:: NAT_LAT semantic presentation
definition
canceled;canceled;func hcflat -> BinOp of
NAT means :
Def3:
:: NAT_LAT:def 3
for
m,
n being
Nat holds
it . m,
n = m hcf n;
existence
ex b1 being BinOp of NAT st
for m, n being Nat holds b1 . m,n = m hcf n
uniqueness
for b1, b2 being BinOp of NAT st ( for m, n being Nat holds b1 . m,n = m hcf n ) & ( for m, n being Nat holds b2 . m,n = m hcf n ) holds
b1 = b2
func lcmlat -> BinOp of
NAT means :
Def4:
:: NAT_LAT:def 4
for
m,
n being
Nat holds
it . m,
n = m lcm n;
existence
ex b1 being BinOp of NAT st
for m, n being Nat holds b1 . m,n = m lcm n
uniqueness
for b1, b2 being BinOp of NAT st ( for m, n being Nat holds b1 . m,n = m lcm n ) & ( for m, n being Nat holds b2 . m,n = m lcm n ) holds
b1 = b2
end;
:: deftheorem Def1 NAT_LAT:def 1 :
canceled;
:: deftheorem Def2 NAT_LAT:def 2 :
canceled;
:: deftheorem Def3 defines hcflat NAT_LAT:def 3 :
:: deftheorem Def4 defines lcmlat NAT_LAT:def 4 :
:: deftheorem Def5 defines @ NAT_LAT:def 5 :
theorem Th1: :: NAT_LAT:1
canceled;
theorem Th2: :: NAT_LAT:2
canceled;
theorem Th3: :: NAT_LAT:3
canceled;
theorem Th4: :: NAT_LAT:4
canceled;
theorem Th5: :: NAT_LAT:5
canceled;
theorem Th6: :: NAT_LAT:6
canceled;
theorem Th7: :: NAT_LAT:7
canceled;
theorem Th8: :: NAT_LAT:8
canceled;
theorem Th9: :: NAT_LAT:9
canceled;
theorem Th10: :: NAT_LAT:10
canceled;
theorem Th11: :: NAT_LAT:11
canceled;
theorem Th12: :: NAT_LAT:12
canceled;
theorem Th13: :: NAT_LAT:13
canceled;
theorem Th14: :: NAT_LAT:14
canceled;
theorem Th15: :: NAT_LAT:15
canceled;
theorem Th16: :: NAT_LAT:16
canceled;
theorem Th17: :: NAT_LAT:17
canceled;
theorem Th18: :: NAT_LAT:18
canceled;
theorem Th19: :: NAT_LAT:19
canceled;
theorem Th20: :: NAT_LAT:20
canceled;
theorem Th21: :: NAT_LAT:21
canceled;
theorem Th22: :: NAT_LAT:22
canceled;
theorem Th23: :: NAT_LAT:23
canceled;
theorem Th24: :: NAT_LAT:24
canceled;
theorem Th25: :: NAT_LAT:25
canceled;
theorem Th26: :: NAT_LAT:26
canceled;
theorem Th27: :: NAT_LAT:27
canceled;
theorem Th28: :: NAT_LAT:28
canceled;
theorem Th29: :: NAT_LAT:29
canceled;
theorem Th30: :: NAT_LAT:30
canceled;
theorem Th31: :: NAT_LAT:31
canceled;
theorem Th32: :: NAT_LAT:32
canceled;
theorem Th33: :: NAT_LAT:33
canceled;
theorem Th34: :: NAT_LAT:34
canceled;
theorem Th35: :: NAT_LAT:35
canceled;
theorem Th36: :: NAT_LAT:36
canceled;
theorem Th37: :: NAT_LAT:37
canceled;
theorem Th38: :: NAT_LAT:38
canceled;
theorem Th39: :: NAT_LAT:39
canceled;
theorem Th40: :: NAT_LAT:40
canceled;
theorem Th41: :: NAT_LAT:41
canceled;
theorem Th42: :: NAT_LAT:42
canceled;
theorem Th43: :: NAT_LAT:43
canceled;
theorem Th44: :: NAT_LAT:44
canceled;
theorem Th45: :: NAT_LAT:45
canceled;
theorem Th46: :: NAT_LAT:46
canceled;
theorem Th47: :: NAT_LAT:47
canceled;
theorem Th48: :: NAT_LAT:48
theorem Th49: :: NAT_LAT:49
Lemma41:
for a, b being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds a "\/" b = b "\/" a
Lemma42:
for a, b being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds a "/\" b = b "/\" a
Lemma43:
for a, b, c being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds a "/\" (b "/\" c) = (a "/\" b) "/\" c
Lemma47:
for a, b, c being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds a "\/" (b "\/" c) = (a "\/" b) "\/" c
Lemma48:
for a, b being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds (a "/\" b) "\/" b = b
Lemma49:
for a, b being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds a "/\" (a "\/" b) = a
theorem Th50: :: NAT_LAT:50
canceled;
theorem Th51: :: NAT_LAT:51
canceled;
theorem Th52: :: NAT_LAT:52
:: deftheorem Def6 defines 0_NN NAT_LAT:def 6 :
:: deftheorem Def7 defines 1_NN NAT_LAT:def 7 :
theorem Th53: :: NAT_LAT:53
canceled;
theorem Th54: :: NAT_LAT:54
canceled;
theorem Th55: :: NAT_LAT:55
theorem Th56: :: NAT_LAT:56
:: deftheorem Def8 defines Nat_Lattice NAT_LAT:def 8 :
theorem Th57: :: NAT_LAT:57
canceled;
theorem Th58: :: NAT_LAT:58
canceled;
theorem Th59: :: NAT_LAT:59
canceled;
theorem Th60: :: NAT_LAT:60
canceled;
theorem Th61: :: NAT_LAT:61
theorem Th62: :: NAT_LAT:62
theorem Th63: :: NAT_LAT:63
theorem Th64: :: NAT_LAT:64
for
p,
q,
r being
Element of
Nat_Lattice holds
(
lcmlat . p,
(lcmlat . q,r) = lcmlat . (lcmlat . q,p),
r &
lcmlat . p,
(lcmlat . q,r) = lcmlat . (lcmlat . p,r),
q &
lcmlat . p,
(lcmlat . q,r) = lcmlat . (lcmlat . r,q),
p &
lcmlat . p,
(lcmlat . q,r) = lcmlat . (lcmlat . r,p),
q )
theorem Th65: :: NAT_LAT:65
theorem Th66: :: NAT_LAT:66
for
p,
q,
r being
Element of
Nat_Lattice holds
(
hcflat . p,
(hcflat . q,r) = hcflat . (hcflat . q,p),
r &
hcflat . p,
(hcflat . q,r) = hcflat . (hcflat . p,r),
q &
hcflat . p,
(hcflat . q,r) = hcflat . (hcflat . r,q),
p &
hcflat . p,
(hcflat . q,r) = hcflat . (hcflat . r,p),
q )
theorem Th67: :: NAT_LAT:67
theorem Th68: :: NAT_LAT:68
:: deftheorem Def9 defines NATPLUS NAT_LAT:def 9 :
:: deftheorem Def10 defines @ NAT_LAT:def 10 :
for
k being
Nat st
k > 0 holds
@ k = k;
:: deftheorem Def11 defines @ NAT_LAT:def 11 :
definition
func hcflatplus -> BinOp of
NATPLUS means :
Def12:
:: NAT_LAT:def 12
for
m,
n being
NatPlus holds
it . m,
n = m hcf n;
existence
ex b1 being BinOp of NATPLUS st
for m, n being NatPlus holds b1 . m,n = m hcf n
uniqueness
for b1, b2 being BinOp of NATPLUS st ( for m, n being NatPlus holds b1 . m,n = m hcf n ) & ( for m, n being NatPlus holds b2 . m,n = m hcf n ) holds
b1 = b2
func lcmlatplus -> BinOp of
NATPLUS means :
Def13:
:: NAT_LAT:def 13
for
m,
n being
NatPlus holds
it . m,
n = m lcm n;
existence
ex b1 being BinOp of NATPLUS st
for m, n being NatPlus holds b1 . m,n = m lcm n
uniqueness
for b1, b2 being BinOp of NATPLUS st ( for m, n being NatPlus holds b1 . m,n = m lcm n ) & ( for m, n being NatPlus holds b2 . m,n = m lcm n ) holds
b1 = b2
end;
:: deftheorem Def12 defines hcflatplus NAT_LAT:def 12 :
:: deftheorem Def13 defines lcmlatplus NAT_LAT:def 13 :
:: deftheorem Def14 defines NatPlus_Lattice NAT_LAT:def 14 :
:: deftheorem Def15 defines @ NAT_LAT:def 15 :
theorem Th69: :: NAT_LAT:69
theorem Th70: :: NAT_LAT:70
Lemma76:
for a, b being Element of NatPlus_Lattice holds a "\/" b = b "\/" a
Lemma77:
for a, b, c being Element of NatPlus_Lattice holds a "\/" (b "\/" c) = (a "\/" b) "\/" c
Lemma78:
for a, b being Element of NatPlus_Lattice holds (a "/\" b) "\/" b = b
Lemma79:
for a, b being Element of NatPlus_Lattice holds a "/\" b = b "/\" a
Lemma80:
for a, b, c being Element of NatPlus_Lattice holds a "/\" (b "/\" c) = (a "/\" b) "/\" c
Lemma81:
for a, b being Element of NatPlus_Lattice holds a "/\" (a "\/" b) = a
registration
cluster NatPlus_Lattice -> non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing ;
coherence
( NatPlus_Lattice is join-commutative & NatPlus_Lattice is join-associative & NatPlus_Lattice is meet-commutative & NatPlus_Lattice is meet-associative & NatPlus_Lattice is join-absorbing & NatPlus_Lattice is meet-absorbing )
by , , , , , , LATTICES:def 4, LATTICES:def 5, LATTICES:def 6, LATTICES:def 7, LATTICES:def 8, LATTICES:def 9;
end;
:: deftheorem Def16 defines SubLattice NAT_LAT:def 16 :
theorem Th71: :: NAT_LAT:71
canceled;
theorem Th72: :: NAT_LAT:72
canceled;
theorem Th73: :: NAT_LAT:73
canceled;
theorem Th74: :: NAT_LAT:74
canceled;
theorem Th75: :: NAT_LAT:75
theorem Th76: :: NAT_LAT:76