:: Introduction to Trees
:: by Grzegorz Bancerek
::
:: Received October 25, 1989
:: Copyright (c) 1990-2011 Association of Mizar Users

begin

notation

let p, q be FinSequence;
synonym p is_a_prefix_of q for p c= q;

end;

definition

let p, q be FinSequence;
redefine pred p c= q means :Def1: :: TREES_1:def 1
ex n being Element of NAT st p = q | (Seg n);
compatibility
( p is_a_prefix_of q iff ex n being Element of NAT st p = q | (Seg n) )

proof end;

end;

:: deftheorem Def1 defines is_a_prefix_of TREES_1:def 1 :
for p, q being FinSequence holds
( p is_a_prefix_of q iff ex n being Element of NAT st p = q | (Seg n) );

theorem :: TREES_1:1

canceled;

theorem :: TREES_1:2

canceled;

theorem :: TREES_1:3

canceled;

theorem :: TREES_1:4

canceled;

theorem :: TREES_1:5

canceled;

theorem :: TREES_1:6

canceled;

theorem :: TREES_1:7

canceled;

theorem Th8: :: TREES_1:8

for p, q being FinSequence holds
( p is_a_prefix_of q iff ex r being FinSequence st q = p ^ r )

proof end;

theorem :: TREES_1:9

canceled;

theorem :: TREES_1:10

canceled;

theorem :: TREES_1:11

canceled;

theorem :: TREES_1:12

canceled;

theorem :: TREES_1:13

canceled;

theorem :: TREES_1:14

canceled;

theorem Th15: :: TREES_1:15

for p, q being FinSequence st p is_a_prefix_of q & len p = len q holds
p = q

proof end;

theorem Th16: :: TREES_1:16

for x, y being set holds
( <*x*> is_a_prefix_of <*y*> iff x = y )

proof end;

notation

let p, q be FinSequence;
synonym p is_a_proper_prefix_of q for p c< q;

end;

Lm1: for A, B being finite set st A c= B & card A = card B holds
A = B

proof end;

theorem :: TREES_1:17

canceled;

theorem :: TREES_1:18

canceled;

theorem Th19: :: TREES_1:19

for p, q being finite set st p,q are_c=-comparable & card p = card q holds
p = q

proof end;

theorem :: TREES_1:20

canceled;

theorem :: TREES_1:21

canceled;

theorem :: TREES_1:22

canceled;

theorem Th23: :: TREES_1:23

for x, y being set st <*x*>,<*y*> are_c=-comparable holds
x = y

proof end;

theorem Th24: :: TREES_1:24

for p, q being finite set st p c< q holds
card p < card q

proof end;

theorem :: TREES_1:25

canceled;

theorem :: TREES_1:26

canceled;

theorem :: TREES_1:27

canceled;

theorem :: TREES_1:28

canceled;

theorem :: TREES_1:29

canceled;

theorem Th30: :: TREES_1:30

for x being set
for p1, p2 being FinSequence st p1 ^ <*x*> is_a_prefix_of p2 holds
p1 is_a_proper_prefix_of p2

proof end;

theorem Th31: :: TREES_1:31

for x being set
for p1, p2 being FinSequence st p1 is_a_prefix_of p2 holds
p1 is_a_proper_prefix_of p2 ^ <*x*>

proof end;

theorem Th32: :: TREES_1:32

for x being set
for p1, p2 being FinSequence st p1 is_a_proper_prefix_of p2 ^ <*x*> holds
p1 is_a_prefix_of p2

proof end;

theorem :: TREES_1:33

for p2, p1 being FinSequence st ( {} is_a_proper_prefix_of p2 or {} <> p2 ) holds
p1 is_a_proper_prefix_of p1 ^ p2

proof end;

definition

let p be FinSequence;
canceled;
canceled;
func ProperPrefixes p -> set means :Def4: :: TREES_1:def 4
for x being set holds
( x in it iff ex q being FinSequence st
( x = q & q is_a_proper_prefix_of p ) );
existence
ex b₁ being set st
for x being set holds
( x in b₁ iff ex q being FinSequence st
( x = q & q is_a_proper_prefix_of p ) )

proof end;

uniqueness
for b₁, b₂ being set st ( for x being set holds
( x in b₁ iff ex q being FinSequence st
( x = q & q is_a_proper_prefix_of p ) ) ) & ( for x being set holds
( x in b₂ iff ex q being FinSequence st
( x = q & q is_a_proper_prefix_of p ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem TREES_1:def 2 :
canceled;

:: deftheorem TREES_1:def 3 :
canceled;

:: deftheorem Def4 defines ProperPrefixes TREES_1:def 4 :
for p being FinSequence
for b₂ being set holds
( b₂ = ProperPrefixes p iff for x being set holds
( x in b₂ iff ex q being FinSequence st
( x = q & q is_a_proper_prefix_of p ) ) );

theorem :: TREES_1:34

canceled;

theorem Th35: :: TREES_1:35

for x being set
for p being FinSequence st x in ProperPrefixes p holds
x is FinSequence

proof end;

theorem Th36: :: TREES_1:36

for p, q being FinSequence holds
( p in ProperPrefixes q iff p is_a_proper_prefix_of q )

proof end;

theorem Th37: :: TREES_1:37

for p, q being FinSequence st p in ProperPrefixes q holds
len p < len q

proof end;

theorem :: TREES_1:38

for p, q, r being FinSequence st q ^ r in ProperPrefixes p holds
q in ProperPrefixes p

proof end;

theorem Th39: :: TREES_1:39

ProperPrefixes {} = {}

proof end;

set D = {{}};

theorem Th40: :: TREES_1:40

for x being set holds ProperPrefixes <*x*> = {{}}

proof end;

theorem :: TREES_1:41

for p, q being FinSequence st p is_a_prefix_of q holds
ProperPrefixes p c= ProperPrefixes q

proof end;

theorem Th42: :: TREES_1:42

for p, q, r being FinSequence st q in ProperPrefixes p & r in ProperPrefixes p holds
q,r are_c=-comparable

proof end;

definition

let X be set ;
attr X is Tree-like means :Def5: :: TREES_1:def 5
( X c= NAT * & ( for p being FinSequence of NAT st p in X holds
ProperPrefixes p c= X ) & ( for p being FinSequence of NAT
for k, n being Element of NAT st p ^ <*k*> in X & n <= k holds
p ^ <*n*> in X ) );

end;

:: deftheorem Def5 defines Tree-like TREES_1:def 5 :
for X being set holds
( X is Tree-like iff ( X c= NAT * & ( for p being FinSequence of NAT st p in X holds
ProperPrefixes p c= X ) & ( for p being FinSequence of NAT
for k, n being Element of NAT st p ^ <*k*> in X & n <= k holds
p ^ <*n*> in X ) ) );

registration

cluster {{}} -> Tree-like ;
coherence
{{}} is Tree-like

proof end;

end;

registration

cluster non empty Tree-like set ;
existence
ex b₁ being set st
( not b₁ is empty & b₁ is Tree-like )

proof end;

end;

definition

mode Tree is non empty Tree-like set ;

end;

theorem :: TREES_1:43

canceled;

theorem Th44: :: TREES_1:44

for x being set
for T being Tree st x in T holds
x is FinSequence of NAT

proof end;

definition

let T be Tree;
:: original: Element
redefine mode Element of T -> FinSequence of NAT ;
coherence
for b₁ being Element of T holds b₁ is FinSequence of NAT by Th44;

end;

theorem Th45: :: TREES_1:45

for T being Tree
for p, q being FinSequence st p in T & q is_a_prefix_of p holds
q in T

proof end;

theorem Th46: :: TREES_1:46

for q being FinSequence of NAT
for T being Tree
for r being FinSequence st q ^ r in T holds
q in T

proof end;

theorem Th47: :: TREES_1:47

for T being Tree holds
( {} in T & <*> NAT in T )

proof end;

theorem :: TREES_1:48

{{}} is Tree ;

registration

let T, T1 be Tree;
cluster T \/ T1 -> Tree-like ;
coherence
T \/ T1 is Tree-like

proof end;

cluster T /\ T1 -> non empty Tree-like ;
coherence
( T /\ T1 is Tree-like & not T /\ T1 is empty )

proof end;

end;

theorem :: TREES_1:49

for T, T1 being Tree holds T \/ T1 is Tree ;

theorem :: TREES_1:50

for T, T1 being Tree holds T /\ T1 is Tree ;

registration

cluster non empty finite Tree-like set ;
existence
ex b₁ being Tree st b₁ is finite

proof end;

end;

theorem :: TREES_1:51

canceled;

theorem :: TREES_1:52

for fT, fT1 being finite Tree holds fT \/ fT1 is finite Tree ;

theorem :: TREES_1:53

for T being Tree
for fT being finite Tree holds fT /\ T is finite Tree ;

definition

let n be Element of NAT ;
canceled;
func elementary_tree n -> Tree equals :: TREES_1:def 7
{ <*k*> where k is Element of NAT : k < n } \/ {{}};
correctness
coherence
{ <*k*> where k is Element of NAT : k < n } \/ {{}} is Tree;

proof end;

end;

:: deftheorem TREES_1:def 6 :
canceled;

:: deftheorem defines elementary_tree TREES_1:def 7 :
for n being Element of NAT holds elementary_tree n = { <*k*> where k is Element of NAT : k < n } \/ {{}};

registration

let n be Element of NAT ;
cluster elementary_tree n -> finite ;
coherence
elementary_tree n is finite

proof end;

end;

theorem :: TREES_1:54

canceled;

theorem Th55: :: TREES_1:55

for k, n being Element of NAT st k < n holds
<*k*> in elementary_tree n

proof end;

theorem Th56: :: TREES_1:56

elementary_tree 0 = {{}}

proof end;

theorem :: TREES_1:57

for n being Element of NAT
for p being FinSequence of NAT holds
( not p in elementary_tree n or p = {} or ex k being Element of NAT st
( k < n & p = <*k*> ) )

proof end;

definition

let T be Tree;
func Leaves T -> Subset of T means :Def8: :: TREES_1:def 8
for p being FinSequence of NAT holds
( p in it iff ( p in T & ( for q being FinSequence of NAT holds
( not q in T or not p is_a_proper_prefix_of q ) ) ) );
existence
ex b₁ being Subset of T st
for p being FinSequence of NAT holds
( p in b₁ iff ( p in T & ( for q being FinSequence of NAT holds
( not q in T or not p is_a_proper_prefix_of q ) ) ) )

proof end;

uniqueness
for b₁, b₂ being Subset of T st ( for p being FinSequence of NAT holds
( p in b₁ iff ( p in T & ( for q being FinSequence of NAT holds
( not q in T or not p is_a_proper_prefix_of q ) ) ) ) ) & ( for p being FinSequence of NAT holds
( p in b₂ iff ( p in T & ( for q being FinSequence of NAT holds
( not q in T or not p is_a_proper_prefix_of q ) ) ) ) ) holds
b₁ = b₂

proof end;

let p be FinSequence of NAT ;
assume A11: p in T ;
func T | p -> Tree means :Def9: :: TREES_1:def 9
for q being FinSequence of NAT holds
( q in it iff p ^ q in T );
existence
ex b₁ being Tree st
for q being FinSequence of NAT holds
( q in b₁ iff p ^ q in T )

proof end;

uniqueness
for b₁, b₂ being Tree st ( for q being FinSequence of NAT holds
( q in b₁ iff p ^ q in T ) ) & ( for q being FinSequence of NAT holds
( q in b₂ iff p ^ q in T ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def8 defines Leaves TREES_1:def 8 :
for T being Tree
for b₂ being Subset of T holds
( b₂ = Leaves T iff for p being FinSequence of NAT holds
( p in b₂ iff ( p in T & ( for q being FinSequence of NAT holds
( not q in T or not p is_a_proper_prefix_of q ) ) ) ) );

:: deftheorem Def9 defines | TREES_1:def 9 :
for T being Tree
for p being FinSequence of NAT st p in T holds
for b₃ being Tree holds
( b₃ = T | p iff for q being FinSequence of NAT holds
( q in b₃ iff p ^ q in T ) );

theorem :: TREES_1:58

canceled;

theorem :: TREES_1:59

canceled;

theorem :: TREES_1:60

for T being Tree holds T | (<*> NAT) = T

proof end;

registration

let T be finite Tree;
let p be Element of T;
cluster T | p -> finite ;
coherence
T | p is finite

proof end;

end;

definition

let T be Tree;
assume A1: Leaves T <> {} ;
mode Leaf of T -> Element of T means :: TREES_1:def 10
it in Leaves T;
existence
ex b₁ being Element of T st b₁ in Leaves T

proof end;

end;

:: deftheorem defines Leaf TREES_1:def 10 :
for T being Tree st Leaves T <> {} holds
for b₂ being Element of T holds
( b₂ is Leaf of T iff b₂ in Leaves T );

definition

let T be Tree;
mode Subtree of T -> Tree means :: TREES_1:def 11
ex p being Element of T st it = T | p;
existence
ex b₁ being Tree ex p being Element of T st b₁ = T | p

proof end;

end;

:: deftheorem defines Subtree TREES_1:def 11 :
for T, b₂ being Tree holds
( b₂ is Subtree of T iff ex p being Element of T st b₂ = T | p );

definition

let T be Tree;
let p be FinSequence of NAT ;
let T1 be Tree;
assume A1: p in T ;
func T with-replacement (p,T1) -> Tree means :Def12: :: TREES_1:def 12
for q being FinSequence of NAT holds
( q in it iff ( ( q in T & not p is_a_proper_prefix_of q ) or ex r being FinSequence of NAT st
( r in T1 & q = p ^ r ) ) );
existence
ex b₁ being Tree st
for q being FinSequence of NAT holds
( q in b₁ iff ( ( q in T & not p is_a_proper_prefix_of q ) or ex r being FinSequence of NAT st
( r in T1 & q = p ^ r ) ) )

proof end;

uniqueness
for b₁, b₂ being Tree st ( for q being FinSequence of NAT holds
( q in b₁ iff ( ( q in T & not p is_a_proper_prefix_of q ) or ex r being FinSequence of NAT st
( r in T1 & q = p ^ r ) ) ) ) & ( for q being FinSequence of NAT holds
( q in b₂ iff ( ( q in T & not p is_a_proper_prefix_of q ) or ex r being FinSequence of NAT st
( r in T1 & q = p ^ r ) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def12 defines with-replacement TREES_1:def 12 :
for T being Tree
for p being FinSequence of NAT
for T1 being Tree st p in T holds
for b₄ being Tree holds
( b₄ = T with-replacement (p,T1) iff for q being FinSequence of NAT holds
( q in b₄ iff ( ( q in T & not p is_a_proper_prefix_of q ) or ex r being FinSequence of NAT st
( r in T1 & q = p ^ r ) ) ) );

theorem :: TREES_1:61

canceled;

theorem :: TREES_1:62

canceled;

theorem :: TREES_1:63

canceled;

theorem Th64: :: TREES_1:64

for p being FinSequence of NAT
for T, T1 being Tree st p in T holds
T with-replacement (p,T1) = { t1 where t1 is Element of T : not p is_a_proper_prefix_of t1 } \/ { (p ^ s) where s is Element of T1 : s = s }

proof end;

theorem :: TREES_1:65

canceled;

theorem :: TREES_1:66

for p being FinSequence of NAT
for T, T1 being Tree st p in T holds
T1 = (T with-replacement (p,T1)) | p

proof end;

registration

let T be finite Tree;
let t be Element of T;
let T1 be finite Tree;
cluster T with-replacement (t,T1) -> finite ;
coherence
T with-replacement (t,T1) is finite

proof end;

end;

theorem Th67: :: TREES_1:67

for p being FinSequence holds ProperPrefixes p, dom p are_equipotent

proof end;

registration

let p be FinSequence;
cluster ProperPrefixes p -> finite ;
coherence
ProperPrefixes p is finite

proof end;

end;

theorem :: TREES_1:68

for p being FinSequence holds card (ProperPrefixes p) = len p

proof end;

definition

let IT be set ;
attr IT is AntiChain_of_Prefixes-like means :Def13: :: TREES_1:def 13
( ( for x being set st x in IT holds
x is FinSequence ) & ( for p1, p2 being FinSequence st p1 in IT & p2 in IT & p1 <> p2 holds
not p1,p2 are_c=-comparable ) );

end;

:: deftheorem Def13 defines AntiChain_of_Prefixes-like TREES_1:def 13 :
for IT being set holds
( IT is AntiChain_of_Prefixes-like iff ( ( for x being set st x in IT holds
x is FinSequence ) & ( for p1, p2 being FinSequence st p1 in IT & p2 in IT & p1 <> p2 holds
not p1,p2 are_c=-comparable ) ) );

registration

cluster AntiChain_of_Prefixes-like set ;
existence
ex b₁ being set st b₁ is AntiChain_of_Prefixes-like

proof end;

end;

definition

mode AntiChain_of_Prefixes is AntiChain_of_Prefixes-like set ;

end;

theorem :: TREES_1:69

canceled;

theorem Th70: :: TREES_1:70

for w being FinSequence holds {w} is AntiChain_of_Prefixes-like

proof end;

theorem Th71: :: TREES_1:71

for p1, p2 being FinSequence st not p1,p2 are_c=-comparable holds
{p1,p2} is AntiChain_of_Prefixes-like

proof end;

definition

let T be Tree;
mode AntiChain_of_Prefixes of T -> AntiChain_of_Prefixes means :Def14: :: TREES_1:def 14
it c= T;
existence
ex b₁ being AntiChain_of_Prefixes st b₁ c= T

proof end;

end;

:: deftheorem Def14 defines AntiChain_of_Prefixes TREES_1:def 14 :
for T being Tree
for b₂ being AntiChain_of_Prefixes holds
( b₂ is AntiChain_of_Prefixes of T iff b₂ c= T );

theorem :: TREES_1:72

canceled;

theorem Th73: :: TREES_1:73

for T being Tree holds
( {} is AntiChain_of_Prefixes of T & {{}} is AntiChain_of_Prefixes of T )

proof end;

theorem :: TREES_1:74

for T being Tree
for t being Element of T holds {t} is AntiChain_of_Prefixes of T

proof end;

theorem :: TREES_1:75

for T being Tree
for t1, t2 being Element of T st not t1,t2 are_c=-comparable holds
{t1,t2} is AntiChain_of_Prefixes of T

proof end;

registration

let T be finite Tree;
cluster -> finite AntiChain_of_Prefixes of T;
coherence
for b₁ being AntiChain_of_Prefixes of T holds b₁ is finite

proof end;

end;

definition

let T be finite Tree;
func height T -> Element of NAT means :Def15: :: TREES_1:def 15
( ex p being FinSequence of NAT st
( p in T & len p = it ) & ( for p being FinSequence of NAT st p in T holds
len p <= it ) );
existence
ex b₁ being Element of NAT st
( ex p being FinSequence of NAT st
( p in T & len p = b₁ ) & ( for p being FinSequence of NAT st p in T holds
len p <= b₁ ) )

proof end;

uniqueness
for b₁, b₂ being Element of NAT st ex p being FinSequence of NAT st
( p in T & len p = b₁ ) & ( for p being FinSequence of NAT st p in T holds
len p <= b₁ ) & ex p being FinSequence of NAT st
( p in T & len p = b₂ ) & ( for p being FinSequence of NAT st p in T holds
len p <= b₂ ) holds
b₁ = b₂

proof end;

func width T -> Element of NAT means :Def16: :: TREES_1:def 16
ex X being AntiChain_of_Prefixes of T st
( it = card X & ( for Y being AntiChain_of_Prefixes of T holds card Y <= card X ) );
existence
ex b₁ being Element of NAT ex X being AntiChain_of_Prefixes of T st
( b₁ = card X & ( for Y being AntiChain_of_Prefixes of T holds card Y <= card X ) )

proof end;

uniqueness
for b₁, b₂ being Element of NAT st ex X being AntiChain_of_Prefixes of T st
( b₁ = card X & ( for Y being AntiChain_of_Prefixes of T holds card Y <= card X ) ) & ex X being AntiChain_of_Prefixes of T st
( b₂ = card X & ( for Y being AntiChain_of_Prefixes of T holds card Y <= card X ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def15 defines height TREES_1:def 15 :
for T being finite Tree
for b₂ being Element of NAT holds
( b₂ = height T iff ( ex p being FinSequence of NAT st
( p in T & len p = b₂ ) & ( for p being FinSequence of NAT st p in T holds
len p <= b₂ ) ) );

:: deftheorem Def16 defines width TREES_1:def 16 :
for T being finite Tree
for b₂ being Element of NAT holds
( b₂ = width T iff ex X being AntiChain_of_Prefixes of T st
( b₂ = card X & ( for Y being AntiChain_of_Prefixes of T holds card Y <= card X ) ) );

theorem :: TREES_1:76

canceled;

theorem :: TREES_1:77

canceled;

theorem :: TREES_1:78

for fT being finite Tree holds 1 <= width fT

proof end;

theorem :: TREES_1:79

height (elementary_tree 0) = 0

proof end;

theorem :: TREES_1:80

for fT being finite Tree st height fT = 0 holds
fT = elementary_tree 0

proof end;

theorem :: TREES_1:81

for n being Element of NAT holds height (elementary_tree (n + 1)) = 1

proof end;

theorem :: TREES_1:82

width (elementary_tree 0) = 1

proof end;

theorem :: TREES_1:83

for n being Element of NAT holds width (elementary_tree (n + 1)) = n + 1

proof end;

theorem :: TREES_1:84

for fT being finite Tree
for t being Element of fT holds height (fT | t) <= height fT

proof end;

theorem Th85: :: TREES_1:85

for fT being finite Tree
for t being Element of fT st t <> {} holds
height (fT | t) < height fT

proof end;

scheme :: TREES_1:sch 1
TreeInd{ P₁[ Tree] } :

for fT being finite Tree holds P₁[fT]

provided

A1: for fT being finite Tree st ( for n being Element of NAT st <*n*> in fT holds
P₁[fT | <*n*>] ) holds
P₁[fT]

proof end;

begin

theorem :: TREES_1:86

for w, t, s being FinSequence st w ^ t is_a_proper_prefix_of w ^ s holds
t is_a_proper_prefix_of s

proof end;

theorem :: TREES_1:87

for n, m being Element of NAT
for s being FinSequence st n <> m holds
not <*n*> is_a_prefix_of <*m*> ^ s

proof end;

theorem :: TREES_1:88

elementary_tree 1 = {{},<*0*>}

proof end;

theorem :: TREES_1:89

for n, m being Element of NAT holds not <*n*> is_a_proper_prefix_of <*m*>

proof end;

theorem :: TREES_1:90

elementary_tree 2 = {{},<*0*>,<*1*>}

proof end;

theorem :: TREES_1:91

for T being Tree
for t being Element of T holds
( t in Leaves T iff not t ^ <*0*> in T )

proof end;

theorem :: TREES_1:92

for T being Tree
for t being Element of T holds
( t in Leaves T iff for n being Element of NAT holds not t ^ <*n*> in T )

proof end;

definition

func TrivialInfiniteTree -> set equals :: TREES_1:def 17
{ (k |-> 0) where k is Element of NAT : verum } ;
coherence
{ (k |-> 0) where k is Element of NAT : verum } is set ;

end;

:: deftheorem defines TrivialInfiniteTree TREES_1:def 17 :
TrivialInfiniteTree = { (k |-> 0) where k is Element of NAT : verum } ;

registration

cluster TrivialInfiniteTree -> non empty Tree-like ;
coherence
( not TrivialInfiniteTree is empty & TrivialInfiniteTree is Tree-like )

proof end;

end;

theorem Th93: :: TREES_1:93

NAT , TrivialInfiniteTree are_equipotent

proof end;

registration

cluster TrivialInfiniteTree -> infinite ;
coherence
not TrivialInfiniteTree is finite by Th93, CARD_1:68;

end;