:: Replacement of Subtrees in a Tree
:: by Oleg Okhotnikov
::
:: Received October 1, 1995
:: Copyright (c) 1995-2011 Association of Mizar Users

begin

theorem Th1: :: TREES_A:1

for p, q, r, s being FinSequence st p ^ q = s ^ r holds
p,s are_c=-comparable

proof end;

definition

let T, T1 be Tree;
let P be AntiChain_of_Prefixes of T;
assume A1: P <> {} ;
func tree (T,P,T1) -> Tree means :Def1: :: TREES_A:def 1
for q being FinSequence of NAT holds
( q in it iff ( ( q in T & ( for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of q ) ) or ex p, r being FinSequence of NAT st
( p in P & 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 & ( for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of q ) ) or ex p, r being FinSequence of NAT st
( p in P & 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 & ( for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of q ) ) or ex p, r being FinSequence of NAT st
( p in P & r in T1 & q = p ^ r ) ) ) ) & ( for q being FinSequence of NAT holds
( q in b₂ iff ( ( q in T & ( for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of q ) ) or ex p, r being FinSequence of NAT st
( p in P & r in T1 & q = p ^ r ) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines tree TREES_A:def 1 :
for T, T1 being Tree
for P being AntiChain_of_Prefixes of T st P <> {} holds
for b₄ being Tree holds
( b₄ = tree (T,P,T1) iff for q being FinSequence of NAT holds
( q in b₄ iff ( ( q in T & ( for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of q ) ) or ex p, r being FinSequence of NAT st
( p in P & r in T1 & q = p ^ r ) ) ) );

theorem Th2: :: TREES_A:2

for T, T1 being Tree
for P being AntiChain_of_Prefixes of T st P <> {} holds
tree (T,P,T1) = { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } \/ { (p ^ s) where p is Element of T, s is Element of T1 : p in P }

proof end;

theorem Th3: :: TREES_A:3

for T being Tree
for P being AntiChain_of_Prefixes of T holds { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } c= { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 }

proof end;

theorem Th4: :: TREES_A:4

for T being Tree
for P being AntiChain_of_Prefixes of T holds P c= { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 }

proof end;

theorem Th5: :: TREES_A:5

for T being Tree
for P being AntiChain_of_Prefixes of T holds { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } \ { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } = P

proof end;

theorem Th6: :: TREES_A:6

for T, T1 being Tree
for P being AntiChain_of_Prefixes of T holds P c= { (p ^ s) where p is Element of T, s is Element of T1 : p in P }

proof end;

theorem Th7: :: TREES_A:7

for T, T1 being Tree
for P being AntiChain_of_Prefixes of T st P <> {} holds
tree (T,P,T1) = { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } \/ { (p ^ s) where p is Element of T, s is Element of T1 : p in P }

proof end;

theorem :: TREES_A:8

canceled;

theorem :: TREES_A:9

for T1, T being Tree
for P being AntiChain_of_Prefixes of T
for p being FinSequence of NAT st p in P holds
T1 = (tree (T,P,T1)) | p

proof end;

registration

let T be Tree;
cluster non empty AntiChain_of_Prefixes-like AntiChain_of_Prefixes of T;
existence
not for b₁ being AntiChain_of_Prefixes of T holds b₁ is empty

proof end;

end;

definition

let T be Tree;
let t be Element of T;
:: original: {
redefine func {t} -> AntiChain_of_Prefixes of T;
correctness
coherence
{t} is AntiChain_of_Prefixes of T;
by TREES_1:74;

end;

theorem Th10: :: TREES_A:10

for T, T1 being Tree
for t being Element of T holds tree (T,{t},T1) = T with-replacement (t,T1)

proof end;

definition

let T be DecoratedTree;
let P be AntiChain_of_Prefixes of dom T;
let T1 be DecoratedTree;
assume A1: P <> {} ;
func tree (T,P,T1) -> DecoratedTree means :Def2: :: TREES_A:def 2
( dom it = tree ((dom T),P,(dom T1)) & ( for q being FinSequence of NAT holds
( not q in tree ((dom T),P,(dom T1)) or for p being FinSequence of NAT st p in P holds
( not p is_a_prefix_of q & it . q = T . q ) or ex p, r being FinSequence of NAT st
( p in P & r in dom T1 & q = p ^ r & it . q = T1 . r ) ) ) );
existence
ex b₁ being DecoratedTree st
( dom b₁ = tree ((dom T),P,(dom T1)) & ( for q being FinSequence of NAT holds
( not q in tree ((dom T),P,(dom T1)) or for p being FinSequence of NAT st p in P holds
( not p is_a_prefix_of q & b₁ . q = T . q ) or ex p, r being FinSequence of NAT st
( p in P & r in dom T1 & q = p ^ r & b₁ . q = T1 . r ) ) ) )

proof end;

uniqueness
for b₁, b₂ being DecoratedTree st dom b₁ = tree ((dom T),P,(dom T1)) & ( for q being FinSequence of NAT holds
( not q in tree ((dom T),P,(dom T1)) or for p being FinSequence of NAT st p in P holds
( not p is_a_prefix_of q & b₁ . q = T . q ) or ex p, r being FinSequence of NAT st
( p in P & r in dom T1 & q = p ^ r & b₁ . q = T1 . r ) ) ) & dom b₂ = tree ((dom T),P,(dom T1)) & ( for q being FinSequence of NAT holds
( not q in tree ((dom T),P,(dom T1)) or for p being FinSequence of NAT st p in P holds
( not p is_a_prefix_of q & b₂ . q = T . q ) or ex p, r being FinSequence of NAT st
( p in P & r in dom T1 & q = p ^ r & b₂ . q = T1 . r ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines tree TREES_A:def 2 :
for T being DecoratedTree
for P being AntiChain_of_Prefixes of dom T
for T1 being DecoratedTree st P <> {} holds
for b₄ being DecoratedTree holds
( b₄ = tree (T,P,T1) iff ( dom b₄ = tree ((dom T),P,(dom T1)) & ( for q being FinSequence of NAT holds
( not q in tree ((dom T),P,(dom T1)) or for p being FinSequence of NAT st p in P holds
( not p is_a_prefix_of q & b₄ . q = T . q ) or ex p, r being FinSequence of NAT st
( p in P & r in dom T1 & q = p ^ r & b₄ . q = T1 . r ) ) ) ) );

theorem :: TREES_A:11

canceled;

theorem :: TREES_A:12

canceled;

theorem Th13: :: TREES_A:13

for T, T1 being DecoratedTree
for P being AntiChain_of_Prefixes of dom T st P <> {} holds
for q being FinSequence of NAT holds
( not q in dom (tree (T,P,T1)) or for p being FinSequence of NAT st p in P holds
( not p is_a_prefix_of q & (tree (T,P,T1)) . q = T . q ) or ex p, r being FinSequence of NAT st
( p in P & r in dom T1 & q = p ^ r & (tree (T,P,T1)) . q = T1 . r ) )

proof end;

theorem Th14: :: TREES_A:14

for p being FinSequence of NAT
for T, T1 being DecoratedTree st p in dom T holds
for q being FinSequence of NAT holds
( not q in dom (T with-replacement (p,T1)) or ( not p is_a_prefix_of q & (T with-replacement (p,T1)) . q = T . q ) or ex r being FinSequence of NAT st
( r in dom T1 & q = p ^ r & (T with-replacement (p,T1)) . q = T1 . r ) )

proof end;

theorem Th15: :: TREES_A:15

for T, T1 being DecoratedTree
for P being AntiChain_of_Prefixes of dom T st P <> {} holds
for q being FinSequence of NAT st q in dom (tree (T,P,T1)) & q in { t1 where t1 is Element of dom T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } holds
(tree (T,P,T1)) . q = T . q

proof end;

theorem Th16: :: TREES_A:16

for p being FinSequence of NAT
for T, T1 being DecoratedTree st p in dom T holds
for q being FinSequence of NAT st q in dom (T with-replacement (p,T1)) & q in { t1 where t1 is Element of dom T : not p is_a_prefix_of t1 } holds
(T with-replacement (p,T1)) . q = T . q

proof end;

theorem Th17: :: TREES_A:17

for T, T1 being DecoratedTree
for P being AntiChain_of_Prefixes of dom T
for q being FinSequence of NAT st q in dom (tree (T,P,T1)) & q in { (p ^ s) where p is Element of dom T, s is Element of dom T1 : p in P } holds
ex p9 being Element of dom T ex r being Element of dom T1 st
( q = p9 ^ r & p9 in P & (tree (T,P,T1)) . q = T1 . r )

proof end;

theorem Th18: :: TREES_A:18

for p being FinSequence of NAT
for T, T1 being DecoratedTree st p in dom T holds
for q being FinSequence of NAT st q in dom (T with-replacement (p,T1)) & q in { (p ^ s) where s is Element of dom T1 : s = s } holds
ex r being Element of dom T1 st
( q = p ^ r & (T with-replacement (p,T1)) . q = T1 . r )

proof end;

theorem :: TREES_A:19

for T, T1 being DecoratedTree
for t being Element of dom T holds tree (T,{t},T1) = T with-replacement (t,T1)

proof end;

definition

let D be non empty set ;
let T be DecoratedTree of D;
let P be AntiChain_of_Prefixes of dom T;
let T1 be DecoratedTree of D;
assume A1: P <> {} ;
func tree (T,P,T1) -> DecoratedTree of D equals :: TREES_A:def 3
tree (T,P,T1);
coherence
tree (T,P,T1) is DecoratedTree of D

proof end;

end;

:: deftheorem defines tree TREES_A:def 3 :
for D being non empty set
for T being DecoratedTree of D
for P being AntiChain_of_Prefixes of dom T
for T1 being DecoratedTree of D st P <> {} holds
tree (T,P,T1) = tree (T,P,T1);