:: HEYTING2 semantic presentation
theorem Th1: :: HEYTING2:1
theorem Th2: :: HEYTING2:2
theorem Th3: :: HEYTING2:3
Lemma36:
for A, B, C being set st A = B \/ C & A c= B holds
A = B
theorem Th4: :: HEYTING2:4
theorem Th5: :: HEYTING2:5
theorem Th6: :: HEYTING2:6
theorem Th7: :: HEYTING2:7
theorem Th8: :: HEYTING2:8
theorem Th9: :: HEYTING2:9
:: deftheorem Def1 defines Involved HEYTING2:def 1 :
theorem Th10: :: HEYTING2:10
Lemma66:
for V being set
for C being finite set
for A being non empty Element of Fin (PFuncs V,C) holds Involved A is finite
theorem Th11: :: HEYTING2:11
theorem Th12: :: HEYTING2:12
theorem Th13: :: HEYTING2:13
:: deftheorem Def2 defines - HEYTING2:def 2 :
theorem Th14: :: HEYTING2:14
theorem Th15: :: HEYTING2:15
theorem Th16: :: HEYTING2:16
theorem Th17: :: HEYTING2:17
theorem Th18: :: HEYTING2:18
theorem Th19: :: HEYTING2:19
definition
let V be
set ;
let C be
finite set ;
let A be
Element of
Fin (PFuncs V,C),
B be
Element of
Fin (PFuncs V,C);
func c3 =>> c4 -> Element of
Fin (PFuncs a1,a2) equals :: HEYTING2:def 3
(PFuncs V,C) /\ { (union { ((f . i) \ i) where i is Element of PFuncs V,C : i in A } ) where f is Element of PFuncs A,B : dom f = A } ;
coherence
(PFuncs V,C) /\ { (union { ((f . i) \ i) where i is Element of PFuncs V,C : i in A } ) where f is Element of PFuncs A,B : dom f = A } is Element of Fin (PFuncs V,C)
end;
:: deftheorem Def3 defines =>> HEYTING2:def 3 :
theorem Th20: :: HEYTING2:20
Lemma100:
for a, V being set
for C being finite set
for A being Element of Fin (PFuncs V,C)
for K being Element of SubstitutionSet V,C st a in A ^ (A =>> K) holds
ex b being set st
( b in K & b c= a )
theorem Th21: :: HEYTING2:21
Lemma103:
for V being set
for C being finite set
for K being Element of SubstitutionSet V,C
for X being set st X c= K holds
X in SubstitutionSet V,C
theorem Th22: :: HEYTING2:22
definition
let V be
set ;
let C be
finite set ;
func pseudo_compl c1,
c2 -> UnOp of the
carrier of
(SubstLatt a1,a2) means :
Def4:
:: HEYTING2:def 4
for
u being
Element of
(SubstLatt V,C) for
u' being
Element of
SubstitutionSet V,
C st
u' = u holds
it . u = mi (- u');
existence
ex b1 being UnOp of the carrier of (SubstLatt V,C) st
for u being Element of (SubstLatt V,C)
for u' being Element of SubstitutionSet V,C st u' = u holds
b1 . u = mi (- u')
correctness
uniqueness
for b1, b2 being UnOp of the carrier of (SubstLatt V,C) st ( for u being Element of (SubstLatt V,C)
for u' being Element of SubstitutionSet V,C st u' = u holds
b1 . u = mi (- u') ) & ( for u being Element of (SubstLatt V,C)
for u' being Element of SubstitutionSet V,C st u' = u holds
b2 . u = mi (- u') ) holds
b1 = b2;
func StrongImpl c1,
c2 -> BinOp of the
carrier of
(SubstLatt a1,a2) means :
Def5:
:: HEYTING2:def 5
for
u,
v being
Element of
(SubstLatt V,C) for
u',
v' being
Element of
SubstitutionSet V,
C st
u' = u &
v' = v holds
it . u,
v = mi (u' =>> v');
existence
ex b1 being BinOp of the carrier of (SubstLatt V,C) st
for u, v being Element of (SubstLatt V,C)
for u', v' being Element of SubstitutionSet V,C st u' = u & v' = v holds
b1 . u,v = mi (u' =>> v')
correctness
uniqueness
for b1, b2 being BinOp of the carrier of (SubstLatt V,C) st ( for u, v being Element of (SubstLatt V,C)
for u', v' being Element of SubstitutionSet V,C st u' = u & v' = v holds
b1 . u,v = mi (u' =>> v') ) & ( for u, v being Element of (SubstLatt V,C)
for u', v' being Element of SubstitutionSet V,C st u' = u & v' = v holds
b2 . u,v = mi (u' =>> v') ) holds
b1 = b2;
let u be
Element of
(SubstLatt V,C);
func SUB c3 -> Element of
Fin the
carrier of
(SubstLatt a1,a2) equals :: HEYTING2:def 6
bool u;
coherence
bool u is Element of Fin the carrier of (SubstLatt V,C)
correctness
;
func diff c3 -> UnOp of the
carrier of
(SubstLatt a1,a2) means :
Def7:
:: HEYTING2:def 7
for
v being
Element of
(SubstLatt V,C) holds
it . v = u \ v;
existence
ex b1 being UnOp of the carrier of (SubstLatt V,C) st
for v being Element of (SubstLatt V,C) holds b1 . v = u \ v
correctness
uniqueness
for b1, b2 being UnOp of the carrier of (SubstLatt V,C) st ( for v being Element of (SubstLatt V,C) holds b1 . v = u \ v ) & ( for v being Element of (SubstLatt V,C) holds b2 . v = u \ v ) holds
b1 = b2;
end;
:: deftheorem Def4 defines pseudo_compl HEYTING2:def 4 :
:: deftheorem Def5 defines StrongImpl HEYTING2:def 5 :
:: deftheorem Def6 defines SUB HEYTING2:def 6 :
:: deftheorem Def7 defines diff HEYTING2:def 7 :
Lemma119:
for V being set
for C being finite set
for v, u being Element of (SubstLatt V,C) st v in SUB u holds
v "\/" ((diff u) . v) = u
Lemma121:
for V being set
for C being finite set
for u being Element of (SubstLatt V,C)
for a being Element of PFuncs V,C
for K being Element of Fin (PFuncs V,C) st a is finite & K = {a} holds
ex v being Element of SubstitutionSet V,C st
( v in SUB u & v ^ K = {} & ( for b being Element of PFuncs V,C st b in (diff u) . v holds
b tolerates a ) )
definition
let V be
set ;
let C be
finite set ;
func Atom c1,
c2 -> Function of
PFuncs a1,
a2,the
carrier of
(SubstLatt a1,a2) means :
Def8:
:: HEYTING2:def 8
for
a being
Element of
PFuncs V,
C holds
it . a = mi {.a.};
existence
ex b1 being Function of PFuncs V,C,the carrier of (SubstLatt V,C) st
for a being Element of PFuncs V,C holds b1 . a = mi {.a.}
correctness
uniqueness
for b1, b2 being Function of PFuncs V,C,the carrier of (SubstLatt V,C) st ( for a being Element of PFuncs V,C holds b1 . a = mi {.a.} ) & ( for a being Element of PFuncs V,C holds b2 . a = mi {.a.} ) holds
b1 = b2;
end;
:: deftheorem Def8 defines Atom HEYTING2:def 8 :
Lemma128:
for V being set
for C being finite set
for a being Element of PFuncs V,C st a is finite holds
(Atom V,C) . a = {a}
theorem Th23: :: HEYTING2:23
theorem Th24: :: HEYTING2:24
Lemma141:
for V being set
for C being finite set
for u, v being Element of (SubstLatt V,C) st ( for a being set st a in u holds
ex b being set st
( b in v & b c= a ) ) holds
u [= v
theorem Th25: :: HEYTING2:25
theorem Th26: :: HEYTING2:26
theorem Th27: :: HEYTING2:27
theorem Th28: :: HEYTING2:28
Lemma147:
for V being set
for C being finite set
for u, v being Element of (SubstLatt V,C) st u [= v holds
for x being set st x in u holds
ex b being set st
( b in v & b c= x )
theorem Th29: :: HEYTING2:29
theorem Th30: :: HEYTING2:30
deffunc H1( set , set ) -> Relation of [:the carrier of (SubstLatt a1,a2),the carrier of (SubstLatt a1,a2):],the carrier of (SubstLatt a1,a2) = the L_meet of (SubstLatt a1,a2);
theorem Th31: :: HEYTING2:31
theorem Th32: :: HEYTING2:32
E155:
now
let V be
set ;
let C be
finite set ;
let u be
Element of
(SubstLatt V,C);
let v be
Element of
(SubstLatt V,C);
deffunc H2(
Element of
(SubstLatt V,C),
Element of
(SubstLatt V,C))
-> Element of the
carrier of
(SubstLatt V,C) =
FinJoin (SUB a1),
(H1(V,C) .: (pseudo_compl V,C),((StrongImpl V,C) [:] (diff a1),a2));
set Psi =
H1(
V,
C)
.: (pseudo_compl V,C),
((StrongImpl V,C) [:] (diff u),v);
E21:
now
let w be
Element of
(SubstLatt V,C);
set u2 =
(diff u) . w;
set pc =
(pseudo_compl V,C) . w;
set si =
(StrongImpl V,C) . ((diff u) . w),
v;
assume
w in SUB u
;
then E24:
w "\/" ((diff u) . w) = u
by E25;
E25:
w "/\" (((pseudo_compl V,C) . w) "/\" ((StrongImpl V,C) . ((diff u) . w),v)) =
(w "/\" ((pseudo_compl V,C) . w)) "/\" ((StrongImpl V,C) . ((diff u) . w),v)
by LATTICES:def 7
.=
(Bottom (SubstLatt V,C)) "/\" ((StrongImpl V,C) . ((diff u) . w),v)
by
.=
Bottom (SubstLatt V,C)
by LATTICES:40
;
E44:
((diff u) . w) "/\" ((StrongImpl V,C) . ((diff u) . w),v) [= v
by Th6;
(H1(V,C) [;] u,(H1(V,C) .: (pseudo_compl V,C),((StrongImpl V,C) [:] (diff u),v))) . w =
H1(
V,
C)
. u,
((H1(V,C) .: (pseudo_compl V,C),((StrongImpl V,C) [:] (diff u),v)) . w)
by FUNCOP_1:66
.=
u "/\" ((H1(V,C) .: (pseudo_compl V,C),((StrongImpl V,C) [:] (diff u),v)) . w)
by LATTICES:def 2
.=
u "/\" (H1(V,C) . ((pseudo_compl V,C) . w),(((StrongImpl V,C) [:] (diff u),v) . w))
by FUNCOP_1:48
.=
u "/\" (((pseudo_compl V,C) . w) "/\" (((StrongImpl V,C) [:] (diff u),v) . w))
by LATTICES:def 2
.=
u "/\" (((pseudo_compl V,C) . w) "/\" ((StrongImpl V,C) . ((diff u) . w),v))
by FUNCOP_1:60
.=
(w "/\" (((pseudo_compl V,C) . w) "/\" ((StrongImpl V,C) . ((diff u) . w),v))) "\/" (((diff u) . w) "/\" (((pseudo_compl V,C) . w) "/\" ((StrongImpl V,C) . ((diff u) . w),v)))
by Th9, LATTICES:def 11
.=
((diff u) . w) "/\" (((StrongImpl V,C) . ((diff u) . w),v) "/\" ((pseudo_compl V,C) . w))
by E44, LATTICES:39
.=
(((diff u) . w) "/\" ((StrongImpl V,C) . ((diff u) . w),v)) "/\" ((pseudo_compl V,C) . w)
by LATTICES:def 7
;
then
(H1(V,C) [;] u,(H1(V,C) .: (pseudo_compl V,C),((StrongImpl V,C) [:] (diff u),v))) . w [= ((diff u) . w) "/\" ((StrongImpl V,C) . ((diff u) . w),v)
by LATTICES:23;
hence
(H1(V,C) [;] u,(H1(V,C) .: (pseudo_compl V,C),((StrongImpl V,C) [:] (diff u),v))) . w [= v
by , LATTICES:25;
end;
u "/\" H2(
u,
v)
= FinJoin (SUB u),
(H1(V,C) [;] u,(H1(V,C) .: (pseudo_compl V,C),((StrongImpl V,C) [:] (diff u),v)))
by LATTICE2:83;
hence
u "/\" H2(
u,
v)
[= v
by Th8, LATTICE2:70;
let w be
Element of
(SubstLatt V,C);
reconsider v' =
v as
Element of
SubstitutionSet V,
C by SUBSTLAT:def 4;
E46:
v = FinJoin v',
(Atom V,C)
by ;
then E49:
u "/\" v = FinJoin v',
(H1(V,C) [;] u,(Atom V,C))
by LATTICE2:83;
assume E50:
u "/\" v [= w
;
now
let a be
Element of
PFuncs V,
C;
assume E51:
a in v'
;
then
(H1(V,C) [;] u,(Atom V,C)) . a [= w
by , , LATTICE2:46;
then
H1(
V,
C)
. u,
((Atom V,C) . a) [= w
by FUNCOP_1:66;
then E52:
u "/\" ((Atom V,C) . a) [= w
by LATTICES:def 2;
reconsider SA =
{.a.} as
Element of
Fin (PFuncs V,C) ;
E53:
a is
finite
by E46, ;
then reconsider SS =
{a} as
Element of
SubstitutionSet V,
C by ;
consider v being
Element of
SubstitutionSet V,
C such that E54:
v in SUB u
and E56:
v ^ SA = {}
and E57:
for
b being
Element of
PFuncs V,
C st
b in (diff u) . v holds
b tolerates a
by , ;
E58:
v ^ SS = {}
by E50;
reconsider v' =
v as
Element of
(SubstLatt V,C) by SUBSTLAT:def 4;
set dv =
(diff u) . v';
(diff u) . v' [= u
by Th3;
then
((diff u) . v') "/\" ((Atom V,C) . a) [= u "/\" ((Atom V,C) . a)
by LATTICES:27;
then E60:
((diff u) . v') "/\" ((Atom V,C) . a) [= w
by , LATTICES:25;
set pf =
pseudo_compl V,
C;
set sf =
(StrongImpl V,C) [:] (diff u),
w;
E88:
a is
finite
by E46, ;
E90:
(Atom V,C) . a [= (pseudo_compl V,C) . v'
by E52, ;
(Atom V,C) . a [= (StrongImpl V,C) . ((diff u) . v'),
w
by E51, E53, E54, Th5;
then E91:
(Atom V,C) . a [= ((StrongImpl V,C) [:] (diff u),w) . v'
by FUNCOP_1:60;
((pseudo_compl V,C) . v') "/\" (((StrongImpl V,C) [:] (diff u),w) . v') =
H1(
V,
C)
. ((pseudo_compl V,C) . v'),
(((StrongImpl V,C) [:] (diff u),w) . v')
by LATTICES:def 2
.=
(H1(V,C) .: (pseudo_compl V,C),((StrongImpl V,C) [:] (diff u),w)) . v'
by FUNCOP_1:48
;
then
(Atom V,C) . a [= (H1(V,C) .: (pseudo_compl V,C),((StrongImpl V,C) [:] (diff u),w)) . v'
by , E56, FILTER_0:7;
hence
(Atom V,C) . a [= H2(
u,
w)
by E49, LATTICE2:44;
end;
hence
v [= H2(
u,
w)
by Th9, LATTICE2:70;
end;
Lemma166:
for V being set
for C being finite set holds SubstLatt V,C is implicative
theorem Th33: :: HEYTING2:33