:: 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
proof end;
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
proof end;
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
proof end;
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
proof end;
end;

:: deftheorem Def1 NAT_LAT:def 1 :
canceled;

:: deftheorem Def2 NAT_LAT:def 2 :
canceled;

:: deftheorem Def3 defines hcflat NAT_LAT:def 3 :
for b1 being BinOp of NAT holds
( b1 = hcflat iff for m, n being Nat holds b1 . m,n = m hcf n );

:: deftheorem Def4 defines lcmlat NAT_LAT:def 4 :
for b1 being BinOp of NAT holds
( b1 = lcmlat iff for m, n being Nat holds b1 . m,n = m lcm n );

definition
let m be Element of LattStr(# NAT ,lcmlat ,hcflat #);
func @ c1 -> Element of NAT equals :: NAT_LAT:def 5
m;
coherence
m is Element of NAT
;
end;

:: deftheorem Def5 defines @ NAT_LAT:def 5 :
for m being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds @ m = m;

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
for p, q being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds p "\/" q = (@ p) lcm (@ q) by ;

theorem Th49: :: NAT_LAT:49
for p, q being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds p "/\" q = (@ p) hcf (@ q) by ;

Lemma41: for a, b being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds a "\/" b = b "\/" a
proof end;

Lemma42: for a, b being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds a "/\" b = b "/\" a
proof end;

Lemma43: for a, b, c being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds a "/\" (b "/\" c) = (a "/\" b) "/\" c
proof end;

Lemma47: for a, b, c being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds a "\/" (b "\/" c) = (a "\/" b) "\/" c
proof end;

Lemma48: for a, b being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds (a "/\" b) "\/" b = b
proof end;

Lemma49: for a, b being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds a "/\" (a "\/" b) = a
proof end;

theorem Th50: :: NAT_LAT:50
canceled;

theorem Th51: :: NAT_LAT:51
canceled;

theorem Th52: :: NAT_LAT:52
for a, b being Element of LattStr(# NAT ,lcmlat ,hcflat #) st a [= b holds
@ a divides @ b
proof end;

definition
func 0_NN -> Element of LattStr(# NAT ,lcmlat ,hcflat #) equals :: NAT_LAT:def 6
1;
coherence
1 is Element of LattStr(# NAT ,lcmlat ,hcflat #)
;
func 1_NN -> Element of LattStr(# NAT ,lcmlat ,hcflat #) equals :: NAT_LAT:def 7
0;
coherence
0 is Element of LattStr(# NAT ,lcmlat ,hcflat #)
;
end;

:: deftheorem Def6 defines 0_NN NAT_LAT:def 6 :
0_NN = 1;

:: deftheorem Def7 defines 1_NN NAT_LAT:def 7 :
1_NN = 0;

theorem Th53: :: NAT_LAT:53
canceled;

theorem Th54: :: NAT_LAT:54
canceled;

theorem Th55: :: NAT_LAT:55
@ 0_NN = 1 ;

theorem Th56: :: NAT_LAT:56
for a being Element of LattStr(# NAT ,lcmlat ,hcflat #) holds
( 0_NN "/\" a = 0_NN & a "/\" 0_NN = 0_NN )
proof end;

definition
func Nat_Lattice -> Lattice equals :: NAT_LAT:def 8
LattStr(# NAT ,lcmlat ,hcflat #);
coherence
LattStr(# NAT ,lcmlat ,hcflat #) is Lattice
proof end;
end;

:: deftheorem Def8 defines Nat_Lattice NAT_LAT:def 8 :
Nat_Lattice = LattStr(# NAT ,lcmlat ,hcflat #);

registration
cluster Nat_Lattice -> strict ;
coherence
Nat_Lattice is strict
;
end;

theorem Th57: :: NAT_LAT:57
canceled;

theorem Th58: :: NAT_LAT:58
canceled;

theorem Th59: :: NAT_LAT:59
canceled;

theorem Th60: :: NAT_LAT:60
canceled;

registration
cluster Nat_Lattice -> strict lower-bounded ;
coherence
Nat_Lattice is lower-bounded
by , LATTICES:def 13;
end;

theorem Th61: :: NAT_LAT:61
for p, q being Element of Nat_Lattice holds lcmlat . p,q = lcmlat . q,p
proof end;

theorem Th62: :: NAT_LAT:62
for q, p being Element of Nat_Lattice holds hcflat . q,p = hcflat . p,q
proof end;

theorem Th63: :: NAT_LAT:63
for p, q, r being Element of Nat_Lattice holds lcmlat . p,(lcmlat . q,r) = lcmlat . (lcmlat . p,q),r
proof end;

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 )
proof end;

theorem Th65: :: NAT_LAT:65
for p, q, r being Element of Nat_Lattice holds hcflat . p,(hcflat . q,r) = hcflat . (hcflat . p,q),r
proof end;

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 )
proof end;

theorem Th67: :: NAT_LAT:67
for q, p being Element of Nat_Lattice holds
( hcflat . q,(lcmlat . q,p) = q & hcflat . (lcmlat . p,q),q = q & hcflat . q,(lcmlat . p,q) = q & hcflat . (lcmlat . q,p),q = q )
proof end;

theorem Th68: :: NAT_LAT:68
for q, p being Element of Nat_Lattice holds
( lcmlat . q,(hcflat . q,p) = q & lcmlat . (hcflat . p,q),q = q & lcmlat . q,(hcflat . p,q) = q & lcmlat . (hcflat . q,p),q = q )
proof end;

definition
func NATPLUS -> Subset of NAT means :Def9: :: NAT_LAT:def 9
for n being Nat holds
( n in it iff 0 < n );
existence
ex b1 being Subset of NAT st
for n being Nat holds
( n in b1 iff 0 < n )
proof end;
uniqueness
for b1, b2 being Subset of NAT st ( for n being Nat holds
( n in b1 iff 0 < n ) ) & ( for n being Nat holds
( n in b2 iff 0 < n ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def9 defines NATPLUS NAT_LAT:def 9 :
for b1 being Subset of NAT holds
( b1 = NATPLUS iff for n being Nat holds
( n in b1 iff 0 < n ) );

registration
cluster NATPLUS -> non empty ;
coherence
not NATPLUS is empty
proof end;
end;

definition
let D be non empty set ;
let S be non empty Subset of D;
let N be non empty Subset of S;
redefine mode Element as Element of c3 -> Element of a2;
coherence
for b1 being Element of N holds b1 is Element of S
proof end;
end;

registration
let D be Subset of REAL ;
cluster -> real Element of a1;
coherence
for b1 being Element of D holds b1 is real
;
end;

registration
let D be Subset of NAT ;
cluster -> real Element of a1;
coherence
for b1 being Element of D holds b1 is real
;
end;

definition
mode NatPlus is Element of NATPLUS ;
end;

definition
let k be Nat;
assume E25: k > 0 ;
func @ c1 -> Element of NATPLUS equals :Def10: :: NAT_LAT:def 10
k;
coherence
k is Element of NATPLUS
by , ;
end;

:: deftheorem Def10 defines @ NAT_LAT:def 10 :
for k being Nat st k > 0 holds
@ k = k;

definition
let k be Element of NATPLUS ;
func @ c1 -> NatPlus equals :: NAT_LAT:def 11
k;
coherence
k is NatPlus
;
end;

:: deftheorem Def11 defines @ NAT_LAT:def 11 :
for k being Element of NATPLUS holds @ k = k;

registration
cluster -> real natural Element of NATPLUS ;
coherence
for b1 being Element of NATPLUS holds b1 is natural
proof end;
end;

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
proof end;
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
proof end;
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
proof end;
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
proof end;
end;

:: deftheorem Def12 defines hcflatplus NAT_LAT:def 12 :
for b1 being BinOp of NATPLUS holds
( b1 = hcflatplus iff for m, n being NatPlus holds b1 . m,n = m hcf n );

:: deftheorem Def13 defines lcmlatplus NAT_LAT:def 13 :
for b1 being BinOp of NATPLUS holds
( b1 = lcmlatplus iff for m, n being NatPlus holds b1 . m,n = m lcm n );

definition
func NatPlus_Lattice -> strict LattStr equals :: NAT_LAT:def 14
LattStr(# NATPLUS ,lcmlatplus ,hcflatplus #);
coherence
LattStr(# NATPLUS ,lcmlatplus ,hcflatplus #) is strict LattStr
;
end;

:: deftheorem Def14 defines NatPlus_Lattice NAT_LAT:def 14 :
NatPlus_Lattice = LattStr(# NATPLUS ,lcmlatplus ,hcflatplus #);

registration
cluster NatPlus_Lattice -> non empty strict ;
coherence
not NatPlus_Lattice is empty
;
end;

definition
let m be Element of NatPlus_Lattice ;
func @ c1 -> NatPlus equals :: NAT_LAT:def 15
m;
coherence
m is NatPlus
;
end;

:: deftheorem Def15 defines @ NAT_LAT:def 15 :
for m being Element of NatPlus_Lattice holds @ m = m;

theorem Th69: :: NAT_LAT:69
for p, q being Element of NatPlus_Lattice holds p "\/" q = (@ p) lcm (@ q) by ;

theorem Th70: :: NAT_LAT:70
for p, q being Element of NatPlus_Lattice holds p "/\" q = (@ p) hcf (@ q) by ;

Lemma76: for a, b being Element of NatPlus_Lattice holds a "\/" b = b "\/" a
proof end;

Lemma77: for a, b, c being Element of NatPlus_Lattice holds a "\/" (b "\/" c) = (a "\/" b) "\/" c
proof end;

Lemma78: for a, b being Element of NatPlus_Lattice holds (a "/\" b) "\/" b = b
proof end;

Lemma79: for a, b being Element of NatPlus_Lattice holds a "/\" b = b "/\" a
proof end;

Lemma80: for a, b, c being Element of NatPlus_Lattice holds a "/\" (b "/\" c) = (a "/\" b) "/\" c
proof end;

Lemma81: for a, b being Element of NatPlus_Lattice holds a "/\" (a "\/" b) = a
proof end;

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;

E82: now
let L be Lattice;
thus the L_join of L = the L_join of L || the carrier of L
proof
[:the carrier of L,the carrier of L:] = dom the L_join of L by FUNCT_2:def 1;
hence the L_join of L = the L_join of L || the carrier of L by RELAT_1:97;
end;
thus the L_meet of L = the L_meet of L || the carrier of L
proof
[:the carrier of L,the carrier of L:] = dom the L_meet of L by FUNCT_2:def 1;
hence the L_meet of L = the L_meet of L || the carrier of L by RELAT_1:97;
end;
end;

definition
let L be Lattice;
mode SubLattice of c1 -> Lattice means :Def16: :: NAT_LAT:def 16
( the carrier of it c= the carrier of L & the L_join of it = the L_join of L || the carrier of it & the L_meet of it = the L_meet of L || the carrier of it );
existence
ex b1 being Lattice st
( the carrier of b1 c= the carrier of L & the L_join of b1 = the L_join of L || the carrier of b1 & the L_meet of b1 = the L_meet of L || the carrier of b1 )
proof end;
end;

:: deftheorem Def16 defines SubLattice NAT_LAT:def 16 :
for L, b2 being Lattice holds
( b2 is SubLattice of L iff ( the carrier of b2 c= the carrier of L & the L_join of b2 = the L_join of L || the carrier of b2 & the L_meet of b2 = the L_meet of L || the carrier of b2 ) );

registration
let L be Lattice;
cluster strict SubLattice of a1;
existence
ex b1 being SubLattice of L st b1 is strict
proof end;
end;

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
for L being Lattice holds L is SubLattice of L
proof end;

theorem Th76: :: NAT_LAT:76
NatPlus_Lattice is SubLattice of Nat_Lattice
proof end;