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Theorem List for Metamath Proof Explorer - 12601-12700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsadcl 12601 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 12602 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 12603* Lemma for sadadd 12606. (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 12604 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 12605* Lemma for sadadd 12606. (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 12606 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 12607 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 12608 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 12609 Lemma for sadass 12610. (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 12610 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 12611 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 12612 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 12613 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 12614* 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 12615* 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 12616* 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 12617* 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 12618* 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 12619* 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 12620* 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 12621* 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 ) ) )
 
Theoremsmupvallem 12622* If  A only has elements less than  N, then all elements of the partial sum sequence past  N already equal the final value. (Contributed by Mario Carneiro, 20-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  ->  A 
 C_  ( 0..^ N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  N ) )   =>    |-  ( ph  ->  ( P `  M )  =  ( A smul  B ) )
 
Theoremsmucl 12623 The product of two sequences is a sequence. (Contributed by Mario Carneiro, 19-Sep-2016.)
 |-  ( ( A  C_  NN0  /\  B  C_  NN0 )  ->  ( A smul  B )  C_  NN0 )
 
Theoremsmu01lem 12624* Lemma for smu01 12625 and smu02 12626. (Contributed by Mario Carneiro, 19-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ( ph  /\  ( k  e.  NN0  /\  n  e.  NN0 )
 )  ->  -.  (
 k  e.  A  /\  ( n  -  k
 )  e.  B ) )   =>    |-  ( ph  ->  ( A smul  B )  =  (/) )
 
Theoremsmu01 12625 Multiplication of a sequence by  0 on the right. (Contributed by Mario Carneiro, 19-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( A smul  (/) )  =  (/) )
 
Theoremsmu02 12626 Multiplication of a sequence by  0 on the left. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( (/) smul  A )  =  (/) )
 
Theoremsmupval 12627* Rewrite the elements of the partial sum sequence in terms of sequence multiplication. (Contributed by Mario Carneiro, 20-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 )  =  ( ( A  i^i  ( 0..^ N ) ) smul 
 B ) )
 
Theoremsmup1 12628* Rewrite smupp1 12619 using only smul instead of the internal recursive function  P. (Contributed by Mario Carneiro, 20-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( A  i^i  (
 0..^ ( N  +  1 ) ) ) smul 
 B )  =  ( ( ( A  i^i  ( 0..^ N ) ) smul 
 B ) sadd  { n  e.  NN0  |  ( N  e.  A  /\  ( n  -  N )  e.  B ) } )
 )
 
Theoremsmueqlem 12629* Any element of a sequence multiplication only depends on the values of the argument sequences up to and including that point. (Contributed by Mario Carneiro, 20-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  N  e.  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 ) ) ) )   &    |-  Q  =  seq  0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  ( B  i^i  ( 0..^ N ) ) ) } )
 ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  (
 ( A smul  B )  i^i  ( 0..^ N ) )  =  ( ( ( A  i^i  (
 0..^ N ) ) smul 
 ( B  i^i  (
 0..^ N ) ) )  i^i  ( 0..^ N ) ) )
 
Theoremsmueq 12630 Any element of a sequence multiplication only depends on the values of the argument sequences up to and including that point. (Contributed by Mario Carneiro, 20-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( A smul  B )  i^i  ( 0..^ N ) )  =  ( ( ( A  i^i  (
 0..^ N ) ) smul 
 ( B  i^i  (
 0..^ N ) ) )  i^i  ( 0..^ N ) ) )
 
Theoremsmumullem 12631 Lemma for smumul 12632. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( (bits `  A )  i^i  ( 0..^ N ) ) smul  (bits `  B ) )  =  (bits `  ( ( A  mod  ( 2 ^ N ) )  x.  B ) ) )
 
Theoremsmumul 12632 For sequences that correspond to valid integers, the sequence multiplication function produces the sequence for the product. This is effectively a proof of the correctness of the multiplication process, implemented in terms of logic gates for df-sad 12590, whose correctness is verified in sadadd 12606.

Outside this range, the sequences cannot be representing integers, but the smul function still "works". This extended function is best interpreted in terms of the ring structure of the 2-adic integers. (Contributed by Mario Carneiro, 22-Sep-2016.)

 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( (bits `  A ) smul  (bits `  B ) )  =  (bits `  ( A  x.  B ) ) )
 
6.1.6  The greatest common divisor operator
 
Syntaxcgcd 12633 Extend the definition of a class to include the greatest common divisor operator.
 class  gcd
 
Definitiondf-gcd 12634* Define the  gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
 |- 
 gcd  =  ( x  e.  ZZ ,  y  e. 
 ZZ  |->  if ( ( x  =  0  /\  y  =  0 ) ,  0 ,  sup ( { n  e.  ZZ  |  ( n  ||  x  /\  n  ||  y ) } ,  RR ,  <  ) ) )
 
Theoremgcdval 12635* The value of the  gcd operator.  ( M  gcd  N ) is the greatest common divisor of  M and  N. If  M and  N are both  0, the result is defined conventionally as  0. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  =  if (
 ( M  =  0 
 /\  N  =  0 ) ,  0 , 
 sup ( { n  e.  ZZ  |  ( n 
 ||  M  /\  n  ||  N ) } ,  RR ,  <  ) ) )
 
Theoremgcd0val 12636 The value, by convention, of the 
gcd operator when both operands are 0. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( 0  gcd  0
 )  =  0
 
Theoremgcdn0val 12637* The value of the  gcd operator when at least one operand is nonzero. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N ) } ,  RR ,  <  ) )
 
Theoremgcdcllem1 12638* Lemma for gcdn0cl 12641, gcddvds 12642 and dvdslegcd 12643. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  S  =  { z  e.  ZZ  |  A. n  e.  A  z  ||  n }   =>    |-  ( ( A  C_  ZZ  /\  E. n  e.  A  n  =/=  0
 )  ->  ( S  =/= 
 (/)  /\  E. x  e. 
 ZZ  A. y  e.  S  y  <_  x ) )
 
Theoremgcdcllem2 12639* Lemma for gcdn0cl 12641, gcddvds 12642 and dvdslegcd 12643. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  S  =  { z  e.  ZZ  |  A. n  e.  { M ,  N } z  ||  n }   &    |-  R  =  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N ) }   =>    |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  R  =  S )
 
Theoremgcdcllem3 12640* Lemma for gcdn0cl 12641, gcddvds 12642 and dvdslegcd 12643. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  S  =  { z  e.  ZZ  |  A. n  e.  { M ,  N } z  ||  n }   &    |-  R  =  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N ) }   =>    |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( sup ( R ,  RR ,  <  )  e.  NN  /\  ( sup ( R ,  RR ,  <  )  ||  M  /\  sup ( R ,  RR ,  <  ) 
 ||  N )  /\  ( ( K  e.  ZZ  /\  K  ||  M  /\  K  ||  N )  ->  K  <_  sup ( R ,  RR ,  <  ) ) ) )
 
Theoremgcdn0cl 12641 Closure of the  gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  e.  NN )
 
Theoremgcddvds 12642 The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M 
 gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N ) )
 
Theoremdvdslegcd 12643 An integer which divides both operands of the  gcd operator is bounded by it. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 )
 )  ->  ( ( K  ||  M  /\  K  ||  N )  ->  K  <_  ( M  gcd  N ) ) )
 
Theoremgcdcl 12644 Closure of the  gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  NN0 )
 
Theoremgcdcld 12645 Closure of the  gcd operator. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( M  gcd  N )  e.  NN0 )
 
Theoremgcdf 12646 Domain and codomain of the  gcd operator. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 16-Nov-2013.)
 |- 
 gcd  : ( ZZ  X.  ZZ ) --> NN0
 
Theoremgcdcom 12647 The  gcd operator is commutative. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  =  ( N 
 gcd  M ) )
 
Theoremgcdeq0 12648 The gcd of two integers is zero iff they are both zero. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M 
 gcd  N )  =  0  <-> 
 ( M  =  0 
 /\  N  =  0 ) ) )
 
Theoremgcdn0gt0 12649 The gcd of two integers is positive (nonzero) iff they are not both zero. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  N  =  0 )  <->  0  <  ( M  gcd  N ) ) )
 
Theoremgcd0id 12650 The gcd of 0 and an integer is the integer's absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  ( 0  gcd  N )  =  ( abs `  N ) )
 
Theoremgcdid0 12651 The gcd of an integer and 0 is the integer's absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( N  e.  ZZ  ->  ( N  gcd  0
 )  =  ( abs `  N ) )
 
Theoremnn0gcdid0 12652 The gcd of a nonnegative integer with 0 is itself. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( N  e.  NN0  ->  ( N  gcd  0 )  =  N )
 
Theoremgcdneg 12653 Negating one operand of the  gcd operator does not alter the result. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  -u N )  =  ( M  gcd  N ) )
 
Theoremneggcd 12654 Negating one operand of the  gcd operator does not alter the result. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M  gcd  N )  =  ( M  gcd  N ) )
 
Theoremgcdaddmlem 12655 Lemma for gcdaddm 12656. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  K  e.  ZZ   &    |-  M  e.  ZZ   &    |-  N  e.  ZZ   =>    |-  ( M  gcd  N )  =  ( M  gcd  (
 ( K  x.  M )  +  N )
 )
 
Theoremgcdaddm 12656 Adding a multiple of one operand of the  gcd operator to the other does not alter the result. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  =  ( M  gcd  ( N  +  ( K  x.  M ) ) ) )
 
Theoremgcdadd 12657 The GCD of two numbers is the same as the GCD of the left and their sum. (Contributed by Scott Fenton, 20-Apr-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  =  ( M 
 gcd  ( N  +  M ) ) )
 
Theoremgcdid 12658 The gcd of a number and itself is its absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( N  e.  ZZ  ->  ( N  gcd  N )  =  ( abs `  N ) )
 
Theoremgcd1 12659 The gcd of a number with 1 is 1. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  ( M  e.  ZZ  ->  ( M  gcd  1
 )  =  1 )
 
Theoremgcdabs 12660 The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M )  gcd  ( abs `  N ) )  =  ( M  gcd  N ) )
 
Theoremgcdabs1 12661  gcd of the absolute value of the first operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( abs `  N )  gcd  M )  =  ( N  gcd  M ) )
 
Theoremgcdabs2 12662  gcd of the absolute value of the second operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  ( abs `  M )
 )  =  ( N 
 gcd  M ) )
 
Theoremmodgcd 12663 The gcd remains unchanged if one operand is replaced with its remainder modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M 
 mod  N )  gcd  N )  =  ( M  gcd  N ) )
 
Theorem1gcd 12664 The GCD of one and an integer is one. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( M  e.  ZZ  ->  ( 1  gcd  M )  =  1 )
 
6.1.7  Bézout's identity
 
Theorembezoutlem1 12665* Lemma for bezout 12669. (Contributed by Mario Carneiro, 15-Mar-2014.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   =>    |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A )  e.  M ) )
 
Theorembezoutlem2 12666* Lemma for bezout 12669. (Contributed by Mario Carneiro, 15-Mar-2014.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  G  =  sup ( M ,  RR ,  `'  <  )   &    |-  ( ph  ->  -.  ( A  =  0 
 /\  B  =  0 ) )   =>    |-  ( ph  ->  G  e.  M )
 
Theorembezoutlem3 12667* Lemma for bezout 12669. (Contributed by Mario Carneiro, 22-Feb-2014.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  G  =  sup ( M ,  RR ,  `'  <  )   &    |-  ( ph  ->  -.  ( A  =  0 
 /\  B  =  0 ) )   =>    |-  ( ph  ->  ( C  e.  M  ->  G 
 ||  C ) )
 
Theorembezoutlem4 12668* Lemma for bezout 12669. (Contributed by Mario Carneiro, 22-Feb-2014.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  G  =  sup ( M ,  RR ,  `'  <  )   &    |-  ( ph  ->  -.  ( A  =  0 
 /\  B  =  0 ) )   =>    |-  ( ph  ->  ( A  gcd  B )  e.  M )
 
Theorembezout 12669* Bézout's identity: For any integers  A and 
B, there are integers  x ,  y such that  ( A  gcd  B )  =  A  x.  x  +  B  x.  y. (Contributed by Mario Carneiro, 22-Feb-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( A  gcd  B )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
 
Theoremdvdsgcd 12670 An integer which divides each of two others also divides their gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 30-May-2014.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  M  /\  K  ||  N )  ->  K  ||  ( M  gcd  N ) ) )
 
Theoremdvdsgcdb 12671 Biconditional form of dvdsgcd 12670. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  M  /\  K  ||  N ) 
 <->  K  ||  ( M  gcd  N ) ) )
 
Theoremgcdass 12672 Associative law for  gcd operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( N  gcd  M )  gcd  P )  =  ( N  gcd  ( M  gcd  P ) ) )
 
Theoremmulgcd 12673 Distribute multiplication by a nonnegative integer over gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 30-May-2014.)
 |-  ( ( K  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N ) )  =  ( K  x.  ( M  gcd  N ) ) )
 
Theoremabsmulgcd 12674 Distribute absolute value of multiplication over gcd. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N ) )  =  ( abs `  ( K  x.  ( M  gcd  N ) ) ) )
 
Theoremmulgcdr 12675 Reverse distribution law for the 
gcd operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( ( A  x.  C )  gcd  ( B  x.  C ) )  =  ( ( A 
 gcd  B )  x.  C ) )
 
Theoremgcddiv 12676 Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  ( C 
 ||  A  /\  C  ||  B ) )  ->  ( ( A  gcd  B )  /  C )  =  ( ( A 
 /  C )  gcd  ( B  /  C ) ) )
 
Theoremgcdmultiple 12677 The GCD of a multiple of a number is the number itself. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  gcd  ( M  x.  N ) )  =  M )
 
Theoremgcdmultiplez 12678 Extend gcdmultiple 12677 so  N can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N ) )  =  M )
 
Theoremgcdeq 12679  A is equal to its gcd with  B if and only if  A divides  B. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A 
 gcd  B )  =  A  <->  A 
 ||  B ) )
 
Theoremdvdssqim 12680 Unidirectional form of dvdssq 12687. (Contributed by Scott Fenton, 19-Apr-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( M ^
 2 )  ||  ( N ^ 2 ) ) )
 
Theoremdvdsmulgcd 12681 A divisibility equivalent for odmulg 14817. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  ( B  x.  C ) 
 <->  A  ||  ( B  x.  ( C  gcd  A ) ) ) )
 
Theoremrpmulgcd 12682 If  K and  M are relatively prime, then the GCD of  K and  M  x.  N is the GCD of  K and  N. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K 
 gcd  M )  =  1 )  ->  ( K  gcd  ( M  x.  N ) )  =  ( K  gcd  N ) )
 
Theoremrplpwr 12683 If  A and  B are relatively prime, then so are  A ^ N and  B. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  ( ( A  gcd  B )  =  1  ->  ( ( A ^ N )  gcd  B )  =  1 ) )
 
Theoremrppwr 12684 If  A and  B are relatively prime, then so are  A ^ N and  B ^ N. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  ( ( A  gcd  B )  =  1  ->  ( ( A ^ N )  gcd  ( B ^ N ) )  =  1 ) )
 
Theoremsqgcd 12685 Square distributes over GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( ( M 
 gcd  N ) ^ 2
 )  =  ( ( M ^ 2 ) 
 gcd  ( N ^
 2 ) ) )
 
Theoremdvdssqlem 12686 Lemma for dvdssq 12687. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  ||  N 
 <->  ( M ^ 2
 )  ||  ( N ^ 2 ) ) )
 
Theoremdvdssq 12687 Two numbers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  ( M ^ 2
 )  ||  ( N ^ 2 ) ) )
 
6.1.8  Algorithms
 
Theoremnn0seqcvgd 12688* A strictly-decreasing nonnegative integer sequence with initial term  N reaches zero by the  N th term. Deduction version. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ph  ->  F : NN0 --> NN0 )   &    |-  ( ph  ->  N  =  ( F `  0 ) )   &    |-  (
 ( ph  /\  k  e. 
 NN0 )  ->  (
 ( F `  (
 k  +  1 ) )  =/=  0  ->  ( F `  ( k  +  1 ) )  <  ( F `  k ) ) )   =>    |-  ( ph  ->  ( F `  N )  =  0 )
 
Theoremseq1st 12689 A sequence whose iteration function ignores the second argument is only affected by the first point of the initial value function. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  R  =  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
 ) )   =>    |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  R  =  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } )
 )
 
Theoremalgr0 12690 The value of the algorithm iterator 
R at  0 is the initial state  A. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  R  =  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
 ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   =>    |-  ( ph  ->  ( R `  M )  =  A )
 
Theoremalgrf 12691 An algorithm is step a function  F : S --> S on a state space  S. An algorithm acts on an initial state  A  e.  S by iteratively applying  F to give  A,  ( F `
 A ),  ( F `  ( F `
 A ) ) and so on. An algorithm is said to halt if a fixed point of  F is reached after a finite number of iterations.

The algorithm iterator  R : NN0 --> S "runs" the algorithm  F so that  ( R `  k ) is the state after  k iterations of  F on the initial state  A.

Domain and codomain of the algorithm iterator  R. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

 |-  Z  =  ( ZZ>= `  M )   &    |-  R  =  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
 ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  F : S --> S )   =>    |-  ( ph  ->  R : Z --> S )
 
Theoremalgrp1 12692 The value of the algorithm iterator 
R at  ( K  + 
1 ). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  R  =  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
 ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  F : S --> S )   =>    |-  ( ( ph  /\  K  e.  Z ) 
 ->  ( R `  ( K  +  1 )
 )  =  ( F `
  ( R `  K ) ) )
 
Theoremalginv 12693* If  I is an invariant of  F, its value is unchanged after any number of iterations of  F. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  R  =  seq  0
 ( ( F  o.  1st ) ,  ( NN0  X. 
 { A } )
 )   &    |-  F : S --> S   &    |-  I  Fn  S   &    |-  ( x  e.  S  ->  ( I `  ( F `  x ) )  =  ( I `  x ) )   =>    |-  ( ( A  e.  S  /\  K  e.  NN0 )  ->  ( I `  ( R `  K ) )  =  ( I `
  ( R `  0 ) ) )
 
Theoremalgcvg 12694* One way to prove that an algorithm halts is to construct a countdown function  C : S --> NN0 whose value is guaranteed to decrease for each iteration of  F until it reaches  0. That is, if  X  e.  S is not a fixed point of  F, then  ( C `  ( F `  X ) )  <  ( C `
 X ).

If  C is a countdown function for algorithm  F, the sequence  ( C `  ( R `  k ) ) reaches  0 after at most  N steps, where  N is the value of  C for the initial state  A. (Contributed by Paul Chapman, 22-Jun-2011.)

 |-  F : S --> S   &    |-  R  =  seq  0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) )   &    |-  C : S --> NN0   &    |-  ( z  e.  S  ->  ( ( C `  ( F `  z ) )  =/=  0  ->  ( C `  ( F `
  z ) )  <  ( C `  z ) ) )   &    |-  N  =  ( C `  A )   =>    |-  ( A  e.  S  ->  ( C `  ( R `  N ) )  =  0 )
 
Theoremalgcvgblem 12695 Lemma for algcvgb 12696. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( ( N  =/=  0  ->  N  <  M ) 
 <->  ( ( M  =/=  0  ->  N  <  M )  /\  ( M  =  0  ->  N  =  0 ) ) ) )
 
Theoremalgcvgb 12696 Two ways of expressing that  C is a countdown function for algorithm  F. The first is used in these theorems. The second states the condition more intuitively as a conjunction: if the countdown function's value is currently non-zero, it must decrease at the next step; if it has reached zero, it must remain zero at the next step. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  F : S --> S   &    |-  C : S --> NN0   =>    |-  ( X  e.  S  ->  ( ( ( C `
  ( F `  X ) )  =/=  0  ->  ( C `  ( F `  X ) )  <  ( C `
  X ) )  <-> 
 ( ( ( C `
  X )  =/=  0  ->  ( C `  ( F `  X ) )  <  ( C `
  X ) ) 
 /\  ( ( C `
  X )  =  0  ->  ( C `  ( F `  X ) )  =  0
 ) ) ) )
 
Theoremalgcvga 12697* The countdown function  C remains  0 after  N steps. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F : S --> S   &    |-  R  =  seq  0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) )   &    |-  C : S --> NN0   &    |-  ( z  e.  S  ->  ( ( C `  ( F `  z ) )  =/=  0  ->  ( C `  ( F `
  z ) )  <  ( C `  z ) ) )   &    |-  N  =  ( C `  A )   =>    |-  ( A  e.  S  ->  ( K  e.  ( ZZ>=
 `  N )  ->  ( C `  ( R `
  K ) )  =  0 ) )
 
Theoremalgfx 12698* If  F reaches a fixed point when the countdown function 
C reaches  0,  F remains fixed after  N steps. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F : S --> S   &    |-  R  =  seq  0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) )   &    |-  C : S --> NN0   &    |-  ( z  e.  S  ->  ( ( C `  ( F `  z ) )  =/=  0  ->  ( C `  ( F `
  z ) )  <  ( C `  z ) ) )   &    |-  N  =  ( C `  A )   &    |-  ( z  e.  S  ->  ( ( C `  z )  =  0  ->  ( F `  z )  =  z ) )   =>    |-  ( A  e.  S  ->  ( K  e.  ( ZZ>=
 `  N )  ->  ( R `  K )  =  ( R `  N ) ) )
 
6.1.9  Euclid's Algorithm
 
Theoremeucalgval2 12699* The value of the step function  E for Euclid's Algorithm on an ordered pair. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
 |-  E  =  ( x  e.  NN0 ,  y  e. 
 NN0  |->  if ( y  =  0 ,  <. x ,  y >. ,  <. y ,  ( x  mod  y
 ) >. ) )   =>    |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M E N )  =  if ( N  =  0 ,  <. M ,  N >. ,  <. N ,  ( M  mod  N ) >. ) )
 
Theoremeucalgval 12700* Euclid's Algorithm eucalg 12705 computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0.

The value of the step function  E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

 |-  E  =  ( x  e.  NN0 ,  y  e. 
 NN0  |->  if ( y  =  0 ,  <. x ,  y >. ,  <. y ,  ( x  mod  y
 ) >. ) )   =>    |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `  X )  =  if (
 ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X ) >. ) )
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