:: SUBSTLAT semantic presentation
:: deftheorem Def1 defines SubstitutionSet SUBSTLAT:def 1 :
Lemma20:
for V, C, a, b being set st b in SubstitutionSet V,C & a in b holds
a is finite
theorem Th1: :: SUBSTLAT:1
theorem Th2: :: SUBSTLAT:2
:: deftheorem Def2 defines mi SUBSTLAT:def 2 :
definition
let V be
set ;
let C be
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 :: SUBSTLAT:def 3
{ (s \/ t) where s is Element of PFuncs V,C, t is Element of PFuncs V,C : ( s in A & t in B & s tolerates t ) } ;
coherence
{ (s \/ t) where s is Element of PFuncs V,C, t is Element of PFuncs V,C : ( s in A & t in B & s tolerates t ) } is Element of Fin (PFuncs V,C)
end;
:: deftheorem Def3 defines ^ SUBSTLAT:def 3 :
theorem Th3: :: SUBSTLAT:3
theorem Th4: :: SUBSTLAT:4
theorem Th5: :: SUBSTLAT:5
theorem Th6: :: SUBSTLAT:6
Lemma55:
for V, C being set
for A, B being Element of Fin (PFuncs V,C) st ( for a being set st a in A holds
a in B ) holds
A c= B
theorem Th7: :: SUBSTLAT:7
theorem Th8: :: SUBSTLAT:8
theorem Th9: :: SUBSTLAT:9
theorem Th10: :: SUBSTLAT:10
theorem Th11: :: SUBSTLAT:11
theorem Th12: :: SUBSTLAT:12
theorem Th13: :: SUBSTLAT:13
theorem Th14: :: SUBSTLAT:14
theorem Th15: :: SUBSTLAT:15
theorem Th16: :: SUBSTLAT:16
Lemma68:
for V, C being set
for A, B being Element of Fin (PFuncs V,C)
for a being finite set st a in A ^ B holds
ex b being finite set st
( b c= a & b in (mi A) ^ B )
theorem Th17: :: SUBSTLAT:17
theorem Th18: :: SUBSTLAT:18
theorem Th19: :: SUBSTLAT:19
theorem Th20: :: SUBSTLAT:20
theorem Th21: :: SUBSTLAT:21
theorem Th22: :: SUBSTLAT:22
Lemma94:
for V, C being set
for A, B being Element of Fin (PFuncs V,C)
for a being set st a in A ^ B holds
ex c being set st
( c in B & c c= a )
Lemma95:
for V, C being set
for K, L being Element of Fin (PFuncs V,C) holds mi ((K ^ L) \/ L) = mi L
theorem Th23: :: SUBSTLAT:23
theorem Th24: :: SUBSTLAT:24
theorem Th25: :: SUBSTLAT:25
definition
let V be
set ;
let C be
set ;
func SubstLatt c1,
c2 -> strict LattStr means :
Def4:
:: SUBSTLAT:def 4
( the
carrier of
it = SubstitutionSet V,
C & ( for
A,
B being
Element of
SubstitutionSet V,
C holds
( the
L_join of
it . A,
B = mi (A \/ B) & the
L_meet of
it . A,
B = mi (A ^ B) ) ) );
existence
ex b1 being strict LattStr st
( the carrier of b1 = SubstitutionSet V,C & ( for A, B being Element of SubstitutionSet V,C holds
( the L_join of b1 . A,B = mi (A \/ B) & the L_meet of b1 . A,B = mi (A ^ B) ) ) )
uniqueness
for b1, b2 being strict LattStr st the carrier of b1 = SubstitutionSet V,C & ( for A, B being Element of SubstitutionSet V,C holds
( the L_join of b1 . A,B = mi (A \/ B) & the L_meet of b1 . A,B = mi (A ^ B) ) ) & the carrier of b2 = SubstitutionSet V,C & ( for A, B being Element of SubstitutionSet V,C holds
( the L_join of b2 . A,B = mi (A \/ B) & the L_meet of b2 . A,B = mi (A ^ B) ) ) holds
b1 = b2
end;
:: deftheorem Def4 defines SubstLatt SUBSTLAT:def 4 :
Lemma105:
for V, C being set
for a, b being Element of (SubstLatt V,C) holds a "\/" b = b "\/" a
Lemma107:
for V, C being set
for a, b, c being Element of (SubstLatt V,C) holds a "\/" (b "\/" c) = (a "\/" b) "\/" c
Lemma108:
for V, C being set
for K, L being Element of SubstitutionSet V,C holds the L_join of (SubstLatt V,C) . (the L_meet of (SubstLatt V,C) . K,L),L = L
Lemma109:
for V, C being set
for a, b being Element of (SubstLatt V,C) holds (a "/\" b) "\/" b = b
Lemma110:
for V, C being set
for a, b being Element of (SubstLatt V,C) holds a "/\" b = b "/\" a
Lemma111:
for V, C being set
for a, b, c being Element of (SubstLatt V,C) holds a "/\" (b "/\" c) = (a "/\" b) "/\" c
Lemma112:
for V, C being set
for K, L, M being Element of SubstitutionSet V,C holds the L_meet of (SubstLatt V,C) . K,(the L_join of (SubstLatt V,C) . L,M) = the L_join of (SubstLatt V,C) . (the L_meet of (SubstLatt V,C) . K,L),(the L_meet of (SubstLatt V,C) . K,M)
Lemma116:
for V, C being set
for a, b being Element of (SubstLatt V,C) holds a "/\" (a "\/" b) = a
theorem Th26: :: SUBSTLAT:26
theorem Th27: :: SUBSTLAT:27