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Theorem List for Metamath Proof Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremreuccatpfxs1lem 14101* Lemma for reuccatpfxs1 14102. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Revised by AV, 9-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈𝑋) ∧ ∀𝑠𝑉 ((𝑊 ++ ⟨“𝑠”⟩) ∈ 𝑋𝑆 = 𝑠) ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (𝑊 = (𝑈 prefix (♯‘𝑊)) → 𝑈 = (𝑊 ++ ⟨“𝑆”⟩)))

Theoremreuccatpfxs1 14102* There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Revised by AV, 13-Oct-2022.)
𝑣𝑋       ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))

Theoremreuccatpfxs1v 14103* There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Revised by AV, 10-May-2022.) (Proof shortened by AV, 13-Oct-2022.)
((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))

5.7.11  Splicing words (substring replacement)

Syntaxcsplice 14104 Syntax for the word splicing operator.
class splice

Definitiondf-splice 14105* Define an operation which replaces portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by AV, 14-Oct-2022.)
splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st ‘(1st𝑏))) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (♯‘𝑠)⟩)))

Theoremsplval 14106 Value of the substring replacement operator. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by AV, 11-May-2020.) (Revised by AV, 15-Oct-2022.)
((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))

Theoremsplcl 14107 Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)
((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)

Theoremsplid 14108 Splicing a subword for the same subword makes no difference. (Contributed by Stefan O'Rear, 20-Aug-2015.) (Proof shortened by AV, 14-Oct-2022.)
((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = 𝑆)

Theoremspllen 14109 The length of a splice. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)
(𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝐹 ∈ (0...𝑇))    &   (𝜑𝑇 ∈ (0...(♯‘𝑆)))    &   (𝜑𝑅 ∈ Word 𝐴)       (𝜑 → (♯‘(𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)) = ((♯‘𝑆) + ((♯‘𝑅) − (𝑇𝐹))))

Theoremsplfv1 14110 Symbols to the left of a splice are unaffected. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)
(𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝐹 ∈ (0...𝑇))    &   (𝜑𝑇 ∈ (0...(♯‘𝑆)))    &   (𝜑𝑅 ∈ Word 𝐴)    &   (𝜑𝑋 ∈ (0..^𝐹))       (𝜑 → ((𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)‘𝑋) = (𝑆𝑋))

Theoremsplfv2a 14111 Symbols within the replacement region of a splice, expressed using the coordinates of the replacement region. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)
(𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝐹 ∈ (0...𝑇))    &   (𝜑𝑇 ∈ (0...(♯‘𝑆)))    &   (𝜑𝑅 ∈ Word 𝐴)    &   (𝜑𝑋 ∈ (0..^(♯‘𝑅)))       (𝜑 → ((𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)‘(𝐹 + 𝑋)) = (𝑅𝑋))

Theoremsplval2 14112 Value of a splice, assuming the input word 𝑆 has already been decomposed into its pieces. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 15-Oct-2022.)
(𝜑𝐴 ∈ Word 𝑋)    &   (𝜑𝐵 ∈ Word 𝑋)    &   (𝜑𝐶 ∈ Word 𝑋)    &   (𝜑𝑅 ∈ Word 𝑋)    &   (𝜑𝑆 = ((𝐴 ++ 𝐵) ++ 𝐶))    &   (𝜑𝐹 = (♯‘𝐴))    &   (𝜑𝑇 = (𝐹 + (♯‘𝐵)))       (𝜑 → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = ((𝐴 ++ 𝑅) ++ 𝐶))

5.7.12  Reversing words

Syntaxcreverse 14113 Syntax for the word reverse operator.
class reverse

Definitiondf-reverse 14114* Define an operation which reverses the order of symbols in a word. This operation is also known as "word reversal" and "word mirroring". (Contributed by Stefan O'Rear, 26-Aug-2015.)
reverse = (𝑠 ∈ V ↦ (𝑥 ∈ (0..^(♯‘𝑠)) ↦ (𝑠‘(((♯‘𝑠) − 1) − 𝑥))))

Theoremrevval 14115* Value of the word reversing function. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(𝑊𝑉 → (reverse‘𝑊) = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ (𝑊‘(((♯‘𝑊) − 1) − 𝑥))))

Theoremrevcl 14116 The reverse of a word is a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(𝑊 ∈ Word 𝐴 → (reverse‘𝑊) ∈ Word 𝐴)

Theoremrevlen 14117 The reverse of a word has the same length as the original. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(𝑊 ∈ Word 𝐴 → (♯‘(reverse‘𝑊)) = (♯‘𝑊))

Theoremrevfv 14118 Reverse of a word at a point. (Contributed by Stefan O'Rear, 26-Aug-2015.)
((𝑊 ∈ Word 𝐴𝑋 ∈ (0..^(♯‘𝑊))) → ((reverse‘𝑊)‘𝑋) = (𝑊‘(((♯‘𝑊) − 1) − 𝑋)))

Theoremrev0 14119 The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(reverse‘∅) = ∅

Theoremrevs1 14120 Singleton words are their own reverses. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(reverse‘⟨“𝑆”⟩) = ⟨“𝑆”⟩

Theoremrevccat 14121 Antiautomorphic property of the reversal operation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑆 ∈ Word 𝐴𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) = ((reverse‘𝑇) ++ (reverse‘𝑆)))

Theoremrevrev 14122 Reversal is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.)
(𝑊 ∈ Word 𝐴 → (reverse‘(reverse‘𝑊)) = 𝑊)

5.7.13  Repeated symbol words

Syntaxcreps 14123 Extend class notation with words consisting of one repeated symbol.
class repeatS

Definitiondf-reps 14124* Definition to construct a word consisting of one repeated symbol, often called "repeated symbol word" for short in the following. (Contributed by Alexander van der Vekens, 4-Nov-2018.)
repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))

Theoremreps 14125* Construct a function mapping a half-open range of nonnegative integers to a constant. (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))

Theoremrepsundef 14126 A function mapping a half-open range of nonnegative integers with an upper bound not being a nonnegative integer to a constant is the empty set (in the meaning of "undefined"). (Contributed by AV, 5-Nov-2018.)
(𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅)

Theoremrepsconst 14127 Construct a function mapping a half-open range of nonnegative integers to a constant, see also fconstmpt 5578. (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = ((0..^𝑁) × {𝑆}))

Theoremrepsf 14128 The constructed function mapping a half-open range of nonnegative integers to a constant is a function. (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝑉)

Theoremrepswsymb 14129 The symbols of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0𝐼 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝐼) = 𝑆)

Theoremrepsw 14130 A function mapping a half-open range of nonnegative integers to a constant is a word consisting of one symbol repeated several times ("repeated symbol word"). (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) ∈ Word 𝑉)

Theoremrepswlen 14131 The length of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (♯‘(𝑆 repeatS 𝑁)) = 𝑁)

Theoremrepsw0 14132 The "repeated symbol word" of length 0. (Contributed by AV, 4-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 0) = ∅)

Theoremrepsdf2 14133* Alternative definition of a "repeated symbol word". (Contributed by AV, 7-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (𝑊 = (𝑆 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝑆)))

Theoremrepswsymball 14134* All the symbols of a "repeated symbol word" are the same. (Contributed by AV, 10-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑆𝑉) → (𝑊 = (𝑆 repeatS (♯‘𝑊)) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) = 𝑆))

Theoremrepswsymballbi 14135* A word is a "repeated symbol word" iff each of its symbols equals the first symbol of the word. (Contributed by AV, 10-Nov-2018.)
(𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)) ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) = (𝑊‘0)))

Theoremrepswfsts 14136 The first symbol of a nonempty "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ) → ((𝑆 repeatS 𝑁)‘0) = 𝑆)

Theoremrepswlsw 14137 The last symbol of a nonempty "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ) → (lastS‘(𝑆 repeatS 𝑁)) = 𝑆)

Theoremrepsw1 14138 The "repeated symbol word" of length 1. (Contributed by AV, 4-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 1) = ⟨“𝑆”⟩)

Theoremrepswswrd 14139 A subword of a "repeated symbol word" is again a "repeated symbol word". The assumption 𝑁𝐿 is required, because otherwise (𝐿 < 𝑁): ((𝑆 repeatS 𝐿) substr ⟨𝑀, 𝑁⟩) = ∅, but for M < N (𝑆 repeatS (𝑁𝑀))) ≠ ∅! The proof is relatively long because the border cases (𝑀 = 𝑁, ¬ (𝑀..^𝑁) ⊆ (0..^𝐿) must have been considered. (Contributed by AV, 6-Nov-2018.)
(((𝑆𝑉𝐿 ∈ ℕ0) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ 𝑁𝐿) → ((𝑆 repeatS 𝐿) substr ⟨𝑀, 𝑁⟩) = (𝑆 repeatS (𝑁𝑀)))

Theoremrepswpfx 14140 A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.)
((𝑆𝑉𝑁 ∈ ℕ0𝐿 ∈ (0...𝑁)) → ((𝑆 repeatS 𝑁) prefix 𝐿) = (𝑆 repeatS 𝐿))

Theoremrepswccat 14141 The concatenation of two "repeated symbol words" with the same symbol is again a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → ((𝑆 repeatS 𝑁) ++ (𝑆 repeatS 𝑀)) = (𝑆 repeatS (𝑁 + 𝑀)))

Theoremrepswrevw 14142 The reverse of a "repeated symbol word". (Contributed by AV, 6-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (reverse‘(𝑆 repeatS 𝑁)) = (𝑆 repeatS 𝑁))

5.7.14  Cyclical shifts of words

A word/string can be regarded as "necklace" by connecting the two ends of the word/string together (see Wikipedia "Necklace (combinatorics)", https://en.wikipedia.org/wiki/Necklace_(combinatorics)).

Two strings are regarded as the same necklace if one string can be rotated/circularly shifted/cyclically shifted to obtain the second string. To cope with words in the sense of necklaces, the rotation/cyclic shift cyclShift is defined as the basic operation, see df-csh 14144. The main theorems in this section are about counting the number of different necklaces resulting from cyclically shifting a given word, see cshwrepswhash1 16430 for words consisting of identical symbols and cshwshash 16432 for words having lengths which are prime numbers.

Syntaxccsh 14143 Extend class notation with Cyclical Shifts.
class cyclShift

Definitiondf-csh 14144* Perform a cyclical shift for an arbitrary class. Meaningful only for words 𝑤 ∈ Word 𝑆 or at least functions over half-open ranges of nonnegative integers. (Contributed by Alexander van der Vekens, 20-May-2018.) (Revised by Mario Carneiro/Alexander van der Vekens/ Gerard Lang, 17-Nov-2018.) (Revised by AV, 4-Nov-2022.)
cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))))

Theoremcshfn 14145* Perform a cyclical shift for a function over a half-open range of nonnegative integers. (Contributed by AV, 20-May-2018.) (Revised by AV, 17-Nov-2018.) (Revised by AV, 4-Nov-2022.)
((𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))))

Theoremcshword 14146 Perform a cyclical shift for a word. (Contributed by Alexander van der Vekens, 20-May-2018.) (Revised by AV, 12-Oct-2022.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))))

Theoremcshnz 14147 A cyclical shift is the empty set if the number of shifts is not an integer. (Contributed by Alexander van der Vekens, 21-May-2018.) (Revised by AV, 17-Nov-2018.)
𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) = ∅)

Theorem0csh0 14148 Cyclically shifting an empty set/word always results in the empty word/set. (Contributed by AV, 25-Oct-2018.) (Revised by AV, 17-Nov-2018.)
(∅ cyclShift 𝑁) = ∅

Theoremcshw0 14149 A word cyclically shifted by 0 is the word itself. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 26-Oct-2018.)
(𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)

Theoremcshwmodn 14150 Cyclically shifting a word is invariant regarding modulo the word's length. (Contributed by AV, 26-Oct-2018.) (Proof shortened by AV, 16-Oct-2022.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (♯‘𝑊))))

Theoremcshwsublen 14151 Cyclically shifting a word is invariant regarding subtraction of the word's length. (Contributed by AV, 3-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 − (♯‘𝑊))))

Theoremcshwn 14152 A word cyclically shifted by its length is the word itself. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 26-Oct-2018.)
(𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (♯‘𝑊)) = 𝑊)

Theoremcshwcl 14153 A cyclically shifted word is a word over the same set as for the original word. (Contributed by AV, 16-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 27-Oct-2018.)
(𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑁) ∈ Word 𝑉)

Theoremcshwlen 14154 The length of a cyclically shifted word is the same as the length of the original word. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 27-Oct-2018.) (Proof shortened by AV, 16-Oct-2022.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (♯‘(𝑊 cyclShift 𝑁)) = (♯‘𝑊))

Theoremcshwf 14155 A cyclically shifted word is a function from a half-open range of integers of the same length as the word as domain to the set of symbols for the word. (Contributed by AV, 12-Nov-2018.)
((𝑊 ∈ Word 𝐴𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁):(0..^(♯‘𝑊))⟶𝐴)

Theoremcshwfn 14156 A cyclically shifted word is a function with a half-open range of integers of the same length as the word as domain. (Contributed by AV, 12-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) Fn (0..^(♯‘𝑊)))

Theoremcshwrn 14157 The range of a cyclically shifted word is a subset of the set of symbols for the word. (Contributed by AV, 12-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → ran (𝑊 cyclShift 𝑁) ⊆ 𝑉)

Theoremcshwidxmod 14158 The symbol at a given index of a cyclically shifted nonempty word is the symbol at the shifted index of the original word. (Contributed by AV, 13-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.) (Proof shortened by AV, 12-Oct-2022.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘𝐼) = (𝑊‘((𝐼 + 𝑁) mod (♯‘𝑊))))

Theoremcshwidxmodr 14159 The symbol at a given index of a cyclically shifted nonempty word is the symbol at the shifted index of the original word. (Contributed by AV, 17-Mar-2021.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((𝐼𝑁) mod (♯‘𝑊))) = (𝑊𝐼))

Theoremcshwidx0mod 14160 The symbol at index 0 of a cyclically shifted nonempty word is the symbol at index N (modulo the length of the word) of the original word. (Contributed by AV, 30-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑁)‘0) = (𝑊‘(𝑁 mod (♯‘𝑊))))

Theoremcshwidx0 14161 The symbol at index 0 of a cyclically shifted nonempty word is the symbol at index N of the original word. (Contributed by AV, 15-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘0) = (𝑊𝑁))

Theoremcshwidxm1 14162 The symbol at index ((n-N)-1) of a word of length n (not 0) cyclically shifted by N positions is the symbol at index (n-1) of the original word. (Contributed by AV, 23-Mar-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘(((♯‘𝑊) − 𝑁) − 1)) = (𝑊‘((♯‘𝑊) − 1)))

Theoremcshwidxm 14163 The symbol at index (n-N) of a word of length n (not 0) cyclically shifted by N positions (not 0) is the symbol at index 0 of the original word. (Contributed by AV, 18-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (1...(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((♯‘𝑊) − 𝑁)) = (𝑊‘0))

Theoremcshwidxn 14164 The symbol at index (n-1) of a word of length n (not 0) cyclically shifted by N positions (not 0) is the symbol at index (N-1) of the original word. (Contributed by AV, 18-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (1...(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((♯‘𝑊) − 1)) = (𝑊‘(𝑁 − 1)))

Theoremcshf1 14165 Cyclically shifting a word which contains a symbol at most once results in a word which contains a symbol at most once. (Contributed by AV, 14-Mar-2021.)
((𝐹:(0..^(♯‘𝐹))–1-1𝐴𝑆 ∈ ℤ ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(♯‘𝐹))–1-1𝐴)

Theoremcshinj 14166 If a word is injectiv (regarded as function), the cyclically shifted word is also injective. (Contributed by AV, 14-Mar-2021.)
((𝐹 ∈ Word 𝐴 ∧ Fun 𝐹𝑆 ∈ ℤ) → (𝐺 = (𝐹 cyclShift 𝑆) → Fun 𝐺))

Theoremrepswcshw 14167 A cyclically shifted "repeated symbol word". (Contributed by Alexander van der Vekens, 7-Nov-2018.) (Proof shortened by AV, 16-Oct-2022.)
((𝑆𝑉𝑁 ∈ ℕ0𝐼 ∈ ℤ) → ((𝑆 repeatS 𝑁) cyclShift 𝐼) = (𝑆 repeatS 𝑁))

Theorem2cshw 14168 Cyclically shifting a word two times. (Contributed by AV, 7-Apr-2018.) (Revised by AV, 4-Jun-2018.) (Revised by AV, 31-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑀) cyclShift 𝑁) = (𝑊 cyclShift (𝑀 + 𝑁)))

Theorem2cshwid 14169 Cyclically shifting a word two times resulting in the word itself. (Contributed by AV, 7-Apr-2018.) (Revised by AV, 5-Jun-2018.) (Revised by AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑁) cyclShift ((♯‘𝑊) − 𝑁)) = 𝑊)

Theoremlswcshw 14170 The last symbol of a word cyclically shifted by N positions is the symbol at index (N-1) of the original word. (Contributed by AV, 21-Mar-2018.) (Revised by AV, 5-Jun-2018.) (Revised by AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (1...(♯‘𝑊))) → (lastS‘(𝑊 cyclShift 𝑁)) = (𝑊‘(𝑁 − 1)))

Theorem2cshwcom 14171 Cyclically shifting a word two times is commutative. (Contributed by AV, 21-Apr-2018.) (Revised by AV, 5-Jun-2018.) (Revised by Mario Carneiro/AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑊 cyclShift 𝑁) cyclShift 𝑀) = ((𝑊 cyclShift 𝑀) cyclShift 𝑁))

Theoremcshwleneq 14172 If the results of cyclically shifting two words are equal, the length of the two words was equal. (Contributed by AV, 21-Apr-2018.) (Revised by AV, 5-Jun-2018.) (Revised by AV, 1-Nov-2018.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (♯‘𝑊) = (♯‘𝑈))

Theorem3cshw 14173 Cyclically shifting a word three times results in a once cyclically shifted word under certain circumstances. (Contributed by AV, 6-Jun-2018.) (Revised by AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (((𝑊 cyclShift 𝑀) cyclShift 𝑁) cyclShift ((♯‘𝑊) − 𝑀)))

Theoremcshweqdif2 14174 If cyclically shifting two words (of the same length) results in the same word, cyclically shifting one of the words by the difference of the numbers of shifts results in the other word. (Contributed by AV, 21-Apr-2018.) (Revised by AV, 6-Jun-2018.) (Revised by AV, 1-Nov-2018.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀) → (𝑈 cyclShift (𝑀𝑁)) = 𝑊))

Theoremcshweqdifid 14175 If cyclically shifting a word by two positions results in the same word, cyclically shifting the word by the difference of these two positions results in the original word itself. (Contributed by AV, 21-Apr-2018.) (Revised by AV, 7-Jun-2018.) (Revised by AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑊 cyclShift 𝑁) = (𝑊 cyclShift 𝑀) → (𝑊 cyclShift (𝑀𝑁)) = 𝑊))

Theoremcshweqrep 14176* If cyclically shifting a word by L position results in the word itself, the symbol at any position is repeated at multiples of L (modulo the length of the word) positions in the word. (Contributed by AV, 13-May-2018.) (Revised by AV, 7-Jun-2018.) (Revised by AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉𝐿 ∈ ℤ) → (((𝑊 cyclShift 𝐿) = 𝑊𝐼 ∈ (0..^(♯‘𝑊))) → ∀𝑗 ∈ ℕ0 (𝑊𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊)))))

Theoremcshw1 14177* If cyclically shifting a word by 1 position results in the word itself, the word is build of identical symbols. Remark: also "valid" for an empty word! (Contributed by AV, 13-May-2018.) (Revised by AV, 7-Jun-2018.) (Proof shortened by AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) = (𝑊‘0))

Theoremcshw1repsw 14178 If cyclically shifting a word by 1 position results in the word itself, the word is a "repeated symbol word". Remark: also "valid" for an empty word! (Contributed by AV, 8-Nov-2018.) (Proof shortened by AV, 10-Nov-2018.)
((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)))

Theoremcshwsexa 14179* The class of (different!) words resulting by cyclically shifting something (not necessarily a word) is a set. (Contributed by AV, 8-Jun-2018.) (Revised by Mario Carneiro/AV, 25-Oct-2018.)
{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V

Theorem2cshwcshw 14180* If a word is a cyclically shifted word, and a second word is the result of cyclically shifting the same word, then the second word is the result of cyclically shifting the first word. (Contributed by AV, 11-May-2018.) (Revised by AV, 12-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → ((𝐾 ∈ (0...𝑁) ∧ 𝑋 = (𝑌 cyclShift 𝐾) ∧ ∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))

Theoremscshwfzeqfzo 14181* For a nonempty word the sets of shifted words, expressd by a finite interval of integers or by a half-open integer range are identical. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)})

Theoremcshwcshid 14182* A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 27813 and erclwwlknsym 27862. (Contributed by AV, 8-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
(𝜑𝑦 ∈ Word 𝑉)    &   (𝜑 → (♯‘𝑥) = (♯‘𝑦))       (𝜑 → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))

Theoremcshwcsh2id 14183* A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 27814 and erclwwlkntr 27863. (Contributed by AV, 9-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
(𝜑𝑧 ∈ Word 𝑉)    &   (𝜑 → ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))       (𝜑 → (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))

Theoremcshimadifsn 14184 The image of a cyclically shifted word under its domain without its left bound is the image of a cyclically shifted word under its domain without the number of shifted symbols. (Contributed by AV, 19-Mar-2021.)
((𝐹 ∈ Word 𝑆𝑁 = (♯‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift 𝐽) “ (1..^𝑁)))

Theoremcshimadifsn0 14185 The image of a cyclically shifted word under its domain without its upper bound is the image of a cyclically shifted word under its domain without the number of shifted symbols. (Contributed by AV, 19-Mar-2021.)
((𝐹 ∈ Word 𝑆𝑁 = (♯‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))

5.7.15  Mapping words by a function

Theoremwrdco 14186 Mapping a word by a function. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (𝐹𝑊) ∈ Word 𝐵)

Theoremlenco 14187 Length of a mapped word is unchanged. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (♯‘(𝐹𝑊)) = (♯‘𝑊))

Theorems1co 14188 Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = ⟨“(𝐹𝑆)”⟩)

Theoremrevco 14189 Mapping of words (i.e., a letterwise mapping) commutes with reversal. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (𝐹 ∘ (reverse‘𝑊)) = (reverse‘(𝐹𝑊)))

Theoremccatco 14190 Mapping of words commutes with concatenation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑆 ∈ Word 𝐴𝑇 ∈ Word 𝐴𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 ++ 𝑇)) = ((𝐹𝑆) ++ (𝐹𝑇)))

Theoremcshco 14191 Mapping of words commutes with the "cyclical shift" operation. (Contributed by AV, 12-Nov-2018.)
((𝑊 ∈ Word 𝐴𝑁 ∈ ℤ ∧ 𝐹:𝐴𝐵) → (𝐹 ∘ (𝑊 cyclShift 𝑁)) = ((𝐹𝑊) cyclShift 𝑁))

Theoremswrdco 14192 Mapping of words commutes with the substring operation. (Contributed by AV, 11-Nov-2018.)
((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝐹:𝐴𝐵) → (𝐹 ∘ (𝑊 substr ⟨𝑀, 𝑁⟩)) = ((𝐹𝑊) substr ⟨𝑀, 𝑁⟩))

Theorempfxco 14193 Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.)
((𝑊 ∈ Word 𝐴𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝐹:𝐴𝐵) → (𝐹 ∘ (𝑊 prefix 𝑁)) = ((𝐹𝑊) prefix 𝑁))

Theoremlswco 14194 Mapping of (nonempty) words commutes with the "last symbol" operation. This theorem would not hold if 𝑊 = ∅, (𝐹‘∅) ≠ ∅ and ∅ ∈ 𝐴, because then (lastS‘(𝐹𝑊)) = (lastS‘∅) = ∅ ≠ (𝐹‘∅) = (𝐹(lastS‘𝑊)). (Contributed by AV, 11-Nov-2018.)
((𝑊 ∈ Word 𝐴𝑊 ≠ ∅ ∧ 𝐹:𝐴𝐵) → (lastS‘(𝐹𝑊)) = (𝐹‘(lastS‘𝑊)))

Theoremrepsco 14195 Mapping of words commutes with the "repeated symbol" operation. (Contributed by AV, 11-Nov-2018.)
((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹𝑆) repeatS 𝑁))

5.7.16  Longer string literals

Syntaxcs2 14196 Syntax for the length 2 word constructor.
class ⟨“𝐴𝐵”⟩

Syntaxcs3 14197 Syntax for the length 3 word constructor.
class ⟨“𝐴𝐵𝐶”⟩

Syntaxcs4 14198 Syntax for the length 4 word constructor.
class ⟨“𝐴𝐵𝐶𝐷”⟩

Syntaxcs5 14199 Syntax for the length 5 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸”⟩

Syntaxcs6 14200 Syntax for the length 6 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩

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454 45301-45347
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