Article HENMODEL, MML version 4.99.1005
:: HENMODEL:th 1
theorem
for b1 being Element of NAT
for b2 being non empty finite Element of bool NAT
for b3 being Function-like quasi_total Relation of b1,b2
st (ex b4 being Element of NAT st
succ b4 = b1) &
b3 is one-to-one &
rng b3 = b2 &
(for b4, b5 being Element of NAT
st b5 in dom b3 & b4 in dom b3 & b4 < b5
holds b3 . b4 in b3 . b5)
holds b3 . union b1 = union rng b3;
:: HENMODEL:th 2
theorem
for b1 being non empty finite Element of bool NAT holds
union b1 in b1 &
(for b2 being set
st b2 in b1 & not b2 in union b1
holds b2 = union b1);
:: HENMODEL:th 3
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of NAT,b1
for b3 being finite set
st (for b4, b5 being Element of NAT
st b5 in dom b2 & b4 in dom b2 & b4 < b5
holds b2 . b4 c= b2 . b5) &
b3 c= union rng b2
holds ex b4 being Element of NAT st
b3 c= b2 . b4;
:: HENMODEL:prednot 1 => HENMODEL:pred 1
definition
let a1 be Element of bool CQC-WFF;
let a2 be Element of CQC-WFF;
pred A1 |- A2 means
ex b1 being FinSequence of CQC-WFF st
rng b1 c= a1 & |- b1 ^ <*a2*>;
end;
:: HENMODEL:dfs 1
definiens
let a1 be Element of bool CQC-WFF;
let a2 be Element of CQC-WFF;
To prove
a1 |- a2
it is sufficient to prove
thus ex b1 being FinSequence of CQC-WFF st
rng b1 c= a1 & |- b1 ^ <*a2*>;
:: HENMODEL:def 2
theorem
for b1 being Element of bool CQC-WFF
for b2 being Element of CQC-WFF holds
b1 |- b2
iff
ex b3 being FinSequence of CQC-WFF st
rng b3 c= b1 & |- b3 ^ <*b2*>;
:: HENMODEL:attrnot 1 => HENMODEL:attr 1
definition
let a1 be Element of bool CQC-WFF;
attr a1 is Consistent means
for b1 being Element of CQC-WFF
st a1 |- b1
holds not a1 |- 'not' b1;
end;
:: HENMODEL:dfs 2
definiens
let a1 be Element of bool CQC-WFF;
To prove
a1 is Consistent
it is sufficient to prove
thus for b1 being Element of CQC-WFF
st a1 |- b1
holds not a1 |- 'not' b1;
:: HENMODEL:def 3
theorem
for b1 being Element of bool CQC-WFF holds
b1 is Consistent
iff
for b2 being Element of CQC-WFF
st b1 |- b2
holds not b1 |- 'not' b2;
:: HENMODEL:attrnot 2 => HENMODEL:attr 1
notation
let a1 be Element of bool CQC-WFF;
antonym Inconsistent for Consistent;
end;
:: HENMODEL:attrnot 3 => HENMODEL:attr 2
definition
let a1 be FinSequence of CQC-WFF;
attr a1 is Consistent means
for b1 being Element of CQC-WFF
st |- a1 ^ <*b1*>
holds not |- a1 ^ <*'not' b1*>;
end;
:: HENMODEL:dfs 3
definiens
let a1 be FinSequence of CQC-WFF;
To prove
a1 is Consistent
it is sufficient to prove
thus for b1 being Element of CQC-WFF
st |- a1 ^ <*b1*>
holds not |- a1 ^ <*'not' b1*>;
:: HENMODEL:def 4
theorem
for b1 being FinSequence of CQC-WFF holds
b1 is Consistent
iff
for b2 being Element of CQC-WFF
st |- b1 ^ <*b2*>
holds not |- b1 ^ <*'not' b2*>;
:: HENMODEL:attrnot 4 => HENMODEL:attr 2
notation
let a1 be FinSequence of CQC-WFF;
antonym Inconsistent for Consistent;
end;
:: HENMODEL:th 4
theorem
for b1 being Element of bool CQC-WFF
for b2 being FinSequence of CQC-WFF
st b1 is Consistent & rng b2 c= b1
holds b2 is Consistent;
:: HENMODEL:th 5
theorem
for b1 being Element of CQC-WFF
for b2, b3 being FinSequence of CQC-WFF
st |- b2 ^ <*b1*>
holds |- (b2 ^ b3) ^ <*b1*>;
:: HENMODEL:th 6
theorem
for b1 being Element of bool CQC-WFF holds
b1 is Inconsistent
iff
for b2 being Element of CQC-WFF holds
b1 |- b2;
:: HENMODEL:th 7
theorem
for b1 being Element of bool CQC-WFF
st b1 is Inconsistent
holds ex b2 being Element of bool CQC-WFF st
b2 c= b1 & b2 is finite & b2 is Inconsistent;
:: HENMODEL:th 8
theorem
for b1 being Element of bool CQC-WFF
for b2, b3 being Element of CQC-WFF
st b1 \/ {b2} |- b3
holds ex b4 being FinSequence of CQC-WFF st
rng b4 c= b1 &
|- (b4 ^ <*b2*>) ^ <*b3*>;
:: HENMODEL:th 9
theorem
for b1 being Element of bool CQC-WFF
for b2 being Element of CQC-WFF holds
b1 |- b2
iff
b1 \/ {'not' b2} is Inconsistent;
:: HENMODEL:th 10
theorem
for b1 being Element of bool CQC-WFF
for b2 being Element of CQC-WFF holds
b1 |- 'not' b2
iff
b1 \/ {b2} is Inconsistent;
:: HENMODEL:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,bool CQC-WFF
st for b2, b3 being Element of NAT
st b3 in dom b1 & b2 in dom b1 & b2 < b3
holds b1 . b2 is Consistent & b1 . b2 c= b1 . b3
holds union rng b1 is Consistent;
:: HENMODEL:th 12
theorem
for b1 being Element of bool CQC-WFF
for b2 being non empty set
st b1 is Inconsistent
for b3 being interpretation of b2
for b4 being Element of Valuations_in b2 holds
not b3,b4 |= b1;
:: HENMODEL:th 13
theorem
{VERUM} is Consistent;
:: HENMODEL:exreg 1
registration
cluster Consistent Element of bool CQC-WFF;
end;
:: HENMODEL:funcnot 1 => HENMODEL:func 1
definition
func HCar -> non empty set equals
bound_QC-variables;
end;
:: HENMODEL:def 5
theorem
HCar = bound_QC-variables;
:: HENMODEL:funcnot 2 => HENMODEL:func 2
definition
let a1 be Element of QC-pred_symbols;
let a2 be CQC-variable_list-like QC-variable_list of the_arity_of a1;
redefine func a1 ! a2 -> Element of CQC-WFF;
end;
:: HENMODEL:modenot 1 => HENMODEL:mode 1
definition
let a1 be Consistent Element of bool CQC-WFF;
mode Henkin_interpretation of A1 -> interpretation of HCar means
for b1 being Element of QC-pred_symbols
for b2 being Element of relations_on HCar
st it . b1 = b2
for b3 being set holds
b3 in b2
iff
ex b4 being CQC-variable_list-like QC-variable_list of the_arity_of b1 st
b3 = b4 & a1 |- b1 ! b4;
end;
:: HENMODEL:dfs 5
definiens
let a1 be Consistent Element of bool CQC-WFF;
let a2 be interpretation of HCar;
To prove
a2 is Henkin_interpretation of a1
it is sufficient to prove
thus for b1 being Element of QC-pred_symbols
for b2 being Element of relations_on HCar
st a2 . b1 = b2
for b3 being set holds
b3 in b2
iff
ex b4 being CQC-variable_list-like QC-variable_list of the_arity_of b1 st
b3 = b4 & a1 |- b1 ! b4;
:: HENMODEL:def 6
theorem
for b1 being Consistent Element of bool CQC-WFF
for b2 being interpretation of HCar holds
b2 is Henkin_interpretation of b1
iff
for b3 being Element of QC-pred_symbols
for b4 being Element of relations_on HCar
st b2 . b3 = b4
for b5 being set holds
b5 in b4
iff
ex b6 being CQC-variable_list-like QC-variable_list of the_arity_of b3 st
b5 = b6 & b1 |- b3 ! b6;
:: HENMODEL:funcnot 3 => HENMODEL:func 3
definition
func valH -> Element of Valuations_in HCar equals
id bound_QC-variables;
end;
:: HENMODEL:def 7
theorem
valH = id bound_QC-variables;
:: HENMODEL:th 14
theorem
for b1 being Element of NAT
for b2 being CQC-variable_list-like QC-variable_list of b1 holds
valH *' b2 = b2;
:: HENMODEL:th 15
theorem
for b1 being FinSequence of CQC-WFF holds
|- b1 ^ <*VERUM*>;
:: HENMODEL:th 16
theorem
for b1 being Consistent Element of bool CQC-WFF
for b2 being Henkin_interpretation of b1 holds
b2,valH |= VERUM
iff
b1 |- VERUM;
:: HENMODEL:th 17
theorem
for b1 being Element of NAT
for b2 being Element of b1 -ary_QC-pred_symbols
for b3 being CQC-variable_list-like QC-variable_list of b1
for b4 being Consistent Element of bool CQC-WFF
for b5 being Henkin_interpretation of b4 holds
b5,valH |= b2 ! b3
iff
b4 |- b2 ! b3;