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_a @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_657_mult__right__less__imp__less,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_658_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_659_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_660_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_661_mult__less__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ B ) )
& ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_a @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_662_mult__le__cancel__left__neg,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ C @ zero_zero_a )
=> ( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_eq_a @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_663_mult__le__cancel__left__pos,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_eq_a @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_664_mult__left__le__imp__le,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_665_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_666_mult__right__le__imp__le,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_667_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_668_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_669_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_670_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_671_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_672_not__sum__squares__lt__zero,axiom,
! [X: a,Y: a] :
~ ( ord_less_a @ ( plus_plus_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) ) @ zero_zero_a ) ).
% not_sum_squares_lt_zero
thf(fact_673_less__add__iff1,axiom,
! [A: a,E2: a,C: a,B: a,D: a] :
( ( ord_less_a @ ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ C ) @ ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ D ) )
= ( ord_less_a @ ( plus_plus_a @ ( times_times_a @ ( minus_minus_a @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_674_less__add__iff2,axiom,
! [A: a,E2: a,C: a,B: a,D: a] :
( ( ord_less_a @ ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ C ) @ ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ D ) )
= ( ord_less_a @ C @ ( plus_plus_a @ ( times_times_a @ ( minus_minus_a @ B @ A ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_675_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_a
!= ( numeral_numeral_a @ N ) ) ).
% zero_neq_numeral
thf(fact_676_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_677_real__linear,axiom,
! [A: a,B: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ A @ B )
| ( A = B )
| ( ord_less_a @ B @ A ) ) ) ) ).
% real_linear
thf(fact_678_real__linear,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ A @ B )
| ( A = B )
| ( ord_less_nat @ B @ A ) ) ) ) ).
% real_linear
thf(fact_679_neg__neg__linear,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ( ord_less_a @ A @ B )
| ( A = B )
| ( ord_less_a @ B @ A ) ) ) ) ).
% neg_neg_linear
thf(fact_680_neg__neg__linear,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ( ord_less_nat @ A @ B )
| ( A = B )
| ( ord_less_nat @ B @ A ) ) ) ) ).
% neg_neg_linear
thf(fact_681_pos__pos__linear,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ( ord_less_a @ A @ B )
| ( A = B )
| ( ord_less_a @ B @ A ) ) ) ) ).
% pos_pos_linear
thf(fact_682_pos__pos__linear,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ A @ B )
| ( A = B )
| ( ord_less_nat @ B @ A ) ) ) ) ).
% pos_pos_linear
thf(fact_683_real__linorder__cases,axiom,
! [A: a,B: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ~ ( ord_less_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ B @ A ) ) ) ) ) ).
% real_linorder_cases
thf(fact_684_real__linorder__cases,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ~ ( ord_less_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ B @ A ) ) ) ) ) ).
% real_linorder_cases
thf(fact_685_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_686_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_687_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_688_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_689_field__lbound__gt__zero,axiom,
! [D1: a,D22: a] :
( ( ord_less_a @ zero_zero_a @ D1 )
=> ( ( ord_less_a @ zero_zero_a @ D22 )
=> ? [E3: a] :
( ( ord_less_a @ zero_zero_a @ E3 )
& ( ord_less_a @ E3 @ D1 )
& ( ord_less_a @ E3 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_690_less__numeral__extra_I3_J,axiom,
~ ( ord_less_a @ zero_zero_a @ zero_zero_a ) ).
% less_numeral_extra(3)
thf(fact_691_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_692_verit__comp__simplify1_I3_J,axiom,
! [B5: a,A5: a] :
( ( ~ ( ord_less_eq_a @ B5 @ A5 ) )
= ( ord_less_a @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_693_verit__comp__simplify1_I3_J,axiom,
! [B5: num,A5: num] :
( ( ~ ( ord_less_eq_num @ B5 @ A5 ) )
= ( ord_less_num @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_694_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_695_order__le__imp__less__or__eq,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_696_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_697_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_698_linorder__le__less__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_a @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_699_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_700_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_701_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_702_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_703_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_704_order__less__le__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_705_order__less__le__subst1,axiom,
! [A: num,F: a > num,B: a,C: a] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_706_order__less__le__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_707_order__less__le__subst1,axiom,
! [A: a,F: num > a,B: num,C: num] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_708_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_709_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_710_order__less__le__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_711_order__less__le__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_712_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_713_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_714_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > num,C: num] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_715_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_716_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > a,C: a] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_717_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_718_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_719_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_720_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_721_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_722_order__le__less__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_723_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_724_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_725_order__less__le__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( ord_less_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z2 )
=> ( ord_less_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_726_order__less__le__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_727_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_728_order__le__less__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_a @ Y @ Z2 )
=> ( ord_less_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_729_order__le__less__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_730_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_731_order__neq__le__trans,axiom,
! [A: a,B: a] :
( ( A != B )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_732_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_733_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_734_order__le__neq__trans,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_735_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_736_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_737_order__less__imp__le,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_738_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_739_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_740_linorder__not__less,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_a @ X @ Y ) )
= ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_not_less
thf(fact_741_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_742_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_743_linorder__not__le,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_eq_a @ X @ Y ) )
= ( ord_less_a @ Y @ X ) ) ).
% linorder_not_le
thf(fact_744_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_745_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_746_order__less__le,axiom,
( ord_less_a
= ( ^ [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_747_order__less__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y3: num] :
( ( ord_less_eq_num @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_748_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_749_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_750_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X2: num,Y3: num] :
( ( ord_less_num @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_751_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_752_dual__order_Ostrict__implies__order,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_eq_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_753_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_754_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_755_order_Ostrict__implies__order,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_eq_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_756_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_757_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_758_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B2: a,A3: a] :
( ( ord_less_eq_a @ B2 @ A3 )
& ~ ( ord_less_eq_a @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_759_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B2: num,A3: num] :
( ( ord_less_eq_num @ B2 @ A3 )
& ~ ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_760_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_761_dual__order_Ostrict__trans2,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_762_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_763_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_764_dual__order_Ostrict__trans1,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_765_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_766_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_767_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B2: a,A3: a] :
( ( ord_less_eq_a @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_768_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B2: num,A3: num] :
( ( ord_less_eq_num @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_769_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_770_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B2: a,A3: a] :
( ( ord_less_a @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_771_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A3: num] :
( ( ord_less_num @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_772_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_773_dense__le__bounded,axiom,
! [X: a,Y: a,Z2: a] :
( ( ord_less_a @ X @ Y )
=> ( ! [W2: a] :
( ( ord_less_a @ X @ W2 )
=> ( ( ord_less_a @ W2 @ Y )
=> ( ord_less_eq_a @ W2 @ Z2 ) ) )
=> ( ord_less_eq_a @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_774_dense__ge__bounded,axiom,
! [Z2: a,X: a,Y: a] :
( ( ord_less_a @ Z2 @ X )
=> ( ! [W2: a] :
( ( ord_less_a @ Z2 @ W2 )
=> ( ( ord_less_a @ W2 @ X )
=> ( ord_less_eq_a @ Y @ W2 ) ) )
=> ( ord_less_eq_a @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_775_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A3: a,B2: a] :
( ( ord_less_eq_a @ A3 @ B2 )
& ~ ( ord_less_eq_a @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_776_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
& ~ ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_777_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_778_order_Ostrict__trans2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_779_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_780_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_781_order_Ostrict__trans1,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_782_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_783_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_784_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A3: a,B2: a] :
( ( ord_less_eq_a @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_785_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_786_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_787_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A3: a,B2: a] :
( ( ord_less_a @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_788_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A3: num,B2: num] :
( ( ord_less_num @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_789_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_790_not__le__imp__less,axiom,
! [Y: a,X: a] :
( ~ ( ord_less_eq_a @ Y @ X )
=> ( ord_less_a @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_791_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_792_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_793_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
& ~ ( ord_less_eq_a @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_794_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y3: num] :
( ( ord_less_eq_num @ X2 @ Y3 )
& ~ ( ord_less_eq_num @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_795_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_796_dense__le,axiom,
! [Y: a,Z2: a] :
( ! [X3: a] :
( ( ord_less_a @ X3 @ Y )
=> ( ord_less_eq_a @ X3 @ Z2 ) )
=> ( ord_less_eq_a @ Y @ Z2 ) ) ).
% dense_le
thf(fact_797_dense__ge,axiom,
! [Z2: a,Y: a] :
( ! [X3: a] :
( ( ord_less_a @ Z2 @ X3 )
=> ( ord_less_eq_a @ Y @ X3 ) )
=> ( ord_less_eq_a @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_798_antisym__conv2,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ~ ( ord_less_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_799_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_800_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_801_antisym__conv1,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_802_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_803_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_804_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_805_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_806_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_807_leI,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% leI
thf(fact_808_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_809_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_810_leD,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_a @ X @ Y ) ) ).
% leD
thf(fact_811_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_812_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_813_mult__not__zero,axiom,
! [A: a,B: a] :
( ( ( times_times_a @ A @ B )
!= zero_zero_a )
=> ( ( A != zero_zero_a )
& ( B != zero_zero_a ) ) ) ).
% mult_not_zero
thf(fact_814_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_815_divisors__zero,axiom,
! [A: a,B: a] :
( ( ( times_times_a @ A @ B )
= zero_zero_a )
=> ( ( A = zero_zero_a )
| ( B = zero_zero_a ) ) ) ).
% divisors_zero
thf(fact_816_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_817_no__zero__divisors,axiom,
! [A: a,B: a] :
( ( A != zero_zero_a )
=> ( ( B != zero_zero_a )
=> ( ( times_times_a @ A @ B )
!= zero_zero_a ) ) ) ).
% no_zero_divisors
thf(fact_818_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_819_mult__left__cancel,axiom,
! [C: a,A: a,B: a] :
( ( C != zero_zero_a )
=> ( ( ( times_times_a @ C @ A )
= ( times_times_a @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_820_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_821_mult__right__cancel,axiom,
! [C: a,A: a,B: a] :
( ( C != zero_zero_a )
=> ( ( ( times_times_a @ A @ C )
= ( times_times_a @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_822_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_823_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_824_add__mono__thms__linordered__field_I5_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_a @ I @ J )
& ( ord_less_a @ K @ L ) )
=> ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_825_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_826_add__mono__thms__linordered__field_I2_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( I = J )
& ( ord_less_a @ K @ L ) )
=> ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_827_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_828_add__mono__thms__linordered__field_I1_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_a @ I @ J )
& ( K = L ) )
=> ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_829_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_830_add__strict__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_831_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_832_add__strict__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_833_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_834_add__strict__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_835_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_836_add__less__imp__less__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
=> ( ord_less_a @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_837_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_838_add__less__imp__less__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
=> ( ord_less_a @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_839_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_840_diff__strict__mono,axiom,
! [A: a,B: a,D: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ D @ C )
=> ( ord_less_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_841_diff__eq__diff__less,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D ) )
=> ( ( ord_less_a @ A @ B )
= ( ord_less_a @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_842_diff__strict__left__mono,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_a @ ( minus_minus_a @ C @ A ) @ ( minus_minus_a @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_843_diff__strict__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_844_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_845_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_846_crossproduct__eq,axiom,
! [W: a,Y: a,X: a,Z2: a] :
( ( ( plus_plus_a @ ( times_times_a @ W @ Y ) @ ( times_times_a @ X @ Z2 ) )
= ( plus_plus_a @ ( times_times_a @ W @ Z2 ) @ ( times_times_a @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_847_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_848_crossproduct__noteq,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) )
!= ( plus_plus_a @ ( times_times_a @ A @ D ) @ ( times_times_a @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_849_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_850_ring__class_Oring__distribs_I2_J,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_851_ring__class_Oring__distribs_I1_J,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ A @ ( plus_plus_a @ B @ C ) )
= ( plus_plus_a @ ( times_times_a @ A @ B ) @ ( times_times_a @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_852_comm__semiring__class_Odistrib,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_853_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_854_distrib__left,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ A @ ( plus_plus_a @ B @ C ) )
= ( plus_plus_a @ ( times_times_a @ A @ B ) @ ( times_times_a @ A @ C ) ) ) ).
% distrib_left
thf(fact_855_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_856_distrib__right,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ).
% distrib_right
thf(fact_857_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_858_combine__common__factor,axiom,
! [A: a,E2: a,B: a,C: a] :
( ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ C ) )
= ( plus_plus_a @ ( times_times_a @ ( plus_plus_a @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_859_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_860_left__diff__distrib,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ ( minus_minus_a @ A @ B ) @ C )
= ( minus_minus_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_861_right__diff__distrib,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ A @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( times_times_a @ A @ B ) @ ( times_times_a @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_862_left__diff__distrib_H,axiom,
! [B: a,C: a,A: a] :
( ( times_times_a @ ( minus_minus_a @ B @ C ) @ A )
= ( minus_minus_a @ ( times_times_a @ B @ A ) @ ( times_times_a @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_863_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_864_right__diff__distrib_H,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ A @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( times_times_a @ A @ B ) @ ( times_times_a @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_865_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_866_mult__less__cancel__right2,axiom,
! [A: a,C: a] :
( ( ord_less_a @ ( times_times_a @ A @ C ) @ C )
= ( ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ one_one_a ) )
& ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_a @ one_one_a @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_867_mult__less__cancel__right1,axiom,
! [C: a,B: a] :
( ( ord_less_a @ C @ ( times_times_a @ B @ C ) )
= ( ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ one_one_a @ B ) )
& ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_a @ B @ one_one_a ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_868_mult__less__cancel__left2,axiom,
! [C: a,A: a] :
( ( ord_less_a @ ( times_times_a @ C @ A ) @ C )
= ( ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ one_one_a ) )
& ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_a @ one_one_a @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_869_mult__less__cancel__left1,axiom,
! [C: a,B: a] :
( ( ord_less_a @ C @ ( times_times_a @ C @ B ) )
= ( ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ one_one_a @ B ) )
& ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_a @ B @ one_one_a ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_870_mult__le__cancel__right2,axiom,
! [A: a,C: a] :
( ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ C )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ one_one_a ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ one_one_a @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_871_mult__le__cancel__right1,axiom,
! [C: a,B: a] :
( ( ord_less_eq_a @ C @ ( times_times_a @ B @ C ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ one_one_a @ B ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ B @ one_one_a ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_872_mult__le__cancel__left2,axiom,
! [C: a,A: a] :
( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ C )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ one_one_a ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ one_one_a @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_873_mult__le__cancel__left1,axiom,
! [C: a,B: a] :
( ( ord_less_eq_a @ C @ ( times_times_a @ C @ B ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ one_one_a @ B ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ B @ one_one_a ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_874_is__num__normalize_I6_J,axiom,
! [X: a,Y: a] :
( ( neg_numeral_is_num_a @ X )
=> ( ( neg_numeral_is_num_a @ Y )
=> ( neg_numeral_is_num_a @ ( plus_plus_a @ X @ Y ) ) ) ) ).
% is_num_normalize(6)
thf(fact_875_is__num__add__commute,axiom,
! [X: a,Y: a] :
( ( neg_numeral_is_num_a @ X )
=> ( ( neg_numeral_is_num_a @ Y )
=> ( ( plus_plus_a @ X @ Y )
= ( plus_plus_a @ Y @ X ) ) ) ) ).
% is_num_add_commute
thf(fact_876_is__num__add__left__commute,axiom,
! [X: a,Y: a,Z2: a] :
( ( neg_numeral_is_num_a @ X )
=> ( ( neg_numeral_is_num_a @ Y )
=> ( ( plus_plus_a @ X @ ( plus_plus_a @ Y @ Z2 ) )
= ( plus_plus_a @ Y @ ( plus_plus_a @ X @ Z2 ) ) ) ) ) ).
% is_num_add_left_commute
thf(fact_877_convex__bound__lt,axiom,
! [X: a,A: a,Y: a,U: a,V: a] :
( ( ord_less_a @ X @ A )
=> ( ( ord_less_a @ Y @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ U )
=> ( ( ord_less_eq_a @ zero_zero_a @ V )
=> ( ( ( plus_plus_a @ U @ V )
= one_one_a )
=> ( ord_less_a @ ( plus_plus_a @ ( times_times_a @ U @ X ) @ ( times_times_a @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_878_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_a @ zero_zero_a @ ( numeral_numeral_a @ N ) ) ).
% zero_le_numeral
thf(fact_879_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_880_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_a @ ( numeral_numeral_a @ N ) @ zero_zero_a ) ).
% not_numeral_le_zero
thf(fact_881_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_882_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_a @ one_one_a @ ( numeral_numeral_a @ N ) ) ).
% one_le_numeral
thf(fact_883_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_884_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_a @ one_one_a @ ( numeral_numeral_a @ X ) )
= ( plus_plus_a @ ( numeral_numeral_a @ X ) @ one_one_a ) ) ).
% one_plus_numeral_commute
thf(fact_885_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_886_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_a
!= ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_887_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_a @ ( numeral_numeral_a @ M ) @ ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_888_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) @ ( numeral_numeral_a @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_889_not__le__real,axiom,
! [A: a,B: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ~ ( ord_less_eq_a @ B @ A ) )
= ( ord_less_a @ A @ B ) ) ) ) ).
% not_le_real
thf(fact_890_not__le__real,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ~ ( ord_less_eq_nat @ B @ A ) )
= ( ord_less_nat @ A @ B ) ) ) ) ).
% not_le_real
thf(fact_891_not__less__real,axiom,
! [A: a,B: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ~ ( ord_less_a @ B @ A ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ).
% not_less_real
thf(fact_892_not__less__real,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ~ ( ord_less_nat @ B @ A ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% not_less_real
thf(fact_893_nonneg__linorder__cases,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ~ ( ord_less_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ B @ A ) ) ) ) ) ).
% nonneg_linorder_cases
thf(fact_894_nonneg__linorder__cases,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ~ ( ord_less_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ B @ A ) ) ) ) ) ).
% nonneg_linorder_cases
thf(fact_895_nonpos__linorder__cases,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ~ ( ord_less_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ B @ A ) ) ) ) ) ).
% nonpos_linorder_cases
thf(fact_896_nonpos__linorder__cases,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ~ ( ord_less_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ B @ A ) ) ) ) ) ).
% nonpos_linorder_cases
thf(fact_897_less__numeral__extra_I1_J,axiom,
ord_less_a @ zero_zero_a @ one_one_a ).
% less_numeral_extra(1)
thf(fact_898_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_899_zero__less__one__class_Ozero__less__one,axiom,
ord_less_a @ zero_zero_a @ one_one_a ).
% zero_less_one_class.zero_less_one
thf(fact_900_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_901_not__one__less__zero,axiom,
~ ( ord_less_a @ one_one_a @ zero_zero_a ) ).
% not_one_less_zero
thf(fact_902_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_903_add__neg__pos__is__real,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a )
| ( ( plus_plus_a @ A @ B )
= zero_zero_a )
| ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ) ).
% add_neg_pos_is_real
thf(fact_904_add__neg__pos__is__real,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat )
| ( ( plus_plus_nat @ A @ B )
= zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ) ).
% add_neg_pos_is_real
thf(fact_905_add__pos__neg__is__real,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a )
| ( ( plus_plus_a @ A @ B )
= zero_zero_a )
| ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ) ).
% add_pos_neg_is_real
thf(fact_906_add__pos__neg__is__real,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat )
| ( ( plus_plus_nat @ A @ B )
= zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ) ).
% add_pos_neg_is_real
thf(fact_907_real__add__less__cancel__left__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( ord_less_a @ A @ B ) ) ) ) ).
% real_add_less_cancel_left_pos
thf(fact_908_real__add__less__cancel__left__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ) ) ).
% real_add_less_cancel_left_pos
thf(fact_909_real__add__less__cancel__right__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( ord_less_a @ A @ B ) ) ) ) ).
% real_add_less_cancel_right_pos
thf(fact_910_real__add__less__cancel__right__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ) ) ).
% real_add_less_cancel_right_pos
thf(fact_911_add__less__zeroD,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ ( plus_plus_a @ X @ Y ) @ zero_zero_a )
=> ( ( ord_less_a @ X @ zero_zero_a )
| ( ord_less_a @ Y @ zero_zero_a ) ) ) ).
% add_less_zeroD
thf(fact_912_add__neg__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_neg_neg
thf(fact_913_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_914_add__pos__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_915_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_916_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_917_pos__add__strict,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_918_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_919_add__mono__thms__linordered__field_I4_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_eq_a @ I @ J )
& ( ord_less_a @ K @ L ) )
=> ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_920_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_921_add__mono__thms__linordered__field_I3_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_a @ I @ J )
& ( ord_less_eq_a @ K @ L ) )
=> ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_922_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_923_add__le__less__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_924_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_925_add__less__le__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_926_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_927_less__iff__diff__less__0,axiom,
( ord_less_a
= ( ^ [A3: a,B2: a] : ( ord_less_a @ ( minus_minus_a @ A3 @ B2 ) @ zero_zero_a ) ) ) ).
% less_iff_diff_less_0
thf(fact_928_mult__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_929_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_930_mult__mono_H,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_931_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_932_zero__le__square,axiom,
! [A: a] : ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ A ) ) ).
% zero_le_square
thf(fact_933_split__mult__pos__le,axiom,
! [A: a,B: a] :
( ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ zero_zero_a @ B ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ B @ zero_zero_a ) ) )
=> ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_934_mult__left__mono__neg,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_935_mult__nonpos__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_936_mult__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_937_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_938_mult__right__mono__neg,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_939_mult__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_940_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_941_mult__le__0__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a )
= ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ B @ zero_zero_a ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ zero_zero_a @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_942_split__mult__neg__le,axiom,
! [A: a,B: a] :
( ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ B @ zero_zero_a ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ zero_zero_a @ B ) ) )
=> ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ).
% split_mult_neg_le
thf(fact_943_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_944_mult__nonneg__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_945_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_946_mult__nonneg__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).
% mult_nonneg_nonpos
thf(fact_947_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_948_mult__nonpos__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).
% mult_nonpos_nonneg
thf(fact_949_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_950_mult__nonneg__nonpos2,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ B @ A ) @ zero_zero_a ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_951_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_952_zero__le__mult__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ B ) )
= ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ zero_zero_a @ B ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ B @ zero_zero_a ) ) ) ) ).
% zero_le_mult_iff
thf(fact_953_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_954_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_955_add__mono1,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( plus_plus_a @ A @ one_one_a ) @ ( plus_plus_a @ B @ one_one_a ) ) ) ).
% add_mono1
thf(fact_956_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_957_less__add__one,axiom,
! [A: a] : ( ord_less_a @ A @ ( plus_plus_a @ A @ one_one_a ) ) ).
% less_add_one
thf(fact_958_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_959_add__scale__eq__noteq,axiom,
! [R: a,A: a,B: a,C: a,D: a] :
( ( R != zero_zero_a )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_a @ A @ ( times_times_a @ R @ C ) )
!= ( plus_plus_a @ B @ ( times_times_a @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_960_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_961_diff__less__eq,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ ( minus_minus_a @ A @ B ) @ C )
= ( ord_less_a @ A @ ( plus_plus_a @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_962_less__diff__eq,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ A @ ( minus_minus_a @ C @ B ) )
= ( ord_less_a @ ( plus_plus_a @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_963_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: a,B: a] :
( ~ ( ord_less_a @ A @ B )
=> ( ( plus_plus_a @ B @ ( minus_minus_a @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_964_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_965_eq__add__iff1,axiom,
! [A: a,E2: a,C: a,B: a,D: a] :
( ( ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ C )
= ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ D ) )
= ( ( plus_plus_a @ ( times_times_a @ ( minus_minus_a @ A @ B ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_966_eq__add__iff2,axiom,
! [A: a,E2: a,C: a,B: a,D: a] :
( ( ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ C )
= ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ D ) )
= ( C
= ( plus_plus_a @ ( times_times_a @ ( minus_minus_a @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_967_square__diff__square__factored,axiom,
! [X: a,Y: a] :
( ( minus_minus_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) )
= ( times_times_a @ ( plus_plus_a @ X @ Y ) @ ( minus_minus_a @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_968_mult__diff__mult,axiom,
! [X: a,Y: a,A: a,B: a] :
( ( minus_minus_a @ ( times_times_a @ X @ Y ) @ ( times_times_a @ A @ B ) )
= ( plus_plus_a @ ( times_times_a @ X @ ( minus_minus_a @ Y @ B ) ) @ ( times_times_a @ ( minus_minus_a @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_969_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) @ zero_zero_a ) ).
% neg_numeral_le_zero
thf(fact_970_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_971_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_a @ one_one_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_972_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_a @ ( numeral_numeral_a @ M ) @ ( uminus_uminus_a @ one_one_a ) ) ).
% not_numeral_le_neg_one
thf(fact_973_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) @ ( uminus_uminus_a @ one_one_a ) ) ).
% neg_numeral_le_neg_one
thf(fact_974_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_a @ ( uminus_uminus_a @ one_one_a ) @ ( numeral_numeral_a @ M ) ) ).
% neg_one_le_numeral
thf(fact_975_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) @ one_one_a ) ).
% neg_numeral_le_one
thf(fact_976_real__add__le__cancel__left__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ).
% real_add_le_cancel_left_pos
thf(fact_977_real__add__le__cancel__left__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% real_add_le_cancel_left_pos
thf(fact_978_real__add__le__cancel__right__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ).
% real_add_le_cancel_right_pos
thf(fact_979_real__add__le__cancel__right__pos,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% real_add_le_cancel_right_pos
thf(fact_980_add__neg__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_neg_nonpos
thf(fact_981_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_982_add__nonneg__pos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_983_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_984_add__nonpos__neg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_nonpos_neg
thf(fact_985_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_986_add__pos__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_987_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_988_add__strict__increasing,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_989_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_990_add__strict__increasing2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_991_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_992_zero__less__two,axiom,
ord_less_a @ zero_zero_a @ ( plus_plus_a @ one_one_a @ one_one_a ) ).
% zero_less_two
thf(fact_993_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_994_less__minus__one__simps_I1_J,axiom,
ord_less_a @ ( uminus_uminus_a @ one_one_a ) @ zero_zero_a ).
% less_minus_one_simps(1)
thf(fact_995_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_a @ zero_zero_a @ ( uminus_uminus_a @ one_one_a ) ) ).
% less_minus_one_simps(3)
thf(fact_996_mult__left__le__one__le,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ord_less_eq_a @ Y @ one_one_a )
=> ( ord_less_eq_a @ ( times_times_a @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_997_mult__right__le__one__le,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ord_less_eq_a @ Y @ one_one_a )
=> ( ord_less_eq_a @ ( times_times_a @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_998_mult__le__one,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ one_one_a )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ B @ one_one_a )
=> ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ one_one_a ) ) ) ) ).
% mult_le_one
thf(fact_999_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1000_mult__left__le,axiom,
! [C: a,A: a] :
( ( ord_less_eq_a @ C @ one_one_a )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1001_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1002_sum__squares__ge__zero,axiom,
! [X: a,Y: a] : ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_1003_ordered__ring__class_Ole__add__iff1,axiom,
! [A: a,E2: a,C: a,B: a,D: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ C ) @ ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ D ) )
= ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ ( minus_minus_a @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1004_ordered__ring__class_Ole__add__iff2,axiom,
! [A: a,E2: a,C: a,B: a,D: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ C ) @ ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ D ) )
= ( ord_less_eq_a @ C @ ( plus_plus_a @ ( times_times_a @ ( minus_minus_a @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1005_square__diff__one__factored,axiom,
! [X: a] :
( ( minus_minus_a @ ( times_times_a @ X @ X ) @ one_one_a )
= ( times_times_a @ ( plus_plus_a @ X @ one_one_a ) @ ( minus_minus_a @ X @ one_one_a ) ) ) ).
% square_diff_one_factored
thf(fact_1006_is__num_Osimps,axiom,
( neg_numeral_is_num_a
= ( ^ [A3: a] :
( ( A3 = one_one_a )
| ? [X2: a] :
( ( A3
= ( uminus_uminus_a @ X2 ) )
& ( neg_numeral_is_num_a @ X2 ) )
| ? [X2: a,Y3: a] :
( ( A3
= ( plus_plus_a @ X2 @ Y3 ) )
& ( neg_numeral_is_num_a @ X2 )
& ( neg_numeral_is_num_a @ Y3 ) ) ) ) ) ).
% is_num.simps
thf(fact_1007_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ V ) ) @ ( plus_plus_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ W ) ) @ Y ) )
= ( plus_plus_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(167)
thf(fact_1008_sum__squares__eq__zero__iff,axiom,
! [X: a,Y: a] :
( ( ( plus_plus_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) )
= zero_zero_a )
= ( ( X = zero_zero_a )
& ( Y = zero_zero_a ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1009_field__le__mult__one__interval,axiom,
! [X: a,Y: a] :
( ! [Z3: a] :
( ( ord_less_a @ zero_zero_a @ Z3 )
=> ( ( ord_less_a @ Z3 @ one_one_a )
=> ( ord_less_eq_a @ ( times_times_a @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_1010_field__le__epsilon,axiom,
! [X: a,Y: a] :
( ! [E3: a] :
( ( ord_less_a @ zero_zero_a @ E3 )
=> ( ord_less_eq_a @ X @ ( plus_plus_a @ Y @ E3 ) ) )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_1011_sum__squares__le__zero__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) ) @ zero_zero_a )
= ( ( X = zero_zero_a )
& ( Y = zero_zero_a ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1012_sum__squares__gt__zero__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) ) )
= ( ( X != zero_zero_a )
| ( Y != zero_zero_a ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1013_mult__hom_Ohom__zero,axiom,
! [C: a] :
( ( times_times_a @ C @ zero_zero_a )
= zero_zero_a ) ).
% mult_hom.hom_zero
thf(fact_1014_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_1015_less__1__mult_H,axiom,
! [A: a,B: a] :
( ( ord_less_a @ one_one_a @ A )
=> ( ( ord_less_eq_a @ one_one_a @ B )
=> ( ord_less_a @ one_one_a @ ( times_times_a @ A @ B ) ) ) ) ).
% less_1_mult'
thf(fact_1016_less__1__mult_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ one_one_nat @ B )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% less_1_mult'
thf(fact_1017_ordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_mono
thf(fact_1018_ordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_mono
thf(fact_1019_mult__hom_Ohom__add,axiom,
! [C: a,X: a,Y: a] :
( ( times_times_a @ C @ ( plus_plus_a @ X @ Y ) )
= ( plus_plus_a @ ( times_times_a @ C @ X ) @ ( times_times_a @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_1020_mult__hom_Ohom__add,axiom,
! [C: nat,X: nat,Y: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_1021_ordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1022_ordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1023_ordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1024_ordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1025_ordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( times_times_a @ B @ A ) @ zero_zero_a ) ) ) ).
% ordered_semiring_strict_class.mult_pos_neg2
thf(fact_1026_ordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% ordered_semiring_strict_class.mult_pos_neg2
thf(fact_1027_ordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% ordered_semiring_strict_class.mult_pos_pos
thf(fact_1028_ordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% ordered_semiring_strict_class.mult_pos_pos
thf(fact_1029_ordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).
% ordered_semiring_strict_class.mult_pos_neg
thf(fact_1030_ordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% ordered_semiring_strict_class.mult_pos_neg
thf(fact_1031_ordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).
% ordered_semiring_strict_class.mult_neg_pos
thf(fact_1032_ordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% ordered_semiring_strict_class.mult_neg_pos
thf(fact_1033_mult__hom_Ohom__add__eq__zero,axiom,
! [X: a,Y: a,C: a] :
( ( ( plus_plus_a @ X @ Y )
= zero_zero_a )
=> ( ( plus_plus_a @ ( times_times_a @ C @ X ) @ ( times_times_a @ C @ Y ) )
= zero_zero_a ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_1034_mult__hom_Ohom__add__eq__zero,axiom,
! [X: nat,Y: nat,C: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_1035_ordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1036_ordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1037_ordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1038_ordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1039_ordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_mono'
thf(fact_1040_ordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% ordered_semiring_strict_class.mult_strict_mono'
thf(fact_1041_square__less__square,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ord_less_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) )
= ( ord_less_a @ X @ Y ) ) ) ) ).
% square_less_square
thf(fact_1042_mult__le__cancel__iff2,axiom,
! [Z2: a,X: a,Y: a] :
( ( ord_less_a @ zero_zero_a @ Z2 )
=> ( ( ord_less_eq_a @ ( times_times_a @ Z2 @ X ) @ ( times_times_a @ Z2 @ Y ) )
= ( ord_less_eq_a @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1043_mult__le__cancel__iff1,axiom,
! [Z2: a,X: a,Y: a] :
( ( ord_less_a @ zero_zero_a @ Z2 )
=> ( ( ord_less_eq_a @ ( times_times_a @ X @ Z2 ) @ ( times_times_a @ Y @ Z2 ) )
= ( ord_less_eq_a @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1044_square__lesseq__square,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ord_less_eq_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) )
= ( ord_less_eq_a @ X @ Y ) ) ) ) ).
% square_lesseq_square
thf(fact_1045_mult__less__iff1,axiom,
! [Z2: a,X: a,Y: a] :
( ( ord_less_a @ zero_zero_a @ Z2 )
=> ( ( ord_less_a @ ( times_times_a @ X @ Z2 ) @ ( times_times_a @ Y @ Z2 ) )
= ( ord_less_a @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_1046_poly__cancel__eq__conv,axiom,
! [X: a,A: a,Y: a,B: a] :
( ( X = zero_zero_a )
=> ( ( A != zero_zero_a )
=> ( ( Y = zero_zero_a )
= ( ( minus_minus_a @ ( times_times_a @ A @ Y ) @ ( times_times_a @ B @ X ) )
= zero_zero_a ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_1047_less__eq__fract__respect,axiom,
! [B: a,B5: a,D: a,D3: a,A: a,A5: a,C: a,C5: a] :
( ( B != zero_zero_a )
=> ( ( B5 != zero_zero_a )
=> ( ( D != zero_zero_a )
=> ( ( D3 != zero_zero_a )
=> ( ( ( times_times_a @ A @ B5 )
= ( times_times_a @ A5 @ B ) )
=> ( ( ( times_times_a @ C @ D3 )
= ( times_times_a @ C5 @ D ) )
=> ( ( ord_less_eq_a @ ( times_times_a @ ( times_times_a @ A @ D ) @ ( times_times_a @ B @ D ) ) @ ( times_times_a @ ( times_times_a @ C @ B ) @ ( times_times_a @ B @ D ) ) )
= ( ord_less_eq_a @ ( times_times_a @ ( times_times_a @ A5 @ D3 ) @ ( times_times_a @ B5 @ D3 ) ) @ ( times_times_a @ ( times_times_a @ C5 @ B5 ) @ ( times_times_a @ B5 @ D3 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_1048_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_a @ one_one_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) )
= ( numeral_numeral_a @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_1049_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_1050_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_1051_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_a @ ( uminus_uminus_a @ one_one_a ) @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_1052_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_a @ ( numeral_numeral_a @ N ) @ one_one_a )
= ( numeral_numeral_a @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1053_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1054_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_a @ one_one_a @ ( numeral_numeral_a @ N ) )
= ( numeral_numeral_a @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1055_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1056_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_a @ ( numeral_numeral_a @ N ) @ one_one_a )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1057_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1058_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_1059_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) @ one_one_a )
= ( uminus_uminus_a @ ( numeral_numeral_a @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_1060_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1061_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1062_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_1063_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_1064_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_1065_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_1066_minus__sub__one__diff__one,axiom,
! [M: num] :
( ( minus_minus_a @ ( uminus_uminus_a @ ( neg_numeral_sub_a @ M @ one ) ) @ one_one_a )
= ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) ) ).
% minus_sub_one_diff_one
thf(fact_1067_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_minus_a @ ( uminus_uminus_a @ one_one_a ) @ ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) )
= ( neg_numeral_sub_a @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_1068_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1069_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_a @ one @ one )
= zero_zero_a ) ).
% sub_num_simps(1)
thf(fact_1070_diff__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( minus_minus_a @ ( numeral_numeral_a @ M ) @ ( numeral_numeral_a @ N ) )
= ( neg_numeral_sub_a @ M @ N ) ) ).
% diff_numeral_simps(1)
thf(fact_1071_add__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( plus_plus_a @ ( numeral_numeral_a @ M ) @ ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) )
= ( neg_numeral_sub_a @ M @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_1072_add__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( plus_plus_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) @ ( numeral_numeral_a @ N ) )
= ( neg_numeral_sub_a @ N @ M ) ) ).
% add_neg_numeral_simps(2)
thf(fact_1073_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ V ) ) @ ( plus_plus_a @ ( numeral_numeral_a @ W ) @ Y ) )
= ( plus_plus_a @ ( neg_numeral_sub_a @ W @ V ) @ Y ) ) ).
% semiring_norm(166)
thf(fact_1074_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y: a] :
( ( plus_plus_a @ ( numeral_numeral_a @ V ) @ ( plus_plus_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ W ) ) @ Y ) )
= ( plus_plus_a @ ( neg_numeral_sub_a @ V @ W ) @ Y ) ) ).
% semiring_norm(165)
thf(fact_1075_diff__numeral__simps_I4_J,axiom,
! [M: num,N: num] :
( ( minus_minus_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) @ ( uminus_uminus_a @ ( numeral_numeral_a @ N ) ) )
= ( neg_numeral_sub_a @ N @ M ) ) ).
% diff_numeral_simps(4)
thf(fact_1076_diff__numeral__special_I2_J,axiom,
! [M: num] :
( ( minus_minus_a @ ( numeral_numeral_a @ M ) @ one_one_a )
= ( neg_numeral_sub_a @ M @ one ) ) ).
% diff_numeral_special(2)
thf(fact_1077_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_minus_a @ one_one_a @ ( numeral_numeral_a @ N ) )
= ( neg_numeral_sub_a @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_1078_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_plus_a @ ( uminus_uminus_a @ one_one_a ) @ ( numeral_numeral_a @ N ) )
= ( neg_numeral_sub_a @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_1079_add__neg__numeral__special_I3_J,axiom,
! [M: num] :
( ( plus_plus_a @ ( numeral_numeral_a @ M ) @ ( uminus_uminus_a @ one_one_a ) )
= ( neg_numeral_sub_a @ M @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_1080_add__neg__numeral__special_I2_J,axiom,
! [M: num] :
( ( plus_plus_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) @ one_one_a )
= ( neg_numeral_sub_a @ one @ M ) ) ).
% add_neg_numeral_special(2)
thf(fact_1081_add__neg__numeral__special_I1_J,axiom,
! [M: num] :
( ( plus_plus_a @ one_one_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) )
= ( neg_numeral_sub_a @ one @ M ) ) ).
% add_neg_numeral_special(1)
thf(fact_1082_diff__numeral__special_I8_J,axiom,
! [M: num] :
( ( minus_minus_a @ ( uminus_uminus_a @ ( numeral_numeral_a @ M ) ) @ ( uminus_uminus_a @ one_one_a ) )
= ( neg_numeral_sub_a @ one @ M ) ) ).
% diff_numeral_special(8)
thf(fact_1083_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1084_neg__numeral__class_Osub__def,axiom,
( neg_numeral_sub_a
= ( ^ [K2: num,L2: num] : ( minus_minus_a @ ( numeral_numeral_a @ K2 ) @ ( numeral_numeral_a @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_1085_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_a @ ( bit1 @ N ) )
= ( plus_plus_a @ ( plus_plus_a @ ( numeral_numeral_a @ N ) @ ( numeral_numeral_a @ N ) ) @ one_one_a ) ) ).
% numeral_Bit1
thf(fact_1086_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_1087_sub__non__negative,axiom,
! [N: num,M: num] :
( ( ord_less_eq_a @ zero_zero_a @ ( neg_numeral_sub_a @ N @ M ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% sub_non_negative
thf(fact_1088_sub__non__positive,axiom,
! [N: num,M: num] :
( ( ord_less_eq_a @ ( neg_numeral_sub_a @ N @ M ) @ zero_zero_a )
= ( ord_less_eq_num @ N @ M ) ) ).
% sub_non_positive
thf(fact_1089_sub__negative,axiom,
! [N: num,M: num] :
( ( ord_less_a @ ( neg_numeral_sub_a @ N @ M ) @ zero_zero_a )
= ( ord_less_num @ N @ M ) ) ).
% sub_negative
thf(fact_1090_sub__positive,axiom,
! [N: num,M: num] :
( ( ord_less_a @ zero_zero_a @ ( neg_numeral_sub_a @ N @ M ) )
= ( ord_less_num @ M @ N ) ) ).
% sub_positive
thf(fact_1091_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1092_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1093_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1094_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1095_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1096_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1097_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1098_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1099_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1100_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1101_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1102_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1103_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1104_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1105_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1106_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_1107_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1108_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1109_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1110_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1111_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1112_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1113_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1114_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1115_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1116_eq__iff__iszero__diff,axiom,
( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
= ( ^ [X2: a,Y3: a] : ( ring_1_iszero_a @ ( minus_minus_a @ X2 @ Y3 ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_1117_iszero__0,axiom,
ring_1_iszero_a @ zero_zero_a ).
% iszero_0
thf(fact_1118_iszero__def,axiom,
( ring_1_iszero_a
= ( ^ [Z4: a] : ( Z4 = zero_zero_a ) ) ) ).
% iszero_def
thf(fact_1119_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1120_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_1121_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_1122_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1123_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1124_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1125_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1126_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1127_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1128_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1129_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1130_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1131_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1132_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1133_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1134_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1135_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1136_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1137_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1138_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1139_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1140_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1141_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1142_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1143_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1144_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K3 )
=> ~ ( P @ I2 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1145_eq__numeral__iff__iszero_I9_J,axiom,
! [X: num] :
( ( ( numeral_numeral_a @ X )
= zero_zero_a )
= ( ring_1_iszero_a @ ( numeral_numeral_a @ X ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_1146_eq__numeral__iff__iszero_I10_J,axiom,
! [Y: num] :
( ( zero_zero_a
= ( numeral_numeral_a @ Y ) )
= ( ring_1_iszero_a @ ( numeral_numeral_a @ Y ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_1147_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1148_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1149_eq__numeral__iff__iszero_I12_J,axiom,
! [Y: num] :
( ( zero_zero_a
= ( uminus_uminus_a @ ( numeral_numeral_a @ Y ) ) )
= ( ring_1_iszero_a @ ( numeral_numeral_a @ Y ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_1150_eq__numeral__iff__iszero_I11_J,axiom,
! [X: num] :
( ( ( uminus_uminus_a @ ( numeral_numeral_a @ X ) )
= zero_zero_a )
= ( ring_1_iszero_a @ ( numeral_numeral_a @ X ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_1151_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1152_forall__finite_I1_J,axiom,
! [P: nat > $o,I2: nat] :
( ( ord_less_nat @ I2 @ zero_zero_nat )
=> ( P @ I2 ) ) ).
% forall_finite(1)
thf(fact_1153_inf__period_I1_J,axiom,
! [P: a > $o,D2: a,Q: a > $o] :
( ! [X3: a,K3: a] :
( ( P @ X3 )
= ( P @ ( minus_minus_a @ X3 @ ( times_times_a @ K3 @ D2 ) ) ) )
=> ( ! [X3: a,K3: a] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_a @ X3 @ ( times_times_a @ K3 @ D2 ) ) ) )
=> ! [X5: a,K4: a] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_a @ X5 @ ( times_times_a @ K4 @ D2 ) ) )
& ( Q @ ( minus_minus_a @ X5 @ ( times_times_a @ K4 @ D2 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1154_minf_I8_J,axiom,
! [T: a] :
? [Z3: a] :
! [X5: a] :
( ( ord_less_a @ X5 @ Z3 )
=> ~ ( ord_less_eq_a @ T @ X5 ) ) ).
% minf(8)
thf(fact_1155_minf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% minf(8)
thf(fact_1156_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_1157_minf_I6_J,axiom,
! [T: a] :
? [Z3: a] :
! [X5: a] :
( ( ord_less_a @ X5 @ Z3 )
=> ( ord_less_eq_a @ X5 @ T ) ) ).
% minf(6)
thf(fact_1158_minf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( ord_less_eq_num @ X5 @ T ) ) ).
% minf(6)
thf(fact_1159_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_1160_pinf_I8_J,axiom,
! [T: a] :
? [Z3: a] :
! [X5: a] :
( ( ord_less_a @ Z3 @ X5 )
=> ( ord_less_eq_a @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1161_pinf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( ord_less_eq_num @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1162_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1163_pinf_I6_J,axiom,
! [T: a] :
? [Z3: a] :
! [X5: a] :
( ( ord_less_a @ Z3 @ X5 )
=> ~ ( ord_less_eq_a @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1164_pinf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1165_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1166_inf__period_I2_J,axiom,
! [P: a > $o,D2: a,Q: a > $o] :
( ! [X3: a,K3: a] :
( ( P @ X3 )
= ( P @ ( minus_minus_a @ X3 @ ( times_times_a @ K3 @ D2 ) ) ) )
=> ( ! [X3: a,K3: a] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_a @ X3 @ ( times_times_a @ K3 @ D2 ) ) ) )
=> ! [X5: a,K4: a] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_a @ X5 @ ( times_times_a @ K4 @ D2 ) ) )
| ( Q @ ( minus_minus_a @ X5 @ ( times_times_a @ K4 @ D2 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1167_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1168_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1169_one__add__one,axiom,
( ( plus_plus_a @ one_one_a @ one_one_a )
= ( numeral_numeral_a @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1170_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1171_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_a @ ( uminus_uminus_a @ one_one_a ) @ ( uminus_uminus_a @ one_one_a ) )
= ( uminus_uminus_a @ ( numeral_numeral_a @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_1172_diff__numeral__special_I10_J,axiom,
( ( minus_minus_a @ ( uminus_uminus_a @ one_one_a ) @ one_one_a )
= ( uminus_uminus_a @ ( numeral_numeral_a @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_1173_diff__numeral__special_I11_J,axiom,
( ( minus_minus_a @ one_one_a @ ( uminus_uminus_a @ one_one_a ) )
= ( numeral_numeral_a @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_1174_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1175_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1176_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_1177_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_a @ ( bit0 @ N ) )
= ( plus_plus_a @ ( numeral_numeral_a @ N ) @ ( numeral_numeral_a @ N ) ) ) ).
% numeral_Bit0
thf(fact_1178_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_1179_left__add__twice,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ A @ ( plus_plus_a @ A @ B ) )
= ( plus_plus_a @ ( times_times_a @ ( numeral_numeral_a @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_1180_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_1181_mult__2__right,axiom,
! [Z2: a] :
( ( times_times_a @ Z2 @ ( numeral_numeral_a @ ( bit0 @ one ) ) )
= ( plus_plus_a @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_1182_mult__2__right,axiom,
! [Z2: nat] :
( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_1183_mult__2,axiom,
! [Z2: a] :
( ( times_times_a @ ( numeral_numeral_a @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_a @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_1184_mult__2,axiom,
! [Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_1185_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1186_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1187_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1188_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_a @ zero_zero_a )
= zero_zero_a ) ).
% dbl_simps(2)
thf(fact_1189_dbl__def,axiom,
( neg_numeral_dbl_a
= ( ^ [X2: a] : ( plus_plus_a @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_1190_plog2__simps_I3_J,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( rBT_plog2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( rBT_plog2 @ N ) ) ) ) ).
% plog2_simps(3)
thf(fact_1191_division__ring__divide__zero,axiom,
! [A: a] :
( ( divide_divide_a @ A @ zero_zero_a )
= zero_zero_a ) ).
% division_ring_divide_zero
thf(fact_1192_divide__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ( divide_divide_a @ A @ C )
= ( divide_divide_a @ B @ C ) )
= ( ( C = zero_zero_a )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_1193_divide__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ( divide_divide_a @ C @ A )
= ( divide_divide_a @ C @ B ) )
= ( ( C = zero_zero_a )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_1194_divide__eq__0__iff,axiom,
! [A: a,B: a] :
( ( ( divide_divide_a @ A @ B )
= zero_zero_a )
= ( ( A = zero_zero_a )
| ( B = zero_zero_a ) ) ) ).
% divide_eq_0_iff
thf(fact_1195_div__by__0,axiom,
! [A: a] :
( ( divide_divide_a @ A @ zero_zero_a )
= zero_zero_a ) ).
% div_by_0
thf(fact_1196_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1197_div__0,axiom,
! [A: a] :
( ( divide_divide_a @ zero_zero_a @ A )
= zero_zero_a ) ).
% div_0
thf(fact_1198_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_1199_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1200_plog2__simps_I1_J,axiom,
( ( rBT_plog2 @ zero_zero_nat )
= zero_zero_nat ) ).
% plog2_simps(1)
thf(fact_1201_div__self,axiom,
! [A: a] :
( ( A != zero_zero_a )
=> ( ( divide_divide_a @ A @ A )
= one_one_a ) ) ).
% div_self
thf(fact_1202_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1203_divide__eq__1__iff,axiom,
! [A: a,B: a] :
( ( ( divide_divide_a @ A @ B )
= one_one_a )
= ( ( B != zero_zero_a )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_1204_one__eq__divide__iff,axiom,
! [A: a,B: a] :
( ( one_one_a
= ( divide_divide_a @ A @ B ) )
= ( ( B != zero_zero_a )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_1205_divide__self,axiom,
! [A: a] :
( ( A != zero_zero_a )
=> ( ( divide_divide_a @ A @ A )
= one_one_a ) ) ).
% divide_self
thf(fact_1206_divide__self__if,axiom,
! [A: a] :
( ( ( A = zero_zero_a )
=> ( ( divide_divide_a @ A @ A )
= zero_zero_a ) )
& ( ( A != zero_zero_a )
=> ( ( divide_divide_a @ A @ A )
= one_one_a ) ) ) ).
% divide_self_if
thf(fact_1207_divide__eq__eq__1,axiom,
! [B: a,A: a] :
( ( ( divide_divide_a @ B @ A )
= one_one_a )
= ( ( A != zero_zero_a )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_1208_eq__divide__eq__1,axiom,
! [B: a,A: a] :
( ( one_one_a
= ( divide_divide_a @ B @ A ) )
= ( ( A != zero_zero_a )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_1209_one__divide__eq__0__iff,axiom,
! [A: a] :
( ( ( divide_divide_a @ one_one_a @ A )
= zero_zero_a )
= ( A = zero_zero_a ) ) ).
% one_divide_eq_0_iff
thf(fact_1210_zero__eq__1__divide__iff,axiom,
! [A: a] :
( ( zero_zero_a
= ( divide_divide_a @ one_one_a @ A ) )
= ( A = zero_zero_a ) ) ).
% zero_eq_1_divide_iff
thf(fact_1211_nonzero__mult__div__cancel__right,axiom,
! [B: a,A: a] :
( ( B != zero_zero_a )
=> ( ( divide_divide_a @ ( times_times_a @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1212_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1213_nonzero__mult__div__cancel__left,axiom,
! [A: a,B: a] :
( ( A != zero_zero_a )
=> ( ( divide_divide_a @ ( times_times_a @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1214_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1215_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: a,A: a,B: a] :
( ( C != zero_zero_a )
=> ( ( divide_divide_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ C @ B ) )
= ( divide_divide_a @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1216_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: a,A: a,B: a] :
( ( C != zero_zero_a )
=> ( ( divide_divide_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( divide_divide_a @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1217_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: a,A: a,B: a] :
( ( C != zero_zero_a )
=> ( ( divide_divide_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ B @ C ) )
= ( divide_divide_a @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1218_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( C != zero_zero_a )
=> ( ( divide_divide_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( divide_divide_a @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1219_mult__divide__mult__cancel__left__if,axiom,
! [C: a,A: a,B: a] :
( ( ( C = zero_zero_a )
=> ( ( divide_divide_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= zero_zero_a ) )
& ( ( C != zero_zero_a )
=> ( ( divide_divide_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( divide_divide_a @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1220_divide__le__0__1__iff,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( divide_divide_a @ one_one_a @ A ) @ zero_zero_a )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% divide_le_0_1_iff
thf(fact_1221_zero__le__divide__1__iff,axiom,
! [A: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( divide_divide_a @ one_one_a @ A ) )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_1222_divide__less__0__1__iff,axiom,
! [A: a] :
( ( ord_less_a @ ( divide_divide_a @ one_one_a @ A ) @ zero_zero_a )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% divide_less_0_1_iff
thf(fact_1223_divide__less__eq__1__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ ( divide_divide_a @ B @ A ) @ one_one_a )
= ( ord_less_a @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1224_divide__less__eq__1__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ ( divide_divide_a @ B @ A ) @ one_one_a )
= ( ord_less_a @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1225_less__divide__eq__1__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ one_one_a @ ( divide_divide_a @ B @ A ) )
= ( ord_less_a @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1226_less__divide__eq__1__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ one_one_a @ ( divide_divide_a @ B @ A ) )
= ( ord_less_a @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1227_zero__less__divide__1__iff,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ ( divide_divide_a @ one_one_a @ A ) )
= ( ord_less_a @ zero_zero_a @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_1228_nonzero__divide__mult__cancel__left,axiom,
! [A: a,B: a] :
( ( A != zero_zero_a )
=> ( ( divide_divide_a @ A @ ( times_times_a @ A @ B ) )
= ( divide_divide_a @ one_one_a @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1229_nonzero__divide__mult__cancel__right,axiom,
! [B: a,A: a] :
( ( B != zero_zero_a )
=> ( ( divide_divide_a @ B @ ( times_times_a @ A @ B ) )
= ( divide_divide_a @ one_one_a @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1230_divide__eq__eq__numeral1_I1_J,axiom,
! [B: a,W: num,A: a] :
( ( ( divide_divide_a @ B @ ( numeral_numeral_a @ W ) )
= A )
= ( ( ( ( numeral_numeral_a @ W )
!= zero_zero_a )
=> ( B
= ( times_times_a @ A @ ( numeral_numeral_a @ W ) ) ) )
& ( ( ( numeral_numeral_a @ W )
= zero_zero_a )
=> ( A = zero_zero_a ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1231_eq__divide__eq__numeral1_I1_J,axiom,
! [A: a,B: a,W: num] :
( ( A
= ( divide_divide_a @ B @ ( numeral_numeral_a @ W ) ) )
= ( ( ( ( numeral_numeral_a @ W )
!= zero_zero_a )
=> ( ( times_times_a @ A @ ( numeral_numeral_a @ W ) )
= B ) )
& ( ( ( numeral_numeral_a @ W )
= zero_zero_a )
=> ( A = zero_zero_a ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1232_divide__le__eq__numeral1_I1_J,axiom,
! [B: a,W: num,A: a] :
( ( ord_less_eq_a @ ( divide_divide_a @ B @ ( numeral_numeral_a @ W ) ) @ A )
= ( ord_less_eq_a @ B @ ( times_times_a @ A @ ( numeral_numeral_a @ W ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_1233_le__divide__eq__numeral1_I1_J,axiom,
! [A: a,B: a,W: num] :
( ( ord_less_eq_a @ A @ ( divide_divide_a @ B @ ( numeral_numeral_a @ W ) ) )
= ( ord_less_eq_a @ ( times_times_a @ A @ ( numeral_numeral_a @ W ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_1234_le__divide__eq__1__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ one_one_a @ ( divide_divide_a @ B @ A ) )
= ( ord_less_eq_a @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1235_le__divide__eq__1__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ one_one_a @ ( divide_divide_a @ B @ A ) )
= ( ord_less_eq_a @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1236_divide__le__eq__1__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ ( divide_divide_a @ B @ A ) @ one_one_a )
= ( ord_less_eq_a @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1237_divide__le__eq__1__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ ( divide_divide_a @ B @ A ) @ one_one_a )
= ( ord_less_eq_a @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1238_divide__eq__eq__numeral1_I2_J,axiom,
! [B: a,W: num,A: a] :
( ( ( divide_divide_a @ B @ ( uminus_uminus_a @ ( numeral_numeral_a @ W ) ) )
= A )
= ( ( ( ( uminus_uminus_a @ ( numeral_numeral_a @ W ) )
!= zero_zero_a )
=> ( B
= ( times_times_a @ A @ ( uminus_uminus_a @ ( numeral_numeral_a @ W ) ) ) ) )
& ( ( ( uminus_uminus_a @ ( numeral_numeral_a @ W ) )
= zero_zero_a )
=> ( A = zero_zero_a ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1239_eq__divide__eq__numeral1_I2_J,axiom,
! [A: a,B: a,W: num] :
( ( A
= ( divide_divide_a @ B @ ( uminus_uminus_a @ ( numeral_numeral_a @ W ) ) ) )
= ( ( ( ( uminus_uminus_a @ ( numeral_numeral_a @ W ) )
!= zero_zero_a )
=> ( ( times_times_a @ A @ ( uminus_uminus_a @ ( numeral_numeral_a @ W ) ) )
= B ) )
& ( ( ( uminus_uminus_a @ ( numeral_numeral_a @ W ) )
= zero_zero_a )
=> ( A = zero_zero_a ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1240_le__divide__eq__numeral1_I2_J,axiom,
! [A: a,B: a,W: num] :
( ( ord_less_eq_a @ A @ ( divide_divide_a @ B @ ( uminus_uminus_a @ ( numeral_numeral_a @ W ) ) ) )
= ( ord_less_eq_a @ B @ ( times_times_a @ A @ ( uminus_uminus_a @ ( numeral_numeral_a @ W ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1241_divide__le__eq__numeral1_I2_J,axiom,
! [B: a,W: num,A: a] :
( ( ord_less_eq_a @ ( divide_divide_a @ B @ ( uminus_uminus_a @ ( numeral_numeral_a @ W ) ) ) @ A )
= ( ord_less_eq_a @ ( times_times_a @ A @ ( uminus_uminus_a @ ( numeral_numeral_a @ W ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1242_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_1243_right__inverse__eq,axiom,
! [B: a,A: a] :
( ( B != zero_zero_a )
=> ( ( ( divide_divide_a @ A @ B )
= one_one_a )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_1244_nonzero__eq__divide__eq,axiom,
! [C: a,A: a,B: a] :
( ( C != zero_zero_a )
=> ( ( A
= ( divide_divide_a @ B @ C ) )
= ( ( times_times_a @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1245_nonzero__divide__eq__eq,axiom,
! [C: a,B: a,A: a] :
( ( C != zero_zero_a )
=> ( ( ( divide_divide_a @ B @ C )
= A )
= ( B
= ( times_times_a @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1246_eq__divide__imp,axiom,
! [C: a,A: a,B: a] :
( ( C != zero_zero_a )
=> ( ( ( times_times_a @ A @ C )
= B )
=> ( A
= ( divide_divide_a @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_1247_divide__eq__imp,axiom,
! [C: a,B: a,A: a] :
( ( C != zero_zero_a )
=> ( ( B
= ( times_times_a @ A @ C ) )
=> ( ( divide_divide_a @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_1248_eq__divide__eq,axiom,
! [A: a,B: a,C: a] :
( ( A
= ( divide_divide_a @ B @ C ) )
= ( ( ( C != zero_zero_a )
=> ( ( times_times_a @ A @ C )
= B ) )
& ( ( C = zero_zero_a )
=> ( A = zero_zero_a ) ) ) ) ).
% eq_divide_eq
thf(fact_1249_divide__eq__eq,axiom,
! [B: a,C: a,A: a] :
( ( ( divide_divide_a @ B @ C )
= A )
= ( ( ( C != zero_zero_a )
=> ( B
= ( times_times_a @ A @ C ) ) )
& ( ( C = zero_zero_a )
=> ( A = zero_zero_a ) ) ) ) ).
% divide_eq_eq
thf(fact_1250_frac__eq__eq,axiom,
! [Y: a,Z2: a,X: a,W: a] :
( ( Y != zero_zero_a )
=> ( ( Z2 != zero_zero_a )
=> ( ( ( divide_divide_a @ X @ Y )
= ( divide_divide_a @ W @ Z2 ) )
= ( ( times_times_a @ X @ Z2 )
= ( times_times_a @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1251_nonzero__minus__divide__divide,axiom,
! [B: a,A: a] :
( ( B != zero_zero_a )
=> ( ( divide_divide_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_a @ B ) )
= ( divide_divide_a @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_1252_nonzero__minus__divide__right,axiom,
! [B: a,A: a] :
( ( B != zero_zero_a )
=> ( ( uminus_uminus_a @ ( divide_divide_a @ A @ B ) )
= ( divide_divide_a @ A @ ( uminus_uminus_a @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_1253_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1254_divide__le__0__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( divide_divide_a @ A @ B ) @ zero_zero_a )
= ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ B @ zero_zero_a ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ zero_zero_a @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_1255_divide__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( divide_divide_a @ A @ C ) @ ( divide_divide_a @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_1256_zero__le__divide__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( divide_divide_a @ A @ B ) )
= ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ zero_zero_a @ B ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ B @ zero_zero_a ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1257_divide__nonneg__nonneg,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ord_less_eq_a @ zero_zero_a @ ( divide_divide_a @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1258_divide__nonneg__nonpos,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ Y @ zero_zero_a )
=> ( ord_less_eq_a @ ( divide_divide_a @ X @ Y ) @ zero_zero_a ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1259_divide__nonpos__nonneg,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ zero_zero_a )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ord_less_eq_a @ ( divide_divide_a @ X @ Y ) @ zero_zero_a ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1260_divide__nonpos__nonpos,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ zero_zero_a )
=> ( ( ord_less_eq_a @ Y @ zero_zero_a )
=> ( ord_less_eq_a @ zero_zero_a @ ( divide_divide_a @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1261_divide__right__mono__neg,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ ( divide_divide_a @ B @ C ) @ ( divide_divide_a @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1262_diff__divide__distrib,axiom,
! [A: a,B: a,C: a] :
( ( divide_divide_a @ ( minus_minus_a @ A @ B ) @ C )
= ( minus_minus_a @ ( divide_divide_a @ A @ C ) @ ( divide_divide_a @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_1263_add__divide__distrib,axiom,
! [A: a,B: a,C: a] :
( ( divide_divide_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ ( divide_divide_a @ A @ C ) @ ( divide_divide_a @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_1264_divide__neg__neg,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ zero_zero_a )
=> ( ( ord_less_a @ Y @ zero_zero_a )
=> ( ord_less_a @ zero_zero_a @ ( divide_divide_a @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_1265_divide__neg__pos,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ zero_zero_a )
=> ( ( ord_less_a @ zero_zero_a @ Y )
=> ( ord_less_a @ ( divide_divide_a @ X @ Y ) @ zero_zero_a ) ) ) ).
% divide_neg_pos
thf(fact_1266_divide__pos__neg,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ zero_zero_a @ X )
=> ( ( ord_less_a @ Y @ zero_zero_a )
=> ( ord_less_a @ ( divide_divide_a @ X @ Y ) @ zero_zero_a ) ) ) ).
% divide_pos_neg
thf(fact_1267_divide__pos__pos,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ zero_zero_a @ X )
=> ( ( ord_less_a @ zero_zero_a @ Y )
=> ( ord_less_a @ zero_zero_a @ ( divide_divide_a @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_1268_divide__less__0__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ ( divide_divide_a @ A @ B ) @ zero_zero_a )
= ( ( ( ord_less_a @ zero_zero_a @ A )
& ( ord_less_a @ B @ zero_zero_a ) )
| ( ( ord_less_a @ A @ zero_zero_a )
& ( ord_less_a @ zero_zero_a @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_1269_divide__less__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ ( divide_divide_a @ A @ C ) @ ( divide_divide_a @ B @ C ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ B ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_a @ B @ A ) )
& ( C != zero_zero_a ) ) ) ).
% divide_less_cancel
thf(fact_1270_zero__less__divide__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ ( divide_divide_a @ A @ B ) )
= ( ( ( ord_less_a @ zero_zero_a @ A )
& ( ord_less_a @ zero_zero_a @ B ) )
| ( ( ord_less_a @ A @ zero_zero_a )
& ( ord_less_a @ B @ zero_zero_a ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1271_divide__strict__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( divide_divide_a @ A @ C ) @ ( divide_divide_a @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1272_divide__strict__right__mono__neg,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_a @ ( divide_divide_a @ A @ C ) @ ( divide_divide_a @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ ( scalar_prod_a @ u @ b ) @ ( scalar_prod_a @ c @ ( minus_minus_vec_a @ v @ w ) ) ) ).
%------------------------------------------------------------------------------