Volume 5, 1993

University of Bialystok

Copyright (c) 1993 Association of Mizar Users

### The abstract of the Mizar article:

### Joining of Decorated Trees

**by****Grzegorz Bancerek**- Received October 8, 1993
- MML identifier: TREES_4

- [ Mizar article, MML identifier index ]

environ vocabulary TREES_2, RELAT_1, FINSEQ_1, FUNCT_1, TREES_1, BOOLE, FUNCOP_1, TREES_3, FINSEQ_2, FUNCT_6, FINSEQ_4, FINSET_1, MCART_1, FUNCT_3, PARTFUN1, TREES_4; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, XREAL_0, NAT_1, RELAT_1, FUNCT_1, FINSET_1, DOMAIN_1, FUNCOP_1, FUNCT_3, FINSEQ_1, FINSEQ_2, TREES_1, TREES_2, FUNCT_6, TREES_3; constructors FINSEQ_2, NAT_1, DOMAIN_1, FUNCT_6, TREES_3, XREAL_0, MEMBERED, XBOOLE_0; clusters SUBSET_1, TREES_2, TREES_3, FINSEQ_1, RELSET_1, ARYTM_3, MEMBERED, ZFMISC_1, XBOOLE_0, ORDINAL2; requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM; begin :: Joining of decorated trees definition let T be DecoratedTree; mode Node of T is Element of dom T; end; reserve x, y, z for set, i, j, n for Nat, p, q, r for FinSequence, v for FinSequence of NAT; definition let T1, T2 be DecoratedTree; redefine pred T1 = T2 means :: TREES_4:def 1 dom T1 = dom T2 & for p being Node of T1 holds T1.p = T2.p; end; theorem :: TREES_4:1 for i,j being Nat st elementary_tree i c= elementary_tree j holds i <= j; theorem :: TREES_4:2 for i,j being Nat st elementary_tree i = elementary_tree j holds i = j; definition let x; func root-tree x -> DecoratedTree equals :: TREES_4:def 2 (elementary_tree 0) --> x; end; definition let D be non empty set, d be Element of D; redefine func root-tree d -> Element of FinTrees D; end; theorem :: TREES_4:3 dom root-tree x = elementary_tree 0 & (root-tree x).{} = x; theorem :: TREES_4:4 root-tree x = root-tree y implies x = y; theorem :: TREES_4:5 for T being DecoratedTree st dom T = elementary_tree 0 holds T = root-tree (T.{}); theorem :: TREES_4:6 root-tree x = {[{},x]}; definition let x; let p be FinSequence; func x-flat_tree(p) -> DecoratedTree means :: TREES_4:def 3 dom it = elementary_tree len p & it.{} = x & for n st n < len p holds it.<*n*> = p.(n+1); end; theorem :: TREES_4:7 x-flat_tree p = y-flat_tree q implies x = y & p = q; theorem :: TREES_4:8 j < i implies (elementary_tree i)|<*j*> = elementary_tree 0; theorem :: TREES_4:9 i < len p implies (x-flat_tree p)|<*i*> = root-tree (p.(i+1)); definition let x, p such that p is DTree-yielding; func x-tree(p) -> DecoratedTree means :: TREES_4:def 4 (ex q being DTree-yielding FinSequence st p = q & dom it = tree(doms q)) & it.{} = x & for n st n < len p holds it|<*n*> = p.(n+1); end; definition let x; let T be DecoratedTree; func x-tree T -> DecoratedTree equals :: TREES_4:def 5 x-tree <*T*>; end; definition let x; let T1, T2 be DecoratedTree; func x-tree (T1,T2) -> DecoratedTree equals :: TREES_4:def 6 x-tree <*T1,T2*>; end; theorem :: TREES_4:10 for p being DTree-yielding FinSequence holds dom (x-tree(p)) = tree(doms p); theorem :: TREES_4:11 for p being DTree-yielding FinSequence holds y in dom (x-tree(p)) iff y = {} or ex i being Nat, T being DecoratedTree, q being Node of T st i < len p & T = p.(i+1) & y = <*i*>^q; theorem :: TREES_4:12 for p being DTree-yielding FinSequence for i being Nat, T being DecoratedTree, q being Node of T st i < len p & T = p.(i+1) holds (x-tree p).(<*i*>^q) = T.q; theorem :: TREES_4:13 for T being DecoratedTree holds dom (x-tree T) = ^dom T; theorem :: TREES_4:14 for T1, T2 being DecoratedTree holds dom (x-tree (T1,T2)) = tree(dom T1, dom T2); theorem :: TREES_4:15 for p,q being DTree-yielding FinSequence st x-tree p = y-tree q holds x = y & p = q; theorem :: TREES_4:16 root-tree x = y-flat_tree p implies x = y & p = {}; theorem :: TREES_4:17 root-tree x = y-tree p & p is DTree-yielding implies x = y & p = {}; theorem :: TREES_4:18 x-flat_tree p = y-tree q & q is DTree-yielding implies x = y & len p = len q & for i st i in dom p holds q.i = root-tree (p.i); theorem :: TREES_4:19 for p being DTree-yielding FinSequence, n being Nat, q being FinSequence st <*n*>^q in dom (x-tree(p)) holds (x-tree(p)).(<*n*>^q) = p..(n+1,q); theorem :: TREES_4:20 x-flat_tree({}) = root-tree x & x-tree({}) = root-tree x; theorem :: TREES_4:21 x-flat_tree(<*y*>) = ((elementary_tree 1) --> x) with-replacement (<*0*>, root-tree y); theorem :: TREES_4:22 for T being DecoratedTree holds x-tree(<*T*>) = ((elementary_tree 1) --> x) with-replacement (<*0*>, T); definition let D be non empty set, d be Element of D, p be FinSequence of D; redefine func d-flat_tree(p) -> DecoratedTree of D; end; definition let D be non empty set, F be non empty DTree-set of D; let d be Element of D, p be FinSequence of F; redefine func d-tree(p) -> DecoratedTree of D; end; definition let D be non empty set, d be Element of D, T be DecoratedTree of D; redefine func d-tree T -> DecoratedTree of D; end; definition let D be non empty set, d be Element of D, T1, T2 be DecoratedTree of D; redefine func d-tree(T1, T2) -> DecoratedTree of D; end; definition let D be non empty set; let p be FinSequence of FinTrees D; redefine func doms p -> FinSequence of FinTrees; end; definition let D be non empty set; let d be Element of D, p be FinSequence of FinTrees D; redefine func d-tree p -> Element of FinTrees D; end; definition let D be non empty set, x be Subset of D; redefine mode FinSequence of x -> FinSequence of D; end; definition let D be non empty constituted-DTrees set; let X be Subset of D; cluster -> DTree-yielding FinSequence of X; end; begin :: Expanding of decoreted tree by substitution scheme ExpandTree{T1() -> Tree, T2() -> Tree, P[set]}: ex T being Tree st for p holds p in T iff p in T1() or ex q being Element of T1(), r being Element of T2() st P[q] & p = q^r; definition let T,T' be DecoratedTree; let x be set; func (T,x) <- T' -> DecoratedTree means :: TREES_4:def 7 (for p holds p in dom it iff p in dom T or ex q being Node of T, r being Node of T' st q in Leaves dom T & T.q = x & p = q^r) & (for p being Node of T st not p in Leaves dom T or T.p <> x holds it.p = T.p) & (for p being Node of T, q being Node of T' st p in Leaves dom T & T.p = x holds it.(p^q) = T'.q); end; definition let D be non empty set; let T,T' be DecoratedTree of D; let x be set; redefine func (T,x) <- T' -> DecoratedTree of D; end; reserve T,T' for DecoratedTree, x,y for set; theorem :: TREES_4:23 not x in rng T or not x in Leaves T implies (T,x) <- T' = T; begin :: Double decoreted trees reserve D1, D2 for non empty set, T for DecoratedTree of D1,D2, d1 for Element of D1, d2 for Element of D2, F for non empty DTree-set of D1,D2, F1 for non empty (DTree-set of D1), F2 for non empty DTree-set of D2; theorem :: TREES_4:24 for D1, D2, T holds dom T`1 = dom T & dom T`2 = dom T; theorem :: TREES_4:25 (root-tree [d1,d2])`1 = root-tree d1 & (root-tree [d1,d2])`2 = root-tree d2; theorem :: TREES_4:26 <:root-tree x, root-tree y:> = root-tree [x,y]; theorem :: TREES_4:27 for D1,D2, d1,d2, F,F1 for p being FinSequence of F, p1 being FinSequence of F1 st dom p1 = dom p & for i st i in dom p for T st T = p.i holds p1.i = T`1 holds ([d1,d2]-tree p)`1 = d1-tree p1; theorem :: TREES_4:28 for D1,D2, d1,d2, F,F2 for p being FinSequence of F, p2 being FinSequence of F2 st dom p2 = dom p & for i st i in dom p for T st T = p.i holds p2.i = T`2 holds ([d1,d2]-tree p)`2 = d2-tree p2; theorem :: TREES_4:29 for D1,D2, d1,d2, F for p being FinSequence of F ex p1 being FinSequence of Trees D1 st dom p1 = dom p & (for i st i in dom p ex T being Element of F st T = p.i & p1.i = T`1) & ([d1,d2]-tree p)`1 = d1-tree p1; theorem :: TREES_4:30 for D1,D2, d1,d2, F for p being FinSequence of F ex p2 being FinSequence of Trees D2 st dom p2 = dom p & (for i st i in dom p ex T being Element of F st T = p.i & p2.i = T`2) & ([d1,d2]-tree p)`2 = d2-tree p2; theorem :: TREES_4:31 for D1,D2, d1,d2 for p being FinSequence of FinTrees [:D1,D2:] ex p1 being FinSequence of FinTrees D1 st dom p1 = dom p & (for i st i in dom p ex T being Element of FinTrees [:D1,D2:] st T = p.i & p1.i = T`1) & ([d1,d2]-tree p)`1 = d1-tree p1; theorem :: TREES_4:32 for D1,D2, d1,d2 for p being FinSequence of FinTrees [:D1,D2:] ex p2 being FinSequence of FinTrees D2 st dom p2 = dom p & (for i st i in dom p ex T being Element of FinTrees [:D1,D2:] st T = p.i & p2.i = T`2) & ([d1,d2]-tree p)`2 = d2-tree p2;

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