:: QC_LANG1 semantic presentation
theorem Th1: :: QC_LANG1:1
theorem Th2: :: QC_LANG1:2
:: deftheorem Def1 defines QC-variables QC_LANG1:def 1 :
theorem Th3: :: QC_LANG1:3
canceled;
theorem Th4: :: QC_LANG1:4
definition
mode QC-variable is Element of
QC-variables ;
func bound_QC-variables -> Subset of
QC-variables equals :: QC_LANG1:def 2
[:{4},NAT :];
coherence
[:{4},NAT :] is Subset of QC-variables
func fixed_QC-variables -> Subset of
QC-variables equals :: QC_LANG1:def 3
[:{5},NAT :];
coherence
[:{5},NAT :] is Subset of QC-variables
func free_QC-variables -> Subset of
QC-variables equals :: QC_LANG1:def 4
[:{6},NAT :];
coherence
[:{6},NAT :] is Subset of QC-variables
func QC-pred_symbols -> set equals :: QC_LANG1:def 5
{ [k,l] where k is Element of NAT , l is Element of NAT : 7 <= k } ;
coherence
{ [k,l] where k is Element of NAT , l is Element of NAT : 7 <= k } is set
;
end;
:: deftheorem Def2 defines bound_QC-variables QC_LANG1:def 2 :
:: deftheorem Def3 defines fixed_QC-variables QC_LANG1:def 3 :
:: deftheorem Def4 defines free_QC-variables QC_LANG1:def 4 :
:: deftheorem Def5 defines QC-pred_symbols QC_LANG1:def 5 :
theorem Th5: :: QC_LANG1:5
canceled;
theorem Th6: :: QC_LANG1:6
canceled;
theorem Th7: :: QC_LANG1:7
canceled;
theorem Th8: :: QC_LANG1:8
canceled;
theorem Th9: :: QC_LANG1:9
canceled;
theorem Th10: :: QC_LANG1:10
:: deftheorem Def6 defines the_arity_of QC_LANG1:def 6 :
:: deftheorem Def7 defines -ary_QC-pred_symbols QC_LANG1:def 7 :
:: deftheorem Def8 defines QC-variable_list QC_LANG1:def 8 :
:: deftheorem Def9 defines QC-closed QC_LANG1:def 9 :
Lemma50:
for k, l being Element of NAT holds <*[k,l]*> is FinSequence of [:NAT ,NAT :]
Lemma51:
for k being Element of NAT
for p being QC-pred_symbol of k
for ll being QC-variable_list of k holds <*p*> ^ ll is FinSequence of [:NAT ,NAT :]
Lemma52:
for x being bound_QC-variable
for p being FinSequence of [:NAT ,NAT :] holds (<*[3,0]*> ^ <*x*>) ^ p is FinSequence of [:NAT ,NAT :]
:: deftheorem Def10 defines QC-WFF QC_LANG1:def 10 :
theorem Th11: :: QC_LANG1:11
canceled;
theorem Th12: :: QC_LANG1:12
canceled;
theorem Th13: :: QC_LANG1:13
canceled;
theorem Th14: :: QC_LANG1:14
canceled;
theorem Th15: :: QC_LANG1:15
canceled;
theorem Th16: :: QC_LANG1:16
canceled;
theorem Th17: :: QC_LANG1:17
canceled;
theorem Th18: :: QC_LANG1:18
canceled;
theorem Th19: :: QC_LANG1:19
canceled;
theorem Th20: :: QC_LANG1:20
canceled;
theorem Th21: :: QC_LANG1:21
:: deftheorem Def11 defines ! QC_LANG1:def 11 :
theorem Th22: :: QC_LANG1:22
canceled;
theorem Th23: :: QC_LANG1:23
:: deftheorem Def12 defines @ QC_LANG1:def 12 :
:: deftheorem Def13 defines VERUM QC_LANG1:def 13 :
:: deftheorem Def14 defines 'not' QC_LANG1:def 14 :
:: deftheorem Def15 defines '&' QC_LANG1:def 15 :
:: deftheorem Def16 defines All QC_LANG1:def 16 :
:: deftheorem Def17 defines atomic QC_LANG1:def 17 :
:: deftheorem Def18 defines negative QC_LANG1:def 18 :
:: deftheorem Def19 defines conjunctive QC_LANG1:def 19 :
:: deftheorem Def20 defines universal QC_LANG1:def 20 :
theorem Th24: :: QC_LANG1:24
canceled;
theorem Th25: :: QC_LANG1:25
canceled;
theorem Th26: :: QC_LANG1:26
canceled;
theorem Th27: :: QC_LANG1:27
canceled;
theorem Th28: :: QC_LANG1:28
canceled;
theorem Th29: :: QC_LANG1:29
canceled;
theorem Th30: :: QC_LANG1:30
canceled;
theorem Th31: :: QC_LANG1:31
canceled;
theorem Th32: :: QC_LANG1:32
canceled;
theorem Th33: :: QC_LANG1:33
theorem Th34: :: QC_LANG1:34
theorem Th35: :: QC_LANG1:35
theorem Th36: :: QC_LANG1:36
theorem Th37: :: QC_LANG1:37
:: deftheorem Def21 defines the_pred_symbol_of QC_LANG1:def 21 :
:: deftheorem Def22 defines the_arguments_of QC_LANG1:def 22 :
:: deftheorem Def23 defines the_argument_of QC_LANG1:def 23 :
:: deftheorem Def24 defines the_left_argument_of QC_LANG1:def 24 :
:: deftheorem Def25 defines the_right_argument_of QC_LANG1:def 25 :
definition
let F be
Element of
QC-WFF ;
assume E32:
F is
universal
;
func bound_in c1 -> bound_QC-variable means :
Def26:
:: QC_LANG1:def 26
ex
p being
Element of
QC-WFF st
F = All it,
p;
existence
ex b1 being bound_QC-variable ex p being Element of QC-WFF st F = All b1,p
by Th10, ;
uniqueness
for b1, b2 being bound_QC-variable st ex p being Element of QC-WFF st F = All b1,p & ex p being Element of QC-WFF st F = All b2,p holds
b1 = b2
func the_scope_of c1 -> QC-formula means :
Def27:
:: QC_LANG1:def 27
ex
x being
bound_QC-variable st
F = All x,
it;
existence
ex b1 being QC-formula ex x being bound_QC-variable st F = All x,b1
uniqueness
for b1, b2 being QC-formula st ex x being bound_QC-variable st F = All x,b1 & ex x being bound_QC-variable st F = All x,b2 holds
b1 = b2
end;
:: deftheorem Def26 defines bound_in QC_LANG1:def 26 :
:: deftheorem Def27 defines the_scope_of QC_LANG1:def 27 :
theorem Th38: :: QC_LANG1:38
canceled;
theorem Th39: :: QC_LANG1:39
canceled;
theorem Th40: :: QC_LANG1:40
canceled;
theorem Th41: :: QC_LANG1:41
canceled;
theorem Th42: :: QC_LANG1:42
canceled;
theorem Th43: :: QC_LANG1:43
canceled;
theorem Th44: :: QC_LANG1:44
canceled;
theorem Th45: :: QC_LANG1:45
theorem Th46: :: QC_LANG1:46
theorem Th47: :: QC_LANG1:47
theorem Th48: :: QC_LANG1:48
theorem Th49: :: QC_LANG1:49
theorem Th50: :: QC_LANG1:50
theorem Th51: :: QC_LANG1:51
:: deftheorem Def28 defines still_not-bound_in QC_LANG1:def 28 :
:: deftheorem Def29 defines still_not-bound_in QC_LANG1:def 29 :
:: deftheorem Def30 defines closed QC_LANG1:def 30 :