TPTP Problem File: ITP129^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP129^2 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer NthRoot_Impl problem prob_162__3274372_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : NthRoot_Impl/prob_162__3274372_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 375 ( 131 unt; 56 typ; 0 def)
% Number of atoms : 761 ( 512 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 2922 ( 77 ~; 21 |; 35 &;2522 @)
% ( 0 <=>; 267 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 26 ( 26 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 54 usr; 6 con; 0-4 aty)
% Number of variables : 766 ( 11 ^; 705 !; 1 ?; 766 :)
% ( 49 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:25:54.921
%------------------------------------------------------------------------------
% Could-be-implicit typings (2)
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
% Explicit typings (54)
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__bits,type,
semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_11004092258visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
semiri134348788visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri1923998003cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid191655569cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid1852923125th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : $o ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
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_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_NthRoot__Impl__Mirabelle__yrjxwmcnbt_Ofixed__root,type,
nthRoo220209705d_root: nat > nat > $o ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_v_n,type,
n: int ).
thf(sy_v_p,type,
p: nat ).
thf(sy_v_pm,type,
pm: nat ).
thf(sy_v_x,type,
x: int ).
% Relevant facts (253)
thf(fact_0_xn,axiom,
( ( power_power @ int @ x @ p )
= n ) ).
% xn
thf(fact_1_of__nat__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% of_nat_power
thf(fact_2_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [B2: nat,W: nat,X: nat] :
( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W )
= ( semiring_1_of_nat @ A @ X ) )
= ( ( power_power @ nat @ B2 @ W )
= X ) ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X: nat,B2: nat,W: nat] :
( ( ( semiring_1_of_nat @ A @ X )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( X
= ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_add
thf(fact_5_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mult
thf(fact_6_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C: A,A2: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C ) @ A2 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C ) ) ) ).
% times_divide_eq_left
thf(fact_7_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,C: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C )
= ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% divide_divide_eq_left
thf(fact_8_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,C: A] :
( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_9_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C ) ) ) ).
% times_divide_eq_right
thf(fact_10_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid1852923125th_nat @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( divide_divide @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_11_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_Suc
thf(fact_12_fixed__root__axioms,axiom,
nthRoo220209705d_root @ p @ pm ).
% fixed_root_axioms
thf(fact_13_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_14_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_15_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( semiring_1_of_nat @ A @ N ) )
= ( M = N ) ) ) ).
% of_nat_eq_iff
thf(fact_16_p,axiom,
( p
= ( suc @ pm ) ) ).
% p
thf(fact_17_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_18_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ M @ ( suc @ N ) )
= ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_19_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_20_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_21_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_22_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide @ nat @ M @ ( times_times @ nat @ N @ Q ) )
= ( divide_divide @ nat @ ( divide_divide @ nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_23_power__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( power_power @ A @ A2 @ ( times_times @ nat @ M @ N ) )
= ( power_power @ A @ ( power_power @ A @ A2 @ M ) @ N ) ) ) ).
% power_mult
thf(fact_24_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_25_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A2: nat] :
( ( A3
= ( plus_plus @ nat @ K @ A2 ) )
=> ( ( suc @ A3 )
= ( plus_plus @ nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_26_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_27_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M )
= ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_28_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( power_power @ A @ A2 @ ( plus_plus @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_add
thf(fact_29_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_30_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_31_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C )
= ( divide_divide @ A @ A2 @ ( times_times @ A @ C @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_32_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z: A,W: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ W ) @ ( times_times @ A @ Y @ Z ) ) ) ) ).
% divide_divide_times_eq
thf(fact_33_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z: A,W: A] :
( ( times_times @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ W ) ) ) ) ).
% times_divide_times_eq
thf(fact_34_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B2 @ C ) ) ) ) ).
% add_divide_distrib
thf(fact_35_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ Y )
= ( times_times @ A @ Y @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_36_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( times_times @ A @ A2 @ B2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_mult_distrib
thf(fact_37_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ A2 @ N ) @ A2 )
= ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_commutes
thf(fact_38_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% mult_of_nat_commute
thf(fact_39_power__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_divide
thf(fact_40_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( euclid1852923125th_nat @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
= ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_41_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ N ) @ A2 ) ) ) ).
% power_Suc2
thf(fact_42_int__ops_I7_J,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A2 @ B2 ) )
= ( times_times @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_43_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_44_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_45_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N @ M ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% int_plus
thf(fact_46_int__ops_I5_J,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A2 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_47_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add_right_cancel
thf(fact_48_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add_left_cancel
thf(fact_49_zdiv__int,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A2 @ B2 ) )
= ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zdiv_int
thf(fact_50_int__distrib_I1_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z2 ) @ W )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z2 @ W ) ) ) ).
% int_distrib(1)
thf(fact_51_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times @ int @ W @ ( plus_plus @ int @ Z1 @ Z2 ) )
= ( plus_plus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z2 ) ) ) ).
% int_distrib(2)
thf(fact_52_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_53_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ( A2 != B2 )
& ( C != D ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ D ) )
!= ( plus_plus @ A @ ( times_times @ A @ A2 @ D ) @ ( times_times @ A @ B2 @ C ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_54_fixed__root_Op,axiom,
! [P: nat,Pm: nat] :
( ( nthRoo220209705d_root @ P @ Pm )
=> ( P
= ( suc @ Pm ) ) ) ).
% fixed_root.p
thf(fact_55_fixed__root__def,axiom,
( nthRoo220209705d_root
= ( ^ [P2: nat,Pm2: nat] :
( P2
= ( suc @ Pm2 ) ) ) ) ).
% fixed_root_def
thf(fact_56_fixed__root_Ointro,axiom,
! [P: nat,Pm: nat] :
( ( P
= ( suc @ Pm ) )
=> ( nthRoo220209705d_root @ P @ Pm ) ) ).
% fixed_root.intro
thf(fact_57_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_58_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% mult.assoc
thf(fact_59_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% mult.commute
thf(fact_60_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A2: A,C: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% mult.left_commute
thf(fact_61_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_62_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ 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_63_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_64_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: A,K: A,B2: A,A2: A] :
( ( B4
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B4 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_65_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.assoc
thf(fact_66_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add.left_cancel
thf(fact_67_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add.right_cancel
thf(fact_68_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% add.commute
thf(fact_69_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.left_commute
thf(fact_70_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
=> ( B2 = C ) ) ) ).
% add_left_imp_eq
thf(fact_71_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
=> ( B2 = C ) ) ) ).
% add_right_imp_eq
thf(fact_72_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ A4 )
= ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_73_int__if,axiom,
! [P3: $o,A2: nat,B2: nat] :
( ( P3
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P3 @ A2 @ B2 ) )
= ( semiring_1_of_nat @ int @ A2 ) ) )
& ( ~ P3
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P3 @ A2 @ B2 ) )
= ( semiring_1_of_nat @ int @ B2 ) ) ) ) ).
% int_if
thf(fact_74_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ int @ M )
= ( semiring_1_of_nat @ int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_75_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A )
=> ! [W: A,Y: A,X: A,Z: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y ) @ ( times_times @ A @ X @ Z ) )
= ( plus_plus @ A @ ( times_times @ A @ W @ Z ) @ ( times_times @ A @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ) ).
% crossproduct_eq
thf(fact_76_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_77_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_78_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_79_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% distrib_left
thf(fact_80_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% distrib_right
thf(fact_81_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,E: A,B2: A,C: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E ) @ C ) ) ) ).
% combine_common_factor
thf(fact_82__092_060open_062_092_060And_062n_O_Ax_A_L_Ax_A_K_An_A_061_Ax_A_K_A_I1_A_L_An_J_092_060close_062,axiom,
! [N: int] :
( ( plus_plus @ int @ x @ ( times_times @ int @ x @ N ) )
= ( times_times @ int @ x @ ( plus_plus @ int @ ( one_one @ int ) @ N ) ) ) ).
% \<open>\<And>n. x + x * n = x * (1 + n)\<close>
thf(fact_83_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_84_mult_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult.left_neutral
thf(fact_85_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% div_by_1
thf(fact_86_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_87_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( one_one @ A ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_eq_1_iff
thf(fact_88_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( one_one @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_1_eq_iff
thf(fact_89_of__nat__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% of_nat_1
thf(fact_90_of__nat__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ M ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) ) ) ).
% of_nat_Suc
thf(fact_91_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_92_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.comm_neutral
thf(fact_93_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_94_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_95_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_one_over
thf(fact_96_int__Suc,axiom,
! [N: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ N ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ).
% int_Suc
thf(fact_97_int__ops_I4_J,axiom,
! [A2: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_98_bits__div__by__1,axiom,
! [A: $tType] :
( ( semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% bits_div_by_1
thf(fact_99_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,C: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C )
=> ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_100_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,A2: A,B2: A,C: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A2 )
=> ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_101_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_102_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_103_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_104_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_105_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_106_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_107_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_108_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_109_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_110_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ( times_times @ A @ A2 @ C )
= ( times_times @ A @ B2 @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_111_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ( times_times @ A @ C @ A2 )
= ( times_times @ A @ C @ B2 ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_112_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_113_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_114_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_115_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_116_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_117_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_118_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_119_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_120_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_121_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_122_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_123_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_124_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_125_bits__div__by__0,axiom,
! [A: $tType] :
( ( semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_126_bits__div__0,axiom,
! [A: $tType] :
( ( semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_127_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_128_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ( divide_divide @ A @ C @ A2 )
= ( divide_divide @ A @ C @ B2 ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_129_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C )
= ( divide_divide @ A @ B2 @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_130_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_131_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_132_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_133_of__nat__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_0
thf(fact_134_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( ( zero_zero @ nat )
= N ) ) ) ).
% of_nat_0_eq_iff
thf(fact_135_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_136_power__Suc0__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% power_Suc0_right
thf(fact_137_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_138_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_139_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% power_one_right
thf(fact_140_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_141_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_142_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_143_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_144_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power @ nat @ X @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_145_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_146_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_147_gbinomial__Suc0,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% gbinomial_Suc0
thf(fact_148_gbinomial__1,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% gbinomial_1
thf(fact_149_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [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_150_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C: A,B2: A] :
( ( C
= ( times_times @ A @ C @ B2 ) )
= ( ( C
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_151_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C: A,A2: A] :
( ( ( times_times @ A @ C @ A2 )
= C )
= ( ( C
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_152_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C: A,B2: A] :
( ( C
= ( times_times @ A @ B2 @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_153_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [A2: A,C: A] :
( ( ( times_times @ A @ A2 @ C )
= C )
= ( ( C
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_154_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ( C
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_155_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C: A,A2: A,B2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_156_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C: A,A2: A,B2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ B2 @ C ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_157_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C: A,A2: A,B2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_158_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C: A,A2: A,B2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ C @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_159_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid191655569cancel @ A )
=> ! [C: A,A2: A,B2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_160_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid191655569cancel @ A )
=> ! [C: A,A2: A,B2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_161_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid191655569cancel @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ( C
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_162_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_163_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_164_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_165_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A2 @ B2 ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_166_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_167_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ 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_168_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ A2 )
= ( one_one @ A ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_169_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B2 @ A2 ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_170_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_171_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_172_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_173_power__0__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
= ( zero_zero @ A ) ) ) ).
% power_0_Suc
thf(fact_174_gbinomial__0_I2_J,axiom,
! [B: $tType] :
( ( ( semiring_char_0 @ B )
& ( semidom_divide @ B ) )
=> ! [K: nat] :
( ( gbinomial @ B @ ( zero_zero @ B ) @ ( suc @ K ) )
= ( zero_zero @ B ) ) ) ).
% gbinomial_0(2)
thf(fact_175_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid191655569cancel @ A )
=> ! [B2: A,A2: A,C: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_176_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid191655569cancel @ A )
=> ! [B2: A,A2: A,C: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) @ B2 )
= ( plus_plus @ A @ C @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_177_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid191655569cancel @ A )
=> ! [B2: A,C: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C @ B2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_178_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid191655569cancel @ A )
=> ! [B2: A,C: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_179_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_180_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_181_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_182_of__nat__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat,K: nat] :
( ( semiring_1_of_nat @ A @ ( gbinomial @ nat @ N @ K ) )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% of_nat_gbinomial
thf(fact_183_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_184_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_185_power__0__left,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% power_0_left
thf(fact_186_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C: A,A2: A,B2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C )
= ( times_times @ A @ B2 @ C ) )
= ( A2 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_187_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C: A,A2: A,B2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C @ A2 )
= ( times_times @ A @ C @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_188_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_189_divisors__zero,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_190_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_191_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_192_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_193_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_194_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_195_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A )
=> ! [B2: A,A2: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_196_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_197_power__not__zero,axiom,
! [A: $tType] :
( ( semiri134348788visors @ A )
=> ! [A2: A,N: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_not_zero
thf(fact_198_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_199_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_200_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_201_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_202_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_203_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_204_nat__induct,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) )
=> ( P3 @ N ) ) ) ).
% nat_induct
thf(fact_205_diff__induct,axiom,
! [P3: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P3 @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y4: nat] : ( P3 @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P3 @ X3 @ Y4 )
=> ( P3 @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P3 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_206_zero__induct,axiom,
! [P3: nat > $o,K: nat] :
( ( P3 @ K )
=> ( ! [N2: nat] :
( ( P3 @ ( suc @ N2 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_207_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_208_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_209_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_210_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_211_old_Onat_Oinducts,axiom,
! [P3: nat > $o,Nat: nat] :
( ( P3 @ ( zero_zero @ nat ) )
=> ( ! [Nat4: nat] :
( ( P3 @ Nat4 )
=> ( P3 @ ( suc @ Nat4 ) ) )
=> ( P3 @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_212_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_213_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_214_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_215_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_216_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_217_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_218_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_219_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
= L ) ).
% plus_int_code(2)
thf(fact_220_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_221_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_222_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A )
=> ! [R: A,A2: A,B2: A,C: A,D: A] :
( ( R
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B2 )
& ( C != D ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R @ C ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R @ D ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_223_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y )
= ( divide_divide @ A @ W @ Z ) )
= ( ( times_times @ A @ X @ Z )
= ( times_times @ A @ W @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_224_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ C )
= A2 )
= ( ( ( C
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ C ) ) )
& ( ( C
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_225_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C: A] :
( ( A2
= ( divide_divide @ A @ B2 @ C ) )
= ( ( ( C
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C )
= B2 ) )
& ( ( C
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_226_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C: A,B2: A,A2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A2 @ C ) )
=> ( ( divide_divide @ A @ B2 @ C )
= A2 ) ) ) ) ).
% divide_eq_imp
thf(fact_227_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C: A,A2: A,B2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C )
= B2 )
=> ( A2
= ( divide_divide @ A @ B2 @ C ) ) ) ) ) ).
% eq_divide_imp
thf(fact_228_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C: A,B2: A,A2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C )
= A2 )
= ( B2
= ( times_times @ A @ A2 @ C ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_229_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C: A,A2: A,B2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( A2
= ( divide_divide @ A @ B2 @ C ) )
= ( ( times_times @ A @ A2 @ C )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_230_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( A2 = B2 ) ) ) ) ).
% right_inverse_eq
thf(fact_231_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_232_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_233_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_234_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_235_of__nat__neq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ N ) )
!= ( zero_zero @ A ) ) ) ).
% of_nat_neq_0
thf(fact_236_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M @ N ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_237_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_238_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_239_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_240_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ A2 @ ( gbinomial @ A @ A2 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_241_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_242_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= B2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_243_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= A2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_244_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_245_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_246_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z @ ( divide_divide @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_247_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ Y ) @ Z ) ) ) ) ).
% add_divide_eq_iff
thf(fact_248_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% divide_add_eq_iff
thf(fact_249_div__add__self1,axiom,
! [A: $tType] :
( ( euclid191655569cancel @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_250_div__add__self2,axiom,
! [A: $tType] :
( ( euclid191655569cancel @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_251_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_252_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid191655569cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
% Type constructors (62)
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid1852923125th_nat @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid191655569cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri1923998003cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri134348788visors @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri1193490041visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord581940658strict @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__bits,axiom,
semiring_bits @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
mult_zero @ int ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_1,axiom,
semiri456707255roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_2,axiom,
euclid1852923125th_nat @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_3,axiom,
euclid191655569cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_4,axiom,
semiri1923998003cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5,axiom,
semiri134348788visors @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_6,axiom,
semiri1193490041visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_7,axiom,
ordere779506340up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_8,axiom,
cancel1352612707id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_9,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_10,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_11,axiom,
comm_monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_12,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_13,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_14,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_15,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__bits_16,axiom,
semiring_bits @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_17,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_18,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_19,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_20,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_21,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_22,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_23,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_24,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_25,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Power_Opower_26,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_27,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_28,axiom,
one @ nat ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( divide_divide @ int @ ( plus_plus @ int @ ( divide_divide @ int @ ( power_power @ int @ x @ ( suc @ pm ) ) @ ( power_power @ int @ x @ pm ) ) @ ( times_times @ int @ x @ ( semiring_1_of_nat @ int @ pm ) ) ) @ ( semiring_1_of_nat @ int @ ( suc @ pm ) ) )
= x ) ).
%------------------------------------------------------------------------------