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Theorem List for Metamath Proof Explorer - 13001-13100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsadcadd 13001* Non-recursive definition of the carry sequence. (Contributed by Mario Carneiro, 8-Sep-2016.)

Theoremsadcl 13005 The sum of two sequences is a sequence. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremsadcom 13006 The adder sequence function is commutative. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremsaddisj 13008 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.)

Theoremsadadd 13010 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 1390 and df-cad 1391.

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.)

Theoremsadid1 13011 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.)

Theoremsadid2 13012 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.)

Theoremsadeq 13015 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.)

Theorembitsres 13016 Restrict the bits of a number to an upper integer set. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits bits

Theorembitsuz 13017 The bits of a number are all at least iff the number is divisible by . (Contributed by Mario Carneiro, 21-Sep-2016.)
bits

Theorembitsshft 13018* 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.)
bits bits

Definitiondf-smu 13019* Define the multiplication of two bit sequences, using repeated sequence addition. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmufval 13020* Define the addition of two bit sequences, using df-had 1390 and df-cad 1391 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmupf 13021* The sequence of partial sums of the sequence multiplication. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmup0 13022* The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmupp1 13023* The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmuval 13024* Define the addition of two bit sequences, using df-had 1390 and df-cad 1391 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmuval2 13025* The partial sum sequence stabilizes at after the -th element of the sequence; this stable value is the value of the sequence multiplication. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmupvallem 13026* If only has elements less than , then all elements of the partial sum sequence past already equal the final value. (Contributed by Mario Carneiro, 20-Sep-2016.)

Theoremsmucl 13027 The product of two sequences is a sequence. (Contributed by Mario Carneiro, 19-Sep-2016.)
smul

Theoremsmu01lem 13028* Lemma for smu01 13029 and smu02 13030. (Contributed by Mario Carneiro, 19-Sep-2016.)
smul

Theoremsmu01 13029 Multiplication of a sequence by on the right. (Contributed by Mario Carneiro, 19-Sep-2016.)
smul

Theoremsmu02 13030 Multiplication of a sequence by on the left. (Contributed by Mario Carneiro, 9-Sep-2016.)
smul

Theoremsmupval 13031* Rewrite the elements of the partial sum sequence in terms of sequence multiplication. (Contributed by Mario Carneiro, 20-Sep-2016.)

Theoremsmup1 13032* Rewrite smupp1 13023 using only smul instead of the internal recursive function . (Contributed by Mario Carneiro, 20-Sep-2016.)

Theoremsmueqlem 13033* 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.)

Theoremsmueq 13034 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.)
smul ..^ ..^ smul ..^ ..^

Theoremsmumullem 13035 Lemma for smumul 13036. (Contributed by Mario Carneiro, 22-Sep-2016.)
bits ..^ smul bits bits

Theoremsmumul 13036 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 12994, whose correctness is verified in sadadd 13010.

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.)

bits smul bits bits

6.1.6  The greatest common divisor operator

Syntaxcgcd 13037 Extend the definition of a class to include the greatest common divisor operator.

Definitiondf-gcd 13038* Define the operator. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdval 13039* The value of the operator. is the greatest common divisor of and . If and are both , the result is defined conventionally as . (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremgcd0val 13040 The value, by convention, of the operator when both operands are 0. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdn0val 13041* The value of the operator when at least one operand is nonzero. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcllem1 13042* Lemma for gcdn0cl 13045, gcddvds 13046 and dvdslegcd 13047. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcllem2 13043* Lemma for gcdn0cl 13045, gcddvds 13046 and dvdslegcd 13047. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcllem3 13044* Lemma for gcdn0cl 13045, gcddvds 13046 and dvdslegcd 13047. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdn0cl 13045 Closure of the operator. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcddvds 13046 The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdslegcd 13047 An integer which divides both operands of the operator is bounded by it. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcl 13048 Closure of the operator. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcld 13049 Closure of the operator. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremgcdf 13050 Domain and codomain of the operator. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremgcdcom 13051 The operator is commutative. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdeq0 13052 The gcd of two integers is zero iff they are both zero. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdn0gt0 13053 The gcd of two integers is positive (nonzero) iff they are not both zero. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremgcd0id 13054 The gcd of 0 and an integer is the integer's absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdid0 13055 The gcd of an integer and 0 is the integer's absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremnn0gcdid0 13056 The gcd of a nonnegative integer with 0 is itself. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcdneg 13057 Negating one operand of the operator does not alter the result. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremneggcd 13058 Negating one operand of the operator does not alter the result. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremgcdaddm 13060 Adding a multiple of one operand of the operator to the other does not alter the result. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcdadd 13061 The GCD of two numbers is the same as the GCD of the left and their sum. (Contributed by Scott Fenton, 20-Apr-2014.)

Theoremgcdid 13062 The gcd of a number and itself is its absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcd1 13063 The gcd of a number with 1 is 1. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremgcdabs 13064 The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcdabs1 13065 of the absolute value of the first operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdabs2 13066 of the absolute value of the second operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremmodgcd 13067 The gcd remains unchanged if one operand is replaced with its remainder modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.)

Theorem1gcd 13068 The GCD of one and an integer is one. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

6.1.7  Bézout's identity

Theorembezoutlem1 13069* Lemma for bezout 13073. (Contributed by Mario Carneiro, 15-Mar-2014.)

Theorembezoutlem2 13070* Lemma for bezout 13073. (Contributed by Mario Carneiro, 15-Mar-2014.)

Theorembezoutlem3 13071* Lemma for bezout 13073. (Contributed by Mario Carneiro, 22-Feb-2014.)

Theorembezoutlem4 13072* Lemma for bezout 13073. (Contributed by Mario Carneiro, 22-Feb-2014.)

Theorembezout 13073* Bézout's identity: For any integers and , there are integers such that . (Contributed by Mario Carneiro, 22-Feb-2014.)

Theoremdvdsgcd 13074 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.)

Theoremdvdsgcdb 13075 Biconditional form of dvdsgcd 13074. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdass 13076 Associative law for operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremmulgcd 13077 Distribute multiplication by a nonnegative integer over gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 30-May-2014.)

Theoremabsmulgcd 13078 Distribute absolute value of multiplication over gcd. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremmulgcdr 13079 Reverse distribution law for the operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcddiv 13080 Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdmultiple 13081 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.)

Theoremgcdmultiplez 13082 Extend gcdmultiple 13081 so can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdeq 13083 is equal to its gcd with if and only if divides . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremdvdssqim 13084 Unidirectional form of dvdssq 13091. (Contributed by Scott Fenton, 19-Apr-2014.)

Theoremdvdsmulgcd 13085 A divisibility equivalent for odmulg 15223. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Theoremrpmulgcd 13086 If and are relatively prime, then the GCD of and is the GCD of and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremrplpwr 13087 If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremrppwr 13088 If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremsqgcd 13089 Square distributes over GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdvdssqlem 13090 Lemma for dvdssq 13091. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdvdssq 13091 Two numbers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

6.1.8  Algorithms

Theoremnn0seqcvgd 13092* A strictly-decreasing nonnegative integer sequence with initial term reaches zero by the th term. Deduction version. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremseq1st 13093 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.)

Theoremalgr0 13094 The value of the algorithm iterator at is the initial state . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Theoremalgrf 13095 An algorithm is step a function on a state space . An algorithm acts on an initial state by iteratively applying to give , , and so on. An algorithm is said to halt if a fixed point of is reached after a finite number of iterations.

The algorithm iterator "runs" the algorithm so that is the state after iterations of on the initial state .

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

Theoremalgrp1 13096 The value of the algorithm iterator at . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremalginv 13097* If is an invariant of , its value is unchanged after any number of iterations of . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremalgcvg 13098* One way to prove that an algorithm halts is to construct a countdown function whose value is guaranteed to decrease for each iteration of until it reaches . That is, if is not a fixed point of , then .

If is a countdown function for algorithm , the sequence reaches after at most steps, where is the value of for the initial state . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremalgcvgblem 13099 Lemma for algcvgb 13100. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremalgcvgb 13100 Two ways of expressing that is a countdown function for algorithm . 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.)

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