P. 111 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

hashen 11001

Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

|- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) = (♯ ` B ) <-> A ~~ B ) )

Theorem

hasheqf1o 11002*

The size of two finite sets is equal if and only if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 17-Dec-2017.)

|- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) = (♯ ` B ) <-> E. f f : A -1-1-onto-> B ) )

Theorem

fiinfnf1o 11003*

There is no bijection between a finite set and an infinite set. By infnfi 7053 the theorem would also hold if "infinite" were expressed as om ~<_ B. (Contributed by Alexander van der Vekens, 25-Dec-2017.)

|- ( ( A e. Fin /\ -. B e. Fin ) -> -. E. f f : A -1-1-onto-> B )

Theorem

fihasheqf1oi 11004

The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)

|- ( ( A e. Fin /\ F : A -1-1-onto-> B ) -> (♯ ` A ) = (♯ ` B ) )

Theorem

fihashf1rn 11005

The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.)

|- ( ( A e. Fin /\ F : A -1-1-> B ) -> (♯ ` F ) = (♯ ` ran F ) )

Theorem

fihasheqf1od 11006

The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)

|- ( ph -> A e. Fin )   &    |- ( ph -> F : A -1-1-onto-> B )   =>    |- ( ph -> (♯ ` A ) = (♯ ` B ) )

Theorem

fz1eqb 11007

Two possibly-empty 1-based finite sets of sequential integers are equal iff their endpoints are equal. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.)

|- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 1 ... M ) = ( 1 ... N ) <-> M = N ) )

Theorem

filtinf 11008

The size of an infinite set is greater than the size of a finite set. (Contributed by Jim Kingdon, 21-Feb-2022.)

|- ( ( A e. Fin /\ om ~<_ B ) -> (♯ ` A ) < (♯ ` B ) )

Theorem

isfinite4im 11009

A finite set is equinumerous to the range of integers from one up to the hash value of the set. (Contributed by Jim Kingdon, 22-Feb-2022.)

|- ( A e. Fin -> ( 1 ... (♯ ` A ) ) ~~ A )

Theorem

fihasheq0 11010

Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)

|- ( A e. Fin -> ( (♯ ` A ) = 0 <-> A = (/) ) )

Theorem

fihashneq0 11011

Two ways of saying a finite set is not empty. Also, "A is inhabited" would be equivalent by fin0 7043. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)

|- ( A e. Fin -> ( 0 < (♯ ` A ) <-> A =/= (/) ) )

Theorem

hashnncl 11012

Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)

|- ( A e. Fin -> ( (♯ ` A ) e. NN <-> A =/= (/) ) )

Theorem

hash0 11013

The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)

|- (♯ ` (/) ) = 0

Theorem

fihashelne0d 11014

A finite set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023.)

|- ( ph -> B e. A )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> -. (♯ ` A ) = 0 )

Theorem

hashsng 11015

The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)

|- ( A e. V -> (♯ ` { A } ) = 1 )

Theorem

fihashen1 11016

A finite set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)

|- ( A e. Fin -> ( (♯ ` A ) = 1 <-> A ~~ 1o ) )

Theorem

fihashfn 11017

A function on a finite set is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) (Intuitionized by Jim Kingdon, 24-Feb-2022.)

|- ( ( F Fn A /\ A e. Fin ) -> (♯ ` F ) = (♯ ` A ) )

Theorem

fseq1hash 11018

The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)

|- ( ( N e. NN0 /\ F Fn ( 1 ... N ) ) -> (♯ ` F ) = N )

Theorem

omgadd 11019

Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)

|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( ( A e. om /\ B e. om ) -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) )

Theorem

fihashdom 11020

Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.)

|- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> A ~<_ B ) )

Theorem

hashunlem 11021

Lemma for hashun 11022. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.)

|- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ph -> ( A i^i B ) = (/) )   &    |- ( ph -> N e. om )   &    |- ( ph -> M e. om )   &    |- ( ph -> A ~~ N )   &    |- ( ph -> B ~~ M )   =>    |- ( ph -> ( A u. B ) ~~ ( N +o M ) )

Theorem

hashun 11022

The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)

|- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> (♯ ` ( A u. B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )

Theorem

1elfz0hash 11023

1 is an element of the finite set of sequential nonnegative integers bounded by the size of a nonempty finite set. (Contributed by AV, 9-May-2020.)

|- ( ( A e. Fin /\ A =/= (/) ) -> 1 e. ( 0 ... (♯ ` A ) ) )

Theorem

hashunsng 11024

The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)

|- ( B e. V -> ( ( A e. Fin /\ -. B e. A ) -> (♯ ` ( A u. { B } ) ) = ( (♯ ` A ) + 1 ) ) )

Theorem

hashprg 11025

The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.)

|- ( ( A e. V /\ B e. W ) -> ( A =/= B <-> (♯ ` { A , B } ) = 2 ) )

Theorem

prhash2ex 11026

There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 11032, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)

|- (♯ ` { 0 , 1 } ) = 2

Theorem

hashp1i 11027

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

|- A e. om   &    |- B = suc A   &    |- (♯ ` A ) = M   &    |- ( M + 1 ) = N   =>    |- (♯ ` B ) = N

Theorem

hash1 11028

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

|- (♯ ` 1o ) = 1

Theorem

hash2 11029

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

|- (♯ ` 2o ) = 2

Theorem

hash3 11030

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

|- (♯ ` 3o ) = 3

Theorem

hash4 11031

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

|- (♯ ` 4o ) = 4

Theorem

pr0hash2ex 11032

There is (at least) one set with two different elements: the unordered pair containing the empty set and the singleton containing the empty set. (Contributed by AV, 29-Jan-2020.)

|- (♯ ` { (/) , { (/) } } ) = 2

Theorem

fihashss 11033

The size of a subset is less than or equal to the size of its superset. (Contributed by Alexander van der Vekens, 14-Jul-2018.)

|- ( ( A e. Fin /\ B e. Fin /\ B C_ A ) -> (♯ ` B ) <_ (♯ ` A ) )

Theorem

fiprsshashgt1 11034

The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.)

|- ( ( ( A e. V /\ B e. W /\ A =/= B ) /\ C e. Fin ) -> ( { A , B } C_ C -> 2 <_ (♯ ` C ) ) )

Theorem

fihashssdif 11035

The size of the difference of a finite set and a finite subset is the set's size minus the subset's. (Contributed by Jim Kingdon, 31-May-2022.)

|- ( ( A e. Fin /\ B e. Fin /\ B C_ A ) -> (♯ ` ( A \ B ) ) = ( (♯ ` A ) - (♯ ` B ) ) )

Theorem

hashdifsn 11036

The size of the difference of a finite set and a singleton subset is the set's size minus 1. (Contributed by Alexander van der Vekens, 6-Jan-2018.)

|- ( ( A e. Fin /\ B e. A ) -> (♯ ` ( A \ { B } ) ) = ( (♯ ` A ) - 1 ) )

Theorem

hashdifpr 11037

The size of the difference of a finite set and a proper ordered pair subset is the set's size minus 2. (Contributed by AV, 16-Dec-2020.)

|- ( ( A e. Fin /\ ( B e. A /\ C e. A /\ B =/= C ) ) -> (♯ ` ( A \ { B , C } ) ) = ( (♯ ` A ) - 2 ) )

Theorem

hashfz 11038

Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)

|- ( B e. ( ZZ>= ` A ) -> (♯ ` ( A ... B ) ) = ( ( B - A ) + 1 ) )

Theorem

hashfzo 11039

Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)

|- ( B e. ( ZZ>= ` A ) -> (♯ ` ( A..^ B ) ) = ( B - A ) )

Theorem

hashfzo0 11040

Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)

|- ( B e. NN0 -> (♯ ` ( 0..^ B ) ) = B )

Theorem

hashfzp1 11041

Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)

|- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( A + 1 ) ... B ) ) = ( B - A ) )

Theorem

hashfz0 11042

Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.)

|- ( B e. NN0 -> (♯ ` ( 0 ... B ) ) = ( B + 1 ) )

Theorem

hashxp 11043

The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)

|- ( ( A e. Fin /\ B e. Fin ) -> (♯ ` ( A X. B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) )

Theorem

fimaxq 11044*

A finite set of rational numbers has a maximum. (Contributed by Jim Kingdon, 6-Sep-2022.)

|- ( ( A C_ QQ /\ A e. Fin /\ A =/= (/) ) -> E. x e. A A. y e. A y <_ x )

Theorem

fiubm 11045*

Lemma for fiubz 11046 and fiubnn 11047. A general form of those theorems. (Contributed by Jim Kingdon, 29-Oct-2024.)

|- ( ph -> A C_ B )   &    |- ( ph -> B C_ QQ )   &    |- ( ph -> C e. B )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> E. x e. B A. y e. A y <_ x )

Theorem

fiubz 11046*

A finite set of integers has an upper bound which is an integer. (Contributed by Jim Kingdon, 29-Oct-2024.)

|- ( ( A C_ ZZ /\ A e. Fin ) -> E. x e. ZZ A. y e. A y <_ x )

Theorem

fiubnn 11047*

A finite set of natural numbers has an upper bound which is a a natural number. (Contributed by Jim Kingdon, 29-Oct-2024.)

|- ( ( A C_ NN /\ A e. Fin ) -> E. x e. NN A. y e. A y <_ x )

Theorem

resunimafz0 11048

The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)

|- ( ph -> Fun I )   &    |- ( ph -> F : ( 0..^ (♯ ` F ) ) --> dom I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   =>    |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )

Theorem

fnfz0hash 11049

The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.)

|- ( ( N e. NN0 /\ F Fn ( 0 ... N ) ) -> (♯ ` F ) = ( N + 1 ) )

Theorem

ffz0hash 11050

The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Proof shortened by AV, 11-Apr-2021.)

|- ( ( N e. NN0 /\ F : ( 0 ... N ) --> B ) -> (♯ ` F ) = ( N + 1 ) )

Theorem

ffzo0hash 11051

The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.)

|- ( ( N e. NN0 /\ F Fn ( 0..^ N ) ) -> (♯ ` F ) = N )

Theorem

fnfzo0hash 11052

The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.)

|- ( ( N e. NN0 /\ F : ( 0..^ N ) --> B ) -> (♯ ` F ) = N )

Theorem

hashfacen 11053*

The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)

|- ( ( A ~~ B /\ C ~~ D ) -> { f | f : A -1-1-onto-> C } ~~ { f | f : B -1-1-onto-> D } )

Theorem

leisorel 11054

Version of isorel 5931 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)

|- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( C <_ D <-> ( F ` C ) <_ ( F ` D ) ) )

Theorem

zfz1isolemsplit 11055

Lemma for zfz1iso 11058. Removing one element from an integer range. (Contributed by Jim Kingdon, 8-Sep-2022.)

|- ( ph -> X e. Fin )   &    |- ( ph -> M e. X )   =>    |- ( ph -> ( 1 ... (♯ ` X ) ) = ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) )

Theorem

zfz1isolemiso 11056*

Lemma for zfz1iso 11058. Adding one element to the order isomorphism. (Contributed by Jim Kingdon, 8-Sep-2022.)

|- ( ph -> X e. Fin )   &    |- ( ph -> X C_ ZZ )   &    |- ( ph -> M e. X )   &    |- ( ph -> A. z e. X z <_ M )   &    |- ( ph -> G Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) )   &    |- ( ph -> A e. ( 1 ... (♯ ` X ) ) )   &    |- ( ph -> B e. ( 1 ... (♯ ` X ) ) )   =>    |- ( ph -> ( A < B <-> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) ) )

Theorem

zfz1isolem1 11057*

Lemma for zfz1iso 11058. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)

|- ( ph -> K e. om )   &    |- ( ph -> A. y ( ( ( y C_ ZZ /\ y e. Fin ) /\ y ~~ K ) -> E. f f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) ) )   &    |- ( ph -> X C_ ZZ )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> X ~~ suc K )   &    |- ( ph -> M e. X )   &    |- ( ph -> A. z e. X z <_ M )   =>    |- ( ph -> E. f f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) )

Theorem

zfz1iso 11058*

A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)

|- ( ( A C_ ZZ /\ A e. Fin ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )

Theorem

seq3coll 11059*

The function F contains a sparse set of nonzero values to be summed. The function G is an order isomorphism from the set of nonzero values of F to a 1-based finite sequence, and H collects these nonzero values together. Under these conditions, the sum over the values in H yields the same result as the sum over the original set F. (Contributed by Mario Carneiro, 2-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)

|- ( ( ph /\ k e. S ) -> ( Z .+ k ) = k )   &    |- ( ( ph /\ k e. S ) -> ( k .+ Z ) = k )   &    |- ( ( ph /\ ( k e. S /\ n e. S ) ) -> ( k .+ n ) e. S )   &    |- ( ph -> Z e. S )   &    |- ( ph -> G Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )   &    |- ( ph -> N e. ( 1 ... (♯ ` A ) ) )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` 1 ) ) -> ( H ` k ) e. S )   &    |- ( ( ph /\ k e. ( ( M ... ( G ` (♯ ` A ) ) ) \ A ) ) -> ( F ` k ) = Z )   &    |- ( ( ph /\ n e. ( 1 ... (♯ ` A ) ) ) -> ( H ` n ) = ( F ` ( G ` n ) ) )   =>    |- ( ph -> ( seq M ( .+ , F ) ` ( G ` N ) ) = ( seq 1 ( .+ , H ) ` N ) )

4.6.10.1  Proper unordered pairs and triples (sets of size 2 and 3)

Theorem

hash2en 11060

Two equivalent ways to say a set has two elements. (Contributed by Jim Kingdon, 4-Dec-2025.)

|- ( V ~~ 2o <-> ( V e. Fin /\ (♯ ` V ) = 2 ) )

Theorem

hashdmprop2dom 11061

A class which contains two ordered pairs with different first components has at least two elements. (Contributed by AV, 12-Nov-2021.)

|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   &    |- ( ph -> D e. Y )   &    |- ( ph -> F e. Z )   &    |- ( ph -> A =/= B )   &    |- ( ph -> { <. A , C >. , <. B , D >. } C_ F )   =>    |- ( ph -> 2o ~<_ dom F )

4.6.10.2  Functions with a domain containing at least two different elements

Theorem

fundm2domnop0 11062

A function with a domain containing (at least) two different elements is not an ordered pair. This theorem (which requires that ( G \ { (/) } ) needs to be a function instead of G) is useful for proofs for extensible structures, see structn0fun 13040. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Proof shortened by AV, 15-Nov-2021.)

|- ( ( Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> -. G e. ( _V X. _V ) )

Theorem

fundm2domnop 11063

A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 12-Oct-2020.) (Proof shortened by AV, 9-Jun-2021.)

|- ( ( Fun G /\ 2o ~<_ dom G ) -> -. G e. ( _V X. _V ) )

Theorem

fun2dmnop0 11064

A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 11065 (with the less restrictive requirement that ( G \ { (/) } ) needs to be a function instead of G) is useful for proofs for extensible structures, see structn0fun 13040. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.)

|- A e. _V   &    |- B e. _V   =>    |- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> -. G e. ( _V X. _V ) )

Theorem

fun2dmnop 11065

A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 9-Jun-2021.)

|- A e. _V   &    |- B e. _V   =>    |- ( ( Fun G /\ A =/= B /\ { A , B } C_ dom G ) -> -. G e. ( _V X. _V ) )

4.7  Words over a set

This section is about words (or strings) over a set (alphabet) defined as finite sequences of symbols (or characters) being elements of the alphabet. Although it is often required that the underlying set/alphabet be nonempty, finite and not a proper class, these restrictions are not made in the current definition df-word 11067. Note that the empty word (/) (i.e., the empty set) is the only word over an empty alphabet, see 0wrd0 11092. The set Word S of words over S is the free monoid over S, where the monoid law is concatenation and the monoid unit is the empty word. Besides the definition of words themselves, several operations on words are defined in this section:

Name	Reference	Explanation	Example	Remarks
Length (or size) of a word	df-ihash 10993: (♯ ` W )	Operation which determines the number of the symbols within the word.	W : ( 0..^ N ) --> S -> ( W e. Word S /\ (♯ ` W ) = N	This is not a special definition for words, but for arbitrary sets.
First symbol of a word	df-fv 5325: ( W ` 0 )	Operation which extracts the first symbol of a word. The result is the symbol at the first place in the sequence representing the word.	W : ( 0..^ 1 ) --> S -> ( W e. Word S /\ ( W ` 0 ) e. S	This is not a special definition for words, but for arbitrary functions with a half-open range of nonnegative integers as domain.
Last symbol of a word	df-lsw 11112: (lastS ` W )	Operation which extracts the last symbol of a word. The result is the symbol at the last place in the sequence representing the word.	W : ( 0..^ 3 ) --> S -> ( W e. Word S /\ (lastS ` W ) = ( W ` 2 )	Note that the index of the last symbol is less by 1 than the length of the word.
Subword (or substring) of a word	df-substr 11173: ( W substr <. I , J >. )	Operation which extracts a portion of a word. The result is a consecutive, reindexed part of the sequence representing the word.	W : ( 0..^ 3 ) --> S -> ( W e. Word S /\ ( W substr <. 1 , 2 >. ) e. Word S /\ (♯ ` ( W substr <. 1 , 2 >. ) ) = 1	Note that the last index of the range defining the subword is greater by 1 than the index of the last symbol of the subword in the sequence of the original word.
Concatenation of two words	df-concat 11121: ( W ++ U )	Operation combining two words to one new word. The result is a combined, reindexed sequence build from the sequences representing the two words.	( W e. Word S /\ U e. Word S ) -> (♯ ` ( W ++ U ) ) = ( (♯ ` W ) + (♯ ` U ) )	Note that the index of the first symbol of the second concatenated word is the length of the first word in the concatenation.
Singleton word	df-s1 11144: <" S ">	Constructor building a word of length 1 from a symbol.	(♯ ` <" S "> ) = 1

Conventions:

• Words are usually represented by class variable W, or if two words are involved, by W and U or by A and B.
• The alphabets are usually represented by class variable V (because any arbitrary set can be an alphabet), sometimes also by S (especially as codomain of the function representing a word, because the alphabet is the set of symbols).
• The symbols are usually represented by class variables S or A, or if two symbols are involved, by S and T or by A and B.
• The indices of the sequence representing a word are usually represented by class variable I, if two indices are involved (especially for subwords) by I and J, or by M and N.
• The length of a word is usually represented by class variables N or L.
• The number of positions by which to cyclically shift a word is usually represented by class variables N or L.

4.7.1  Definitions and basic theorems

Syntax

cword 11066

Syntax for the Word operator.

class Word S

Definition

df-word 11067*

Define the class of words over a set. A word (sometimes also called a string) is a finite sequence of symbols from a set (alphabet) S. Definition in Section 9.1 of [AhoHopUll] p. 318. The domain is forced to be an initial segment of NN0 so that two words with the same symbols in the same order be equal. The set Word S is sometimes denoted by S*, using the Kleene star, although the Kleene star, or Kleene closure, is sometimes reserved to denote an operation on languages. The set Word S equipped with concatenation is the free monoid over S, and the monoid unit is the empty word. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 14-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

|- Word S = { w | E. l e. NN0 w : ( 0..^ l ) --> S }

Theorem

iswrd 11068*

Property of being a word over a set with an existential quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)

|- ( W e. Word S <-> E. l e. NN0 W : ( 0..^ l ) --> S )

Theorem

wrdval 11069*

Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

|- ( S e. V -> Word S = U_ l e. NN0 ( S ^m ( 0..^ l ) ) )

Theorem

lencl 11070

The length of a word is a nonnegative integer. This corresponds to the definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 27-Aug-2015.)

|- ( W e. Word S -> (♯ ` W ) e. NN0 )

Theorem

iswrdinn0 11071

A zero-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 16-Aug-2025.)

|- ( ( W : ( 0..^ L ) --> S /\ L e. NN0 ) -> W e. Word S )

Theorem

wrdf 11072

A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)

|- ( W e. Word S -> W : ( 0..^ (♯ ` W ) ) --> S )

Theorem

iswrdiz 11073

A zero-based sequence is a word. In iswrdinn0 11071 we can specify a length as an nonnegative integer. However, it will occasionally be helpful to allow a negative length, as well as zero, to specify an empty sequence. (Contributed by Jim Kingdon, 18-Aug-2025.)

|- ( ( W : ( 0..^ L ) --> S /\ L e. ZZ ) -> W e. Word S )

Theorem

wrddm 11074

The indices of a word (i.e. its domain regarded as function) are elements of an open range of nonnegative integers (of length equal to the length of the word). (Contributed by AV, 2-May-2020.)

|- ( W e. Word S -> dom W = ( 0..^ (♯ ` W ) ) )

Theorem

sswrd 11075

The set of words respects ordering on the base set. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)

|- ( S C_ T -> Word S C_ Word T )

Theorem

snopiswrd 11076

A singleton of an ordered pair (with 0 as first component) is a word. (Contributed by AV, 23-Nov-2018.) (Proof shortened by AV, 18-Apr-2021.)

|- ( S e. V -> { <. 0 , S >. } e. Word V )

Theorem

wrdexg 11077

The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by JJ, 18-Nov-2022.)

|- ( S e. V -> Word S e. _V )

Theorem

wrdexb 11078

The set of words over a set is a set, bidirectional version. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 23-Nov-2018.)

|- ( S e. _V <-> Word S e. _V )

Theorem

wrdexi 11079

The set of words over a set is a set, inference form. (Contributed by AV, 23-May-2021.)

|- S e. _V   =>    |- Word S e. _V

Theorem

wrdsymbcl 11080

A symbol within a word over an alphabet belongs to the alphabet. (Contributed by Alexander van der Vekens, 28-Jun-2018.)

|- ( ( W e. Word V /\ I e. ( 0..^ (♯ ` W ) ) ) -> ( W ` I ) e. V )

Theorem

wrdfn 11081

A word is a function with a zero-based sequence of integers as domain. (Contributed by Alexander van der Vekens, 13-Apr-2018.)

|- ( W e. Word S -> W Fn ( 0..^ (♯ ` W ) ) )

Theorem

wrdv 11082

A word over an alphabet is a word over the universal class. (Contributed by AV, 8-Feb-2021.) (Proof shortened by JJ, 18-Nov-2022.)

|- ( W e. Word V -> W e. Word _V )

Theorem

wrdlndm 11083

The length of a word is not in the domain of the word (regarded as a function). (Contributed by AV, 3-Mar-2021.) (Proof shortened by JJ, 18-Nov-2022.)

|- ( W e. Word V -> (♯ ` W ) e/ dom W )

Theorem

iswrdsymb 11084*

An arbitrary word is a word over an alphabet if all of its symbols belong to the alphabet. (Contributed by AV, 23-Jan-2021.)

|- ( ( W e. Word _V /\ A. i e. ( 0..^ (♯ ` W ) ) ( W ` i ) e. V ) -> W e. Word V )

Theorem

wrdfin 11085

A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.) (Proof shortened by AV, 18-Nov-2018.)

|- ( W e. Word S -> W e. Fin )

Theorem

lennncl 11086

The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)

|- ( ( W e. Word S /\ W =/= (/) ) -> (♯ ` W ) e. NN )

Theorem

wrdffz 11087

A word is a function from a finite interval of integers. (Contributed by AV, 10-Feb-2021.)

|- ( W e. Word S -> W : ( 0 ... ( (♯ ` W ) - 1 ) ) --> S )

Theorem

wrdeq 11088

Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)

|- ( S = T -> Word S = Word T )

Theorem

wrdeqi 11089

Equality theorem for the set of words, inference form. (Contributed by AV, 23-May-2021.)

|- S = T   =>    |- Word S = Word T

Theorem

iswrddm0 11090

A function with empty domain is a word. (Contributed by AV, 13-Oct-2018.)

|- ( W : (/) --> S -> W e. Word S )

Theorem

wrd0 11091

The empty set is a word (the empty word, frequently denoted ε in this context). This corresponds to the definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 13-May-2020.)

|- (/) e. Word S

Theorem

0wrd0 11092

The empty word is the only word over an empty alphabet. (Contributed by AV, 25-Oct-2018.)

|- ( W e. Word (/) <-> W = (/) )

Theorem

ffz0iswrdnn0 11093

A sequence with zero-based indices is a word. (Contributed by AV, 31-Jan-2018.) (Proof shortened by AV, 13-Oct-2018.) (Proof shortened by JJ, 18-Nov-2022.)

|- ( ( W : ( 0 ... L ) --> S /\ L e. NN0 ) -> W e. Word S )

Theorem

wrdsymb 11094

A word is a word over the symbols it consists of. (Contributed by AV, 1-Dec-2022.)

|- ( S e. Word A -> S e. Word ( S " ( 0..^ (♯ ` S ) ) ) )

Theorem

nfwrd 11095

Hypothesis builder for Word S. (Contributed by Mario Carneiro, 26-Feb-2016.)

|- F/_ x S   =>    |- F/_ xWord S

Theorem

csbwrdg 11096*

Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

|- ( S e. V -> [_ S / x ]_Word x = Word S )

Theorem

wrdnval 11097*

Words of a fixed length are mappings from a fixed half-open integer interval. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Proof shortened by AV, 13-May-2020.)

|- ( ( V e. X /\ N e. NN0 ) -> { w e. Word V | (♯ ` w ) = N } = ( V ^m ( 0..^ N ) ) )

Theorem

wrdmap 11098

Words as a mapping. (Contributed by Thierry Arnoux, 4-Mar-2020.)

|- ( ( V e. X /\ N e. NN0 ) -> ( ( W e. Word V /\ (♯ ` W ) = N ) <-> W e. ( V ^m ( 0..^ N ) ) ) )

Theorem

wrdsymb0 11099

A symbol at a position "outside" of a word. (Contributed by Alexander van der Vekens, 26-May-2018.) (Proof shortened by AV, 2-May-2020.)

|- ( ( W e. Word V /\ I e. ZZ ) -> ( ( I < 0 \/ (♯ ` W ) <_ I ) -> ( W ` I ) = (/) ) )

Theorem

wrdlenge1n0 11100

A word with length at least 1 is not empty. (Contributed by AV, 14-Oct-2018.)

|- ( W e. Word V -> ( W =/= (/) <-> 1 <_ (♯ ` W ) ) )