TPTP Problem File: DAT239^1.p
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%------------------------------------------------------------------------------
% File : DAT239^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Red-black trees 1450
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : rbt_impl__1450.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.67 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 346 ( 97 unt; 45 typ; 0 def)
% Number of atoms : 855 ( 257 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 3434 ( 64 ~; 22 |; 37 &;2983 @)
% ( 0 <=>; 328 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 54 ( 54 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 42 usr; 2 con; 0-3 aty)
% Number of variables : 652 ( 9 ^; 597 !; 8 ?; 652 :)
% ( 38 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:46:57.593
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Num_Onum,type,
num: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
%----Explicit typings (41)
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Divides_Osemiring__div,type,
semiring_div:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Divides_Osemiring__div__parity,type,
semiring_div_parity:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Divides_Osemiring__numeral__div,type,
semiring_numeral_div:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_v_na____,type,
na: nat ).
%----Relevant facts (251)
thf(fact_0_ns,axiom,
~ ( ( na
= ( zero_zero @ nat ) )
| ( na
= ( one_one @ nat ) ) ) ).
% ns
thf(fact_1_False,axiom,
~ ( ord_less_eq @ nat @ na @ ( one_one @ nat ) ) ).
% False
thf(fact_2_Divides_Odiv__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% Divides.div_less
thf(fact_3_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_4_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% half_gt_zero_iff
thf(fact_5_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_6_div__0,axiom,
! [A: $tType] :
( ( semiring_div @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_7_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ( divide_divide @ A @ A2 @ B )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_8_div__by__0,axiom,
! [A: $tType] :
( ( semiring_div @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_9_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A @ ( type @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ( divide_divide @ A @ C @ A2 )
= ( divide_divide @ A @ C @ B ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B ) ) ) ) ).
% divide_cancel_left
thf(fact_10_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A @ ( type @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ( divide_divide @ A @ A2 @ C )
= ( divide_divide @ A @ B @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B ) ) ) ) ).
% divide_cancel_right
thf(fact_11_divide__zero,axiom,
! [A: $tType] :
( ( semidom_divide @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% divide_zero
thf(fact_12_divide__zero__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% divide_zero_left
thf(fact_13_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit0 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_14_num_Oinject_I1_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% num.inject(1)
thf(fact_15_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type @ A ) )
=> ! [M: num,N: num] :
( ( ( numeral_numeral @ A @ M )
= ( numeral_numeral @ A @ N ) )
= ( M = N ) ) ) ).
% numeral_eq_iff
thf(fact_16_numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [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_17_numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [M: num,N: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ M @ N ) ) ) ).
% numeral_less_iff
thf(fact_18_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_19_divide__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% divide_1
thf(fact_20_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_21_one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type @ A ) )
=> ! [N: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ N ) )
= ( one2 = N ) ) ) ).
% one_eq_numeral_iff
thf(fact_22_numeral__eq__one__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type @ A ) )
=> ! [N: num] :
( ( ( numeral_numeral @ A @ N )
= ( one_one @ A ) )
= ( N = one2 ) ) ) ).
% numeral_eq_one_iff
thf(fact_23_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_24_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_25_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B @ A2 ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B ) ) ) ) ).
% eq_divide_eq_1
thf(fact_26_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( ( divide_divide @ A @ B @ A2 )
= ( one_one @ A ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B ) ) ) ) ).
% divide_eq_eq_1
thf(fact_27_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_28_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_29_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A2 @ B ) )
= ( ( B
!= ( zero_zero @ A ) )
& ( A2 = B ) ) ) ) ).
% one_eq_divide_iff
thf(fact_30_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_31_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ( divide_divide @ A @ A2 @ B )
= ( one_one @ A ) )
= ( ( B
!= ( zero_zero @ A ) )
& ( A2 = B ) ) ) ) ).
% divide_eq_1_iff
thf(fact_32_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_33_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_34_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_35_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_36_one__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: num] :
( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ one2 @ N ) ) ) ).
% one_less_numeral_iff
thf(fact_37_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_38_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
= ( ord_less @ A @ A2 @ B ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_39_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
= ( ord_less @ A @ B @ A2 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_40_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B @ A2 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_41_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A2 @ B ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_42_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_43_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_44_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
= ( ord_less_eq @ A @ B @ A2 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_45_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B @ A2 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_46_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_47_one__div__two__eq__zero,axiom,
! [A: $tType] :
( ( semiring_div_parity @ A @ ( type @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% one_div_two_eq_zero
thf(fact_48_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_49_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_50_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_51_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_52_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_53_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_54_not__one__le__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type @ A ) )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_le_zero
thf(fact_55_zero__le__one,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_le_one
thf(fact_56_le__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(1)
thf(fact_57_le__numeral__extra_I2_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(2)
thf(fact_58_one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% one_le_numeral
thf(fact_59_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_60_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_61_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_62_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_63_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( M != N )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_64_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less @ nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_65_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_66_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_67_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_68_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_69_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_70_not__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_le_zero
thf(fact_71_zero__le__numeral,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% zero_le_numeral
thf(fact_72_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B @ C ) @ ( divide_divide @ A @ A2 @ C ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_73_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [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_74_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [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_75_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [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_76_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [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_77_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_78_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% divide_right_mono
thf(fact_79_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_80_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( P @ N )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_81_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type @ A ) )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_82_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_83_div__by__1,axiom,
! [A: $tType] :
( ( semiring_div @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% div_by_1
thf(fact_84_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ A2 @ B ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_85_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B @ A2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A2 @ B ) )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_86_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_87_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_88_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_89_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_90_semiring__numeral__div__class_Odiv__positive,axiom,
! [A: $tType] :
( ( semiring_numeral_div @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B )
=> ( ( ord_less_eq @ A @ B @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) ) ) ) ) ).
% semiring_numeral_div_class.div_positive
thf(fact_91_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ A2 @ B ) )
& ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_92_semiring__numeral__div__class_Odiv__less,axiom,
! [A: $tType] :
( ( semiring_numeral_div @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ B )
=> ( ( divide_divide @ A @ A2 @ B )
= ( zero_zero @ A ) ) ) ) ) ).
% semiring_numeral_div_class.div_less
thf(fact_93_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [X: A,Y: A,W: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_94_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [X: A,Y: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% frac_less
thf(fact_95_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [X: A,Y: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% frac_le
thf(fact_96_Divides_Odiv__positive,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) ) ) ) ).
% Divides.div_positive
thf(fact_97_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_98_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_99_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [X3: A] :
? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% linordered_field_no_ub
thf(fact_100_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [X3: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% linordered_field_no_lb
thf(fact_101_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V2: A > nat,X: A] :
( ! [X4: A] :
( ~ ( P @ X4 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X4 ) )
& ~ ( P @ Y4 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_102_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y4: A] :
( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_103_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_104_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_105_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_106_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_107_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y4: A] :
( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_108_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_109_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_110_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_111_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_112_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_113_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A @ ( type @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_114_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_115_less__numeral__extra_I2_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(2)
thf(fact_116_not__numeral__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% not_numeral_less_one
thf(fact_117_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_118_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( B
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B )
= ( one_one @ A ) )
= ( A2 = B ) ) ) ) ).
% right_inverse_eq
thf(fact_119_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_120_div__eq__dividend__iff,axiom,
! [A2: nat,B: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ( ( ( divide_divide @ nat @ A2 @ B )
= A2 )
= ( B
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_121_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ B ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B @ A2 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_122_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B @ A2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A2 @ B ) )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_123_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_124_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_125_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type @ A ) )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_126_num_Odistinct_I1_J,axiom,
! [X2: num] :
( one2
!= ( bit0 @ X2 ) ) ).
% num.distinct(1)
thf(fact_127_infinite__descent0__measure,axiom,
! [A: $tType,V2: A > nat,P: A > $o,X: A] :
( ! [X4: A] :
( ( ( V2 @ X4 )
= ( zero_zero @ nat ) )
=> ( P @ X4 ) )
=> ( ! [X4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X4 ) )
=> ( ~ ( P @ X4 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X4 ) )
& ~ ( P @ Y4 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_128_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_129_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_130_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_131_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_132_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_133_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_134_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_135_not__numeral__less__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_less_zero
thf(fact_136_zero__less__numeral,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% zero_less_numeral
thf(fact_137_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_138_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_139_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_140_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less @ A @ A2 @ B ) )
& ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B @ A2 ) )
& ( C
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_141_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_142_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [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_143_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [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_144_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [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_145_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [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_146_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% divide_numeral_1
thf(fact_147_div__eq__0__iff,axiom,
! [A2: nat,B: nat] :
( ( ( divide_divide @ nat @ A2 @ B )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ A2 @ B )
| ( B
= ( zero_zero @ nat ) ) ) ) ).
% div_eq_0_iff
thf(fact_148_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_149_zero__not__eq__two,axiom,
! [A: $tType] :
( ( semiring_div_parity @ A @ ( type @ A ) )
=> ( ( zero_zero @ A )
!= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% zero_not_eq_two
thf(fact_150_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_151_semiring__norm_I83_J,axiom,
! [N: num] :
( one2
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_152_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one2 ) ).
% semiring_norm(85)
thf(fact_153_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_154_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_155_dbl__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type @ A ) )
=> ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% dbl_simps(3)
thf(fact_156_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_157_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_158_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_159_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_160_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_161_dbl__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type @ A ) )
=> ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% dbl_simps(2)
thf(fact_162_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_163_dbl__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type @ A ) )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% dbl_simps(5)
thf(fact_164_Divides_Otransfer__nat__int__function__closures_I1_J,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ X @ Y ) ) ) ) ).
% Divides.transfer_nat_int_function_closures(1)
thf(fact_165_zdiv__mono1,axiom,
! [A2: int,A3: int,B: int] :
( ( ord_less_eq @ int @ A2 @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( divide_divide @ int @ A3 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_166_zdiv__mono2,axiom,
! [A2: int,B2: int,B: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ B2 @ B )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_167_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide @ int @ I @ K )
= ( zero_zero @ int ) )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
& ( ord_less @ int @ I @ K ) )
| ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_168_zdiv__mono1__neg,axiom,
! [A2: int,A3: int,B: int] :
( ( ord_less_eq @ int @ A2 @ A3 )
=> ( ( ord_less @ int @ B @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B ) @ ( divide_divide @ int @ A2 @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_169_zdiv__mono2__neg,axiom,
! [A2: int,B2: int,B: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ B2 @ B )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_170_div__neg__pos__less0,axiom,
! [A2: int,B: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
=> ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% div_neg_pos_less0
thf(fact_171_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_172_div__nonneg__neg__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ B @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_173_div__nonpos__pos__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_174_div__neg__neg__trivial,axiom,
! [A2: int,B: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B @ A2 )
=> ( ( divide_divide @ int @ A2 @ B )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_175_div__pos__pos__trivial,axiom,
! [A2: int,B: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ A2 @ B )
=> ( ( divide_divide @ int @ A2 @ B )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_176_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less @ int @ B @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_177_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_178_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K ) )
= ( ord_less_eq @ int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_179_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less @ int @ B @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
= ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_180_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_181_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
= ( ( ord_less_eq @ int @ B @ A2 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_182_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq @ num @ X @ one2 )
= ( X = one2 ) ) ).
% le_num_One_iff
thf(fact_183_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_184_one__reorient,axiom,
! [A: $tType] :
( ( one @ A @ ( type @ A ) )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_185_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_186_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type @ A ) )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_187_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type @ A ) )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_188_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type @ A ) )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_189_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type @ A ) )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_190_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ).
% Nat_Transfer.transfer_nat_int_function_closures(7)
thf(fact_191_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_192_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_193_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add_right_cancel
thf(fact_194_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add_left_cancel
thf(fact_195_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_cancel_right
thf(fact_196_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_cancel_left
thf(fact_197_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_198_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_199_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_200_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_201_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( ( plus_plus @ A @ B @ A2 )
= A2 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_202_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ( plus_plus @ A @ A2 @ B )
= A2 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_203_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( A2
= ( plus_plus @ A @ B @ A2 ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_204_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_205_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_left
thf(fact_206_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_right
thf(fact_207_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ! [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_208_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ! [V: num,W: num,Z: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) @ Z ) ) ) ).
% add_numeral_left
thf(fact_209_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_210_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_211_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_212_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_213_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_214_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_215_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% le_add_same_cancel1
thf(fact_216_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% le_add_same_cancel2
thf(fact_217_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_218_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_219_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_220_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_221_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% less_add_same_cancel2
thf(fact_222_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% less_add_same_cancel1
thf(fact_223_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_224_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_225_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_226_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ! [N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_227_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ! [N: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% one_plus_numeral
thf(fact_228_div__add__self2,axiom,
! [A: $tType] :
( ( semiring_div @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( B
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_229_div__add__self1,axiom,
! [A: $tType] :
( ( semiring_div @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( B
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_230_one__add__one,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% one_add_one
thf(fact_231_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= M ) ).
% add_self_div_2
thf(fact_232_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ).
% Nat_Transfer.transfer_nat_int_function_closures(6)
thf(fact_233_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% Nat_Transfer.transfer_nat_int_function_closures(5)
thf(fact_234_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type @ A ) )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X5: A] : ( plus_plus @ A @ X5 @ X5 ) ) ) ) ).
% dbl_def
thf(fact_235_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
=> ( B = C ) ) ) ).
% add_right_imp_eq
thf(fact_236_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
=> ( B = C ) ) ) ).
% add_left_imp_eq
thf(fact_237_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add.left_commute
thf(fact_238_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% add.commute
thf(fact_239_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add.right_cancel
thf(fact_240_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add.left_cancel
thf(fact_241_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add.assoc
thf(fact_242_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type @ A ) )
=> ! [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_243_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_244_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_imp_le_right
thf(fact_245_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_imp_le_left
thf(fact_246_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
? [C2: A] :
( B3
= ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% le_iff_add
thf(fact_247_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add_right_mono
thf(fact_248_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% add_left_mono
thf(fact_249_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_mono
thf(fact_250_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type @ A ) )
=> ! [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)
%----Type constructors (49)
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Divides_Osemiring__numeral__div,axiom,
semiring_numeral_div @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Divides_Osemiring__div__parity,axiom,
semiring_div_parity @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Divides_Osemiring__div,axiom,
semiring_div @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Orderings_Opreorder,axiom,
preorder @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int @ ( type @ int ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1,axiom,
ordere516151231imp_le @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_2,axiom,
ordere236663937imp_le @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_3,axiom,
linord1659791738miring @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_4,axiom,
ordere779506340up_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_5,axiom,
cancel1352612707id_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Divides_Osemiring__numeral__div_6,axiom,
semiring_numeral_div @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_7,axiom,
cancel_semigroup_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Divides_Osemiring__div__parity_8,axiom,
semiring_div_parity @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_9,axiom,
linordered_semidom @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_10,axiom,
ab_semigroup_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_11,axiom,
semidom_divide @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_12,axiom,
semigroup_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Divides_Osemiring__div_13,axiom,
semiring_div @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_14,axiom,
zero_less_one @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_15,axiom,
semiring_char_0 @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_16,axiom,
zero_neq_one @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_17,axiom,
preorder @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_18,axiom,
monoid_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Num_Onumeral_19,axiom,
numeral @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero_20,axiom,
zero @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oone_21,axiom,
one @ nat @ ( type @ nat ) ).
thf(tcon_Num_Onum___Orderings_Opreorder_22,axiom,
preorder @ num @ ( type @ num ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_23,axiom,
preorder @ $o @ ( type @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
%------------------------------------------------------------------------------