:: TREES_A semantic presentation
theorem Th1: :: TREES_A:1
:: deftheorem Def1 defines tree TREES_A:def 1 :
theorem Th2: :: TREES_A:2
theorem Th3: :: TREES_A:3
theorem Th4: :: TREES_A:4
theorem Th5: :: TREES_A:5
theorem Th6: :: TREES_A:6
theorem Th7: :: TREES_A:7
theorem Th8: :: TREES_A:8
canceled;
theorem Th9: :: TREES_A:9
theorem Th10: :: TREES_A:10
definition
let c
1 be
DecoratedTree;
let c
2 be
AntiChain_of_Prefixes of
dom c
1;
let c
3 be
DecoratedTree;
assume E10:
c
2 <> {}
;
func tree c
1,c
2,c
3 -> DecoratedTree means :
Def2:
:: TREES_A:def 2
(
dom a
4 = tree (dom a1),a
2,
(dom a3) & ( for b
1 being
FinSequence of
NAT holds
not ( b
1 in tree (dom a1),a
2,
(dom a3) & ex b
2 being
FinSequence of
NAT st
( b
2 in a
2 & not ( not b
2 is_a_prefix_of b
1 & a
4 . b
1 = a
1 . b
1 ) ) & ( for b
2, b
3 being
FinSequence of
NAT holds
not ( b
2 in a
2 & b
3 in dom a
3 & b
1 = b
2 ^ b
3 & a
4 . b
1 = a
3 . b
3 ) ) ) ) );
existence
ex b1 being DecoratedTree st
( dom b1 = tree (dom c1),c2,(dom c3) & ( for b2 being FinSequence of NAT holds
not ( b2 in tree (dom c1),c2,(dom c3) & ex b3 being FinSequence of NAT st
( b3 in c2 & not ( not b3 is_a_prefix_of b2 & b1 . b2 = c1 . b2 ) ) & ( for b3, b4 being FinSequence of NAT holds
not ( b3 in c2 & b4 in dom c3 & b2 = b3 ^ b4 & b1 . b2 = c3 . b4 ) ) ) ) )
uniqueness
for b1, b2 being DecoratedTree holds
( dom b1 = tree (dom c1),c2,(dom c3) & ( for b3 being FinSequence of NAT holds
not ( b3 in tree (dom c1),c2,(dom c3) & ex b4 being FinSequence of NAT st
( b4 in c2 & not ( not b4 is_a_prefix_of b3 & b1 . b3 = c1 . b3 ) ) & ( for b4, b5 being FinSequence of NAT holds
not ( b4 in c2 & b5 in dom c3 & b3 = b4 ^ b5 & b1 . b3 = c3 . b5 ) ) ) ) & dom b2 = tree (dom c1),c2,(dom c3) & ( for b3 being FinSequence of NAT holds
not ( b3 in tree (dom c1),c2,(dom c3) & ex b4 being FinSequence of NAT st
( b4 in c2 & not ( not b4 is_a_prefix_of b3 & b2 . b3 = c1 . b3 ) ) & ( for b4, b5 being FinSequence of NAT holds
not ( b4 in c2 & b5 in dom c3 & b3 = b4 ^ b5 & b2 . b3 = c3 . b5 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines tree TREES_A:def 2 :
theorem Th11: :: TREES_A:11
canceled;
theorem Th12: :: TREES_A:12
canceled;
theorem Th13: :: TREES_A:13
theorem Th14: :: TREES_A:14
theorem Th15: :: TREES_A:15
theorem Th16: :: TREES_A:16
theorem Th17: :: TREES_A:17
theorem Th18: :: TREES_A:18
theorem Th19: :: TREES_A:19
:: deftheorem Def3 defines tree TREES_A:def 3 :