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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | csplice 14101 | Syntax for the word splicing operator. |
class splice | ||
Definition | df-splice 14102* | 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 ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) ++ (𝑠 substr 〈(2nd ‘(1st ‘𝑏)), (♯‘𝑠)〉))) | ||
Theorem | splval 14103 | 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.) |
⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (𝑆 splice 〈𝐹, 𝑇, 𝑅〉) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr 〈𝑇, (♯‘𝑆)〉))) | ||
Theorem | splcl 14104 | Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.) |
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 splice 〈𝐹, 𝑇, 𝑅〉) ∈ Word 𝐴) | ||
Theorem | splid 14105 | 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) | ||
Theorem | spllen 14106 | The length of a splice. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.) |
⊢ (𝜑 → 𝑆 ∈ Word 𝐴) & ⊢ (𝜑 → 𝐹 ∈ (0...𝑇)) & ⊢ (𝜑 → 𝑇 ∈ (0...(♯‘𝑆))) & ⊢ (𝜑 → 𝑅 ∈ Word 𝐴) ⇒ ⊢ (𝜑 → (♯‘(𝑆 splice 〈𝐹, 𝑇, 𝑅〉)) = ((♯‘𝑆) + ((♯‘𝑅) − (𝑇 − 𝐹)))) | ||
Theorem | splfv1 14107 | Symbols to the left of a splice are unaffected. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.) |
⊢ (𝜑 → 𝑆 ∈ Word 𝐴) & ⊢ (𝜑 → 𝐹 ∈ (0...𝑇)) & ⊢ (𝜑 → 𝑇 ∈ (0...(♯‘𝑆))) & ⊢ (𝜑 → 𝑅 ∈ Word 𝐴) & ⊢ (𝜑 → 𝑋 ∈ (0..^𝐹)) ⇒ ⊢ (𝜑 → ((𝑆 splice 〈𝐹, 𝑇, 𝑅〉)‘𝑋) = (𝑆‘𝑋)) | ||
Theorem | splfv2a 14108 | 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.) |
⊢ (𝜑 → 𝑆 ∈ Word 𝐴) & ⊢ (𝜑 → 𝐹 ∈ (0...𝑇)) & ⊢ (𝜑 → 𝑇 ∈ (0...(♯‘𝑆))) & ⊢ (𝜑 → 𝑅 ∈ Word 𝐴) & ⊢ (𝜑 → 𝑋 ∈ (0..^(♯‘𝑅))) ⇒ ⊢ (𝜑 → ((𝑆 splice 〈𝐹, 𝑇, 𝑅〉)‘(𝐹 + 𝑋)) = (𝑅‘𝑋)) | ||
Theorem | splval2 14109 | 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.) |
⊢ (𝜑 → 𝐴 ∈ Word 𝑋) & ⊢ (𝜑 → 𝐵 ∈ Word 𝑋) & ⊢ (𝜑 → 𝐶 ∈ Word 𝑋) & ⊢ (𝜑 → 𝑅 ∈ Word 𝑋) & ⊢ (𝜑 → 𝑆 = ((𝐴 ++ 𝐵) ++ 𝐶)) & ⊢ (𝜑 → 𝐹 = (♯‘𝐴)) & ⊢ (𝜑 → 𝑇 = (𝐹 + (♯‘𝐵))) ⇒ ⊢ (𝜑 → (𝑆 splice 〈𝐹, 𝑇, 𝑅〉) = ((𝐴 ++ 𝑅) ++ 𝐶)) | ||
Syntax | creverse 14110 | Syntax for the word reverse operator. |
class reverse | ||
Definition | df-reverse 14111* | 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)))) | ||
Theorem | revval 14112* | Value of the word reversing function. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
⊢ (𝑊 ∈ 𝑉 → (reverse‘𝑊) = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ (𝑊‘(((♯‘𝑊) − 1) − 𝑥)))) | ||
Theorem | revcl 14113 | The reverse of a word is a word. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
⊢ (𝑊 ∈ Word 𝐴 → (reverse‘𝑊) ∈ Word 𝐴) | ||
Theorem | revlen 14114 | 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)) | ||
Theorem | revfv 14115 | Reverse of a word at a point. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑋 ∈ (0..^(♯‘𝑊))) → ((reverse‘𝑊)‘𝑋) = (𝑊‘(((♯‘𝑊) − 1) − 𝑋))) | ||
Theorem | rev0 14116 | The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
⊢ (reverse‘∅) = ∅ | ||
Theorem | revs1 14117 | Singleton words are their own reverses. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
⊢ (reverse‘〈“𝑆”〉) = 〈“𝑆”〉 | ||
Theorem | revccat 14118 | Antiautomorphic property of the reversal operation. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) = ((reverse‘𝑇) ++ (reverse‘𝑆))) | ||
Theorem | revrev 14119 | Reversal is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ (𝑊 ∈ Word 𝐴 → (reverse‘(reverse‘𝑊)) = 𝑊) | ||
Syntax | creps 14120 | Extend class notation with words consisting of one repeated symbol. |
class repeatS | ||
Definition | df-reps 14121* | 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)) | ||
Theorem | reps 14122* | Construct a function mapping a half-open range of nonnegative integers to a constant. (Contributed by AV, 4-Nov-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆)) | ||
Theorem | repsundef 14123 | 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) = ∅) | ||
Theorem | repsconst 14124 | Construct a function mapping a half-open range of nonnegative integers to a constant, see also fconstmpt 5608. (Contributed by AV, 4-Nov-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = ((0..^𝑁) × {𝑆})) | ||
Theorem | repsf 14125 | The constructed function mapping a half-open range of nonnegative integers to a constant is a function. (Contributed by AV, 4-Nov-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝑉) | ||
Theorem | repswsymb 14126 | The symbols of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝐼) = 𝑆) | ||
Theorem | repsw 14127 | 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.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) ∈ Word 𝑉) | ||
Theorem | repswlen 14128 | The length of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (♯‘(𝑆 repeatS 𝑁)) = 𝑁) | ||
Theorem | repsw0 14129 | The "repeated symbol word" of length 0. (Contributed by AV, 4-Nov-2018.) |
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 0) = ∅) | ||
Theorem | repsdf2 14130* | Alternative definition of a "repeated symbol word". (Contributed by AV, 7-Nov-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑊 = (𝑆 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆))) | ||
Theorem | repswsymball 14131* | All the symbols of a "repeated symbol word" are the same. (Contributed by AV, 10-Nov-2018.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉) → (𝑊 = (𝑆 repeatS (♯‘𝑊)) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = 𝑆)) | ||
Theorem | repswsymballbi 14132* | 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))(𝑊‘𝑖) = (𝑊‘0))) | ||
Theorem | repswfsts 14133 | The first symbol of a nonempty "repeated symbol word". (Contributed by AV, 4-Nov-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑆 repeatS 𝑁)‘0) = 𝑆) | ||
Theorem | repswlsw 14134 | The last symbol of a nonempty "repeated symbol word". (Contributed by AV, 4-Nov-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (lastS‘(𝑆 repeatS 𝑁)) = 𝑆) | ||
Theorem | repsw1 14135 | The "repeated symbol word" of length 1. (Contributed by AV, 4-Nov-2018.) |
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 1) = 〈“𝑆”〉) | ||
Theorem | repswswrd 14136 | A subword of a "repeated symbol word" is again a "repeated symbol word". The assumption 𝑁 ≤ 𝐿 is required, because otherwise (𝐿 < 𝑁): ((𝑆 repeatS 𝐿) substr 〈𝑀, 𝑁〉) = ∅, but for M < N (𝑆 repeatS (𝑁 − 𝑀))) ≠ ∅! 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.) |
⊢ (((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → ((𝑆 repeatS 𝐿) substr 〈𝑀, 𝑁〉) = (𝑆 repeatS (𝑁 − 𝑀))) | ||
Theorem | repswpfx 14137 | A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → ((𝑆 repeatS 𝑁) prefix 𝐿) = (𝑆 repeatS 𝐿)) | ||
Theorem | repswccat 14138 | The concatenation of two "repeated symbol words" with the same symbol is again a "repeated symbol word". (Contributed by AV, 4-Nov-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → ((𝑆 repeatS 𝑁) ++ (𝑆 repeatS 𝑀)) = (𝑆 repeatS (𝑁 + 𝑀))) | ||
Theorem | repswrevw 14139 | The reverse of a "repeated symbol word". (Contributed by AV, 6-Nov-2018.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (reverse‘(𝑆 repeatS 𝑁)) = (𝑆 repeatS 𝑁)) | ||
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 14141. The main theorems in this section are about counting the number of different necklaces resulting from cyclically shifting a given word, see cshwrepswhash1 16426 for words consisting of identical symbols and cshwshash 16428 for words having lengths which are prime numbers. | ||
Syntax | ccsh 14140 | Extend class notation with Cyclical Shifts. |
class cyclShift | ||
Definition | df-csh 14141* | 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)))))) | ||
Theorem | cshfn 14142* | 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)))))) | ||
Theorem | cshword 14143 | Perform a cyclical shift for a word. (Contributed by Alexander van der Vekens, 20-May-2018.) (Revised by AV, 12-Oct-2022.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))) | ||
Theorem | cshnz 14144 | 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) = ∅) | ||
Theorem | 0csh0 14145 | 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) = ∅ | ||
Theorem | cshw0 14146 | 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) | ||
Theorem | cshwmodn 14147 | 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 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (♯‘𝑊)))) | ||
Theorem | cshwsublen 14148 | Cyclically shifting a word is invariant regarding subtraction of the word's length. (Contributed by AV, 3-Nov-2018.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 − (♯‘𝑊)))) | ||
Theorem | cshwn 14149 | 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) | ||
Theorem | cshwcl 14150 | 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) | ||
Theorem | cshwlen 14151 | 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 𝑉 ∧ 𝑁 ∈ ℤ) → (♯‘(𝑊 cyclShift 𝑁)) = (♯‘𝑊)) | ||
Theorem | cshwf 14152 | 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 𝐴 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁):(0..^(♯‘𝑊))⟶𝐴) | ||
Theorem | cshwfn 14153 | 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 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) Fn (0..^(♯‘𝑊))) | ||
Theorem | cshwrn 14154 | 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 𝑉 ∧ 𝑁 ∈ ℤ) → ran (𝑊 cyclShift 𝑁) ⊆ 𝑉) | ||
Theorem | cshwidxmod 14155 | 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 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘𝐼) = (𝑊‘((𝐼 + 𝑁) mod (♯‘𝑊)))) | ||
Theorem | cshwidxmodr 14156 | 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 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((𝐼 − 𝑁) mod (♯‘𝑊))) = (𝑊‘𝐼)) | ||
Theorem | cshwidx0mod 14157 | 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 𝑉 ∧ 𝑊 ≠ ∅ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑁)‘0) = (𝑊‘(𝑁 mod (♯‘𝑊)))) | ||
Theorem | cshwidx0 14158 | 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 𝑉 ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘0) = (𝑊‘𝑁)) | ||
Theorem | cshwidxm1 14159 | 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 𝑉 ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘(((♯‘𝑊) − 𝑁) − 1)) = (𝑊‘((♯‘𝑊) − 1))) | ||
Theorem | cshwidxm 14160 | 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 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((♯‘𝑊) − 𝑁)) = (𝑊‘0)) | ||
Theorem | cshwidxn 14161 | 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 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((♯‘𝑊) − 1)) = (𝑊‘(𝑁 − 1))) | ||
Theorem | cshf1 14162 | 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))–1-1→𝐴 ∧ 𝑆 ∈ ℤ ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(♯‘𝐹))–1-1→𝐴) | ||
Theorem | cshinj 14163 | 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 ◡𝐹 ∧ 𝑆 ∈ ℤ) → (𝐺 = (𝐹 cyclShift 𝑆) → Fun ◡𝐺)) | ||
Theorem | repswcshw 14164 | A cyclically shifted "repeated symbol word". (Contributed by Alexander van der Vekens, 7-Nov-2018.) (Proof shortened by AV, 16-Oct-2022.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℤ) → ((𝑆 repeatS 𝑁) cyclShift 𝐼) = (𝑆 repeatS 𝑁)) | ||
Theorem | 2cshw 14165 | 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 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑀) cyclShift 𝑁) = (𝑊 cyclShift (𝑀 + 𝑁))) | ||
Theorem | 2cshwid 14166 | 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 𝑉 ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑁) cyclShift ((♯‘𝑊) − 𝑁)) = 𝑊) | ||
Theorem | lswcshw 14167 | 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 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → (lastS‘(𝑊 cyclShift 𝑁)) = (𝑊‘(𝑁 − 1))) | ||
Theorem | 2cshwcom 14168 | 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 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑊 cyclShift 𝑁) cyclShift 𝑀) = ((𝑊 cyclShift 𝑀) cyclShift 𝑁)) | ||
Theorem | cshwleneq 14169 | 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 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (♯‘𝑊) = (♯‘𝑈)) | ||
Theorem | 3cshw 14170 | 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 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (((𝑊 cyclShift 𝑀) cyclShift 𝑁) cyclShift ((♯‘𝑊) − 𝑀))) | ||
Theorem | cshweqdif2 14171 | 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 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀) → (𝑈 cyclShift (𝑀 − 𝑁)) = 𝑊)) | ||
Theorem | cshweqdifid 14172 | 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 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑊 cyclShift 𝑁) = (𝑊 cyclShift 𝑀) → (𝑊 cyclShift (𝑀 − 𝑁)) = 𝑊)) | ||
Theorem | cshweqrep 14173* | 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 𝑉 ∧ 𝐿 ∈ ℤ) → (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ∀𝑗 ∈ ℕ0 (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊))))) | ||
Theorem | cshw1 14174* | 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))(𝑊‘𝑖) = (𝑊‘0)) | ||
Theorem | cshw1repsw 14175 | 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))) | ||
Theorem | cshwsexa 14176* | 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 | ||
Theorem | 2cshwcshw 14177* | 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))) | ||
Theorem | scshwfzeqfzo 14178* | 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 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)}) | ||
Theorem | cshwcshid 14179* | A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 27727 and erclwwlknsym 27777. (Contributed by AV, 8-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.) |
⊢ (𝜑 → 𝑦 ∈ Word 𝑉) & ⊢ (𝜑 → (♯‘𝑥) = (♯‘𝑦)) ⇒ ⊢ (𝜑 → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))) | ||
Theorem | cshwcsh2id 14180* | A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 27728 and erclwwlkntr 27778. (Contributed by AV, 9-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.) |
⊢ (𝜑 → 𝑧 ∈ 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))) | ||
Theorem | cshimadifsn 14181 | 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 𝑆 ∧ 𝑁 = (♯‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift 𝐽) “ (1..^𝑁))) | ||
Theorem | cshimadifsn0 14182 | 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 𝑆 ∧ 𝑁 = (♯‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1)))) | ||
Theorem | wrdco 14183 | Mapping a word by a function. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑊) ∈ Word 𝐵) | ||
Theorem | lenco 14184 | Length of a mapped word is unchanged. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑊)) = (♯‘𝑊)) | ||
Theorem | s1co 14185 | Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 〈“𝑆”〉) = 〈“(𝐹‘𝑆)”〉) | ||
Theorem | revco 14186 | Mapping of words (i.e., a letterwise mapping) commutes with reversal. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (reverse‘𝑊)) = (reverse‘(𝐹 ∘ 𝑊))) | ||
Theorem | ccatco 14187 | Mapping of words commutes with concatenation. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 ++ 𝑇)) = ((𝐹 ∘ 𝑆) ++ (𝐹 ∘ 𝑇))) | ||
Theorem | cshco 14188 | Mapping of words commutes with the "cyclical shift" operation. (Contributed by AV, 12-Nov-2018.) |
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑁 ∈ ℤ ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑊 cyclShift 𝑁)) = ((𝐹 ∘ 𝑊) cyclShift 𝑁)) | ||
Theorem | swrdco 14189 | Mapping of words commutes with the substring operation. (Contributed by AV, 11-Nov-2018.) |
⊢ ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑊 substr 〈𝑀, 𝑁〉)) = ((𝐹 ∘ 𝑊) substr 〈𝑀, 𝑁〉)) | ||
Theorem | pfxco 14190 | Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.) |
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑊 prefix 𝑁)) = ((𝐹 ∘ 𝑊) prefix 𝑁)) | ||
Theorem | lswco 14191 | Mapping of (nonempty) words commutes with the "last symbol" operation. This theorem would not hold if 𝑊 = ∅, (𝐹‘∅) ≠ ∅ and ∅ ∈ 𝐴, because then (lastS‘(𝐹 ∘ 𝑊)) = (lastS‘∅) = ∅ ≠ (𝐹‘∅) = (𝐹(lastS‘𝑊)). (Contributed by AV, 11-Nov-2018.) |
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅ ∧ 𝐹:𝐴⟶𝐵) → (lastS‘(𝐹 ∘ 𝑊)) = (𝐹‘(lastS‘𝑊))) | ||
Theorem | repsco 14192 | Mapping of words commutes with the "repeated symbol" operation. (Contributed by AV, 11-Nov-2018.) |
⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹‘𝑆) repeatS 𝑁)) | ||
Syntax | cs2 14193 | Syntax for the length 2 word constructor. |
class 〈“𝐴𝐵”〉 | ||
Syntax | cs3 14194 | Syntax for the length 3 word constructor. |
class 〈“𝐴𝐵𝐶”〉 | ||
Syntax | cs4 14195 | Syntax for the length 4 word constructor. |
class 〈“𝐴𝐵𝐶𝐷”〉 | ||
Syntax | cs5 14196 | Syntax for the length 5 word constructor. |
class 〈“𝐴𝐵𝐶𝐷𝐸”〉 | ||
Syntax | cs6 14197 | Syntax for the length 6 word constructor. |
class 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 | ||
Syntax | cs7 14198 | Syntax for the length 7 word constructor. |
class 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 | ||
Syntax | cs8 14199 | Syntax for the length 8 word constructor. |
class 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”〉 | ||
Definition | df-s2 14200 | Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.) |
⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) |
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