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Theorem List for Metamath Proof Explorer - 12501-12600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvdsle 12501 The divisors of a positive integer are bounded by it. The proof does not use  /. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  ||  N  ->  M  <_  N ) )
 
Theoremdvdsleabs 12502 The divisors of a nonzero integer are bounded by its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 ) 
 ->  ( M  ||  N  ->  M  <_  ( abs `  N ) ) )
 
Theoremdvdseq 12503 If two integers divide each other, they must be equal, up to a difference in sign. (Contributed by Mario Carneiro, 30-May-2014.)
 |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M 
 ||  N  /\  N  ||  M ) )  ->  M  =  N )
 
Theoremdvds1 12504 The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( M  e.  NN0  ->  ( M  ||  1  <->  M  =  1
 ) )
 
Theoremalzdvds 12505* Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  ( A. x  e. 
 ZZ  x  ||  N  <->  N  =  0 ) )
 
Theoremdvdsext 12506* Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  =  B  <->  A. x  e.  NN0  ( A  ||  x  <->  B  ||  x ) ) )
 
Theoremfzm1ndvds 12507 No number between  1 and  M  - 
1 divides  M. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  ( ( M  e.  NN  /\  N  e.  (
 1 ... ( M  -  1 ) ) ) 
 ->  -.  M  ||  N )
 
Theoremfzo0dvdseq 12508 Zero is the only one of the first 
A nonnegative integers that is divisible by  A. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( B  e.  (
 0..^ A )  ->  ( A  ||  B  <->  B  =  0
 ) )
 
Theoremfzocongeq 12509 Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( ( A  e.  ( C..^ D )  /\  B  e.  ( C..^ D ) )  ->  ( ( D  -  C )  ||  ( A  -  B )  <->  A  =  B ) )
 
Theoremdvdsfac 12510 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
 |-  ( ( K  e.  NN  /\  N  e.  ( ZZ>=
 `  K ) ) 
 ->  K  ||  ( ! `  N ) )
 
Theoremdvdsexp 12511 A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( A  e.  ZZ  /\  M  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  ->  ( A ^ M ) 
 ||  ( A ^ N ) )
 
Theoremdvdsmod 12512 Any number  K whose mod base  N is divisible by a divisor  P of the base is also divisible by  P. This means that primes will also be relatively prime to the base when reduced  mod  N for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P 
 ||  ( K  mod  N )  <->  P  ||  K ) )
 
Theoremodd2np1lem 12513* Lemma for odd2np1 12514. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( N  e.  NN0  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N  \/  E. k  e.  ZZ  (
 k  x.  2 )  =  N ) )
 
Theoremodd2np1 12514* An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( N  e.  ZZ  ->  ( -.  2  ||  N 
 <-> 
 E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
 
Theoremoddm1even 12515 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  ( -.  2  ||  N 
 <->  2  ||  ( N  -  1 ) ) )
 
Theoremoddp1even 12516 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  ( -.  2  ||  N 
 <->  2  ||  ( N  +  1 ) ) )
 
Theoremoexpneg 12517 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)
 |-  ( ( A  e.  CC  /\  N  e.  NN  /\ 
 -.  2  ||  N )  ->  ( -u A ^ N )  =  -u ( A ^ N ) )
 
Theorem3dvds 12518* A rule for divisibility by 3 of a number written in base 10. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.)
 |-  ( ( N  e.  NN0  /\  F : ( 0
 ... N ) --> ZZ )  ->  ( 3  ||  sum_ k  e.  ( 0 ... N ) ( ( F `
  k )  x.  ( 10 ^ k
 ) )  <->  3  ||  sum_ k  e.  ( 0 ... N ) ( F `  k ) ) )
 
6.1.4  The division algorithm
 
Theoremdivalglem0 12519 Lemma for divalg 12529. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   =>    |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( K  x.  ( abs `  D ) ) ) ) ) )
 
Theoremdivalglem1 12520 Lemma for divalg 12529. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   =>    |-  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) )
 
Theoremdivalglem2 12521* Lemma for divalg 12529. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   &    |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }   =>    |-  sup ( S ,  RR ,  `'  <  )  e.  S
 
Theoremdivalglem4 12522* Lemma for divalg 12529. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   &    |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }   =>    |-  S  =  {
 r  e.  NN0  |  E. q  e.  ZZ  N  =  ( (
 q  x.  D )  +  r ) }
 
Theoremdivalglem5 12523* Lemma for divalg 12529. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   &    |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }   &    |-  R  =  sup ( S ,  RR ,  `'  <  )   =>    |-  (
 0  <_  R  /\  R  <  ( abs `  D ) )
 
Theoremdivalglem6 12524 Lemma for divalg 12529. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  A  e.  NN   &    |-  X  e.  ( 0 ... ( A  -  1 ) )   &    |-  K  e.  ZZ   =>    |-  ( K  =/=  0  ->  -.  ( X  +  ( K  x.  A ) )  e.  (
 0 ... ( A  -  1 ) ) )
 
Theoremdivalglem7 12525 Lemma for divalg 12529. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  D  e.  ZZ   &    |-  D  =/=  0   =>    |-  ( ( X  e.  ( 0 ... (
 ( abs `  D )  -  1 ) ) 
 /\  K  e.  ZZ )  ->  ( K  =/=  0  ->  -.  ( X  +  ( K  x.  ( abs `  D ) ) )  e.  ( 0
 ... ( ( abs `  D )  -  1
 ) ) ) )
 
Theoremdivalglem8 12526* Lemma for divalg 12529. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   &    |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }   =>    |-  ( ( ( X  e.  S  /\  Y  e.  S )  /\  ( X  <  ( abs `  D )  /\  Y  <  ( abs `  D ) ) )  ->  ( K  e.  ZZ  ->  ( ( K  x.  ( abs `  D )
 )  =  ( Y  -  X )  ->  X  =  Y )
 ) )
 
Theoremdivalglem9 12527* Lemma for divalg 12529. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   &    |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }   &    |-  R  =  sup ( S ,  RR ,  `'  <  )   =>    |-  E! x  e.  S  x  <  ( abs `  D )
 
Theoremdivalglem10 12528* Lemma for divalg 12529. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   &    |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }   =>    |-  E! r  e. 
 ZZ  E. q  e.  ZZ  ( 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r
 ) )
 
Theoremdivalg 12529* The division algorithm (theorem). Dividing an integer  N by a nonzero integer  D produces a (unique) quotient  q and a unique remainder  0  <_  r  <  ( abs `  D
). The proof does not use  /,  |_ or  mod. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 ) 
 ->  E! r  e.  ZZ  E. q  e.  ZZ  (
 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r )
 ) )
 
Theoremdivalgb 12530* Express the division algorithm as stated in divalg 12529 in terms of  ||. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 ) 
 ->  ( E! r  e. 
 ZZ  E. q  e.  ZZ  ( 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r
 ) )  <->  E! r  e.  NN0  ( r  <  ( abs `  D )  /\  D  ||  ( N  -  r
 ) ) ) )
 
Theoremdivalg2 12531* The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E! r  e. 
 NN0  ( r  <  D  /\  D  ||  ( N  -  r ) ) )
 
Theoremdivalgmod 12532* The result of the  mod operator satisfies the requirements for the remainder  r in the division algorithm for a positive divisor (compare divalg2 12531 and divalgb 12530). This demonstration theorem justifies the use of  mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( r  =  ( N  mod  D ) 
 <->  ( r  e.  NN0  /\  ( r  <  D  /\  D  ||  ( N  -  r ) ) ) ) )
 
Theoremndvdssub 12533 Corollary of the division algorithm. If an integer  D greater than  1 divides  N, then it does not divide any of  N  -  1,  N  -  2...  N  -  ( D  -  1 ). (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  ->  ( D  ||  N  ->  -.  D  ||  ( N  -  K ) ) )
 
Theoremndvdsadd 12534 Corollary of the division algorithm. If an integer  D greater than  1 divides  N, then it does not divide any of  N  +  1,  N  +  2...  N  +  ( D  -  1 ). (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  ->  ( D  ||  N  ->  -.  D  ||  ( N  +  K ) ) )
 
Theoremndvdsp1 12535 Special case of ndvdsadd 12534. If an integer  D greater than  1 divides  N, it does not divide  N  +  1. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  1  <  D ) 
 ->  ( D  ||  N  ->  -.  D  ||  ( N  +  1 )
 ) )
 
Theoremndvdsi 12536 A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN   &    |-  Q  e.  NN0   &    |-  R  e.  NN   &    |-  (
 ( A  x.  Q )  +  R )  =  B   &    |-  R  <  A   =>    |-  -.  A  ||  B
 
6.1.5  Bit sequences
 
Syntaxcbits 12537 Define the binary bits of an integer.
 class bits
 
Syntaxcsad 12538 Define the sequence addition on bit sequences.
 class sadd
 
Syntaxcsmu 12539 Define the sequence multiplication on bit sequences.
 class smul
 
Definitiondf-bits 12540* Define the binary bits of an integer. The expression  M  e.  (bits `  N ) means that the  M-th bit of  N is 1 (and its negation means the bit is 0). (Contributed by Mario Carneiro, 4-Sep-2016.)
 |- bits  =  ( n  e.  ZZ  |->  { m  e.  NN0  |  -.  2  ||  ( |_ `  ( n  /  (
 2 ^ m ) ) ) } )
 
Theorembitsfval 12541* Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  (bits `  N )  =  { m  e.  NN0  |  -.  2  ||  ( |_ `  ( N  /  ( 2 ^ m ) ) ) }
 )
 
Theorembitsval 12542 Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( M  e.  (bits `  N )  <->  ( N  e.  ZZ  /\  M  e.  NN0  /\ 
 -.  2  ||  ( |_ `  ( N  /  ( 2 ^ M ) ) ) ) )
 
Theorembitsval2 12543 Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( M  e.  (bits `  N )  <->  -.  2  ||  ( |_ `  ( N  /  ( 2 ^ M ) ) ) ) )
 
Theorembitsss 12544 The set of bits of an integer is a subset of  NN0. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  (bits `  N )  C_ 
 NN0
 
Theorembitsf 12545 The bits function is a function from integers to subsets of nonnegative integers. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |- bits : ZZ --> ~P NN0
 
Theorembits0 12546 Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  ( 0  e.  (bits `  N )  <->  -.  2  ||  N ) )
 
Theorembits0e 12547 The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  -.  0  e.  (bits `  ( 2  x.  N ) ) )
 
Theorembits0o 12548 The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  0  e.  (bits `  ( ( 2  x.  N )  +  1 ) ) )
 
Theorembitsp1 12549 The  M  +  1-th bit of  N is the  M-th bit of  |_ ( N  / 
2 ). (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( ( M  +  1 )  e.  (bits `  N )  <->  M  e.  (bits `  ( |_ `  ( N  / 
 2 ) ) ) ) )
 
Theorembitsp1e 12550 The  M  +  1-th bit of  2 N is the  M-th bit of  N. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( ( M  +  1 )  e.  (bits `  ( 2  x.  N ) )  <->  M  e.  (bits `  N ) ) )
 
Theorembitsp1o 12551 The  M  +  1-th bit of  2 N  +  1 is the  M-th bit of  N. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( ( M  +  1 )  e.  (bits `  ( (
 2  x.  N )  +  1 ) )  <->  M  e.  (bits `  N ) ) )
 
Theorembitsfzolem 12552* Lemma for bitsfzo 12553. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  (bits `  N )  C_  (
 0..^ M ) )   &    |-  S  =  sup ( { n  e.  NN0  |  N  <  ( 2 ^ n ) } ,  RR ,  `'  <  )   =>    |-  ( ph  ->  N  e.  ( 0..^ ( 2 ^ M ) ) )
 
Theorembitsfzo 12553 The bits of a number are all less than  M iff the number is nonnegative and less than  2 ^ M. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( N  e.  ( 0..^ ( 2 ^ M ) )  <->  (bits `  N )  C_  ( 0..^ M ) ) )
 
Theorembitsmod 12554 Truncating the bit sequence after some  M is equivalent to reducing the argument  mod  2 ^ M. (Contributed by Mario Carneiro, 6-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  (bits `  ( N  mod  ( 2 ^ M ) ) )  =  ( (bits `  N )  i^i  ( 0..^ M ) ) )
 
Theorembitsfi 12555 Every number is associated to a finite set of bits. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  NN0  ->  (bits `  N )  e. 
 Fin )
 
Theorembitscmp 12556 The bit complement of  N is  -u N  - 
1. (Thus, by bitsfi 12555, all negative numbers have cofinite bits representations.) (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  ( NN0  \  (bits `  N ) )  =  (bits `  ( -u N  -  1 ) ) )
 
Theorem0bits 12557 The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.)
 |-  (bits `  0 )  =  (/)
 
Theoremm1bits 12558 The bits of negative one. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  (bits `  -u 1 )  =  NN0
 
Theorembitsinv1lem 12559 Lemma for bitsinv1 12560. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( N  mod  ( 2 ^ ( M  +  1 )
 ) )  =  ( ( N  mod  (
 2 ^ M ) )  +  if ( M  e.  (bits `  N ) ,  ( 2 ^ M ) ,  0 ) ) )
 
Theorembitsinv1 12560* There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 12556), part 1. (Contributed by Mario Carneiro, 7-Sep-2016.)
 |-  ( N  e.  NN0  ->  sum_ n  e.  (bits `  N ) ( 2 ^ n )  =  N )
 
Theorembitsinv2 12561* There is an explicit inverse to the bits function for nonnegative integers, part 2. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( A  e.  ( ~P NN0  i^i  Fin )  ->  (bits `  sum_ n  e.  A  ( 2 ^ n ) )  =  A )
 
Theorembitsf1ocnv 12562* The bits function restricted to nonnegative integers is a bijection from the integers to the finite sets of integers. It is in fact the inverse of the Ackermann bijection ackbijnn 12216. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( (bits  |`  NN0 ) : NN0
 -1-1-onto-> ( ~P NN0  i^i  Fin )  /\  `' (bits  |`  NN0 )  =  ( x  e.  ( ~P
 NN0  i^i  Fin )  |->  sum_ n  e.  x  ( 2 ^ n ) ) )
 
Theorembitsf1o 12563 The bits function restricted to nonnegative integers is a bijection from the integers to the finite sets of integers. It is in fact the inverse of the Ackermann bijection ackbijnn 12216. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  (bits  |`  NN0 ) : NN0 -1-1-onto-> ( ~P NN0  i^i  Fin )
 
Theorembitsf1 12564 The bits function is an injection from  ZZ to  ~P NN0. It is obviously not a bijection (by Cantor's theorem canth2 6947), and in fact its range is the set of finite and cofinite subsets of  NN0. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |- bits : ZZ -1-1-> ~P NN0
 
Theorem2ebits 12565 The bits of a power of two. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  NN0  ->  (bits `  ( 2 ^ N ) )  =  { N } )
 
Theorembitsinv 12566* The inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  K  =  `' (bits  |` 
 NN0 )   =>    |-  ( A  e.  ( ~P NN0  i^i  Fin )  ->  ( K `  A )  =  sum_ k  e.  A  ( 2 ^ k
 ) )
 
Theorembitsinvp1 12567 Recursive definition of the inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  K  =  `' (bits  |` 
 NN0 )   =>    |-  ( ( A  C_  NN0  /\  N  e.  NN0 )  ->  ( K `  ( A  i^i  ( 0..^ ( N  +  1 ) ) ) )  =  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  if ( N  e.  A ,  (
 2 ^ N ) ,  0 ) ) )
 
Theoremsadadd2lem2 12568 The core of the proof of sadadd2 12578. The intuitive justification for this is that cadd is true if at least two arguments are true, and hadd is true if an odd number of arguments are true, so altogether the result is  n  x.  A where  n is the number of true arguments, which is equivalently obtained by adding together one  A for each true argument, on the right side. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( A  e.  CC  ->  ( if (hadd ( ph ,  ps ,  ch ) ,  A , 
 0 )  +  if (cadd ( ph ,  ps ,  ch ) ,  (
 2  x.  A ) ,  0 ) )  =  ( ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  if ( ch ,  A ,  0 ) ) )
 
Definitiondf-sad 12569* Define the addition of two bit sequences, using df-had 1376 and df-cad 1377 bit operations. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |- sadd  =  ( x  e.  ~P NN0
 ,  y  e.  ~P NN0  |->  { k  e.  NN0  | hadd ( k  e.  x ,  k  e.  y ,  (/)  e.  (  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  x ,  m  e.  y ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) ) `  k ) ) } )
 
Theoremsadfval 12570* Define the addition of two bit sequences, using df-had 1376 and df-cad 1377 bit operations. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  ( A sadd  B )  =  {
 k  e.  NN0  | hadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `
  k ) ) } )
 
Theoremsadcf 12571* The carry sequence is a sequence of elements of  2o encoding a "sequence of wffs". (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  C : NN0 --> 2o )
 
Theoremsadc0 12572* The initial element of the carry sequence is  F.. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  -.  (/)  e.  ( C `  0 ) )
 
Theoremsadcp1 12573* The carry sequence (which is a sequence of wffs, encoded as  1o and  (/)) is defined recursively as the carry operation applied to the previous carry and the two current inputs. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( (/) 
 e.  ( C `  ( N  +  1
 ) )  <-> cadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
  N ) ) ) )
 
Theoremsadval 12574* The full adder sequence is the half adder function applied to the inputs and the carry sequence. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( N  e.  ( A sadd  B )  <-> hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
  N ) ) ) )
 
Theoremsadcaddlem 12575* Lemma for sadcadd 12576. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   &    |-  ( ph  ->  ( (/)  e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) ) )   =>    |-  ( ph  ->  ( (/) 
 e.  ( C `  ( N  +  1
 ) )  <->  ( 2 ^
 ( N  +  1 ) )  <_  (
 ( K `  ( A  i^i  ( 0..^ ( N  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( N  +  1 ) ) ) ) ) ) )
 
Theoremsadcadd 12576* Non-recursive definition of the carry sequence. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   =>    |-  ( ph  ->  ( (/) 
 e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) ) )
 
Theoremsadadd2lem 12577* Lemma for sadadd2 12578. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   &    |-  ( ph  ->  ( ( K `  (
 ( A sadd  B )  i^i  ( 0..^ N ) ) )  +  if ( (/)  e.  ( C `
  N ) ,  ( 2 ^ N ) ,  0 )
 )  =  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) )   =>    |-  ( ph  ->  (
 ( K `  (
 ( A sadd  B )  i^i  ( 0..^ ( N  +  1 ) ) ) )  +  if ( (/)  e.  ( C `
  ( N  +  1 ) ) ,  ( 2 ^ ( N  +  1 )
 ) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ ( N  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( N  +  1 ) ) ) ) ) )
 
Theoremsadadd2 12578* Sum of initial segments of the sadd sequence. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   =>    |-  ( ph  ->  (
 ( K `  (
 ( A sadd  B )  i^i  ( 0..^ N ) ) )  +  if ( (/)  e.  ( C `
  N ) ,  ( 2 ^ N ) ,  0 )
 )  =  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) )
 
Theoremsadadd3 12579* Sum of initial segments of the sadd sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   =>    |-  ( ph  ->  (
 ( K `  (
 ( A sadd  B )  i^i  ( 0..^ N ) ) )  mod  (
 2 ^ N ) )  =  ( ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) 
 mod  ( 2 ^ N ) ) )
 
Theoremsadcl 12580 The sum of two sequences is a sequence. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( A  C_  NN0  /\  B  C_  NN0 )  ->  ( A sadd  B )  C_  NN0 )
 
Theoremsadcom 12581 The adder sequence function is commutative. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( A  C_  NN0  /\  B  C_  NN0 )  ->  ( A sadd  B )  =  ( B sadd  A ) )
 
Theoremsaddisjlem 12582* Lemma for sadadd 12585. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( N  e.  ( A sadd  B )  <->  N  e.  ( A  u.  B ) ) )
 
Theoremsaddisj 12583 The sum of disjoint sequences is the union of the sequences. (In this case, there are no carried bits.) (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   =>    |-  ( ph  ->  ( A sadd  B )  =  ( A  u.  B ) )
 
Theoremsadaddlem 12584* Lemma for sadadd 12585. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  C  =  seq  0
 ( ( c  e. 
 2o ,  m  e. 
 NN0  |->  if (cadd ( m  e.  (bits `  A ) ,  m  e.  (bits `  B ) ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e. 
 NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   &    |-  K  =  `' (bits  |`  NN0 )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( (bits `  A ) sadd  (bits `  B )
 )  i^i  ( 0..^ N ) )  =  (bits `  ( ( A  +  B )  mod  ( 2 ^ N ) ) ) )
 
Theoremsadadd 12585 For sequences that correspond to valid integers, the adder sequence function produces the sequence for the sum. This is effectively a proof of the correctness of the ripple carry adder, implemented with logic gates corresponding to df-had 1376 and df-cad 1377.

It is interesting to consider in what sense the sadd function can be said to be "adding" things outside the range of the bits function, that is, when adding sequences that are not eventually constant and so do not denote any integer. The correct interpretation is that the sequences are representations of 2-adic integers, which have a natural ring structure. (Contributed by Mario Carneiro, 9-Sep-2016.)

 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( (bits `  A ) sadd  (bits `  B ) )  =  (bits `  ( A  +  B ) ) )
 
Theoremsadid1 12586 The adder sequence function has a left identity, the empty set, which is the representation of the integer zero. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( A sadd  (/) )  =  A )
 
Theoremsadid2 12587 The adder sequence function has a right identity, the empty set, which is the representation of the integer zero. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( (/) sadd  A )  =  A )
 
Theoremsadasslem 12588 Lemma for sadass 12589. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  C 
 C_  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( ( A sadd  B ) sadd  C )  i^i  (
 0..^ N ) )  =  ( ( A sadd 
 ( B sadd  C )
 )  i^i  ( 0..^ N ) ) )
 
Theoremsadass 12589 Sequence addition is associative. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_ 
 NN0 )  ->  (
 ( A sadd  B ) sadd  C )  =  ( A sadd 
 ( B sadd  C )
 ) )
 
Theoremsadeq 12590 Any element of a sequence sum only depends on the values of the argument sequences up to and including that point. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( A sadd  B )  i^i  ( 0..^ N ) )  =  ( ( ( A  i^i  (
 0..^ N ) ) sadd 
 ( B  i^i  (
 0..^ N ) ) )  i^i  ( 0..^ N ) ) )
 
Theorembitsres 12591 Restrict the bits of a number to an upper integer set. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( (bits `  A )  i^i  ( ZZ>= `  N ) )  =  (bits `  ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) ) ) )
 
Theorembitsuz 12592 The bits of a number are all at least  N iff the number is divisible by  2 ^ N. (Contributed by Mario Carneiro, 21-Sep-2016.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( ( 2 ^ N )  ||  A 
 <->  (bits `  A )  C_  ( ZZ>= `  N )
 ) )
 
Theorembitsshft 12593* Shifting a bit sequence to the left (toward the more significant bits) causes the number to be multiplied by a power of two. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  { n  e. 
 NN0  |  ( n  -  N )  e.  (bits `  A ) }  =  (bits `  ( A  x.  ( 2 ^ N ) ) ) )
 
Definitiondf-smu 12594* Define the multiplication of two bit sequences, using repeated sequence addition. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |- smul  =  ( x  e.  ~P NN0
 ,  y  e.  ~P NN0  |->  { k  e.  NN0  |  k  e.  (  seq  0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  x  /\  ( n  -  m )  e.  y ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ) `  (
 k  +  1 ) ) } )
 
Theoremsmufval 12595* Define the addition of two bit sequences, using df-had 1376 and df-cad 1377 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  P  =  seq  0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  ( A smul  B )  =  {
 k  e.  NN0  |  k  e.  ( P `  ( k  +  1 ) ) } )
 
Theoremsmupf 12596* The sequence of partial sums of the sequence multiplication. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  P  =  seq  0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  P : NN0 --> ~P NN0 )
 
Theoremsmup0 12597* The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  P  =  seq  0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  ( P `  0 )  =  (/) )
 
Theoremsmupp1 12598* The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  P  =  seq  0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( P `  ( N  +  1 ) )  =  ( ( P `  N ) sadd  { n  e. 
 NN0  |  ( N  e.  A  /\  ( n  -  N )  e.  B ) } )
 )
 
Theoremsmuval 12599* Define the addition of two bit sequences, using df-had 1376 and df-cad 1377 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  P  =  seq  0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( N  e.  ( A smul  B )  <->  N  e.  ( P `  ( N  +  1 ) ) ) )
 
Theoremsmuval2 12600* The partial sum sequence stabilizes at  N after the  N  +  1-th element of the sequence; this stable value is the value of the sequence multiplication. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  P  =  seq  0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  ( N  +  1 )
 ) )   =>    |-  ( ph  ->  ( N  e.  ( A smul  B )  <->  N  e.  ( P `  M ) ) )
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