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Theorem List for Metamath Proof Explorer - 11401-11500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfz1isolem 11401* Lemma for fz1iso 11402. (Contributed by Mario Carneiro, 2-Apr-2014.)
OrdIso

Theoremfz1iso 11402* Any finite ordered set has an order isometry to a one-based finite sequence. (Contributed by Mario Carneiro, 2-Apr-2014.)

Theoremseqcoll 11403* The function contains a sparse set of non-zero values to be summed. The function is an order isomorphism from the set of non-zero values of to a 1-based finite sequence, and collects these non-zero values together. Under these conditions, the sum over the values in yields the same result as the sum over the original set . (Contributed by Mario Carneiro, 2-Apr-2014.)

Theoremseqcoll2 11404* The function contains a sparse set of non-zero values to be summed. The function is an order isomorphism from the set of non-zero values of to a 1-based finite sequence, and collects these non-zero values together. Under these conditions, the sum over the values in yields the same result as the sum over the original set . (Contributed by Mario Carneiro, 13-Dec-2014.)

5.6.9  Words over a set

Syntaxcword 11405 Syntax for the Word operator.
Word

Syntaxcconcat 11406 Syntax for the concatenation operator.
concat

Syntaxcs1 11407 Syntax for the singleton word constructor.

Syntaxcsubstr 11408 Syntax for the word slicing operator.
substr

Syntaxcsplice 11409 Syntax for the word splicing operator.
splice

Syntaxcreverse 11410 Syntax for the word reverse operator.
reverse

Definitiondf-word 11411* Define the class of words over a set. A word is a finite sequence of symbols from a set. The domain is forced so that two words with the same symbols in the same order will be the same. This is sometimes denoted with the Kleene star, although properly speaking that is an operator on languages. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 14-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word ..^

Definitiondf-concat 11412* Define the concatenation operator which combines two words. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 15-Aug-2015.)
concat ..^ ..^

Definitiondf-s1 11413 Define the canonical injection from symbols to words. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Definitiondf-substr 11414* Define an operation which extracts portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
substr ..^ ..^

Definitiondf-splice 11415* Define an operation which replaces portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
splice substr concat concat substr

Definitiondf-reverse 11416* Define an operation which reverses the order of symbols in a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
reverse ..^

Theoremiswrd 11417* Property of being a word over a set with a quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word ..^

Theoremwrdval 11418* Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word ..^

Theoremiswrdi 11419 A one-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
..^ Word

Theoremwrd0 11420 The empty set is a word (frequently denoted ε in this context). (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word

Theoremwrdf 11421 A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word ..^

Theoremwrdfin 11422 A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.)
Word

Theoremlencl 11423 The length of a word is a nonnegative integer. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word

Theoremlennncl 11424 The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word

Theoremsswrd 11425 The set of words respects ordering on the base set. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word Word

Theoremwrdeq 11426 Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word Word

Theoremwrdexg 11427 The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theoremnfwrd 11428 Hypothesis builder for Word . (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theoremccatfn 11429 The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.)
concat

Theoremccatfval 11430* Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
concat ..^ ..^

Theoremccatcl 11431 The concatenation of two words is a word. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Stefan O'Rear, 15-Aug-2015.)
Word Word concat Word

Theoremccatlen 11432 The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word Word concat

Theoremccatval1 11433 Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Word Word ..^ concat

Theoremccatval2 11434 Value of a symbol in the right half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Word Word ..^ concat

Theoremccatval3 11435 Value of a symbol in the right half of a concatenated word, using an index relative to the subword. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Word Word ..^ concat

Theoremccatlid 11436 Concatenation of a word by the empty word on the left. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word concat

Theoremccatrid 11437 Concatenation of a word by the empty word on the right. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word concat

Theoremccatass 11438 Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word Word Word concat concat concat concat

Theoremids1 11439 Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems1val 11440 Value of a single-symbol word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems1eq 11441 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems1eqd 11442 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems1cl 11443 A singleton word is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word

Theorems1cld 11444 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems1cli 11445 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems1len 11446 Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems1nz 11447 A singleton is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems1fv 11448 Sole symbol of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theoremeqs1 11449 A word of length 1 is a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Word

Theorems111 11450 The singleton word function is injective. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theoremwrdexb 11451 The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theoremswrdval 11452* Value of a subword. (Contributed by Stefan O'Rear, 15-Aug-2015.)
substr ..^ ..^

Theoremswrd00 11453 A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.)
substr

Theoremswrdcl 11454 Closure of the subword extractor. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word substr Word

Theoremswrdval2 11455* Value of the subword extractor in its intended domain. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word substr ..^

Theoremswrd0val 11456 Value of the subword extractor for left-anchored subwords. (Contributed by Stefan O'Rear, 24-Aug-2015.)
Word substr ..^

Theoremswrd0len 11457 Length of a left-anchored subword. (Contributed by Stefan O'Rear, 24-Aug-2015.)
Word substr

Theoremswrdlen 11458 Length of an extracted subword. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Word substr

Theoremswrdfv 11459 A symbol in an extracted subword, indexed using the subword's indices. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Word ..^ substr

Theoremswrdid 11460 A word is a subword of itself. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Word substr

Theoremccatswrd 11461 Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)
Word substr concat substr substr

Theoremswrdccat1 11462 Recover the left half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word Word concat substr

Theoremswrdccat2 11463 Recover the right half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
Word Word concat substr

Theoremccatopth 11464 An opth 4247-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word Word Word Word concat concat

Theoremccatopth2 11465 An opth 4247-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word Word Word Word concat concat

Theoremccatlcan 11466 Concatenation of words is left-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
Word Word Word concat concat

Theoremccatrcan 11467 Concatenation of words is right-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
Word Word Word concat concat

Theoremsplval 11468 Value of the substring replacement operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
splice substr concat concat substr

Theoremsplcl 11469 Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Word Word splice Word

Theoremsplid 11470 Splicing a subword for the same subword makes no difference. (Contributed by Stefan O'Rear, 20-Aug-2015.)
Word splice substr

Theoremspllen 11471 The length of a splice. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Word                      Word        splice

Theoremsplfv1 11472 Symbols to the left of a splice are unaffected. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Word                      Word        ..^       splice

Theoremsplfv2a 11473 Symbols within the replacement region of a splice, expressed using the coordinates of the replacement region. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Word                      Word        ..^       splice

Theoremsplval2 11474 Value of a splice, assuming the input word has already been decomposed into its pieces. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word        Word        Word        Word        concat concat                      splice concat concat

Theoremswrds1 11475 Extract a single symbol from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Word ..^ substr

Theoremwrdeqcats1 11476 Decompose a non-empty word by separating off the last symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
Word substr concat

Theoremwrdeqs1cat 11477 Decompose a non-empty word by separating off the first symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
Word concat substr

Theoremcats1un 11478 Express a word with an extra symbol as the union of the word and the new value. (Contributed by Mario Carneiro, 28-Feb-2016.)
Word concat

Theoremwrdind 11479* Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
concat                      Word        Word

Theoremrevval 11480* Value of the word reversing function. (Contributed by Stefan O'Rear, 26-Aug-2015.)
reverse ..^

Theoremrevcl 11481 The reverse of a word is a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Word reverse Word

Theoremrevlen 11482 The reverse of a word has the same length as the original. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Word reverse

Theoremrevfv 11483 Reverse of a word at a point. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Word ..^ reverse

Theoremrev0 11484 The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.)
reverse

Theoremrevs1 11485 Singleton words are their own reverses. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
reverse

Theoremrevccat 11486 Antiautomorphic property of the reversal operation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word Word reverse concat reverse concat reverse

Theoremrevrev 11487 Reversion is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word reversereverse

Theoremwrdco 11488 Mapping a word by a function. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word Word

Theoremlenco 11489 Length of a mapped word is unchanged. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word

Theorems1co 11490 Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theoremrevco 11491 Mapping of words commutes with reversal. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word reverse reverse

Theoremccatco 11492 Mapping of words commutes with concatenation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word Word concat concat

5.6.10  Longer string literals

Syntaxcs2 11493 Syntax for the length 2 word constructor.

Syntaxcs3 11494 Syntax for the length 3 word constructor.

Syntaxcs4 11495 Syntax for the length 4 word constructor.

Syntaxcs5 11496 Syntax for the length 5 word constructor.

Syntaxcs6 11497 Syntax for the length 6 word constructor.

Syntaxcs7 11498 Syntax for the length 7 word constructor.

Syntaxcs8 11499 Syntax for the length 8 word constructor.

Definitiondf-s2 11500 Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

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