:: BINTREE1 semantic presentation
:: deftheorem Def1 defines root-label BINTREE1:def 1 :
theorem Th1: :: BINTREE1:1
theorem Th2: :: BINTREE1:2
:: deftheorem Def2 defines binary BINTREE1:def 2 :
theorem Th3: :: BINTREE1:3
canceled;
theorem Th4: :: BINTREE1:4
canceled;
theorem Th5: :: BINTREE1:5
theorem Th6: :: BINTREE1:6
theorem Th7: :: BINTREE1:7
:: deftheorem Def3 defines binary BINTREE1:def 3 :
theorem Th8: :: BINTREE1:8
theorem Th9: :: BINTREE1:9
theorem Th10: :: BINTREE1:10
theorem Th11: :: BINTREE1:11
:: deftheorem Def4 defines binary BINTREE1:def 4 :
theorem Th12: :: BINTREE1:12
scheme :: BINTREE1:sch 3
s3{ F
1()
-> non
empty with_terminals with_nonterminals with_useful_nonterminals binary DTConstrStr , F
2()
-> non
empty set , F
3(
set )
-> Element of F
2(), F
4(
set ,
set ,
set ,
set ,
set )
-> Element of F
2() } :
ex b
1 being
Function of
TS F
1(),F
2() st
( ( for b
2 being
Terminal of F
1() holds b
1 . (root-tree b2) = F
3(b
2) ) & ( for b
2 being
NonTerminal of F
1()
for b
3, b
4 being
Element of
TS F
1()
for b
5, b
6 being
Symbol of F
1() holds
( b
5 = root-label b
3 & b
6 = root-label b
4 & b
2 ==> <*b5,b6*> implies for b
7, b
8 being
Element of F
2() holds
( b
7 = b
1 . b
3 & b
8 = b
1 . b
4 implies b
1 . (b2 -tree b3,b4) = F
4(b
2,b
5,b
6,b
7,b
8) ) ) ) )
scheme :: BINTREE1:sch 4
s4{ F
1()
-> non
empty with_terminals with_nonterminals with_useful_nonterminals binary DTConstrStr , F
2()
-> non
empty set , F
3()
-> Function of
TS F
1(),F
2(), F
4()
-> Function of
TS F
1(),F
2(), F
5(
set )
-> Element of F
2(), F
6(
set ,
set ,
set ,
set ,
set )
-> Element of F
2() } :
provided
E10:
( ( for b
1 being
Terminal of F
1() holds F
3()
. (root-tree b1) = F
5(b
1) ) & ( for b
1 being
NonTerminal of F
1()
for b
2, b
3 being
Element of
TS F
1()
for b
4, b
5 being
Symbol of F
1() holds
( b
4 = root-label b
2 & b
5 = root-label b
3 & b
1 ==> <*b4,b5*> implies for b
6, b
7 being
Element of F
2() holds
( b
6 = F
3()
. b
2 & b
7 = F
3()
. b
3 implies F
3()
. (b1 -tree b2,b3) = F
6(b
1,b
4,b
5,b
6,b
7) ) ) ) )
and
E11:
( ( for b
1 being
Terminal of F
1() holds F
4()
. (root-tree b1) = F
5(b
1) ) & ( for b
1 being
NonTerminal of F
1()
for b
2, b
3 being
Element of
TS F
1()
for b
4, b
5 being
Symbol of F
1() holds
( b
4 = root-label b
2 & b
5 = root-label b
3 & b
1 ==> <*b4,b5*> implies for b
6, b
7 being
Element of F
2() holds
( b
6 = F
4()
. b
2 & b
7 = F
4()
. b
3 implies F
4()
. (b1 -tree b2,b3) = F
6(b
1,b
4,b
5,b
6,b
7) ) ) ) )
definition
let c
1, c
2, c
3 be non
empty set ;
let c
4 be
Element of c
1;
let c
5 be
Element of c
2;
let c
6 be
Element of c
3;
redefine func [ as
[c4,c5,c6] -> Element of
[:a1,a2,a3:];
coherence
[c4,c5,c6] is Element of [:c1,c2,c3:]
by MCART_1:73;
end;
scheme :: BINTREE1:sch 5
s5{ F
1()
-> non
empty with_terminals with_nonterminals with_useful_nonterminals binary DTConstrStr , F
2()
-> non
empty set , F
3()
-> non
empty set , F
4(
set ,
set )
-> Element of F
3(), F
5(
set ,
set ,
set ,
set )
-> Element of F
3() } :
ex b
1 being
Function of
TS F
1(),
Funcs F
2(),F
3() st
( ( for b
2 being
Terminal of F
1() holds
ex b
3 being
Function of F
2(),F
3() st
( b
3 = b
1 . (root-tree b2) & ( for b
4 being
Element of F
2() holds b
3 . b
4 = F
4(b
2,b
4) ) ) ) & ( for b
2 being
NonTerminal of F
1()
for b
3, b
4 being
Element of
TS F
1()
for b
5, b
6 being
Symbol of F
1() holds
not ( b
5 = root-label b
3 & b
6 = root-label b
4 & b
2 ==> <*b5,b6*> & ( for b
7, b
8, b
9 being
Function of F
2(),F
3() holds
not ( b
7 = b
1 . (b2 -tree b3,b4) & b
8 = b
1 . b
3 & b
9 = b
1 . b
4 & ( for b
10 being
Element of F
2() holds b
7 . b
10 = F
5(b
2,b
8,b
9,b
10) ) ) ) ) ) )
scheme :: BINTREE1:sch 6
s6{ F
1()
-> non
empty with_terminals with_nonterminals with_useful_nonterminals binary DTConstrStr , F
2()
-> non
empty set , F
3()
-> non
empty set , F
4()
-> Function of
TS F
1(),
Funcs F
2(),F
3(), F
5()
-> Function of
TS F
1(),
Funcs F
2(),F
3(), F
6(
set ,
set )
-> Element of F
3(), F
7(
set ,
set ,
set ,
set )
-> Element of F
3() } :
provided
E10:
( ( for b
1 being
Terminal of F
1() holds
ex b
2 being
Function of F
2(),F
3() st
( b
2 = F
4()
. (root-tree b1) & ( for b
3 being
Element of F
2() holds b
2 . b
3 = F
6(b
1,b
3) ) ) ) & ( for b
1 being
NonTerminal of F
1()
for b
2, b
3 being
Element of
TS F
1()
for b
4, b
5 being
Symbol of F
1() holds
not ( b
4 = root-label b
2 & b
5 = root-label b
3 & b
1 ==> <*b4,b5*> & ( for b
6, b
7, b
8 being
Function of F
2(),F
3() holds
not ( b
6 = F
4()
. (b1 -tree b2,b3) & b
7 = F
4()
. b
2 & b
8 = F
4()
. b
3 & ( for b
9 being
Element of F
2() holds b
6 . b
9 = F
7(b
1,b
7,b
8,b
9) ) ) ) ) ) )
and
E11:
( ( for b
1 being
Terminal of F
1() holds
ex b
2 being
Function of F
2(),F
3() st
( b
2 = F
5()
. (root-tree b1) & ( for b
3 being
Element of F
2() holds b
2 . b
3 = F
6(b
1,b
3) ) ) ) & ( for b
1 being
NonTerminal of F
1()
for b
2, b
3 being
Element of
TS F
1()
for b
4, b
5 being
Symbol of F
1() holds
not ( b
4 = root-label b
2 & b
5 = root-label b
3 & b
1 ==> <*b4,b5*> & ( for b
6, b
7, b
8 being
Function of F
2(),F
3() holds
not ( b
6 = F
5()
. (b1 -tree b2,b3) & b
7 = F
5()
. b
2 & b
8 = F
5()
. b
3 & ( for b
9 being
Element of F
2() holds b
6 . b
9 = F
7(b
1,b
7,b
8,b
9) ) ) ) ) ) )