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Theorem List for Metamath Proof Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremswrdspsleq 13301* Two words have a common subword (starting at the same position with the same length) iff they have the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Proof shortened by AV, 7-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))

Theoremswrdtrcfvl 13302 The last symbol in a word truncated by one symbol. (Contributed by AV, 16-Jun-2018.) (Proof shortened by Mario Carneiro/AV, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → ( lastS ‘(𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩)) = (𝑊‘((#‘𝑊) − 2)))

Theoremswrds1 13303 Extract a single symbol from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.)
((𝑊 ∈ Word 𝐴𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 substr ⟨𝐼, (𝐼 + 1)⟩) = ⟨“(𝑊𝐼)”⟩)

Theoremswrdlsw 13304 Extract the last single symbol from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑊)”⟩)

Theorem2swrdeqwrdeq 13305 Two words are equal if and only if they have the same prefix and the same suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ((𝑊 substr ⟨0, 𝐼⟩) = (𝑆 substr ⟨0, 𝐼⟩) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))))

Theorem2swrd1eqwrdeq 13306 Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Revised by Mario Carneiro/AV, 23-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))

Theoremdisjxwrd 13307* Sets of words are disjoint if each set contains extensions of distinct words of a fixed length. (Contributed by AV, 29-Jul-2018.) (Proof shortened by AV, 7-May-2020.)
Disj 𝑦𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦}

Theoremccatswrd 13308 Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)
((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 substr ⟨𝑋, 𝑍⟩))

Theoremswrdccat1 13309 Recover the left half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) substr ⟨0, (#‘𝑆)⟩) = 𝑆)

Theoremswrdccat2 13310 Recover the right half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) substr ⟨(#‘𝑆), ((#‘𝑆) + (#‘𝑇))⟩) = 𝑇)

5.7.7  Subwords of subwords

Theoremswrdswrdlem 13311 Lemma for swrdswrd 13312. (Contributed by Alexander van der Vekens, 4-Apr-2018.)
(((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ (𝐾 ∈ (0...(𝑁𝑀)) ∧ 𝐿 ∈ (𝐾...(𝑁𝑀)))) → (𝑊 ∈ Word 𝑉 ∧ (𝑀 + 𝐾) ∈ (0...(𝑀 + 𝐿)) ∧ (𝑀 + 𝐿) ∈ (0...(#‘𝑊))))

Theoremswrdswrd 13312 A subword of a subword. (Contributed by Alexander van der Vekens, 4-Apr-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → ((𝐾 ∈ (0...(𝑁𝑀)) ∧ 𝐿 ∈ (𝐾...(𝑁𝑀))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) substr ⟨𝐾, 𝐿⟩) = (𝑊 substr ⟨(𝑀 + 𝐾), (𝑀 + 𝐿)⟩)))

Theoremswrd0swrd 13313 A prefix of a subword. (Contributed by AV, 2-Apr-2018.) (Proof shortened by AV, 21-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝐿 ∈ (0...(𝑁𝑀)) → ((𝑊 substr ⟨𝑀, 𝑁⟩) substr ⟨0, 𝐿⟩) = (𝑊 substr ⟨𝑀, (𝑀 + 𝐿)⟩)))

Theoremswrdswrd0 13314 A subword of a prefix. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊))) → ((𝐾 ∈ (0...𝑁) ∧ 𝐿 ∈ (𝐾...𝑁)) → ((𝑊 substr ⟨0, 𝑁⟩) substr ⟨𝐾, 𝐿⟩) = (𝑊 substr ⟨𝐾, 𝐿⟩)))

Theoremswrd0swrd0 13315 A prefix of a prefix. (Contributed by Alexander van der Vekens, 7-Apr-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊)) ∧ 𝐿 ∈ (0...𝑁)) → ((𝑊 substr ⟨0, 𝑁⟩) substr ⟨0, 𝐿⟩) = (𝑊 substr ⟨0, 𝐿⟩))

Theoremswrd0swrdid 13316 A prefix of a prefix with the same length is the prefix. (Contributed by AV, 5-Apr-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊))) → ((𝑊 substr ⟨0, 𝑁⟩) substr ⟨0, 𝑁⟩) = (𝑊 substr ⟨0, 𝑁⟩))

5.7.8  Subwords and concatenations

Theoremwrdcctswrd 13317 The concatenation of two parts of a word yields the word itself. (Contributed by AV, 21-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → ((𝑊 substr ⟨0, 𝑀⟩) ++ (𝑊 substr ⟨𝑀, (#‘𝑊)⟩)) = 𝑊)

Theoremlencctswrd 13318 The length of two concatenated parts of a word is the length of the word. (Contributed by AV, 21-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → (#‘((𝑊 substr ⟨0, 𝑀⟩) ++ (𝑊 substr ⟨𝑀, (#‘𝑊)⟩))) = (#‘𝑊))

Theoremlenrevcctswrd 13319 The length of two reversely concatenated parts of a word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → (#‘((𝑊 substr ⟨𝑀, (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, 𝑀⟩))) = (#‘𝑊))

Theoremswrdccatwrd 13320 Reconstruct a nonempty word from its prefix and last symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ++ ⟨“( lastS ‘𝑊)”⟩) = 𝑊)

Theoremccats1swrdeq 13321 The last symbol of a word concatenated with the subword of the word having length less by 1 than the word results in the word itself. (Contributed by Alexander van der Vekens, 24-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) → 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))

Theoremccatopth 13322 An opth 4871-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
(((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremccatopth2 13323 An opth 4871-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
(((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐵) = (#‘𝐷)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremccatlcan 13324 Concatenation of words is left-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋𝐶 ∈ Word 𝑋) → ((𝐶 ++ 𝐴) = (𝐶 ++ 𝐵) ↔ 𝐴 = 𝐵))

Theoremccatrcan 13325 Concatenation of words is right-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋𝐶 ∈ Word 𝑋) → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵))

Theoremwrdeqs1cat 13326 Decompose a nonempty word by separating off the first symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 9-May-2020.)
((𝑊 ∈ Word 𝐴𝑊 ≠ ∅) → 𝑊 = (⟨“(𝑊‘0)”⟩ ++ (𝑊 substr ⟨1, (#‘𝑊)⟩)))

Theoremcats1un 13327 Express a word with an extra symbol as the union of the word and the new value. (Contributed by Mario Carneiro, 28-Feb-2016.)
((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) = (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}))

Theoremwrdind 13328* Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   ((𝑦 ∈ Word 𝐵𝑧𝐵) → (𝜒𝜃))       (𝐴 ∈ Word 𝐵𝜏)

Theoremwrd2ind 13329* Perform induction over the structure of two words of the same length. (Contributed by AV, 23-Jan-2019.)
((𝑥 = ∅ ∧ 𝑤 = ∅) → (𝜑𝜓))    &   ((𝑥 = 𝑦𝑤 = 𝑢) → (𝜑𝜒))    &   ((𝑥 = (𝑦 ++ ⟨“𝑧”⟩) ∧ 𝑤 = (𝑢 ++ ⟨“𝑠”⟩)) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜌𝜏))    &   (𝑤 = 𝐵 → (𝜑𝜌))    &   𝜓    &   (((𝑦 ∈ Word 𝑋𝑧𝑋) ∧ (𝑢 ∈ Word 𝑌𝑠𝑌) ∧ (#‘𝑦) = (#‘𝑢)) → (𝜒𝜃))       ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑌 ∧ (#‘𝐴) = (#‘𝐵)) → 𝜏)

Theoremccats1swrdeqrex 13330* There exists a symbol such that its concatenation with the subword obtained by deleting the last symbol of a nonempty word results in the word itself. (Contributed by AV, 5-Oct-2018.) (Proof shortened by AV, 24-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) → ∃𝑠𝑉 𝑈 = (𝑊 ++ ⟨“𝑠”⟩)))

Theoremreuccats1lem 13331* Lemma for reuccats1 13332. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Proof shortened by AV, 15-Jan-2020.)
(((𝑊 ∈ Word 𝑉𝑈𝑋 ∧ (𝑊 ++ ⟨“𝑆”⟩) ∈ 𝑋) ∧ (∀𝑠𝑉 ((𝑊 ++ ⟨“𝑠”⟩) ∈ 𝑋𝑆 = 𝑠) ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)))) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) → 𝑈 = (𝑊 ++ ⟨“𝑆”⟩)))

Theoremreuccats1 13332* A set of words having the length of a given word increased by 1 contains a unique word with the given word as prefix if there is a unique symbol which extends the given word to be a word of the set. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩)))

5.7.9  Subwords of concatenations

Theoremswrdccatfn 13333 The subword of a concatenation as function. (Contributed by Alexander van der Vekens, 27-May-2018.)
(((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))

Theoremswrdccatin1 13334 The subword of a concatenation of two words within the first of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))

Theoremswrdccatin12lem1 13335 Lemma 1 for swrdccatin12 13342. (Contributed by Alexander van der Vekens, 30-Mar-2018.) (Revised by Alexander van der Vekens, 23-May-2018.)
((𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿𝑀))) → 𝐾 ∈ ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿)))))

Theoremswrdccatin12lem2a 13336 Lemma 1 for swrdccatin12lem2 13340. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...𝑋)) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿𝑀))) → (𝐾 + 𝑀) ∈ (𝐿..^𝑋)))

Theoremswrdccatin12lem2b 13337 Lemma 2 for swrdccatin12lem2 13340. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...𝑋)) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿𝑀))) → (𝐾 − (𝐿𝑀)) ∈ (0..^((𝑁𝐿) − 0))))

Theoremswrdccatin2 13338 The subword of a concatenation of two words within the second of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Revised by Alexander van der Vekens, 27-May-2018.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩)))

Theoremswrdccatin12lem2c 13339 Lemma for swrdccatin12lem2 13340 and swrdccatin12lem3 13341. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
𝐿 = (#‘𝐴)       (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))

Theoremswrdccatin12lem2 13340 Lemma 2 for swrdccatin12 13342. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
𝐿 = (#‘𝐴)       (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝐾) = ((𝐵 substr ⟨0, (𝑁𝐿)⟩)‘(𝐾 − (#‘(𝐴 substr ⟨𝑀, 𝐿⟩))))))

Theoremswrdccatin12lem3 13341 Lemma 3 for swrdccatin12 13342. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
𝐿 = (#‘𝐴)       (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ 𝐾 ∈ (0..^(𝐿𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝐾) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝐾)))

Theoremswrdccatin12 13342 The subword of a concatenation of two words within both of the concatenated words. (Contributed by Alexander van der Vekens, 5-Apr-2018.) (Revised by Alexander van der Vekens, 27-May-2018.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩))))

Theoremswrdccat3 13343 The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 30-Mar-2018.) (Revised by Alexander van der Vekens, 28-May-2018.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = if(𝑁𝐿, (𝐴 substr ⟨𝑀, 𝑁⟩), if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩))))))

Theoremswrdccat 13344 The subword of a concatenation of two words as concatenation of subwords of the two concatenated words. (Contributed by Alexander van der Vekens, 29-May-2018.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, if(𝑁𝐿, 𝑁, 𝐿)⟩) ++ (𝐵 substr ⟨if(0 ≤ (𝑀𝐿), (𝑀𝐿), 0), (𝑁𝐿)⟩))))

Theoremswrdccat3a 13345 A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by Alexander van der Vekens, 29-May-2018.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩)))))

Theoremswrdccat3blem 13346 Lemma for swrdccat3b 13347. (Contributed by AV, 30-May-2018.)
𝐿 = (#‘𝐴)       ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ (𝐿 + (#‘𝐵)) ≤ 𝐿) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩))

Theoremswrdccat3b 13347 A suffix of a concatenation is either a suffix of the second concatenated word or a concatenation of a suffix of the first word with the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by Alexander van der Vekens, 30-May-2018.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵))))

Theoremswrdccatid 13348 A prefix of a concatenation of length of the first concatenated word is the first word itself. (Contributed by Alexander van der Vekens, 20-Sep-2018.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = 𝐴)

Theoremccats1swrdeqbi 13349 A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. (Contributed by AV, 24-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) ↔ 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))

Theoremswrdccatin1d 13350 The subword of a concatenation of two words within the first of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by Mario Carneiro/AV, 21-Oct-2018.)
(𝜑 → (#‘𝐴) = 𝐿)    &   (𝜑 → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))    &   (𝜑𝑀 ∈ (0...𝑁))    &   (𝜑𝑁 ∈ (0...𝐿))       (𝜑 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))

Theoremswrdccatin2d 13351 The subword of a concatenation of two words within the second of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by Mario Carneiro/AV, 21-Oct-2018.)
(𝜑 → (#‘𝐴) = 𝐿)    &   (𝜑 → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))    &   (𝜑𝑀 ∈ (𝐿...𝑁))    &   (𝜑𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))       (𝜑 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩))

Theoremswrdccatin12d 13352 The subword of a concatenation of two words within both of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by Mario Carneiro/AV, 21-Oct-2018.)
(𝜑 → (#‘𝐴) = 𝐿)    &   (𝜑 → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))    &   (𝜑𝑀 ∈ (0...𝐿))    &   (𝜑𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))       (𝜑 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩)))

5.7.10  Splicing words (substring replacement)

Theoremsplval 13353 Value of the substring replacement operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))

Theoremsplcl 13354 Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)
((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)

Theoremsplid 13355 Splicing a subword for the same subword makes no difference. (Contributed by Stefan O'Rear, 20-Aug-2015.)
((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = 𝑆)

Theoremspllen 13356 The length of a splice. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝐹 ∈ (0...𝑇))    &   (𝜑𝑇 ∈ (0...(#‘𝑆)))    &   (𝜑𝑅 ∈ Word 𝐴)       (𝜑 → (#‘(𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)) = ((#‘𝑆) + ((#‘𝑅) − (𝑇𝐹))))

Theoremsplfv1 13357 Symbols to the left of a splice are unaffected. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝐹 ∈ (0...𝑇))    &   (𝜑𝑇 ∈ (0...(#‘𝑆)))    &   (𝜑𝑅 ∈ Word 𝐴)    &   (𝜑𝑋 ∈ (0..^𝐹))       (𝜑 → ((𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)‘𝑋) = (𝑆𝑋))

Theoremsplfv2a 13358 Symbols within the replacement region of a splice, expressed using the coordinates of the replacement region. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(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 13359 Value of a splice, assuming the input word 𝑆 has already been decomposed into its pieces. (Contributed by Mario Carneiro, 1-Oct-2015.)
(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))

5.7.11  Reversing words

Theoremrevval 13360* 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 13361 The reverse of a word is a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(𝑊 ∈ Word 𝐴 → (reverse‘𝑊) ∈ Word 𝐴)

Theoremrevlen 13362 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 13363 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 13364 The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(reverse‘∅) = ∅

Theoremrevs1 13365 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 13366 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 13367 Reversion is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.)
(𝑊 ∈ Word 𝐴 → (reverse‘(reverse‘𝑊)) = 𝑊)

5.7.12  Repeated symbol words

Theoremreps 13368* 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 13369 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 13370 Construct a function mapping a half-open range of nonnegative integers to a constant, see also fconstmpt 5085. (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = ((0..^𝑁) × {𝑆}))

Theoremrepsf 13371 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 13372 The symbols of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0𝐼 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝐼) = 𝑆)

Theoremrepsw 13373 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 13374 The length of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (#‘(𝑆 repeatS 𝑁)) = 𝑁)

Theoremrepsw0 13375 The "repeated symbol word" of length 0. (Contributed by AV, 4-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 0) = ∅)

Theoremrepsdf2 13376* 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 13377* 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 13378* 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 13379 The first symbol of a nonempty "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ) → ((𝑆 repeatS 𝑁)‘0) = 𝑆)

Theoremrepswlsw 13380 The last symbol of a nonempty "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ) → ( lastS ‘(𝑆 repeatS 𝑁)) = 𝑆)

Theoremrepsw1 13381 The "repeated symbol word" of length 1. (Contributed by AV, 4-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 1) = ⟨“𝑆”⟩)

Theoremrepswswrd 13382 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 (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)))

Theoremrepswccat 13383 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 13384 The reverse of a "repeated symbol word". (Contributed by AV, 6-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (reverse‘(𝑆 repeatS 𝑁)) = (𝑆 repeatS 𝑁))

5.7.13  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 13386. The main theorems in this section are about counting the number of different necklaces resulting from cyclically shifting a given word, see cshwrepswhash1 15647 for words consisting of identical symbols and cshwshash 15649 for words having lengths which are prime numbers.

Syntaxccsh 13385 Extend class notation with Cyclical Shifts.
class cyclShift

Definitiondf-csh 13386* 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.)
cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (#‘𝑤)), (#‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (#‘𝑤))⟩))))

Theoremcshfn 13387* 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.)
((𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))

Theoremcshword 13388 Perform a cyclical shift for a word. (Contributed by Alexander van der Vekens, 20-May-2018.) (Revised by AV, 17-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))

Theoremcshnz 13389 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 13390 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 13391 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 13392 Cyclically shifting a word is invariant regarding modulo the word's length. (Contributed by AV, 26-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (#‘𝑊))))

Theoremcshwsublen 13393 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 13394 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 13395 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 13396 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.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))

Theoremcshwf 13397 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 13398 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 13399 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 13400 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.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝑁)‘𝐼) = (𝑊‘((𝐼 + 𝑁) mod (#‘𝑊))))

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