:: TREES_9 semantic presentation
Lemma1:
for b1 being set
for b2 being FinSequence holds
not ( b1 in dom b2 & ( for b3 being Nat holds
not ( b1 = b3 + 1 & b3 < len b2 ) ) )
Lemma3:
for b1 being Nat
for b2 being FinSequence holds
( b1 < len b2 implies ( b1 + 1 in dom b2 & b2 . (b1 + 1) in rng b2 ) )
theorem Th1: :: TREES_9:1
theorem Th2: :: TREES_9:2
theorem Th3: :: TREES_9:3
:: deftheorem Def1 defines root TREES_9:def 1 :
theorem Th4: :: TREES_9:4
theorem Th5: :: TREES_9:5
theorem Th6: :: TREES_9:6
:: deftheorem Def2 defines finite-branching TREES_9:def 2 :
:: deftheorem Def3 defines finite-order TREES_9:def 3 :
:: deftheorem Def4 defines finite-branching TREES_9:def 4 :
scheme :: TREES_9:sch 1
s1{ F
1(
set )
-> set , F
2()
-> finite set } :
for b
1 being
Nat holds
( F
1(b
1)
in F
2() iff b
1 < card F
2() )
provided
E14:
for b
1 being
set holds
not ( b
1 in F
2() & ( for b
2 being
Nat holds
not b
1 = F
1(b
2) ) )
and
E15:
for b
1, b
2 being
Nat holds
( b
1 < b
2 & F
1(b
2)
in F
2() implies F
1(b
1)
in F
2() )
and
E16:
for b
1, b
2 being
Nat holds
( F
1(b
1)
= F
1(b
2) implies b
1 = b
2 )
theorem Th7: :: TREES_9:7
:: deftheorem Def5 defines succ TREES_9:def 5 :
:: deftheorem Def6 defines succ TREES_9:def 6 :
theorem Th8: :: TREES_9:8
Lemma18:
for b1 being finite DecoratedTree
for b2 being Node of b1 holds b1 | b2 is finite
;
theorem Th9: :: TREES_9:9
canceled;
theorem Th10: :: TREES_9:10
:: deftheorem Def7 defines Subtrees TREES_9:def 7 :
theorem Th11: :: TREES_9:11
theorem Th12: :: TREES_9:12
theorem Th13: :: TREES_9:13
theorem Th14: :: TREES_9:14
:: deftheorem Def8 defines FixedSubtrees TREES_9:def 8 :
theorem Th15: :: TREES_9:15
theorem Th16: :: TREES_9:16
theorem Th17: :: TREES_9:17
:: deftheorem Def9 defines -Subtrees TREES_9:def 9 :
theorem Th18: :: TREES_9:18
theorem Th19: :: TREES_9:19
:: deftheorem Def10 defines -ImmediateSubtrees TREES_9:def 10 :
:: deftheorem Def11 defines Subtrees TREES_9:def 11 :
theorem Th20: :: TREES_9:20
theorem Th21: :: TREES_9:21
theorem Th22: :: TREES_9:22
theorem Th23: :: TREES_9:23
theorem Th24: :: TREES_9:24
:: deftheorem Def12 defines -Subtrees TREES_9:def 12 :
theorem Th25: :: TREES_9:25
theorem Th26: :: TREES_9:26
theorem Th27: :: TREES_9:27
theorem Th28: :: TREES_9:28
:: deftheorem Def13 defines -ImmediateSubtrees TREES_9:def 13 :
theorem Th29: :: TREES_9:29
theorem Th30: :: TREES_9:30
theorem Th31: :: TREES_9:31
theorem Th32: :: TREES_9:32