:: TREES_3 semantic presentation
Lemma34:
for n being Element of NAT
for p, q being FinSequence st 1 <= n & n <= len p holds
(p ^ q) . n = p . n
:: deftheorem Def1 defines Trees TREES_3:def 1 :
for
b1 being
set holds
(
b1 = Trees iff for
x being
set holds
(
x in b1 iff
x is
Tree ) );
:: deftheorem Def2 defines FinTrees TREES_3:def 2 :
:: deftheorem Def3 defines constituted-Trees TREES_3:def 3 :
:: deftheorem Def4 defines constituted-FinTrees TREES_3:def 4 :
:: deftheorem Def5 defines constituted-DTrees TREES_3:def 5 :
theorem Th1: :: TREES_3:1
theorem Th2: :: TREES_3:2
theorem Th3: :: TREES_3:3
theorem Th4: :: TREES_3:4
theorem Th5: :: TREES_3:5
theorem Th6: :: TREES_3:6
theorem Th7: :: TREES_3:7
theorem Th8: :: TREES_3:8
theorem Th9: :: TREES_3:9
theorem Th10: :: TREES_3:10
theorem Th11: :: TREES_3:11
theorem Th12: :: TREES_3:12
theorem Th13: :: TREES_3:13
theorem Th14: :: TREES_3:14
theorem Th15: :: TREES_3:15
theorem Th16: :: TREES_3:16
theorem Th17: :: TREES_3:17
theorem Th18: :: TREES_3:18
theorem Th19: :: TREES_3:19
theorem Th20: :: TREES_3:20
theorem Th21: :: TREES_3:21
:: deftheorem Def6 defines DTree-set TREES_3:def 6 :
:: deftheorem Def7 defines Trees TREES_3:def 7 :
:: deftheorem Def8 defines FinTrees TREES_3:def 8 :
theorem Th22: :: TREES_3:22
:: deftheorem Def9 defines Tree-yielding TREES_3:def 9 :
:: deftheorem Def10 defines FinTree-yielding TREES_3:def 10 :
:: deftheorem Def11 defines DTree-yielding TREES_3:def 11 :
theorem Th23: :: TREES_3:23
theorem Th24: :: TREES_3:24
theorem Th25: :: TREES_3:25
theorem Th26: :: TREES_3:26
theorem Th27: :: TREES_3:27
theorem Th28: :: TREES_3:28
theorem Th29: :: TREES_3:29
theorem Th30: :: TREES_3:30
theorem Th31: :: TREES_3:31
theorem Th32: :: TREES_3:32
theorem Th33: :: TREES_3:33
theorem Th34: :: TREES_3:34
theorem Th35: :: TREES_3:35
theorem Th36: :: TREES_3:36
theorem Th37: :: TREES_3:37
theorem Th38: :: TREES_3:38
Lemma88:
for x, y being set holds
( not <*x*> is empty & not <*x,y*> is empty )
theorem Th39: :: TREES_3:39
theorem Th40: :: TREES_3:40
Lemma92:
for D being non empty set
for T being DecoratedTree of D holds T is Function of dom T,D
definition
let D1 be non
empty set ,
D2 be non
empty set ;
redefine func pr1 as
pr1 c1,
c2 -> Function of
[:a1,a2:],
a1;
coherence
pr1 D1,D2 is Function of [:D1,D2:],D1
redefine func pr2 as
pr2 c1,
c2 -> Function of
[:a1,a2:],
a2;
coherence
pr2 D1,D2 is Function of [:D1,D2:],D2
end;
:: deftheorem Def12 defines `1 TREES_3:def 12 :
:: deftheorem Def13 defines `2 TREES_3:def 13 :
theorem Th41: :: TREES_3:41
theorem Th42: :: TREES_3:42
:: deftheorem Def14 defines T-Substitution TREES_3:def 14 :
theorem Th43: :: TREES_3:43
theorem Th44: :: TREES_3:44
Lemma103:
for n being Element of NAT
for p being FinSequence st n < len p holds
( n + 1 in dom p & p . (n + 1) in rng p )
theorem Th45: :: TREES_3:45
canceled;
theorem Th46: :: TREES_3:46
theorem Th47: :: TREES_3:47
theorem Th48: :: TREES_3:48
theorem Th49: :: TREES_3:49
theorem Th50: :: TREES_3:50
for
T1,
T2 being
Tree for
p being
Element of
T1 \/ T2 holds
( (
p in T1 &
p in T2 implies
(T1 \/ T2) | p = (T1 | p) \/ (T2 | p) ) & ( not
p in T1 implies
(T1 \/ T2) | p = T2 | p ) & ( not
p in T2 implies
(T1 \/ T2) | p = T1 | p ) )
:: deftheorem Def15 defines tree TREES_3:def 15 :
:: deftheorem Def16 defines ^ TREES_3:def 16 :
:: deftheorem Def17 defines tree TREES_3:def 17 :
theorem Th51: :: TREES_3:51
theorem Th52: :: TREES_3:52
theorem Th53: :: TREES_3:53
theorem Th54: :: TREES_3:54
theorem Th55: :: TREES_3:55
theorem Th56: :: TREES_3:56
theorem Th57: :: TREES_3:57
theorem Th58: :: TREES_3:58
theorem Th59: :: TREES_3:59
theorem Th60: :: TREES_3:60
theorem Th61: :: TREES_3:61
theorem Th62: :: TREES_3:62
theorem Th63: :: TREES_3:63
theorem Th64: :: TREES_3:64
theorem Th65: :: TREES_3:65
theorem Th66: :: TREES_3:66
for
T1,
T2 being
Tree st
T1 c= T2 holds
^ T1 c= ^ T2
theorem Th67: :: TREES_3:67
for
T1,
T2 being
Tree st
^ T1 = ^ T2 holds
T1 = T2
theorem Th68: :: TREES_3:68
theorem Th69: :: TREES_3:69
theorem Th70: :: TREES_3:70
theorem Th71: :: TREES_3:71
theorem Th72: :: TREES_3:72
theorem Th73: :: TREES_3:73
theorem Th74: :: TREES_3:74
theorem Th75: :: TREES_3:75
theorem Th76: :: TREES_3:76
for
T1,
T2,
W1,
W2 being
Tree st
tree T1,
T2 = tree W1,
W2 holds
(
T1 = W1 &
T2 = W2 )
theorem Th77: :: TREES_3:77
theorem Th78: :: TREES_3:78
theorem Th79: :: TREES_3:79
theorem Th80: :: TREES_3:80
theorem Th81: :: TREES_3:81
theorem Th82: :: TREES_3:82
theorem Th83: :: TREES_3:83
theorem Th84: :: TREES_3:84
:: deftheorem Def18 defines roots TREES_3:def 18 :