:: The Lattice of Natural Numbers and The Sublattice of it. The Set of Prime Numbers
:: by Marek Chmur
::
:: Received April 26, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users


definition
func hcflat -> BinOp of NAT means :Def1: :: NAT_LAT:def 1
 for n, m being Nat holds it . (m,n) = m gcd n;
existence
ex b1 being BinOp of NAT st
for n, m being Nat holds b1 . (m,n) = m gcd n
proof end;
uniqueness
for b1, b2 being BinOp of NAT st ( for n, m being Nat holds b1 . (m,n) = m gcd n ) & ( for n, m being Nat holds b2 . (m,n) = m gcd n ) holds
b1 = b2
proof end;
func lcmlat -> BinOp of NAT means :Def2: :: NAT_LAT:def 2
 for n, m being Nat holds it . (m,n) = m lcm n;
existence
ex b1 being BinOp of NAT st
for n, m being Nat holds b1 . (m,n) = m lcm n
proof end;
uniqueness
for b1, b2 being BinOp of NAT st ( for n, m being Nat holds b1 . (m,n) = m lcm n ) & ( for n, m being Nat holds b2 . (m,n) = m lcm n ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines hcflat NAT_LAT:def 1 :
for b1 being BinOp of NAT holds
( b1 = hcflat iff for n, m being Nat holds b1 . (m,n) = m gcd n );

:: deftheorem Def2 defines lcmlat NAT_LAT:def 2 :
for b1 being BinOp of NAT holds
( b1 = lcmlat iff for n, m being Nat holds b1 . (m,n) = m lcm n );

definition
func Nat_Lattice -> non empty strict LattStr equals :: NAT_LAT:def 3
 LattStr(# NAT,lcmlat,hcflat #);
coherence
LattStr(# NAT,lcmlat,hcflat #) is non empty strict LattStr ;
end;

:: deftheorem defines Nat_Lattice NAT_LAT:def 3 :
Nat_Lattice = LattStr(# NAT,lcmlat,hcflat #);

registration
cluster the carrier of Nat_Lattice -> natural-membered ;
coherence
the carrier of Nat_Lattice is natural-membered ;
end;

registration
let p, q be Element of Nat_Lattice;
identify p "\/" q with p lcm q;
compatibility
p "\/" q = p lcm q by Def2;
identify p "/\" q with p gcd q;
compatibility
p "/\" q = p gcd q by Def1;
end;

theorem :: NAT_LAT:1
for p, q being Element of Nat_Lattice holds p "\/" q = p lcm q ;

theorem :: NAT_LAT:2
for p, q being Element of Nat_Lattice holds p "/\" q = p gcd q ;

theorem :: NAT_LAT:3
for a, b being Element of Nat_Lattice st a [= b holds
a divides b by NEWTON:44;

definition
func 0_NN -> Element of Nat_Lattice equals :: NAT_LAT:def 4
1;
coherence
1 is Element of Nat_Lattice ;
func 1_NN -> Element of Nat_Lattice equals :: NAT_LAT:def 5
 0 ;
coherence
0 is Element of Nat_Lattice ;
end;

:: deftheorem defines 0_NN NAT_LAT:def 4 :
0_NN = 1;

:: deftheorem defines 1_NN NAT_LAT:def 5 :
1_NN = 0 ;

theorem :: NAT_LAT:4
0_NN = 1 ;

Lm1: for a being Element of Nat_Lattice holds
( 0_NN "/\" a = 0_NN & a "/\" 0_NN = 0_NN )
by NEWTON:51;

registration
cluster Nat_Lattice -> non empty strict Lattice-like ;
coherence
Nat_Lattice is Lattice-like
proof end;
end;

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

registration
cluster Nat_Lattice -> non empty strict lower-bounded ;
coherence
Nat_Lattice is lower-bounded by Lm1;
end;

theorem Th5: :: NAT_LAT:5
for p, q being Element of Nat_Lattice holds lcmlat . (p,q) = lcmlat . (q,p)
proof end;

theorem Th6: :: NAT_LAT:6
for p, q being Element of Nat_Lattice holds hcflat . (q,p) = hcflat . (p,q)
proof end;

theorem Th7: :: NAT_LAT:7
for p, q, r being Element of Nat_Lattice holds lcmlat . (p,(lcmlat . (q,r))) = lcmlat . ((lcmlat . (p,q)),r)
proof end;

theorem :: NAT_LAT:8
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 Th9: :: NAT_LAT:9
for p, q, r being Element of Nat_Lattice holds hcflat . (p,(hcflat . (q,r))) = hcflat . ((hcflat . (p,q)),r)
proof end;

theorem :: NAT_LAT:10
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 :: NAT_LAT:11
for p, q 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 :: NAT_LAT:12
for p, q 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;

:: NATPLUS
definition
func NATPLUS -> Subset of NAT means :Def6: :: NAT_LAT:def 6
 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 Def6 defines NATPLUS NAT_LAT:def 6 :
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;
:: original: Element
redefine mode Element of N -> Element of S;
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 for Element of D;
coherence
for b1 being Element of D holds b1 is real ;
end;

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

definition
mode NatPlus is Element of NATPLUS ;
end;

:: LATTICE of NATURAL NUMBERS > 0
definition
let k be Nat;
assume A1: k > 0 ;
func @ k -> NatPlus equals :Def7: :: NAT_LAT:def 7
k;
coherence
k is NatPlus by A1, Def6;
end;

:: deftheorem Def7 defines @ NAT_LAT:def 7 :
for k being Nat st k > 0 holds
@ k = k;

registration
cluster -> natural non zero for Element of NATPLUS ;
coherence
for b1 being NatPlus holds
( b1 is natural & not b1 is zero ) by Def6;
end;

definition
func hcflatplus -> BinOp of NATPLUS means :Def8: :: NAT_LAT:def 8
 for m, n being NatPlus holds it . (m,n) = m gcd n;
existence
ex b1 being BinOp of NATPLUS st
for m, n being NatPlus holds b1 . (m,n) = m gcd n
proof end;
uniqueness
for b1, b2 being BinOp of NATPLUS st ( for m, n being NatPlus holds b1 . (m,n) = m gcd n ) & ( for m, n being NatPlus holds b2 . (m,n) = m gcd n ) holds
b1 = b2
proof end;
func lcmlatplus -> BinOp of NATPLUS means :Def9: :: NAT_LAT:def 9
 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 Def8 defines hcflatplus NAT_LAT:def 8 :
for b1 being BinOp of NATPLUS holds
( b1 = hcflatplus iff for m, n being NatPlus holds b1 . (m,n) = m gcd n );

:: deftheorem Def9 defines lcmlatplus NAT_LAT:def 9 :
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 10
 LattStr(# NATPLUS,lcmlatplus,hcflatplus #);
coherence
LattStr(# NATPLUS,lcmlatplus,hcflatplus #) is strict LattStr ;
end;

:: deftheorem defines NatPlus_Lattice NAT_LAT:def 10 :
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 @ m -> NatPlus equals :: NAT_LAT:def 11
m;
coherence
m is NatPlus ;
end;

:: deftheorem defines @ NAT_LAT:def 11 :
for m being Element of NatPlus_Lattice holds @ m = m;

registration
cluster -> natural non zero for Element of the carrier of NatPlus_Lattice;
coherence
for b1 being Element of NatPlus_Lattice holds
( b1 is natural & not b1 is zero ) ;
end;

registration
let p, q be Element of NatPlus_Lattice;
identify p "\/" q with p lcm q;
compatibility
p "\/" q = p lcm q by Def9;
identify p "/\" q with p gcd q;
compatibility
p "/\" q = p gcd q by Def8;
end;

theorem :: NAT_LAT:13
for p, q being Element of NatPlus_Lattice holds p "\/" q = (@ p) lcm (@ q) ;

theorem :: NAT_LAT:14
for p, q being Element of NatPlus_Lattice holds p "/\" q = (@ p) gcd (@ q) ;

Lm2: ( ( for a, b being Element of NatPlus_Lattice holds a "\/" b = b "\/" a ) & ( for a, b, c being Element of NatPlus_Lattice holds a "\/" (b "\/" c) = (a "\/" b) "\/" c ) )
by NEWTON:43;

Lm3: ( ( for a, b being Element of NatPlus_Lattice holds (a "/\" b) "\/" b = b ) & ( for a, b being Element of NatPlus_Lattice holds a "/\" b = b "/\" a ) )
by NEWTON:53;

Lm4: ( ( for a, b, c being Element of NatPlus_Lattice holds a "/\" (b "/\" c) = (a "/\" b) "/\" c ) & ( for a, b being Element of NatPlus_Lattice holds a "/\" (a "\/" b) = a ) )
by NEWTON:48, NEWTON:54;

registration
cluster NatPlus_Lattice -> 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 Lm2, Lm3, Lm4;
end;

Lm5: now :: thesis: for L being Lattice holds
( the L_join of L = the L_join of L || the carrier of L & the L_meet of L = the L_meet of L || the carrier of L )
let L be Lattice; :: thesis: ( the L_join of L = the L_join of L || the carrier of L & the L_meet of L = the L_meet of L || the carrier of L )
[: 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_1:68; :: thesis: the L_meet of L = the L_meet of L || the carrier of L
[: 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_1:68; :: thesis: verum
end;

definition
let L be Lattice;
mode SubLattice of L -> Lattice means :Def12: :: NAT_LAT:def 12
( 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 Def12 defines SubLattice NAT_LAT:def 12 :
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 non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like for SubLattice of L;
existence
ex b1 being SubLattice of L st b1 is strict
proof end;
end;

theorem :: NAT_LAT:15
for L being Lattice holds L is SubLattice of L
proof end;

theorem :: NAT_LAT:16
NatPlus_Lattice is SubLattice of Nat_Lattice
proof end;