:: HILBERT2 semantic presentation
theorem Th1: :: HILBERT2:1
theorem Th2: :: HILBERT2:2
theorem Th3: :: HILBERT2:3
theorem Th4: :: HILBERT2:4
theorem Th5: :: HILBERT2:5
theorem Th6: :: HILBERT2:6
theorem Th7: :: HILBERT2:7
theorem Th8: :: HILBERT2:8
:: deftheorem Def1 defines prop HILBERT2:def 1 :
:: deftheorem Def2 defines with_VERUM HILBERT2:def 2 :
:: deftheorem Def3 defines with_propositional_variables HILBERT2:def 3 :
:: deftheorem Def4 defines with_implication HILBERT2:def 4 :
:: deftheorem Def5 defines with_conjunction HILBERT2:def 5 :
:: deftheorem Def6 defines conjunctive HILBERT2:def 6 :
:: deftheorem Def7 defines conditional HILBERT2:def 7 :
:: deftheorem Def8 defines simple HILBERT2:def 8 :
theorem Th9: :: HILBERT2:9
theorem Th10: :: HILBERT2:10
theorem Th11: :: HILBERT2:11
theorem Th12: :: HILBERT2:12
theorem Th13: :: HILBERT2:13
theorem Th14: :: HILBERT2:14
theorem Th15: :: HILBERT2:15
theorem Th16: :: HILBERT2:16
theorem Th17: :: HILBERT2:17
theorem Th18: :: HILBERT2:18
for b
1, b
2, b
3, b
4 being
Element of
HP-WFF holds
( b
1 ^ b
2 = b
3 ^ b
4 implies ( b
1 = b
3 & b
2 = b
4 ) )
theorem Th19: :: HILBERT2:19
theorem Th20: :: HILBERT2:20
for b
1, b
2, b
3, b
4 being
Element of
HP-WFF holds
( b
1 => b
2 = b
3 => b
4 implies ( b
1 = b
3 & b
4 = b
2 ) )
theorem Th21: :: HILBERT2:21
theorem Th22: :: HILBERT2:22
theorem Th23: :: HILBERT2:23
theorem Th24: :: HILBERT2:24
theorem Th25: :: HILBERT2:25
theorem Th26: :: HILBERT2:26
theorem Th27: :: HILBERT2:27
theorem Th28: :: HILBERT2:28
theorem Th29: :: HILBERT2:29
scheme :: HILBERT2:sch 3
s3{ F
1()
-> set , F
2(
Nat)
-> set , P
1[
Element of
HP-WFF ,
Element of
HP-WFF ,
set ,
set ,
set ], P
2[
Element of
HP-WFF ,
Element of
HP-WFF ,
set ,
set ,
set ] } :
provided
E31:
for b
1, b
2 being
Element of
HP-WFF for b
3, b
4 being
set holds
ex b
5 being
set st P
1[b
1,b
2,b
3,b
4,b
5]
and
E32:
for b
1, b
2 being
Element of
HP-WFF for b
3, b
4 being
set holds
ex b
5 being
set st P
2[b
1,b
2,b
3,b
4,b
5]
and
E33:
for b
1, b
2 being
Element of
HP-WFF for b
3, b
4, b
5, b
6 being
set holds
( P
1[b
1,b
2,b
3,b
4,b
5] & P
1[b
1,b
2,b
3,b
4,b
6] implies b
5 = b
6 )
and
E34:
for b
1, b
2 being
Element of
HP-WFF for b
3, b
4, b
5, b
6 being
set holds
( P
2[b
1,b
2,b
3,b
4,b
5] & P
2[b
1,b
2,b
3,b
4,b
6] implies b
5 = b
6 )
definition
func HP-Subformulae -> ManySortedSet of
HP-WFF means :
Def9:
:: HILBERT2:def 9
( a
1 . VERUM = root-tree VERUM & ( for b
1 being
Nat holds a
1 . (prop b1) = root-tree (prop b1) ) & ( for b
1, b
2 being
Element of
HP-WFF holds
ex b
3, b
4 being
DecoratedTree of
HP-WFF st
( b
3 = a
1 . b
1 & b
4 = a
1 . b
2 & a
1 . (b1 '&' b2) = (b1 '&' b2) -tree b
3,b
4 & a
1 . (b1 => b2) = (b1 => b2) -tree b
3,b
4 ) ) );
existence
ex b1 being ManySortedSet of HP-WFF st
( b1 . VERUM = root-tree VERUM & ( for b2 being Nat holds b1 . (prop b2) = root-tree (prop b2) ) & ( for b2, b3 being Element of HP-WFF holds
ex b4, b5 being DecoratedTree of HP-WFF st
( b4 = b1 . b2 & b5 = b1 . b3 & b1 . (b2 '&' b3) = (b2 '&' b3) -tree b4,b5 & b1 . (b2 => b3) = (b2 => b3) -tree b4,b5 ) ) )
uniqueness
for b1, b2 being ManySortedSet of HP-WFF holds
( b1 . VERUM = root-tree VERUM & ( for b3 being Nat holds b1 . (prop b3) = root-tree (prop b3) ) & ( for b3, b4 being Element of HP-WFF holds
ex b5, b6 being DecoratedTree of HP-WFF st
( b5 = b1 . b3 & b6 = b1 . b4 & b1 . (b3 '&' b4) = (b3 '&' b4) -tree b5,b6 & b1 . (b3 => b4) = (b3 => b4) -tree b5,b6 ) ) & b2 . VERUM = root-tree VERUM & ( for b3 being Nat holds b2 . (prop b3) = root-tree (prop b3) ) & ( for b3, b4 being Element of HP-WFF holds
ex b5, b6 being DecoratedTree of HP-WFF st
( b5 = b2 . b3 & b6 = b2 . b4 & b2 . (b3 '&' b4) = (b3 '&' b4) -tree b5,b6 & b2 . (b3 => b4) = (b3 => b4) -tree b5,b6 ) ) implies b1 = b2 )
end;
:: deftheorem Def9 defines HP-Subformulae HILBERT2:def 9 :
:: deftheorem Def10 defines Subformulae HILBERT2:def 10 :
theorem Th30: :: HILBERT2:30
theorem Th31: :: HILBERT2:31
theorem Th32: :: HILBERT2:32
theorem Th33: :: HILBERT2:33
theorem Th34: :: HILBERT2:34
theorem Th35: :: HILBERT2:35
theorem Th36: :: HILBERT2:36