HENMODEL: Equivalences of Inconsistency and {H}enkin Models

:: Equivalences of Inconsistency and {H}enkin Models
:: by Patrick Braselmann and Peter Koepke
::
:: Received September 25, 2004
:: Copyright (c) 2004-2011 Association of Mizar Users

begin

theorem Th1: :: HENMODEL:1

for n being Element of NAT
for A being non empty finite Subset of NAT
for f being Function of n,A st ex m being Element of NAT st succ m = n & rng f = A & ( for n, m being Element of NAT st m in dom f & n in dom f & n < m holds
f . n in f . m ) holds
f . (union n) = union (rng f)

proof end;

theorem Th2: :: HENMODEL:2

for A being non empty finite Subset of NAT holds
( union A in A & ( for a being set holds
( not a in A or a in union A or a = union A ) ) )

proof end;

theorem Th3: :: HENMODEL:3

for C being non empty set
for f being Function of NAT,C
for X being finite set st ( for n, m being Element of NAT st m in dom f & n in dom f & n < m holds
f . n c= f . m ) & X c= union (rng f) holds
ex k being Element of NAT st X c= f . k

proof end;

definition

let X be Subset of CQC-WFF;
let p be Element of CQC-WFF ;
canceled;
pred X |- p means :Def2: :: HENMODEL:def 2
ex f being FinSequence of CQC-WFF st
( rng f c= X & |- f ^ <*p*> );

end;

:: deftheorem HENMODEL:def 1 :

:: deftheorem Def2 defines |- HENMODEL:def 2 :

definition

let X be Subset of CQC-WFF;
attr X is Consistent means :Def3: :: HENMODEL:def 3
for p being Element of CQC-WFF holds
( not X |- p or not X |- 'not' p );

end;

:: deftheorem Def3 defines Consistent HENMODEL:def 3 :

notation

let X be Subset of CQC-WFF;
antonym Inconsistent X for Consistent ;

end;

definition

let f be FinSequence of CQC-WFF ;
attr f is Consistent means :Def4: :: HENMODEL:def 4
for p being Element of CQC-WFF holds
( not |- f ^ <*p*> or not |- f ^ <*('not' p)*> );

end;

:: deftheorem Def4 defines Consistent HENMODEL:def 4 :

notation

let f be FinSequence of CQC-WFF ;
antonym Inconsistent f for Consistent ;

end;

theorem Th4: :: HENMODEL:4

for X being Subset of CQC-WFF
for g being FinSequence of CQC-WFF st X is Consistent & rng g c= X holds
g is Consistent

proof end;

theorem Th5: :: HENMODEL:5

for p being Element of CQC-WFF
for f, g being FinSequence of CQC-WFF st |- f ^ <*p*> holds
|- (f ^ g) ^ <*p*>

proof end;

theorem Th6: :: HENMODEL:6

for X being Subset of CQC-WFF holds
( not X is Consistent iff for p being Element of CQC-WFF holds X |- p )

proof end;

theorem Th7: :: HENMODEL:7

for X being Subset of CQC-WFF st not X is Consistent holds
ex Y being Subset of CQC-WFF st
( Y c= X & Y is finite & not Y is Consistent )

proof end;

Lm1: for f, g being FinSequence holds Seg (len (f ^ g)) = (Seg (len f)) \/ (seq ((len f),(len g)))

proof end;

theorem :: HENMODEL:8

for X being Subset of CQC-WFF
for p, q being Element of CQC-WFF st X \/ {p} |- q holds
ex g being FinSequence of CQC-WFF st
( rng g c= X & |- (g ^ <*p*>) ^ <*q*> )

proof end;

theorem :: HENMODEL:9

for X being Subset of CQC-WFF
for p being Element of CQC-WFF holds
( X |- p iff not X \/ {('not' p)} is Consistent )

proof end;

theorem :: HENMODEL:10

for X being Subset of CQC-WFF
for p being Element of CQC-WFF holds
( X |- 'not' p iff not X \/ {p} is Consistent )

proof end;

begin

theorem :: HENMODEL:11

for f being Function of NAT,(bool CQC-WFF) st ( for n, m being Element of NAT st m in dom f & n in dom f & n < m holds
( f . n is Consistent & f . n c= f . m ) ) holds
union (rng f) is Consistent

proof end;

begin

theorem Th12: :: HENMODEL:12

for X being Subset of CQC-WFF
for A being non empty set st not X is Consistent holds
for J being interpretation of A
for v being Element of Valuations_in A holds not J,v |= X

proof end;

theorem Th13: :: HENMODEL:13

{VERUM} is Consistent

proof end;

registration

cluster Consistent Element of bool CQC-WFF;
existence by Th13;

end;

definition

func HCar -> non empty set equals :: HENMODEL:def 5
bound_QC-variables ;
coherence ;

end;

:: deftheorem defines HCar HENMODEL:def 5 :

definition

let P be Element of QC-pred_symbols ;
let ll be CQC-variable_list of the_arity_of P;
:: original: !
redefine func P ! ll -> Element of CQC-WFF ;
coherence

proof end;

end;

definition

let CX be Consistent Subset of CQC-WFF;
mode Henkin_interpretation of CX -> interpretation of HCar means :Def6: :: HENMODEL:def 6
for P being Element of QC-pred_symbols
for r being Element of relations_on HCar st it . P = r holds
for a being set holds
( a in r iff ex ll being CQC-variable_list of the_arity_of P st
( a = ll & CX |- P ! ll ) );
existence

proof end;

end;

:: deftheorem Def6 defines Henkin_interpretation HENMODEL:def 6 :

definition

func valH -> Element of Valuations_in HCar equals :: HENMODEL:def 7
id bound_QC-variables;
coherence by VALUAT_1:def 1;

end;

:: deftheorem defines valH HENMODEL:def 7 :

begin

theorem Th14: :: HENMODEL:14

for k being Element of NAT
for ll being CQC-variable_list of k holds valH *' ll = ll

proof end;

theorem Th15: :: HENMODEL:15

for f being FinSequence of CQC-WFF holds |- f ^ <*VERUM*>

proof end;

theorem :: HENMODEL:16

for CX being Consistent Subset of CQC-WFF
for JH being Henkin_interpretation of CX holds
( JH, valH |= VERUM iff CX |- VERUM )

proof end;

theorem :: HENMODEL:17

for k being Element of NAT
for P being QC-pred_symbol of k
for ll being CQC-variable_list of k
for CX being Consistent Subset of CQC-WFF
for JH being Henkin_interpretation of CX holds
( JH, valH |= P ! ll iff CX |- P ! ll )

proof end;