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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | revfv 14801 | 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 14802 | The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
| ⊢ (reverse‘∅) = ∅ | ||
| Theorem | revs1 14803 | 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 14804 | 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 14805 | Reversal is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.) |
| ⊢ (𝑊 ∈ Word 𝐴 → (reverse‘(reverse‘𝑊)) = 𝑊) | ||
| Syntax | creps 14806 | Extend class notation with words consisting of one repeated symbol. |
| class repeatS | ||
| Definition | df-reps 14807* | 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 14808* | 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 14809 | 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 14810 | Construct a function mapping a half-open range of nonnegative integers to a constant, see also fconstmpt 5747. (Contributed by AV, 4-Nov-2018.) |
| ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = ((0..^𝑁) × {𝑆})) | ||
| Theorem | repsf 14811 | 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 14812 | 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 14813 | 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 14814 | The length of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.) |
| ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (♯‘(𝑆 repeatS 𝑁)) = 𝑁) | ||
| Theorem | repsw0 14815 | The "repeated symbol word" of length 0. (Contributed by AV, 4-Nov-2018.) |
| ⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 0) = ∅) | ||
| Theorem | repsdf2 14816* | 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 14817* | 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 14818* | 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 14819 | 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 14820 | 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 14821 | The "repeated symbol word" of length 1. (Contributed by AV, 4-Nov-2018.) |
| ⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 1) = 〈“𝑆”〉) | ||
| Theorem | repswswrd 14822 | 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 14823 | 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 14824 | 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 14825 | 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 14827. The main theorems in this section are about counting the number of different necklaces resulting from cyclically shifting a given word, see cshwrepswhash1 17140 for words consisting of identical symbols and cshwshash 17142 for words having lengths which are prime numbers. | ||
| Syntax | ccsh 14826 | Extend class notation with Cyclical Shifts. |
| class cyclShift | ||
| Definition | df-csh 14827* | 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 14828* | 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 14829 | 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 14830 | 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 14831 | 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 14832 | 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 14833 | 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 14834 | 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 14835 | 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 14836 | 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 14837 | 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 14838 | 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 14839 | 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 14840 | 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 14841 | 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 14842 | 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 14843 | 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 14844 | 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 14845 | 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 14846 | 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 14847 | 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 14848 | 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 14849 | 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 14850 | 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 14851 | 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 14852 | 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 14853 | 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 14854 | 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 14855 | 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 14856 | 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 14857 | 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 14858 | 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 14859* | 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 14860* | 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 14861 | 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 14862* | 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.) (Proof shortened by SN, 15-Jan-2025.) |
| ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V | ||
| Theorem | cshwsexaOLD 14863* | Obsolete version of cshwsexa 14862 as of 15-Jan-2025. (Contributed by AV, 8-Jun-2018.) (Revised by Mario Carneiro/AV, 25-Oct-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V | ||
| Theorem | 2cshwcshw 14864* | 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 14865* | 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 14866* | A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 30040 and erclwwlknsym 30089. (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 14867* | A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 30041 and erclwwlkntr 30090. (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 14868 | 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 14869 | 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 14870 | 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 14871 | 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 14872 | 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 14873 | 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 14874 | 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 14875 | 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 14876 | 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 14877 | 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 14878 | 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 14879 | 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 14880 | Syntax for the length 2 word constructor. |
| class 〈“𝐴𝐵”〉 | ||
| Syntax | cs3 14881 | Syntax for the length 3 word constructor. |
| class 〈“𝐴𝐵𝐶”〉 | ||
| Syntax | cs4 14882 | Syntax for the length 4 word constructor. |
| class 〈“𝐴𝐵𝐶𝐷”〉 | ||
| Syntax | cs5 14883 | Syntax for the length 5 word constructor. |
| class 〈“𝐴𝐵𝐶𝐷𝐸”〉 | ||
| Syntax | cs6 14884 | Syntax for the length 6 word constructor. |
| class 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 | ||
| Syntax | cs7 14885 | Syntax for the length 7 word constructor. |
| class 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 | ||
| Syntax | cs8 14886 | Syntax for the length 8 word constructor. |
| class 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”〉 | ||
| Definition | df-s2 14887 | Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | ||
| Definition | df-s3 14888 | Define the length 3 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) | ||
| Definition | df-s4 14889 | Define the length 4 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 〈“𝐴𝐵𝐶𝐷”〉 = (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) | ||
| Definition | df-s5 14890 | Define the length 5 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 〈“𝐴𝐵𝐶𝐷𝐸”〉 = (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸”〉) | ||
| Definition | df-s6 14891 | Define the length 6 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 = (〈“𝐴𝐵𝐶𝐷𝐸”〉 ++ 〈“𝐹”〉) | ||
| Definition | df-s7 14892 | Define the length 7 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 ++ 〈“𝐺”〉) | ||
| Definition | df-s8 14893 | Define the length 8 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”〉 = (〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 ++ 〈“𝐻”〉) | ||
| Theorem | cats1cld 14894 | Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 𝑇 = (𝑆 ++ 〈“𝑋”〉) & ⊢ (𝜑 → 𝑆 ∈ Word 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑇 ∈ Word 𝐴) | ||
| Theorem | cats1co 14895 | Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 𝑇 = (𝑆 ++ 〈“𝑋”〉) & ⊢ (𝜑 → 𝑆 ∈ Word 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → (𝐹 ∘ 𝑆) = 𝑈) & ⊢ 𝑉 = (𝑈 ++ 〈“(𝐹‘𝑋)”〉) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝑇) = 𝑉) | ||
| Theorem | cats1cli 14896 | Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 𝑇 = (𝑆 ++ 〈“𝑋”〉) & ⊢ 𝑆 ∈ Word V ⇒ ⊢ 𝑇 ∈ Word V | ||
| Theorem | cats1fvn 14897 | The last symbol of a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 𝑇 = (𝑆 ++ 〈“𝑋”〉) & ⊢ 𝑆 ∈ Word V & ⊢ (♯‘𝑆) = 𝑀 ⇒ ⊢ (𝑋 ∈ 𝑉 → (𝑇‘𝑀) = 𝑋) | ||
| Theorem | cats1fv 14898 | A symbol other than the last in a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 𝑇 = (𝑆 ++ 〈“𝑋”〉) & ⊢ 𝑆 ∈ Word V & ⊢ (♯‘𝑆) = 𝑀 & ⊢ (𝑌 ∈ 𝑉 → (𝑆‘𝑁) = 𝑌) & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑁 < 𝑀 ⇒ ⊢ (𝑌 ∈ 𝑉 → (𝑇‘𝑁) = 𝑌) | ||
| Theorem | cats1len 14899 | The length of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 𝑇 = (𝑆 ++ 〈“𝑋”〉) & ⊢ 𝑆 ∈ Word V & ⊢ (♯‘𝑆) = 𝑀 & ⊢ (𝑀 + 1) = 𝑁 ⇒ ⊢ (♯‘𝑇) = 𝑁 | ||
| Theorem | cats1cat 14900 | Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
| ⊢ 𝑇 = (𝑆 ++ 〈“𝑋”〉) & ⊢ 𝐴 ∈ Word V & ⊢ 𝑆 ∈ Word V & ⊢ 𝐶 = (𝐵 ++ 〈“𝑋”〉) & ⊢ 𝐵 = (𝐴 ++ 𝑆) ⇒ ⊢ 𝐶 = (𝐴 ++ 𝑇) | ||
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