:: TREES_2 semantic presentation
theorem Th1: :: TREES_2:1
theorem Th2: :: TREES_2:2
Lemma3:
for b1 being set
for b2 being FinSequence holds len (b2 ^ <*b1*>) = (len b2) + 1
theorem Th3: :: TREES_2:3
canceled;
theorem Th4: :: TREES_2:4
theorem Th5: :: TREES_2:5
canceled;
theorem Th6: :: TREES_2:6
theorem Th7: :: TREES_2:7
for b
1, b
2 being
Tree holds
( ( for b
3 being
FinSequence of
NAT holds
( b
3 in b
1 iff b
3 in b
2 ) ) implies b
1 = b
2 )
:: deftheorem Def1 defines = TREES_2:def 1 :
theorem Th8: :: TREES_2:8
theorem Th9: :: TREES_2:9
theorem Th10: :: TREES_2:10
:: deftheorem Def2 defines finite-order TREES_2:def 2 :
:: deftheorem Def3 defines Chain TREES_2:def 3 :
:: deftheorem Def4 defines Level TREES_2:def 4 :
:: deftheorem Def5 defines succ TREES_2:def 5 :
theorem Th11: :: TREES_2:11
theorem Th12: :: TREES_2:12
theorem Th13: :: TREES_2:13
:: deftheorem Def6 defines -level TREES_2:def 6 :
theorem Th14: :: TREES_2:14
theorem Th15: :: TREES_2:15
theorem Th16: :: TREES_2:16
theorem Th17: :: TREES_2:17
theorem Th18: :: TREES_2:18
theorem Th19: :: TREES_2:19
theorem Th20: :: TREES_2:20
theorem Th21: :: TREES_2:21
theorem Th22: :: TREES_2:22
:: deftheorem Def7 defines Branch-like TREES_2:def 7 :
theorem Th23: :: TREES_2:23
theorem Th24: :: TREES_2:24
theorem Th25: :: TREES_2:25
theorem Th26: :: TREES_2:26
theorem Th27: :: TREES_2:27
theorem Th28: :: TREES_2:28
theorem Th29: :: TREES_2:29
theorem Th30: :: TREES_2:30
scheme :: TREES_2:sch 4
s4{ P
1[
set ,
Nat], F
1()
-> set } :
ex b
1 being
Function st
(
dom b
1 = F
1() & ( for b
2 being
set holds
not ( b
2 in F
1() & ( for b
3 being
Nat holds
not ( b
1 . b
2 = b
3 & P
1[b
2,b
3] & ( for b
4 being
Nat holds
( P
1[b
2,b
4] implies b
3 <= b
4 ) ) ) ) ) ) )
provided
E23:
for b
1 being
set holds
not ( b
1 in F
1() & ( for b
2 being
Nat holds
not P
1[b
1,b
2] ) )
Lemma23:
for b1 being set holds
not ( b1 is finite & ( for b2 being Nat holds
ex b3 being Nat st
( b3 in b1 & not b3 <= b2 ) ) )
scheme :: TREES_2:sch 5
s5{ F
1()
-> set , F
2()
-> set , P
1[
set ], P
2[
set ,
set ] } :
ex b
1 being
Function st
(
dom b
1 = NAT &
rng b
1 c= F
1() & b
1 . 0
= F
2() & ( for b
2 being
Nat holds
( P
2[b
1 . b
2,b
1 . (b2 + 1)] & P
1[b
1 . b
2] ) ) )
provided
E24:
( F
2()
in F
1() & P
1[F
2()] )
and
E25:
for b
1 being
set holds
not ( b
1 in F
1() & P
1[b
1] & ( for b
2 being
set holds
not ( b
2 in F
1() & P
2[b
1,b
2] & P
1[b
2] ) ) )
theorem Th31: :: TREES_2:31
theorem Th32: :: TREES_2:32
:: deftheorem Def8 defines DecoratedTree-like TREES_2:def 8 :
:: deftheorem Def9 defines ParametrizedSubset TREES_2:def 9 :
theorem Th33: :: TREES_2:33
:: deftheorem Def10 defines Leaves TREES_2:def 10 :
:: deftheorem Def11 defines | TREES_2:def 11 :
theorem Th34: :: TREES_2:34
definition
let c
1 be
DecoratedTree;
let c
2 be
FinSequence of
NAT ;
let c
3 be
DecoratedTree;
assume E30:
c
2 in dom c
1
;
func c
1 with-replacement c
2,c
3 -> DecoratedTree means :: TREES_2:def 12
(
dom a
4 = (dom a1) with-replacement a
2,
(dom a3) & ( for b
1 being
FinSequence of
NAT holds
not ( b
1 in (dom a1) with-replacement a
2,
(dom a3) & not ( not a
2 is_a_prefix_of b
1 & a
4 . b
1 = a
1 . b
1 ) & ( for b
2 being
FinSequence of
NAT holds
not ( b
2 in dom a
3 & b
1 = a
2 ^ b
2 & a
4 . b
1 = a
3 . b
2 ) ) ) ) );
existence
ex b1 being DecoratedTree st
( dom b1 = (dom c1) with-replacement c2,(dom c3) & ( for b2 being FinSequence of NAT holds
not ( b2 in (dom c1) with-replacement c2,(dom c3) & not ( not c2 is_a_prefix_of b2 & b1 . b2 = c1 . b2 ) & ( for b3 being FinSequence of NAT holds
not ( b3 in dom c3 & b2 = c2 ^ b3 & b1 . b2 = c3 . b3 ) ) ) ) )
uniqueness
for b1, b2 being DecoratedTree holds
( dom b1 = (dom c1) with-replacement c2,(dom c3) & ( for b3 being FinSequence of NAT holds
not ( b3 in (dom c1) with-replacement c2,(dom c3) & not ( not c2 is_a_prefix_of b3 & b1 . b3 = c1 . b3 ) & ( for b4 being FinSequence of NAT holds
not ( b4 in dom c3 & b3 = c2 ^ b4 & b1 . b3 = c3 . b4 ) ) ) ) & dom b2 = (dom c1) with-replacement c2,(dom c3) & ( for b3 being FinSequence of NAT holds
not ( b3 in (dom c1) with-replacement c2,(dom c3) & not ( not c2 is_a_prefix_of b3 & b2 . b3 = c1 . b3 ) & ( for b4 being FinSequence of NAT holds
not ( b4 in dom c3 & b3 = c2 ^ b4 & b2 . b3 = c3 . b4 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def12 defines with-replacement TREES_2:def 12 :
theorem Th35: :: TREES_2:35
theorem Th36: :: TREES_2:36
theorem Th37: :: TREES_2:37
theorem Th38: :: TREES_2:38
:: deftheorem Def13 defines branchdeg TREES_2:def 13 :
Lemma34:
for b1 being Function holds (pr1 (dom b1),(rng b1)) .: b1 = dom b1
Lemma35:
for b1 being Function holds
( dom b1 is finite iff b1 is finite )