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Type | Label | Description |
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Statement | ||
Theorem | ccatw2s1len 12801 | The length of the concatenation of a word with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
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Theorem | ccats1val1 12802 | Value of a symbol in the left half of a word concatenated with a single symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.) |
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Theorem | ccats1val2 12803 | Value of the symbol concatenated with a word. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Proof shortened by Alexander van der Vekens, 14-Oct-2018.) |
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Theorem | ccat2s1p1 12804 | Extract the first of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
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Theorem | ccat2s1p2 12805 | Extract the second of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
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Theorem | ccatw2s1ass 12806 | Associative law for a concatenation of a word with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
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Theorem | ccatws1lenrev 12807 | The length of a word concatenated with a singleton word. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
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Theorem | ccatws1n0 12808 | The concatenation of a word with a singleton word is not the empty set. (Contributed by Alexander van der Vekens, 29-Sep-2018.) |
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Theorem | ccatws1ls 12809 | The last symbol of the concatenation of a word with a singleton word is the symbol of the singleton word. (Contributed by AV, 29-Sep-2018.) (Proof shortened by AV, 14-Oct-2018.) |
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Theorem | lswccats1 12810 | The last symbol of a word concatenated with a singleton word is the symbol of the singleton word. (Contributed by AV, 6-Aug-2018.) (Proof shortened by AV, 22-Oct-2018.) |
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Theorem | lswccats1fst 12811 | The last symbol of a nonempty word concatenated with its first symbol is the first symbol. (Contributed by AV, 28-Jun-2018.) (Proof shortened by AV, 1-May-2020.) |
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Theorem | ccatw2s1p1 12812 | Extract the symbol of the first singleton word of a word concatenated with this singleton word and another singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.) |
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Theorem | ccatw2s1p2 12813 | Extract the second of two single symbols concatenated with a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.) |
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Theorem | ccat2s1fvw 12814 | Extract a symbol of a word from the concatenation of the word with two single symbols. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 13-Jan-2020.) (Proof shortened by AV, 1-May-2020.) |
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Theorem | ccat2s1fst 12815 | The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
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Theorem | swrdval 12816* | Value of a subword. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
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Theorem | swrd00 12817 | A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
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Theorem | swrdcl 12818 | Closure of the subword extractor. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
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Theorem | swrdval2 12819* | Value of the subword extractor in its intended domain. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 2-May-2020.) |
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Theorem | swrd0val 12820 | Value of the subword extractor for left-anchored subwords. (Contributed by Stefan O'Rear, 24-Aug-2015.) |
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Theorem | swrd0len 12821 | Length of a left-anchored subword. (Contributed by Stefan O'Rear, 24-Aug-2015.) |
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Theorem | swrdlen 12822 | Length of an extracted subword. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
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Theorem | swrdfv 12823 | A symbol in an extracted subword, indexed using the subword's indices. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
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Theorem | swrdf 12824 | A subword of a word is a function from a half-open range of nonnegative integers of the same length as the subword to the set of symbols for the original word. (Contributed by AV, 13-Nov-2018.) |
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Theorem | swrdvalfn 12825 | Value of the subword extractor as function with domain. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by AV, 2-May-2020.) |
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Theorem | swrd0f 12826 | A left-anchored subword of a word is a function from a half-open range of nonnegative integers of the same length as the subword to the set of symbols for the original word. (Contributed by AV, 2-May-2020.) |
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Theorem | swrdid 12827 | A word is a subword of itself. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Proof shortened by AV, 2-May-2020.) |
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Theorem | swrdrn 12828 | The range of a subword of a word is a subset of the set of symbols for the word. (Contributed by AV, 13-Nov-2018.) |
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Theorem | swrdn0 12829 | A prefixing subword consisting of at least one symbol is not empty. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Proof shortened by AV, 2-May-2020.) |
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Theorem | swrdlend 12830 | The value of the subword extractor is the empty set (undefined) if the range is not valid. (Contributed by Alexander van der Vekens, 16-Mar-2018.) (Proof shortened by AV, 2-May-2020.) |
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Theorem | swrdnd 12831 | The value of the subword extractor is the empty set (undefined) if the range is not valid. (Contributed by Alexander van der Vekens, 16-Mar-2018.) (Proof shortened by AV, 2-May-2020.) |
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Theorem | swrdnd2 12832 | Value of the subword extractor outside its intended domain. (Contributed by Alexander van der Vekens, 24-May-2018.) |
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Theorem | swrd0 12833 | A subword of an empty set is always the empty set. (Contributed by AV, 31-Mar-2018.) (Revised by AV, 20-Oct-2018.) (Proof shortened by AV, 2-May-2020.) |
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Theorem | swrdrlen 12834 | Length of a right-anchored subword. (Contributed by Alexander van der Vekens, 5-Apr-2018.) |
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Theorem | swrd0len0 12835 | Length of a prefix of a word reduced by a single symbol, analogous to swrd0len 12821. (Contributed by AV, 4-Aug-2018.) (Proof shortened by AV, 14-Oct-2018.) |
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Theorem | addlenrevswrd 12836 | The sum of the lengths of two reversed parts of a word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.) |
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Theorem | addlenswrd 12837 | The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 21-Oct-2018.) |
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Theorem | swrd0fv 12838 | A symbol in an left-anchored subword, indexed using the subword's indices. (Contributed by Alexander van der Vekens, 16-Jun-2018.) |
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Theorem | swrd0fv0 12839 | The first symbol in a left-anchored subword. (Contributed by Alexander van der Vekens, 16-Jun-2018.) |
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Theorem | swrdtrcfv 12840 | A symbol in a word truncated by one symbol. (Contributed by Alexander van der Vekens, 16-Jun-2018.) |
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Theorem | swrdtrcfv0 12841 | The first symbol in a word truncated by one symbol. (Contributed by Alexander van der Vekens, 16-Jun-2018.) |
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Theorem | swrd0fvlsw 12842 | The last symbol in a left-anchored subword. (Contributed by Alexander van der Vekens, 24-Jun-2018.) |
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Theorem | swrdeq 12843* | Two subwords of words are equal iff they have the same length and the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.) |
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Theorem | swrdlen2 12844 | Length of an extracted subword. (Contributed by AV, 5-May-2020.) |
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Theorem | swrdfv2 12845 | A symbol in an extracted subword, indexed using the word's indices. (Contributed by AV, 5-May-2020.) |
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Theorem | swrdsb0eq 12846 | Two subwords with the same bounds are equal if the range is not valid. (Contributed by AV, 4-May-2020.) |
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Theorem | swrdsbslen 12847 | Two subwords with the same bounds have the same length. (Contributed by AV, 4-May-2020.) |
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Theorem | swrdspsleq 12848* | 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.) |
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Theorem | swrdtrcfvl 12849 | 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.) |
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Theorem | swrds1 12850 | Extract a single symbol from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
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Theorem | swrdlsw 12851 | Extract the last single symbol from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
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Theorem | 2swrdeqwrdeq 12852 | 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.) |
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Theorem | 2swrd1eqwrdeq 12853 | 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.) |
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Theorem | disjxwrd 12854* | 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.) |
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Theorem | ccatswrd 12855 | Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.) |
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Theorem | swrdccat1 12856 | Recover the left half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.) |
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Theorem | swrdccat2 12857 | Recover the right half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.) |
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Theorem | swrdswrdlem 12858 | Lemma for swrdswrd 12859. (Contributed by Alexander van der Vekens, 4-Apr-2018.) |
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Theorem | swrdswrd 12859 | A subword of a subword. (Contributed by Alexander van der Vekens, 4-Apr-2018.) |
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Theorem | swrd0swrd 12860 | A prefix of a subword. (Contributed by AV, 2-Apr-2018.) (Proof shortened by AV, 21-Oct-2018.) |
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Theorem | swrdswrd0 12861 | A subword of a prefix. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
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Theorem | swrd0swrd0 12862 | A prefix of a prefix. (Contributed by Alexander van der Vekens, 7-Apr-2018.) |
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Theorem | swrd0swrdid 12863 | 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.) |
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Theorem | wrdcctswrd 12864 | The concatenation of two parts of a word yields the word itself. (Contributed by AV, 21-Oct-2018.) |
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Theorem | lencctswrd 12865 | The length of two concatenated parts of a word is the length of the word. (Contributed by AV, 21-Oct-2018.) |
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Theorem | lenrevcctswrd 12866 | 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.) |
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Theorem | swrdccatwrd 12867 | Reconstruct a nonempty word from its prefix and last symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.) |
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Theorem | ccats1swrdeq 12868 | 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.) |
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Theorem | ccatopth 12869 | An opth 4693-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.) |
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Theorem | ccatopth2 12870 | An opth 4693-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.) |
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Theorem | ccatlcan 12871 | Concatenation of words is left-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | ccatrcan 12872 | Concatenation of words is right-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | wrdeqcats1OLD 12873 | Decompose a nonempty word by separating off the last symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 26-Apr-2020.) Obsolete version of swrdccatwrd 12867 as of 9-May-2020. (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | wrdeqs1cat 12874 | 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.) |
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Theorem | cats1un 12875 | Express a word with an extra symbol as the union of the word and the new value. (Contributed by Mario Carneiro, 28-Feb-2016.) |
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Theorem | wrdind 12876* | Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
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Theorem | wrd2ind 12877* | Perform induction over the structure of two words of the same length. (Contributed by AV, 23-Jan-2019.) |
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Theorem | ccats1swrdeqrex 12878* | 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.) |
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Theorem | reuccats1lem 12879* | Lemma for reuccats1 12880. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Proof shortened by AV, 15-Jan-2020.) |
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Theorem | reuccats1 12880* | 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.) |
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Theorem | swrdccatfn 12881 | The subword of a concatenation as function. (Contributed by Alexander van der Vekens, 27-May-2018.) |
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Theorem | swrdccatin1 12882 | The subword of a concatenation of two words within the first of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.) |
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Theorem | swrdccatin12lem1 12883 | Lemma 1 for swrdccatin12 12890. (Contributed by Alexander van der Vekens, 30-Mar-2018.) (Revised by Alexander van der Vekens, 23-May-2018.) |
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Theorem | swrdccatin12lem2a 12884 | Lemma 1 for swrdccatin12lem2 12888. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.) |
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Theorem | swrdccatin12lem2b 12885 | Lemma 2 for swrdccatin12lem2 12888. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.) |
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Theorem | swrdccatin2 12886 | 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.) |
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Theorem | swrdccatin12lem2c 12887 | Lemma for swrdccatin12lem2 12888 and swrdccatin12lem3 12889. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.) |
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Theorem | swrdccatin12lem2 12888 | Lemma 2 for swrdccatin12 12890. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.) |
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Theorem | swrdccatin12lem3 12889 | Lemma 3 for swrdccatin12 12890. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.) |
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Theorem | swrdccatin12 12890 | 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.) |
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Theorem | swrdccat3 12891 | 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.) |
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Theorem | swrdccat 12892 | 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.) |
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Theorem | swrdccat3a 12893 | 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.) |
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Theorem | swrdccat3blem 12894 | Lemma for swrdccat3b 12895. (Contributed by AV, 30-May-2018.) |
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Theorem | swrdccat3b 12895 | 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.) |
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Theorem | swrdccatid 12896 | 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.) |
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Theorem | ccats1swrdeqbi 12897 | 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.) |
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Theorem | swrdccatin1d 12898 | 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.) |
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Theorem | swrdccatin2d 12899 | 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.) |
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Theorem | swrdccatin12d 12900 | 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.) |
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