:: Order Sorted Quotient Algebra
:: by Josef Urban
::
:: Received September 19, 2002
:: Copyright (c) 2002-2017 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, XBOOLE_0, ORDERS_2, MSUALG_4, OSALG_1, RELAT_1,
STRUCT_0, FUNCOP_1, PBOOLE, SUBSET_1, XXREAL_0, FUNCT_1, MSUALG_1,
TARSKI, EQREL_1, NAT_1, MARGREL1, PARTFUN1, FINSEQ_1, ARYTM_3, ARYTM_1,
ZFMISC_1, FINSEQ_5, CARD_1, RELAT_2, ORDINAL4, NATTRA_1, YELLOW15,
SETFAM_1, COH_SP, YELLOW18, WAYBEL_0, CARD_3, MSUALG_3, WELLORD1, SEQM_3,
OSALG_4;
notations ZFMISC_1, TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, RELAT_2, FUNCT_1,
RELSET_1, PARTFUN1, EQREL_1, SETFAM_1, XCMPLX_0, ORDERS_2, ORDERS_3,
ORDINAL1, NUMBERS, FUNCT_2, FINSEQ_1, CARD_3, FINSEQ_2, FINSEQ_4,
FINSEQ_5, FUNCOP_1, PBOOLE, STRUCT_0, WAYBEL_0, MSUALG_1, MSUALG_3,
OSALG_1, OSALG_3, MSUALG_4, XXREAL_0;
constructors REAL_1, NAT_1, NAT_D, EQREL_1, FINSEQ_4, FINSEQ_5, MSUALG_3,
MSUALG_4, ORDERS_3, WAYBEL_0, OSALG_3, RELSET_1;
registrations SUBSET_1, RELAT_1, PARTFUN1, FUNCOP_1, XREAL_0, INT_1, FINSEQ_1,
EQREL_1, FUNCT_1, PBOOLE, STRUCT_0, MSUALG_1, MSUALG_3, MSUALG_4,
ORDERS_3, OSALG_1, ORDINAL1, CARD_3, TOLER_1, RELSET_1, FINSEQ_2;
requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM;
definitions TARSKI, MSUALG_3, XBOOLE_0, OSALG_1, OSALG_3;
equalities XBOOLE_0;
expansions TARSKI, MSUALG_3, XBOOLE_0, OSALG_1, OSALG_3;
theorems XBOOLE_0, XBOOLE_1, FUNCT_1, FUNCT_2, PBOOLE, CARD_3, MSUALG_1,
MSUALG_5, OSALG_1, OSALG_3, RELAT_2, ZFMISC_1, RELAT_1, RELSET_1,
EQREL_1, MSUALG_3, MSUALG_6, MSAFREE, TARSKI, SYSREL, ORDERS_2, FINSEQ_1,
FINSEQ_3, ENUMSET1, FINSEQ_5, INT_1, ORDERS_3, PARTFUN1, WAYBEL_0,
GRFUNC_1, MSUALG_9, MSUALG_4, FINSEQ_2, ORDERS_1, FUNCOP_1, XREAL_1,
XXREAL_0, XTUPLE_0;
schemes FUNCT_1, CLASSES1, NAT_1, FUNCT_2;
begin
registration
let R be non empty Poset;
cluster Relation-yielding for OrderSortedSet of R;
existence
proof
set R1 = the Relation;
set I = the carrier of R;
set f = I --> R1;
reconsider f as ManySortedSet of I;
f is order-sorted
proof
let s1,s2 be Element of R such that
s1 <= s2;
f.s1 = R1 by FUNCOP_1:7;
hence thesis by FUNCOP_1:7;
end;
then reconsider f as OrderSortedSet of R;
take f;
for x be set st x in dom f holds f.x is Relation by FUNCOP_1:7;
hence thesis by FUNCOP_1:def 11;
end;
end;
:: this is a stronger condition for relation than just being order-sorted
definition
let R be non empty Poset;
let A,B be ManySortedSet of the carrier of R;
let IT be ManySortedRelation of A,B;
attr IT is os-compatible means
for s1,s2 being Element of R st s1 <=
s2 for x,y being set st x in A.s1 & y in B.s1 holds [x,y] in IT.s1 iff [x,y] in
IT.s2;
end;
registration
let R be non empty Poset;
let A,B be ManySortedSet of the carrier of R;
cluster os-compatible for ManySortedRelation of A,B;
existence
proof
set I = the carrier of R;
consider R1 be Relation such that
A1: R1 = {};
set f = I --> R1;
reconsider f as ManySortedSet of R;
set f1 = f;
for i be set st i in I holds f.i is Relation of A.i,B.i
proof
let i be set;
assume i in I;
then f.i = {} by A1,FUNCOP_1:7;
hence thesis by RELSET_1:12;
end;
then reconsider f2 = f1 as ManySortedRelation of A,B by MSUALG_4:def 1;
take f;
for s1,s2 being Element of R st s1 <= s2 for x,y being set st x in A.
s1 & y in B.s1 holds [x,y] in f1.s1 iff [x,y] in f1.s2
proof
let s1,s2 be Element of R such that
s1 <= s2;
let x,y be set such that
x in A.s1 and
y in B.s1;
f1.s1 = R1 by FUNCOP_1:7
.= f1.s2 by FUNCOP_1:7;
hence thesis;
end;
then f2 is os-compatible;
hence thesis;
end;
end;
definition
let R be non empty Poset;
let A,B be ManySortedSet of the carrier of R;
mode OrderSortedRelation of A,B is os-compatible ManySortedRelation of A,B;
end;
theorem Th1:
for R being non empty Poset, A,B being ManySortedSet of the
carrier of R, OR being ManySortedRelation of A,B holds OR is os-compatible
implies OR is OrderSortedSet of R
proof
let R be non empty Poset, A,B be ManySortedSet of the carrier of R, OR be
ManySortedRelation of A,B;
set OR1 = OR;
assume
A1: OR is os-compatible;
OR1 is order-sorted
proof
let s1,s2 be Element of R such that
A2: s1 <= s2;
for x,y being object holds [x,y] in OR.s1 implies [x,y] in OR.s2
proof
let x,y be object such that
A3: [x,y] in OR.s1;
x in A.s1 & y in B.s1 by A3,ZFMISC_1:87;
hence thesis by A1,A2,A3;
end;
hence thesis by RELAT_1:def 3;
end;
hence thesis;
end;
registration
let R be non empty Poset;
let A,B be ManySortedSet of R;
cluster os-compatible -> order-sorted for ManySortedRelation of A,B;
coherence by Th1;
end;
definition
let R be non empty Poset;
let A be ManySortedSet of the carrier of R;
mode OrderSortedRelation of A is OrderSortedRelation of A,A;
end;
definition
let S be OrderSortedSign, U1 be OSAlgebra of S;
mode OrderSortedRelation of U1 -> ManySortedRelation of U1 means
:Def2:
it is os-compatible;
existence
proof
reconsider Y = the Sorts of U1 as OrderSortedSet of S by OSALG_1:17;
set X = the OrderSortedRelation of Y;
reconsider X1=X as ManySortedRelation of U1;
take X1;
thus thesis;
end;
end;
:: REVISE: the definition of ManySorted diagonal from MSUALG_6
:: should be moved to MSUALG_4; the "compatible" attr from MSUALG_6
:: should replace the MSCongruence-like
registration
let S be OrderSortedSign, U1 be OSAlgebra of S;
cluster MSEquivalence-like for OrderSortedRelation of U1;
existence
proof
deffunc F(Element of S) = id ((the Sorts of U1).$1);
consider f be Function such that
A1: dom f = the carrier of S & for d be Element of S holds f.d = F(d)
from FUNCT_1:sch 4;
reconsider f as ManySortedSet of the carrier of S
by A1,PARTFUN1:def 2,RELAT_1:def 18;
for i be set st i in the carrier of S holds f.i is Relation of (the
Sorts of U1).i,(the Sorts of U1).i
proof
let i be set;
assume i in the carrier of S;
then f.i = id ((the Sorts of U1).i) by A1;
hence thesis;
end;
then reconsider
f as ManySortedRelation of the Sorts of U1,the Sorts of U1 by
MSUALG_4:def 1;
reconsider f as ManySortedRelation of U1;
set f1 = f;
A2: f1 is os-compatible
proof
reconsider X = the Sorts of U1 as OrderSortedSet of S by OSALG_1:17;
let s1,s2 be Element of S such that
A3: s1 <= s2;
reconsider s3 = s1, s4 = s2 as Element of S;
let x,y be set such that
A4: x in (the Sorts of U1).s1 and
y in (the Sorts of U1).s1;
A5: f1.s1 = id(X.s1) by A1;
A6: f1.s2 = id(X.s2) by A1;
X.s3 c= X.s4 by A3,OSALG_1:def 16;
then id(X.s1) c= id(X.s2) by SYSREL:15;
hence [x,y] in f1.s1 implies [x,y] in f1.s2 by A5,A6;
assume [x,y] in f1.s2;
then x = y by A6,RELAT_1:def 10;
hence thesis by A5,A4,RELAT_1:def 10;
end;
take f;
for i be set, R be Relation of (the Sorts of U1).i st i in the carrier
of S & f.i = R holds R is Equivalence_Relation of (the Sorts of U1).i
proof
let i be set, R be Relation of (the Sorts of U1).i;
assume i in the carrier of S & f.i = R;
then R = id ((the Sorts of U1).i) by A1;
hence thesis;
end;
then f is MSEquivalence_Relation-like by MSUALG_4:def 2;
hence thesis by A2,Def2,MSUALG_4:def 3;
end;
end;
:: REVISE: we need the fact that id has the type,
:: the original prf can be simplified
registration
let S be OrderSortedSign, U1 be non-empty OSAlgebra of S;
cluster MSCongruence-like for MSEquivalence-like OrderSortedRelation of U1;
existence
proof
deffunc F(Element of S) = id ((the Sorts of U1).$1);
consider f be Function such that
A1: dom f = the carrier of S & for d be Element of S holds f.d = F(d)
from FUNCT_1:sch 4;
reconsider f as ManySortedSet of the carrier of S by A1,PARTFUN1:def 2
,RELAT_1:def 18;
for i be set st i in the carrier of S holds f.i is Relation of (the
Sorts of U1).i,(the Sorts of U1).i
proof
let i be set;
assume i in the carrier of S;
then f.i = id ((the Sorts of U1).i) by A1;
hence thesis;
end;
then reconsider
f as ManySortedRelation of the Sorts of U1,the Sorts of U1 by
MSUALG_4:def 1;
reconsider f as ManySortedRelation of U1;
for i be set, R be Relation of (the Sorts of U1).i st i in the carrier
of S & f.i = R holds R is Equivalence_Relation of (the Sorts of U1).i
proof
let i be set, R be Relation of (the Sorts of U1).i;
assume i in the carrier of S & f.i = R;
then R = id ((the Sorts of U1).i) by A1;
hence thesis;
end;
then f is MSEquivalence_Relation-like by MSUALG_4:def 2;
then reconsider f as MSEquivalence-like ManySortedRelation of U1 by
MSUALG_4:def 3;
set f1 = f;
f1 is os-compatible
proof
reconsider X = the Sorts of U1 as OrderSortedSet of S by OSALG_1:17;
let s1,s2 be Element of S such that
A2: s1 <= s2;
reconsider s3 = s1, s4 = s2 as Element of S;
let x,y be set such that
A3: x in (the Sorts of U1).s1 and
y in (the Sorts of U1).s1;
A4: f1.s1 = id(X.s1) by A1;
A5: f1.s2 = id(X.s2) by A1;
X.s3 c= X.s4 by A2,OSALG_1:def 16;
then id(X.s1) c= id(X.s2) by SYSREL:15;
hence [x,y] in f1.s1 implies [x,y] in f1.s2 by A4,A5;
assume [x,y] in f1.s2;
then x = y by A5,RELAT_1:def 10;
hence [x,y] in f1.s1 by A4,A3,RELAT_1:def 10;
end;
then reconsider f = f1 as MSEquivalence-like OrderSortedRelation of U1 by
Def2;
take f;
for o be Element of the carrier' of S for x,y be Element of Args(o,U1
) st (for n be Nat st n in dom x holds [x.n,y.n] in f.((the_arity_of o)/.n))
holds [Den(o,U1).x,Den(o,U1).y] in f.(the_result_sort_of o)
proof
let o be Element of the carrier' of S;
let x,y be Element of Args(o,U1);
A6: dom x = dom (the_arity_of o) by MSUALG_3:6;
assume
A7: for n be Nat st n in dom x holds [x.n,y.n] in f.((the_arity_of o)/.n);
A8: for a be object st a in dom (the_arity_of o) holds x.a = y.a
proof
set ao = the_arity_of o;
let a be object;
assume
A9: a in dom (the_arity_of o);
then reconsider n = a as Nat;
ao.n in rng ao by A9,FUNCT_1:def 3;
then
A10: f.(ao.n) = id ((the Sorts of U1).(ao.n)) by A1;
(the_arity_of o)/.n = (the_arity_of o).n by A9,PARTFUN1:def 6;
then [x.n,y.n] in f.(ao.n) by A7,A6,A9;
hence thesis by A10,RELAT_1:def 10;
end;
set r = the_result_sort_of o;
A11: f.r = id ((the Sorts of U1).r) by A1;
A12: dom (the ResultSort of S) = the carrier' of S by FUNCT_2:def 1;
then
A13: dom ((the Sorts of U1)*(the ResultSort of S)) = dom (the ResultSort
of S) by PARTFUN1:def 2;
A14: Result(o,U1) = ((the Sorts of U1) * the ResultSort of S).o by
MSUALG_1:def 5
.= (the Sorts of U1).((the ResultSort of S).o) by A12,A13,FUNCT_1:12
.= (the Sorts of U1).r by MSUALG_1:def 2;
dom y = dom (the_arity_of o) by MSUALG_3:6;
then Den(o,U1).x = Den(o,U1).y by A6,A8,FUNCT_1:2;
hence thesis by A11,A14,RELAT_1:def 10;
end;
hence thesis by MSUALG_4:def 4;
end;
end;
definition
let S be OrderSortedSign, U1 be non-empty OSAlgebra of S;
mode OSCongruence of U1 is MSCongruence-like MSEquivalence-like
OrderSortedRelation of U1;
end;
:: TODO: a smooth transition between Relations and Graphs would
:: be useful here, the FinSequence approach seemed to me faster than
:: transitive closure of R \/ R" .. maybe not, can be later a theorem
:: all could be done generally for reflexive (or with small
:: modification even non reflexive) Relations
:: I found ex post that attributes disconnected and connected defined
:: in ORDERS_3 have st. in common here, but the theory there is not developed
:: suggested improvements: f connects x,y; x is_connected_with y;
:: connected iff for x,y holds x is_connected_with y
:: finally I found this is the EqCl from MSUALG_5 - should be merged
definition
let R be non empty Poset;
func Path_Rel R -> Equivalence_Relation of the carrier of R means
:Def3:
for x,y being object
holds [x,y] in it iff x in the carrier of R & y in the carrier of
R & ex p being FinSequence of the carrier of R st 1 < len p & p.1 = x & p.(len
p) = y & for n being Nat st 2 <= n & n <= len p holds [p.n,p.(n-1)] in the
InternalRel of R or [p.(n-1),p.n] in the InternalRel of R;
existence
proof
defpred PC[object,object] means
ex p being FinSequence of the carrier of R st 1
< len p & p.1 = $1 & p.(len p) = $2 & for n being Nat st 2 <= n & n <= len p
holds [p.n,p.(n-1)] in the InternalRel of R or [p.(n-1),p.n] in the InternalRel
of R;
set P = { [x,y] where x,y is Element of R: x in the carrier of R & y in
the carrier of R & PC[x,y]};
P c= [:the carrier of R, the carrier of R:]
proof
let z be object;
assume z in P;
then
ex x,y being Element of R st z = [x,y] & x in the carrier of R & y in
the carrier of R & PC[x,y];
hence thesis by ZFMISC_1:87;
end;
then reconsider P1 = P as Relation of the carrier of R;
A1: for x,y being object holds [x,y] in P iff x in the carrier of R & y in
the carrier of R & PC[x,y]
proof
let x,y be object;
hereby
assume [x,y] in P;
then consider x1,y1 being Element of R such that
A2: [x,y] = [x1,y1] and
x1 in the carrier of R and
y1 in the carrier of R and
A3: PC[x1,y1];
x = x1 & y = y1 by A2,XTUPLE_0:1;
hence x in the carrier of R & y in the carrier of R & PC[x,y] by A3;
end;
thus thesis;
end;
now
let x,y be object;
assume that
A4: x in the carrier of R & y in the carrier of R and
A5: [x,y] in P1;
consider p being FinSequence of the carrier of R such that
A6: 1 < len p and
A7: p.1 = x & p.(len p) = y and
A8: for n being Nat st 2 <= n & n <= len p holds [p.n,p.(n-1)] in
the InternalRel of R or [p.(n-1),p.n] in the InternalRel of R by A1,A5;
PC[y,x]
proof
take F = Rev p;
thus 1 < len F by A6,FINSEQ_5:def 3;
A9: len F = len p by FINSEQ_5:def 3;
hence F.1 = y & F.(len F) = x by A7,FINSEQ_5:62;
let n1 be Nat such that
A10: 2 <= n1 and
A11: n1 <= len F;
1 <= n1 by A10,XXREAL_0:2;
then
A12: n1 in dom p by A9,A11,FINSEQ_3:25;
set n2 = (len p - n1) + 2;
0 <= len p - n1 by A9,A11,XREAL_1:48;
then
A13: 2 + 0 <= n2 by XREAL_1:6;
A14: 2 - 1 <= n1 - 1 by A10,XREAL_1:9;
then reconsider n11 = n1 - 1 as Element of NAT by INT_1:3,XXREAL_0:2;
n1 - 1 <= len F - 0 by A11,XREAL_1:13;
then n11 in dom p by A9,A14,FINSEQ_3:25;
then
A15: F.(n1 - 1) = p.(len p - (n1 - 1) + 1) by FINSEQ_5:58
.= p.((len p - n1) + 2);
reconsider n2 = (len p - n1) + 2 as Element of NAT by A13,INT_1:3
,XXREAL_0:2;
len p - n1 <= len p - 2 by A10,XREAL_1:13;
then
A16: (len p - n1) + 2 <= len p - 2 + 2 by XREAL_1:6;
p.(n2-1) = p.((len p - n1) + (2 - 1)) .= F.n1 by A12,FINSEQ_5:58;
hence thesis by A8,A15,A13,A16;
end;
hence [y,x] in P1 by A4;
end;
then
A17: P1 is_symmetric_in the carrier of R by RELAT_2:def 3;
A18: the InternalRel of R is_reflexive_in the carrier of R by ORDERS_2:def 2;
now
let x be object;
assume
A19: x in the carrier of R;
PC[x,x]
proof
set F = <*x,x*>;
rng F = {x,x} by FINSEQ_2:127
.= {x} by ENUMSET1:29;
then rng F c= the carrier of R by A19,ZFMISC_1:31;
then reconsider F1 = F as FinSequence of the carrier of R by
FINSEQ_1:def 4;
take F1;
A20: len F = 2 by FINSEQ_1:44;
hence 1 < len F1;
thus F1.1 = x & F1.(len F1) = x by A20,FINSEQ_1:44;
let n1 be Nat;
assume 2 <= n1 & n1 <= len F1;
then
A21: n1 = 2 by A20,XXREAL_0:1;
F.1 = x & F.2 = x by FINSEQ_1:44;
hence thesis by A18,A19,A21,RELAT_2:def 1;
end;
hence [x,x] in P1 by A19;
end;
then P1 is_reflexive_in the carrier of R by RELAT_2:def 1;
then
A22: dom P1 = the carrier of R & field P1 = the carrier of R by ORDERS_1:13;
now
let x,y,z be object;
assume that
A23: x in the carrier of R and
A24: y in the carrier of R and
A25: z in the carrier of R and
A26: [x,y] in P1 & [y,z] in P1;
( PC[x,y])& PC[y,z] by A1,A26;
then consider p1,p2 being FinSequence of the carrier of R such that
A27: 1 < len p1 and
A28: p1.1 = x and
A29: p1.(len p1) = y and
A30: for n being Nat st 2 <= n & n <= len p1 holds [p1.n,p1.(n-1)]
in the InternalRel of R or [p1.(n-1),p1.n] in the InternalRel of R and
A31: 1 < len p2 and
A32: p2.1 = y and
A33: p2.(len p2) = z and
A34: for n being Nat st 2 <= n & n <= len p2 holds [p2.n,p2.(n-1)]
in the InternalRel of R or [p2.(n-1),p2.n] in the InternalRel of R;
PC[x,z]
proof
take F = p1^p2;
A35: len F = len p1 + len p2 by FINSEQ_1:22;
1 + 1 < len p1 + len p2 by A27,A31,XREAL_1:8;
hence 1 < len F by A35,XXREAL_0:2;
1 in dom p1 by A27,FINSEQ_3:25;
hence F.1 = x by A28,FINSEQ_1:def 7;
len p2 in dom p2 by A31,FINSEQ_3:25;
hence F.(len F) = z by A33,A35,FINSEQ_1:def 7;
let n1 be Nat such that
A36: 2 <= n1 and
A37: n1 <= len F;
A38: 1 <= n1 by A36,XXREAL_0:2;
A39: 2 - 1 <= n1 - 1 by A36,XREAL_1:9;
then reconsider n11 = n1 - 1 as Element of NAT by INT_1:3,XXREAL_0:2;
A40: len p1 + 1 <= n1 implies len p1 + 1 = n1 or len p1 + 1 < n1 by
XXREAL_0:1;
A41: n1 - 1 <= len F - 0 by A37,XREAL_1:13;
per cases by A40,INT_1:7;
suppose
A42: n1 <= len p1;
then n1 - 1 <= len p1 - 0 by XREAL_1:13;
then n11 in dom p1 by A39,FINSEQ_3:25;
then
A43: F.(n1 -1) = p1.(n1 - 1) by FINSEQ_1:def 7;
n1 in dom p1 by A38,A42,FINSEQ_3:25;
then F.n1 = p1.n1 by FINSEQ_1:def 7;
hence thesis by A30,A36,A42,A43;
end;
suppose
A44: len p1 + 1 < n1;
len p1 < len p1 + 1 by XREAL_1:29;
then
A45: len p1 < n1 by A44,XXREAL_0:2;
then reconsider k = n1 - len p1 as Element of NAT by INT_1:3
,XREAL_1:48;
(len p1 + 1) - 1 < n1 - 1 by A44,XREAL_1:9;
then F.n11 = p2.(n11 - len p1) by A41,FINSEQ_1:24;
then
A46: F.(n1 - 1) = p2.(k - 1);
1 < n1 - len p1 by A44,XREAL_1:20;
then
A47: 1 + 1 <= n1 - len p1 by INT_1:7;
n1 <= len p1 + len p2 by A37,FINSEQ_1:22;
then
A48: k <= len p2 by XREAL_1:20;
F.n1 = p2.k by A37,A45,FINSEQ_1:24;
hence thesis by A34,A46,A47,A48;
end;
suppose
A49: len p1 + 1 = n1;
len p1 + 1 <= len p1 + len p2 & (len p1 + 1) - len p1 = 1 by A31,
XREAL_1:8;
then
A50: F.n1 = y by A32,A49,FINSEQ_1:23;
len p1 in dom p1 by A27,FINSEQ_3:25;
then F.(n1 - 1) = y by A29,A49,FINSEQ_1:def 7;
hence thesis by A18,A24,A50,RELAT_2:def 1;
end;
end;
hence [x,z] in P1 by A23,A25;
end;
then P1 is_transitive_in the carrier of R by RELAT_2:def 8;
then reconsider
P1 = P as Equivalence_Relation of the carrier of R by A22,A17,
PARTFUN1:def 2,RELAT_2:def 11,def 16;
take P1;
thus thesis by A1;
end;
uniqueness
proof
let X,Y be Equivalence_Relation of the carrier of R;
defpred PC1[object,object] means
$1 in the carrier of R & $2 in the carrier of R
& ex p being FinSequence of the carrier of R st 1 < len p & p.1 = $1 & p.(len p
) = $2 & for n being Nat st 2 <= n & n <= len p holds [p.n,p.(n-1)] in the
InternalRel of R or [p.(n-1),p.n] in the InternalRel of R;
assume
A51: for x,y being object holds [x,y] in X iff PC1[x,y];
assume
A52: for x,y being object holds [x,y] in Y iff PC1[x,y];
for x,y being object holds [x,y] in X iff [x,y] in Y
proof
let x,y be object;
hereby
assume [x,y] in X;
then PC1[x,y] by A51;
hence [x,y] in Y by A52;
end;
assume [x,y] in Y;
then PC1[x,y] by A52;
hence thesis by A51;
end;
hence thesis by RELAT_1:def 2;
end;
end;
theorem Th2:
for R being non empty Poset, s1,s2 being Element of R st s1 <= s2
holds [s1,s2] in Path_Rel R
proof
let R be non empty Poset, s1,s2 be Element of R such that
A1: s1 <= s2;
set p = <* s1, s2 *>;
A2: len p = 2 by FINSEQ_1:44;
A3: p.1 = s1 by FINSEQ_1:44;
A4: for n being Nat st 2 <= n & n <= len p holds [p.n,p.(n-1)] in the
InternalRel of R or [p.(n-1),p.n] in the InternalRel of R
proof
let n1 be Nat;
assume 2 <= n1 & n1 <= len p;
then
A5: n1 = 2 by A2,XXREAL_0:1;
[s1,s2] in the InternalRel of R by A1,ORDERS_2:def 5;
hence thesis by A3,A5,FINSEQ_1:44;
end;
p.(len p) = s2 by A2,FINSEQ_1:44;
hence thesis by A2,A3,A4,Def3;
end;
:: again, should be defined for Relations probably
definition
let R be non empty Poset;
let s1,s2 be Element of R;
pred s1 ~= s2 means
[s1,s2] in Path_Rel R;
reflexivity
proof
set PR = Path_Rel R;
field PR = the carrier of R by ORDERS_1:12;
then PR is_reflexive_in the carrier of R by RELAT_2:def 9;
hence thesis by RELAT_2:def 1;
end;
symmetry
proof
set PR = Path_Rel R;
field PR = the carrier of R by ORDERS_1:12;
then PR is_symmetric_in the carrier of R by RELAT_2:def 11;
hence thesis by RELAT_2:def 3;
end;
end;
theorem
for R being non empty Poset, s1,s2,s3 being Element of R holds s1 ~=
s2 & s2 ~= s3 implies s1 ~= s3
proof
let R be non empty Poset;
let s1,s2,s3 be Element of R;
set PR = Path_Rel R;
field PR = the carrier of R by ORDERS_1:12;
then
A1: PR is_transitive_in the carrier of R by RELAT_2:def 16;
assume s1 ~= s2 & s2 ~= s3;
then [s1,s2] in PR & [s2,s3] in PR;
then [s1,s3] in PR by A1,RELAT_2:def 8;
hence thesis;
end;
:: do for Relations
definition
let R be non empty Poset;
func Components R -> non empty (Subset-Family of R) equals
Class Path_Rel R;
coherence;
end;
registration
let R be non empty Poset;
cluster -> non empty for Element of Components R;
coherence
proof
let X be Element of Components R;
ex x being object st x in the carrier of R & X = Class( Path_Rel R,x)
by
EQREL_1:def 3;
hence thesis by EQREL_1:20;
end;
end;
definition
let R be non empty Poset;
mode Component of R is Element of Components R;
end;
definition
let R be non empty Poset;
let s1 be Element of R;
func CComp s1 -> Component of R equals
Class(Path_Rel R,s1);
correctness by EQREL_1:def 3;
end;
theorem Th4:
for R being non empty Poset, s1,s2 being Element of R st s1 <= s2
holds CComp(s1) = CComp(s2)
proof
let R be non empty Poset, s1,s2 be Element of R;
assume s1 <= s2;
then [s1,s2] in Path_Rel R by Th2;
hence thesis by EQREL_1:35;
end;
definition
let R be non empty Poset;
let A be ManySortedSet of the carrier of R;
let C be Component of R;
func A-carrier_of C -> set equals
union {A.s where s is Element of R: s in C};
correctness;
end;
theorem Th5:
for R being non empty Poset, A being ManySortedSet of the carrier
of R, s being (Element of R), x being set st x in A.s holds x in A-carrier_of
CComp(s)
proof
let R be non empty Poset;
let A be ManySortedSet of the carrier of R, s be (Element of R), x be set
such that
A1: x in A.s;
s in CComp(s) by EQREL_1:20;
then A.s in {A.s1 where s1 is Element of R: s1 in CComp(s)};
hence thesis by A1,TARSKI:def 4;
end;
definition
let R be non empty Poset;
attr R is locally_directed means
:Def8:
for C being Component of R holds C is directed;
end;
theorem Th6:
for R being discrete non empty Poset holds for x,y being Element
of R st [x,y] in Path_Rel R holds x = y
proof
let R be discrete non empty Poset;
let x,y be Element of R;
assume [x,y] in Path_Rel R;
then consider p being FinSequence of the carrier of R such that
A1: 1 < len p and
A2: p.1 = x and
A3: p.(len p) = y and
A4: for n being Nat st 2 <= n & n <= len p holds [p.n,p.(n-1)] in the
InternalRel of R or [p.(n-1),p.n] in the InternalRel of R by Def3;
for n1 being Nat st 1 <= n1 & n1 <= len p holds p.n1 = x
proof
defpred P[Nat] means p.$1 <> x & 1 <= $1;
let n1 be Nat such that
A5: 1 <= n1 and
A6: n1 <= len p;
assume
A7: p.n1 <> x;
then
A8: ex k being Nat st P[k] by A5;
consider k being Nat such that
A9: P[k] & for n being Nat st P[n] holds k <= n from NAT_1:sch 5(A8);
1 < k by A2,A9,XXREAL_0:1;
then
A10: 1 + 1 <= k by INT_1:7;
then
A11: 1 + 1 - 1 <= k - 1 by XREAL_1:9;
then reconsider k1 = k - 1 as Element of NAT by INT_1:3,XXREAL_0:2;
A12: p.k1 = x by A9,A11,XREAL_1:146;
k <= n1 by A5,A7,A9;
then
A13: k <= len p by A6,XXREAL_0:2;
then k in dom p by A9,FINSEQ_3:25;
then reconsider pk = p.k as Element of R by PARTFUN1:4;
per cases by A4,A10,A13,A12;
suppose
[pk,x] in the InternalRel of R;
then pk <= x by ORDERS_2:def 5;
hence contradiction by A9,ORDERS_3:1;
end;
suppose
[x,pk] in the InternalRel of R;
then x <= pk by ORDERS_2:def 5;
hence contradiction by A9,ORDERS_3:1;
end;
end;
hence thesis by A1,A3;
end;
theorem Th7:
for R being discrete non empty Poset, C being Component of R ex x
being Element of R st C = {x}
proof
let R be discrete non empty Poset, C be Component of R;
consider x being object such that
A1: x in the carrier of R and
A2: C = Class(Path_Rel R,x) by EQREL_1:def 3;
reconsider x1 = x as Element of R by A1;
take x1;
for y being object holds y in C iff y = x1
proof
let y be object;
hereby
assume
A3: y in C;
then reconsider y1 = y as Element of R;
[y,x] in Path_Rel R by A2,A3,EQREL_1:19;
then y1 = x1 by Th6;
hence y = x1;
end;
assume y = x1;
hence thesis by A2,EQREL_1:20;
end;
hence thesis by TARSKI:def 1;
end;
theorem Th8:
for R being discrete non empty Poset holds R is locally_directed
proof
let R be discrete non empty Poset;
let C be Component of R;
consider x being Element of R such that
A1: C = {x} by Th7;
for x,y being Element of R st x in C & y in C ex z being Element of R st
z in C & x <= z & y <= z
proof
let x1,y1 be Element of R such that
A2: x1 in C and
A3: y1 in C;
take x1;
x1 = x by A1,A2,TARSKI:def 1;
hence thesis by A1,A3,TARSKI:def 1;
end;
hence thesis by WAYBEL_0:def 1;
end;
:: the system could generate this one from the following
registration
cluster locally_directed for non empty Poset;
existence
proof
set R = the discrete non empty Poset;
take R;
thus thesis by Th8;
end;
end;
registration
cluster locally_directed for OrderSortedSign;
existence
proof
set R = the discrete OrderSortedSign;
take R;
thus thesis by Th8;
end;
end;
registration
cluster discrete -> locally_directed for non empty Poset;
coherence by Th8;
end;
registration
let S be locally_directed non empty Poset;
cluster -> directed for Component of S;
coherence by Def8;
end;
theorem Th9:
{} is Equivalence_Relation of {} by RELSET_1:12;
:: Much of what follows can be done generally for OrderSortedRelations
:: of OrderSortedSets (and not just OrderSortedRelations of OSAlgebras),
:: unfortunately, multiple inheritence would be needed to widen the
:: latter to the former
:: Classes on connected components
definition
let S be locally_directed OrderSortedSign;
let A be OSAlgebra of S;
let E be MSEquivalence-like OrderSortedRelation of A;
let C be Component of S;
func CompClass(E,C) -> Equivalence_Relation of (the Sorts of A)-carrier_of C
means
:Def9:
for x,y being object holds [x,y] in it iff ex s1 being Element of S
st s1 in C & [x,y] in E.s1;
existence
proof
defpred CC1[object,object] means
ex s1 being Element of S st s1 in C & [$1,$2] in E.s1;
A1: E is os-compatible by Def2;
per cases;
suppose
A2: (the Sorts of A)-carrier_of C is empty;
for x,y being object holds [x,y] in {} iff ex s1 being Element of S st
s1 in C & [x,y] in E.s1
proof
let x,y be object;
thus [x,y] in {} implies ex s1 being Element of S st s1 in C & [x,y]
in E.s1;
assume ex s1 being Element of S st s1 in C & [x,y] in E.s1;
then consider s1 being Element of S such that
A3: s1 in C & [x,y] in E.s1;
x in (the Sorts of A).s1 & (the Sorts of A).s1 in {(the Sorts of A
).s where s is Element of S: s in C} by A3,ZFMISC_1:87;
hence thesis by A2,TARSKI:def 4;
end;
hence thesis by A2,Th9;
end;
suppose
(the Sorts of A)-carrier_of C is non empty;
then reconsider SAC = (the Sorts of A)-carrier_of C as non empty set;
set CC = {[x,y] where x,y is Element of SAC: CC1[x,y]};
CC c= [:SAC, SAC:]
proof
let z be object;
assume z in CC;
then ex x,y being Element of SAC st z = [x,y] & CC1[x,y];
hence thesis by ZFMISC_1:87;
end;
then reconsider P1 = CC as Relation of SAC;
now
let x,y be object such that
A4: x in SAC & y in SAC and
A5: [x,y] in P1;
reconsider x1 = x, y1 = y as Element of SAC by A4;
consider x2,y2 being Element of SAC such that
A6: [x,y] = [x2,y2] and
A7: CC1[x2,y2] by A5;
A8: x = x2 & y = y2 by A6,XTUPLE_0:1;
consider s5 being Element of S such that
A9: s5 in C and
A10: [x2,y2] in E.s5 by A7;
field(E.s5) = (the Sorts of A).s5 by ORDERS_1:12;
then
A11: E.s5 is_symmetric_in (the Sorts of A).s5 by RELAT_2:def 11;
x2 in (the Sorts of A).s5 & y2 in (the Sorts of A).s5 by A10,
ZFMISC_1:87;
then [y,x] in E.s5 by A8,A10,A11,RELAT_2:def 3;
then [y1,x1] in CC by A9;
hence [y,x] in P1;
end;
then
A12: P1 is_symmetric_in SAC by RELAT_2:def 3;
now
let x be object such that
A13: x in SAC;
reconsider x1 = x as Element of SAC by A13;
consider X being set such that
A14: x in X and
A15: X in {(the Sorts of A).s where s is Element of S: s in C} by A13,
TARSKI:def 4;
consider s being Element of S such that
A16: X = (the Sorts of A).s and
A17: s in C by A15;
field(E.s) = (the Sorts of A).s by ORDERS_1:12;
then E.s is_reflexive_in (the Sorts of A).s by RELAT_2:def 9;
then [x,x] in E.s by A14,A16,RELAT_2:def 1;
then [x1,x1] in CC by A17;
hence [x,x] in P1;
end;
then P1 is_reflexive_in SAC by RELAT_2:def 1;
then
A18: dom P1 = SAC & field P1 = SAC by ORDERS_1:13;
now
let x,y,z be object such that
x in SAC and
y in SAC and
z in SAC and
A19: [x,y] in P1 and
A20: [y,z] in P1;
consider x2,y2 being Element of SAC such that
A21: [x,y] = [x2,y2] and
A22: CC1[x2,y2] by A19;
A23: x = x2 by A21,XTUPLE_0:1;
consider y3,z3 being Element of SAC such that
A24: [y,z] = [y3,z3] and
A25: CC1[y3,z3] by A20;
A26: y = y3 by A24,XTUPLE_0:1;
consider s5 being Element of S such that
A27: s5 in C and
A28: [x2,y2] in E.s5 by A22;
consider s6 being Element of S such that
A29: s6 in C and
A30: [y3,z3] in E.s6 by A25;
reconsider s51 = s5, s61 = s6 as Element of S;
consider su being Element of S such that
A31: su in C and
A32: s51 <= su and
A33: s61 <= su by A27,A29,WAYBEL_0:def 1;
y3 in (the Sorts of A).s6 & z3 in (the Sorts of A).s6 by A30,
ZFMISC_1:87;
then
A34: [y3,z3] in E.su by A1,A30,A33;
then
A35: z3 in (the Sorts of A).su by ZFMISC_1:87;
A36: z = z3 by A24,XTUPLE_0:1;
A37: y = y2 by A21,XTUPLE_0:1;
field(E.su) = (the Sorts of A).su by ORDERS_1:12;
then
A38: E.su is_transitive_in (the Sorts of A).su by RELAT_2:def 16;
x2 in (the Sorts of A).s5 & y2 in (the Sorts of A).s5 by A28,
ZFMISC_1:87;
then
A39: [x2,y2] in E.su by A1,A28,A32;
then x2 in (the Sorts of A).su & y2 in (the Sorts of A).su by
ZFMISC_1:87;
then [x2,z3] in E.su by A37,A26,A39,A34,A35,A38,RELAT_2:def 8;
hence [x,z] in P1 by A23,A36,A31;
end;
then P1 is_transitive_in SAC by RELAT_2:def 8;
then reconsider
P1 as Equivalence_Relation of (the Sorts of A)-carrier_of C
by A18,A12,PARTFUN1:def 2,RELAT_2:def 11,def 16;
take P1;
for x,y being object holds [x,y] in CC iff CC1[x,y]
proof
let x,y be object;
hereby
assume [x,y] in CC;
then ex x1,y1 being Element of SAC st [x,y] = [x1,y1] & CC1[x1,y1];
hence CC1[x,y];
end;
assume
A40: CC1[x,y];
then consider s1 being Element of S such that
A41: s1 in C and
A42: [x,y] in E.s1;
A43: y in (the Sorts of A).s1 by A42,ZFMISC_1:87;
(the Sorts of A).s1 in {(the Sorts of A).s where s is Element of
S: s in C} & x in (the Sorts of A).s1 by A41,A42,ZFMISC_1:87;
then reconsider x1 = x, y1 =y as Element of SAC by A43,TARSKI:def 4;
[x1,y1] in CC by A40;
hence thesis;
end;
hence thesis;
end;
end;
uniqueness
proof
let X,Y be Equivalence_Relation of (the Sorts of A)-carrier_of C;
defpred CC1[object,object] means
ex s1 being Element of S st s1 in C & [$1,$2] in E.s1;
assume
A44: for x,y being object holds [x,y] in X iff CC1[x,y];
assume
A45: for x,y being object holds [x,y] in Y iff CC1[x,y];
for x,y being object holds [x,y] in X iff [x,y] in Y
proof
let x,y be object;
hereby
assume [x,y] in X;
then CC1[x,y] by A44;
hence [x,y] in Y by A45;
end;
assume [x,y] in Y;
then CC1[x,y] by A45;
hence thesis by A44;
end;
hence thesis by RELAT_1:def 2;
end;
end;
:: we could give a name to Class CompClass(E,C)
:: restriction of Class CompClass(E,C) to A.s1
definition
let S be locally_directed OrderSortedSign;
let A be OSAlgebra of S;
let E be MSEquivalence-like OrderSortedRelation of A;
let s1 be Element of S;
func OSClass(E,s1) -> Subset of Class(CompClass(E,CComp(s1))) means
:Def10:
for z being set holds z in it iff ex x being set st x in (the Sorts of A).s1 &
z = Class( CompClass(E,CComp(s1)), x);
existence
proof
set SAC = (the Sorts of A)-carrier_of CComp(s1);
set CS = Class(CompClass(E,CComp(s1)));
defpred CC1[set] means ex x being set st x in (the Sorts of A).s1 & $1 =
Class( CompClass(E,CComp(s1)), x);
per cases;
suppose
A1: CS is empty;
reconsider CS1 = {} as Subset of CS by XBOOLE_1:2;
take CS1;
let z be set;
thus z in CS1 implies CC1[z];
assume CC1[z];
then consider x being set such that
A2: x in (the Sorts of A).s1 and
z = Class( CompClass(E,CComp(s1)), x);
x in SAC by A2,Th5;
hence thesis by A1;
end;
suppose
CS is non empty;
then reconsider CS1 = CS as non empty Subset-Family of SAC;
set CC = {x where x is Element of CS1: CC1[x]};
now
let x be object;
assume x in CC;
then ex y being Element of CS1 st x = y & CC1[y];
hence x in CS1;
end;
then reconsider CC2 = CC as Subset of CS by TARSKI:def 3;
take CC2;
for x being set holds x in CC iff CC1[x]
proof
let x be set;
hereby
assume x in CC;
then ex x1 being Element of CS1 st x = x1 & CC1[x1];
hence CC1[x];
end;
assume
A3: CC1[x];
then consider y being set such that
A4: y in (the Sorts of A).s1 and
A5: x = Class( CompClass(E,CComp(s1)), y);
s1 in CComp(s1) by EQREL_1:20;
then (the Sorts of A).s1 in {(the Sorts of A).s where s is Element
of S: s in CComp(s1)};
then y in union {(the Sorts of A).s where s is Element
of S: s in CComp(s1)} by A4,TARSKI:def 4;
then x in Class( CompClass(E,CComp(s1))) by A5,EQREL_1:def 3;
hence thesis by A3;
end;
hence thesis;
end;
end;
uniqueness
proof
let X,Y be Subset of Class(CompClass(E,CComp(s1)));
defpred CC1[object] means
ex x being set st x in (the Sorts of A).s1 & $1 =
Class( CompClass(E,CComp(s1)), x);
assume
A6: for x being set holds x in X iff CC1[x];
assume
A7: for x being set holds x in Y iff CC1[x];
for x being object holds x in X iff x in Y
proof
let x be object;
hereby
assume x in X;
then CC1[x] by A6;
hence x in Y by A7;
end;
assume x in Y;
then CC1[x] by A7;
hence thesis by A6;
end;
hence thesis by TARSKI:2;
end;
end;
registration
let S be locally_directed OrderSortedSign;
let A be non-empty OSAlgebra of S;
let E be MSEquivalence-like OrderSortedRelation of A;
let s1 be Element of S;
cluster OSClass(E,s1) -> non empty;
coherence
proof
consider x being object such that
A1: x in (the Sorts of A).s1 by XBOOLE_0:def 1;
Class( CompClass(E,CComp(s1)), x) in OSClass(E,s1) by A1,Def10;
hence thesis;
end;
end;
theorem Th10:
for S being locally_directed OrderSortedSign, A being OSAlgebra
of S, E being MSEquivalence-like (OrderSortedRelation of A), s1,s2 being
Element of S st s1 <= s2 holds OSClass(E,s1) c= OSClass(E,s2)
proof
let S be locally_directed OrderSortedSign;
let A be OSAlgebra of S;
let E be MSEquivalence-like OrderSortedRelation of A;
let s1,s2 be Element of S;
reconsider s3 = s1, s4 = s2 as Element of S;
assume
A1: s1 <= s2;
then
A2: CComp(s1) = CComp(s2) by Th4;
thus OSClass(E,s1) c= OSClass(E,s2)
proof
reconsider SO = the Sorts of A as OrderSortedSet of S by OSALG_1:17;
let z be object;
assume z in OSClass(E,s1);
then
A3: ex x being set st x in (the Sorts of A).s1 & z = Class( CompClass(E,
CComp(s1)), x) by Def10;
SO.s3 c= SO.s4 by A1,OSALG_1:def 16;
hence thesis by A2,A3,Def10;
end;
end;
:: finally, this is analogy of the Many-Sorted Class E for order-sorted E
:: this definition should work for order-sorted MSCongruence too
:: if non-empty not needed, prove the following cluster
definition
let S be locally_directed OrderSortedSign;
let A be OSAlgebra of S;
let E be MSEquivalence-like OrderSortedRelation of A;
func OSClass E -> OrderSortedSet of S means
:Def11:
for s1 being Element of S holds it.s1 = OSClass(E,s1);
existence
proof
deffunc F(Element of S) = OSClass(E,$1);
consider f being Function such that
A1: dom f = the carrier of S and
A2: for i being Element of S holds f.i = F(i) from FUNCT_1:sch 4;
reconsider f1 = f as ManySortedSet of the carrier of S by A1,PARTFUN1:def 2
,RELAT_1:def 18;
set f2 = f1;
f2 is order-sorted
proof
let s1,s2 be Element of S such that
A3: s1 <= s2;
f2.s1 = OSClass(E,s1) & f2.s2 = OSClass(E,s2) by A2;
hence thesis by A3,Th10;
end;
hence thesis by A2;
end;
uniqueness
proof
let X,Y be OrderSortedSet of S;
assume
A4: for s1 being Element of S holds X.s1 = OSClass(E,s1);
assume
A5: for s1 being Element of S holds Y.s1 = OSClass(E,s1);
now
let s1 be object;
assume s1 in the carrier of S;
then reconsider s2 = s1 as Element of S;
thus X.s1 = OSClass(E,s2) by A4
.= Y.s1 by A5;
end;
hence X = Y by PBOOLE:3;
end;
end;
registration
let S be locally_directed OrderSortedSign;
let A be non-empty OSAlgebra of S;
let E be MSEquivalence-like OrderSortedRelation of A;
cluster OSClass E -> non-empty;
coherence
proof
for i being object st i in the carrier of S
holds (OSClass E).i is non empty
proof
let i be object;
assume i in the carrier of S;
then reconsider s = i as Element of S;
(OSClass E).s = OSClass( E,s) by Def11;
hence thesis;
end;
hence thesis by PBOOLE:def 13;
end;
end;
:: order-sorted equiv of Class(R,x)
definition
let S be locally_directed OrderSortedSign;
let U1 be non-empty OSAlgebra of S;
let E be MSEquivalence-like OrderSortedRelation of U1;
let s be Element of S;
let x be Element of (the Sorts of U1).s;
func OSClass(E,x) -> Element of OSClass(E,s) equals
Class( CompClass(E,
CComp(s)), x);
correctness by Def10;
end;
theorem
for R being locally_directed non empty Poset, x,y being Element of R
st (ex z being Element of R st z <= x & z <= y) holds ex u being Element of R
st x <= u & y <= u
proof
let R be locally_directed non empty Poset, x,y be Element of R;
assume ex z being Element of R st z <= x & z <= y;
then consider z being Element of R such that
A1: z <= x and
A2: z <= y;
reconsider x1 = x,y1 = y as Element of R;
CComp(z) = CComp(y) by A2,Th4;
then
A3: y in CComp(z) by EQREL_1:20;
CComp z = CComp(x) by A1,Th4;
then x in CComp(z) by EQREL_1:20;
then ex u being Element of R st u in CComp(z) & x1 <= u & y1 <= u by A3,
WAYBEL_0:def 1;
hence thesis;
end;
theorem Th12:
for S be locally_directed OrderSortedSign, U1 be non-empty
OSAlgebra of S, E be MSEquivalence-like (OrderSortedRelation of U1),
s be Element of S, x,y be Element of (the Sorts of U1).s holds OSClass(E,x) =
OSClass(E,y) iff [x,y] in E.s
proof
let S be locally_directed OrderSortedSign;
let U1 be non-empty OSAlgebra of S;
let E be MSEquivalence-like OrderSortedRelation of U1;
let s be Element of S;
let x,y be Element of (the Sorts of U1).s;
reconsider SU = the Sorts of U1 as OrderSortedSet of S by OSALG_1:17;
A1: s in CComp(s) by EQREL_1:20;
A2: E is os-compatible by Def2;
A3: x in (the Sorts of U1)-carrier_of CComp(s) by Th5;
hereby
assume OSClass(E,x) = OSClass(E,y);
then [x,y] in CompClass(E, CComp(s)) by A3,EQREL_1:35;
then consider s1 being Element of S such that
A4: s1 in CComp(s) and
A5: [x,y] in E.s1 by Def9;
reconsider sn1 = s, s11 = s1 as Element of S;
consider s2 being Element of S such that
s2 in CComp(s) and
A6: s11 <= s2 and
A7: sn1 <= s2 by A1,A4,WAYBEL_0:def 1;
x in SU.s11 & y in SU.s11 by A5,ZFMISC_1:87;
then [x,y] in E.s2 by A2,A5,A6;
hence [x,y] in E.s by A2,A7;
end;
A8: s in CComp(s) by EQREL_1:20;
A9: x in (the Sorts of U1)-carrier_of CComp(s) by Th5;
assume [x,y] in E.s;
then [x,y] in CompClass(E, CComp(s)) by A8,Def9;
hence thesis by A9,EQREL_1:35;
end;
theorem
for S be locally_directed OrderSortedSign, U1 be non-empty OSAlgebra
of S, E be MSEquivalence-like (OrderSortedRelation of U1), s1,s2 be Element of
S, x be Element of (the Sorts of U1).s1 st s1 <= s2 holds for y being Element
of (the Sorts of U1).s2 st y = x holds OSClass(E,x) = OSClass(E,y) by Th4;
begin :: Order Sorted Quotient Algebra
:: take care (or even prove counterexample) - order-sorted
:: ManySortedFunction generaly does not exist
reserve S for locally_directed OrderSortedSign;
reserve o for Element of the carrier' of S;
:: multiclasses
definition
let S,o;
let A be non-empty OSAlgebra of S;
let R be OSCongruence of A;
let x be Element of Args(o,A);
func R #_os x -> Element of product ((OSClass R) * (the_arity_of o)) means
:
Def13: for n be Nat st n in dom (the_arity_of o) ex y being Element of (the
Sorts of A).((the_arity_of o)/.n) st y = x.n & it.n = OSClass(R, y);
existence
proof
defpred P[object,object] means
for n be Nat st n = $1 holds ex y being Element
of (the Sorts of A).((the_arity_of o)/.n) st y = x.n & $2 = OSClass(R, y);
set ar = the_arity_of o, da = dom ar;
A1: for k be object st k in da ex u be object st P[k,u]
proof
let k be object;
assume
A2: k in da;
then reconsider n = k as Nat;
reconsider y = x.n as Element of (the Sorts of A).((the_arity_of o)/.n)
by A2,MSUALG_6:2;
take OSClass(R,y);
thus thesis;
end;
consider f be Function such that
A3: dom f = da & for x be object st x in da holds P[x,f.x] from CLASSES1:
sch 1(A1);
A4: dom ((OSClass R) * ar) = da by PARTFUN1:def 2;
for y be object st y in dom ((OSClass R) * ar) holds f.y in ((OSClass R)
* ar).y
proof
let y be object;
assume
A5: y in dom ((OSClass R) * ar);
then reconsider n = y as Nat;
ar.y in rng ar by A4,A5,FUNCT_1:def 3;
then reconsider s = ar.y as Element of S;
(ex z being Element of (the Sorts of A).((the_arity_of o)/.n ) st z
= x.n & f.n = OSClass(R, z) )& ((ar)/.n) = ar.n by A3,A4,A5,PARTFUN1:def 6;
then
A6: f.y in OSClass (R,s);
((OSClass R) * ar).y = (OSClass R).(ar.y) by A5,FUNCT_1:12;
hence thesis by A6,Def11;
end;
then reconsider f as Element of product ((OSClass R) * ar) by A3,A4,
CARD_3:9;
take f;
let n be Nat;
assume n in da;
hence thesis by A3;
end;
uniqueness
proof
let F,G be Element of product ((OSClass R) * (the_arity_of o));
assume
A7: ( for n be Nat st n in dom (the_arity_of o) holds ex y being
Element of (the Sorts of A).((the_arity_of o)/.n) st y = x.n & F.n = OSClass(R,
y))& for n be Nat st n in dom (the_arity_of o) holds ex y being Element of (the
Sorts of A).( (the_arity_of o)/.n) st y = x.n & G.n = OSClass(R, y);
A8: for y be object st y in dom (the_arity_of o) holds F.y = G.y
proof
let y be object;
assume
A9: y in dom (the_arity_of o);
then reconsider n = y as Nat;
(ex z being Element of (the Sorts of A).((the_arity_of o)/.n ) st z
= x.n & F.n = OSClass(R, z) )& ex z1 being Element of (the Sorts of A).((
the_arity_of o )/. n) st z1 = x.n & G.n = OSClass(R, z1) by A7,A9;
hence thesis;
end;
A10: dom G = dom (the_arity_of o) by PARTFUN1:def 2;
dom F = dom (the_arity_of o) by PARTFUN1:def 2;
hence thesis by A10,A8,FUNCT_1:2;
end;
end;
:: the quotient will be different for order-sorted;
:: this def seems ok for order-sorted
definition
let S,o;
let A be non-empty OSAlgebra of S;
let R be OSCongruence of A;
func OSQuotRes(R,o) -> Function of ((the Sorts of A) * the ResultSort of S).
o, ((OSClass R) * the ResultSort of S).o means
:Def14:
for x being Element of (
the Sorts of A).(the_result_sort_of o) holds it.x = OSClass(R,x);
existence
proof
set rs = (the_result_sort_of o), D = (the Sorts of A).rs;
deffunc F(Element of D) = OSClass(R,$1);
consider f be Function such that
A1: dom f = D & for d be Element of D holds f.d = F(d) from FUNCT_1:
sch 4;
A2: for x be object st x in D holds f.x in (OSClass R).rs
proof
let x be object;
assume x in D;
then reconsider x1 = x as Element of D;
f.x1 = OSClass(R,x1) by A1;
then f.x1 in OSClass (R,rs);
hence thesis by Def11;
end;
o in the carrier' of S;
then o in dom ((the Sorts of A)*(the ResultSort of S)) by PARTFUN1:def 2;
then
A3: ((the Sorts of A) * the ResultSort of S).o = (the Sorts of A).((the
ResultSort of S).o) by FUNCT_1:12
.= D by MSUALG_1:def 2;
A4: dom (the ResultSort of S) = the carrier' of S by FUNCT_2:def 1;
then
dom ((OSClass R) * the ResultSort of S) = dom (the ResultSort of S) by
PARTFUN1:def 2;
then ((OSClass R) * the ResultSort of S).o = (OSClass R).((the ResultSort
of S).o) by A4,FUNCT_1:12
.= (OSClass R).rs by MSUALG_1:def 2;
then reconsider
f as Function of ((the Sorts of A) * the ResultSort of S).o, ((
OSClass R) * the ResultSort of S).o by A1,A3,A2,FUNCT_2:3;
take f;
thus thesis by A1;
end;
uniqueness
proof
set SA = the Sorts of A, RS = the ResultSort of S, rs = the_result_sort_of
o;
let f,g be Function of ((the Sorts of A) * the ResultSort of S).o, ((
OSClass R) * the ResultSort of S).o;
assume that
A5: for x being Element of SA.rs holds f.x = OSClass(R,x) and
A6: for x being Element of SA.rs holds g.x = OSClass(R,x);
A7: now
let x be object;
assume x in SA.rs;
then reconsider x1 = x as Element of SA.rs;
f.x1 = OSClass(R,x1) by A5;
hence f.x = g.x by A6;
end;
o in the carrier' of S;
then o in dom (SA*RS) by PARTFUN1:def 2;
then (SA * RS).o = SA.(RS.o) by FUNCT_1:12
.= SA.rs by MSUALG_1:def 2;
then dom f = SA.rs & dom g = SA.rs by FUNCT_2:def 1;
hence thesis by A7,FUNCT_1:2;
end;
func OSQuotArgs(R,o) -> Function of ((the Sorts of A)# * the Arity of S).o,
((OSClass R)# * the Arity of S).o means
for x be Element of Args(o,A) holds it. x = R #_os x;
existence
proof
set ao = the_arity_of o;
set D = Args(o,A);
deffunc F(Element of D) = R #_os $1;
consider f be Function such that
A8: dom f = D & for d be Element of D holds f.d = F(d) from FUNCT_1:
sch 4;
A9: o in the carrier' of S;
then o in dom ((the Sorts of A)# * the Arity of S) by PARTFUN1:def 2;
then
A10: ((the Sorts of A)# * the Arity of S).o = (the Sorts of A)#.((the
Arity of S).o) by FUNCT_1:12
.= ((the Sorts of A)#.(the_arity_of o)) by MSUALG_1:def 1;
A11: for x be object st x in (the Sorts of A)#.ao holds f.x in (OSClass R)#.ao
proof
let x be object;
assume x in (the Sorts of A)#.ao;
then reconsider x1 = x as Element of D by A10,MSUALG_1:def 4;
f.x1 = R #_os x1 & (OSClass R)#.ao = product((OSClass R)*ao) by A8,
FINSEQ_2:def 5;
hence thesis;
end;
o in dom ((OSClass R)# * the Arity of S) by A9,PARTFUN1:def 2;
then
((OSClass R)# * the Arity of S).o = (OSClass R)#.((the Arity of S).o)
by FUNCT_1:12
.= (OSClass R)#.ao by MSUALG_1:def 1;
then reconsider
f as Function of ((the Sorts of A)# * the Arity of S).o, ((
OSClass R)# * the Arity of S).o by A8,A10,A11,FUNCT_2:3,MSUALG_1:def 4;
take f;
thus thesis by A8;
end;
uniqueness
proof
set ao = the_arity_of o;
let f,g be Function of ((the Sorts of A)# * the Arity of S).o, ((OSClass R
)# * the Arity of S).o;
assume that
A12: for x be Element of Args(o,A) holds f.x = R #_os x and
A13: for x be Element of Args(o,A) holds g.x = R #_os x;
o in the carrier' of S;
then o in dom ((the Sorts of A)# * the Arity of S) by PARTFUN1:def 2;
then
A14: ((the Sorts of A)# * the Arity of S).o = (the Sorts of A)#.((the
Arity of S).o) by FUNCT_1:12
.= ((the Sorts of A)#.(the_arity_of o)) by MSUALG_1:def 1;
A15: now
let x be object;
assume x in (the Sorts of A)#.ao;
then reconsider x1 = x as Element of Args(o,A) by A14,MSUALG_1:def 4;
f.x1 = R #_os x1 by A12;
hence f.x = g.x by A13;
end;
dom f = (the Sorts of A)#.ao & dom g = (the Sorts of A)#.ao by A14,
FUNCT_2:def 1;
hence thesis by A15,FUNCT_1:2;
end;
end;
definition
let S;
let A be non-empty OSAlgebra of S;
let R be OSCongruence of A;
func OSQuotRes R -> ManySortedFunction of ((the Sorts of A) * the ResultSort
of S), ((OSClass R) * the ResultSort of S) means
for o being OperSymbol of S holds it.o = OSQuotRes(R,o);
existence
proof
defpred P[object,object] means
for o be OperSymbol of S st o = $1 holds $2 =
OSQuotRes(R,o);
set O = the carrier' of S;
A1: for x be object st x in O ex y be object st P[x,y]
proof
let x be object;
assume x in O;
then reconsider x1 = x as OperSymbol of S;
take OSQuotRes(R,x1);
thus thesis;
end;
consider f be Function such that
A2: dom f = O & for x be object st x in O holds P[x,f.x] from CLASSES1:
sch 1( A1);
reconsider f as ManySortedSet of O by A2,PARTFUN1:def 2,RELAT_1:def 18;
for x be object st x in dom f holds f.x is Function
proof
let x be object;
assume x in dom f;
then reconsider x1 = x as OperSymbol of S;
f.x1 = OSQuotRes(R,x1) by A2;
hence thesis;
end;
then reconsider f as ManySortedFunction of O by FUNCOP_1:def 6;
for i be object st i in O holds f.i is Function of ((the Sorts of A) *
the ResultSort of S).i, ((OSClass R) * the ResultSort of S).i
proof
let i be object;
assume i in O;
then reconsider i1 = i as OperSymbol of S;
f.i1 = OSQuotRes(R,i1) by A2;
hence thesis;
end;
then reconsider
f as ManySortedFunction of ((the Sorts of A) * the ResultSort
of S), ((OSClass R) * the ResultSort of S) by PBOOLE:def 15;
take f;
thus thesis by A2;
end;
uniqueness
proof
let f,g be ManySortedFunction of ((the Sorts of A) * the ResultSort of S),
((OSClass R) * the ResultSort of S);
assume that
A3: for o being OperSymbol of S holds f.o = OSQuotRes(R,o) and
A4: for o being OperSymbol of S holds g.o = OSQuotRes(R,o);
now
let i be object;
assume i in the carrier' of S;
then reconsider i1 = i as OperSymbol of S;
f.i1 = OSQuotRes(R,i1) by A3;
hence f.i = g.i by A4;
end;
hence thesis by PBOOLE:3;
end;
func OSQuotArgs R -> ManySortedFunction of (the Sorts of A)# * the Arity of
S, (OSClass R)# * the Arity of S means
for o be OperSymbol of S holds it.o = OSQuotArgs(R,o);
existence
proof
defpred P[object,object] means
for o be OperSymbol of S st o = $1 holds $2 =
OSQuotArgs(R,o);
set O = the carrier' of S;
A5: for x be object st x in O ex y be object st P[x,y]
proof
let x be object;
assume x in O;
then reconsider x1 = x as OperSymbol of S;
take OSQuotArgs(R,x1);
thus thesis;
end;
consider f be Function such that
A6: dom f = O & for x be object st x in O holds P[x,f.x] from CLASSES1:
sch 1( A5);
reconsider f as ManySortedSet of O by A6,PARTFUN1:def 2,RELAT_1:def 18;
for x be object st x in dom f holds f.x is Function
proof
let x be object;
assume x in dom f;
then reconsider x1 = x as OperSymbol of S;
f.x1 = OSQuotArgs(R,x1) by A6;
hence thesis;
end;
then reconsider f as ManySortedFunction of O by FUNCOP_1:def 6;
for i be object st i in O holds f.i is Function of ((the Sorts of A)# *
the Arity of S).i, ((OSClass R)# * the Arity of S).i
proof
let i be object;
assume i in O;
then reconsider i1 = i as OperSymbol of S;
f.i1 = OSQuotArgs(R,i1) by A6;
hence thesis;
end;
then reconsider
f as ManySortedFunction of ((the Sorts of A)# * the Arity of S)
, ((OSClass R)# * the Arity of S) by PBOOLE:def 15;
take f;
thus thesis by A6;
end;
uniqueness
proof
let f,g be ManySortedFunction of ((the Sorts of A)# * the Arity of S), ((
OSClass R)# * the Arity of S);
assume that
A7: for o being OperSymbol of S holds f.o = OSQuotArgs(R,o) and
A8: for o being OperSymbol of S holds g.o = OSQuotArgs(R,o);
now
let i be object;
assume i in the carrier' of S;
then reconsider i1 = i as OperSymbol of S;
f.i1 = OSQuotArgs(R,i1) by A7;
hence f.i = g.i by A8;
end;
hence thesis by PBOOLE:3;
end;
end;
theorem Th14:
for A be non-empty OSAlgebra of S, R be OSCongruence of A, x be
set st x in ((OSClass R)# * the Arity of S).o ex a be Element of Args(o,A) st x
= R #_os a
proof
let A be non-empty OSAlgebra of S, R be OSCongruence of A, x be set;
assume
A1: x in ((OSClass R)# * the Arity of S).o;
set ar = the_arity_of o;
A2: ar = (the Arity of S).o by MSUALG_1:def 1;
then ((OSClass R)# * the Arity of S).o = product ((OSClass R) * ar) by
MSAFREE:1;
then consider f be Function such that
A3: f = x and
A4: dom f = dom ((OSClass R) * ar) and
A5: for y be object st y in dom ((OSClass R) * ar) holds f.y in ((OSClass R
) * ar).y by A1,CARD_3:def 5;
defpred P[object,object] means
$2 in (the Sorts of A).((ar)/.$1) & $2 in f.$1;
A6: dom ((OSClass R) * ar) = dom ar by PARTFUN1:def 2;
A7: for n be Nat st n in dom f holds f.n in OSClass (R,((ar)/.n))
proof
let n be Nat;
reconsider s = ((ar)/.n) as Element of S;
assume
A8: n in dom f;
then ar.n = ((ar)/.n) by A4,A6,PARTFUN1:def 6;
then ((OSClass R) * ar).n = (OSClass R).s by A4,A8,FUNCT_1:12
.= OSClass (R,s) by Def11;
hence thesis by A4,A5,A8;
end;
A9: for a be object st a in dom f ex b be object st P[a,b]
proof
let a be object;
reconsider s = ((ar)/.a) as Element of S;
assume
A10: a in dom f;
then reconsider n = a as Nat by A4;
f.n in OSClass (R,s) by A7,A10;
then consider x being set such that
A11: x in (the Sorts of A).s & f.n = Class( CompClass(R,CComp(s)), x) by Def10;
take x;
thus thesis by A11,Th5,EQREL_1:20;
end;
consider g be Function such that
A12: dom g = dom f & for a be object st a in dom f holds P[a,g.a] from
CLASSES1:sch 1(A9);
dom (the Sorts of A) = the carrier of S by PARTFUN1:def 2;
then rng ar c= dom (the Sorts of A);
then
A13: dom g = dom ((the Sorts of A) * ar) by A4,A6,A12,RELAT_1:27;
A14: for y be object st y in dom ((the Sorts of A) * ar) holds g.y in ((the
Sorts of A) * ar).y
proof
let y be object;
assume
A15: y in dom ((the Sorts of A) * ar);
then reconsider n = y as Nat;
reconsider s = ((ar)/.n) as Element of S;
ar.n = ((ar)/.n) & g.n in (the Sorts of A).s by A4,A6,A12,A13,A15,
PARTFUN1:def 6;
hence thesis by A15,FUNCT_1:12;
end;
Args(o,A) = ((the Sorts of A)# * (the Arity of S)).o by MSUALG_1:def 4
.= product ((the Sorts of A) * ar) by A2,MSAFREE:1;
then reconsider g as Element of Args(o,A) by A13,A14,CARD_3:9;
A16: for x being object st x in dom ar holds f.x = (R #_os g).x
proof
let x be object;
assume
A17: x in dom ar;
then reconsider n = x as Nat;
A18: (ex z being Element of (the Sorts of A).((the_arity_of o)/.n ) st z =
g.n & (R #_os g).n = OSClass(R, z) )& g.n in f.n by A4,A6,A12,A17,Def13;
reconsider s = ((ar)/.n) as Element of S;
f.n in OSClass (R,s) by A4,A6,A7,A17;
then consider u being set such that
A19: u in (the Sorts of A).s and
A20: f.n = Class( CompClass(R,CComp(s)), u) by Def10;
u in (the Sorts of A)-carrier_of CComp(s) by A19,Th5;
hence thesis by A18,A20,EQREL_1:23;
end;
take g;
dom (R #_os g) = dom ((OSClass R) * ar) by CARD_3:9;
hence thesis by A3,A4,A6,A16,FUNCT_1:2;
end;
definition
let S,o;
let A be non-empty OSAlgebra of S;
let R be OSCongruence of A;
func OSQuotCharact(R,o) -> Function of ((OSClass R)# * the Arity of S).o, ((
OSClass R) * the ResultSort of S).o means
:Def18:
for a be Element of Args(o,A)
st R #_os a in ((OSClass R)# * the Arity of S).o holds it.(R #_os a) = ((
OSQuotRes(R,o)) * (Den(o,A))).a;
existence
proof
defpred P[object,object] means
for a be Element of Args(o,A) st $1 = R #_os a
holds $2 = ((OSQuotRes(R,o)) * (Den(o,A))).a;
set Ca = ((OSClass R)# * the Arity of S).o, Cr = ((OSClass R) * the
ResultSort of S).o;
A1: for x be object st x in Ca ex y be object st y in Cr & P[x,y]
proof
set ro = the_result_sort_of o, ar = the_arity_of o;
let x be object;
assume x in Ca;
then consider a be Element of Args(o,A) such that
A2: x = R #_os a by Th14;
take y = ((OSQuotRes(R,o)) * (Den(o,A))).a;
A3: o in the carrier' of S;
then o in dom ((OSClass R) * the ResultSort of S) by PARTFUN1:def 2;
then
A4: ((OSClass R) * the ResultSort of S).o = (OSClass R).((the ResultSort
of S).o) by FUNCT_1:12
.= (OSClass R).ro by MSUALG_1:def 2;
o in dom ((the Sorts of A) * the ResultSort of S) by A3,PARTFUN1:def 2;
then
A5: ((the Sorts of A) * the ResultSort of S).o = (the Sorts of A).((the
ResultSort of S).o) by FUNCT_1:12
.= (the Sorts of A).ro by MSUALG_1:def 2;
then
A6: dom (OSQuotRes(R,o)) = (the Sorts of A).ro & Result(o,A) = (the
Sorts of A). ro by FUNCT_2:def 1,MSUALG_1:def 5;
rng Den(o,A) c= Result(o,A);
then
A7: dom Den(o,A) = Args(o,A) & dom (OSQuotRes(R,o) * Den(o,A)) = dom
Den(o,A) by A6,FUNCT_2:def 1,RELAT_1:27;
OSQuotRes(R,o).(Den(o,A).a) in rng (OSQuotRes(R,o)) by A6,FUNCT_1:def 3;
then OSQuotRes(R,o).(Den(o,A).a) in (OSClass R).ro by A4;
hence y in Cr by A4,A7,FUNCT_1:12;
let b be Element of Args(o,A);
reconsider da = (Den(o,A)).a, db = (Den(o,A)).b as Element of (the Sorts
of A).ro by A5,MSUALG_1:def 5;
A8: ((OSQuotRes(R,o)) * (Den(o,A))).b = (OSQuotRes(R,o)).db by A7,FUNCT_1:12
.= OSClass(R,db) by Def14
.= Class( CompClass(R, CComp(ro)), db);
assume
A9: x = R #_os b;
for n be Nat st n in dom a holds [a.n,b.n] in R.((ar)/.n)
proof
let n be Nat;
A10: dom a = dom ar by MSUALG_3:6;
assume n in dom a;
then (ex ya being Element of (the Sorts of A).((ar)/.n) st ya = a.n &
(R #_os a). n = OSClass(R, ya) )& ex yb being Element of (the Sorts of A).((ar)
/.n) st yb = b.n & (R #_os b).n = OSClass(R, yb) by A10,Def13;
hence thesis by A2,A9,Th12;
end;
then ro in CComp(ro) & [da,db] in R.ro by EQREL_1:20,MSUALG_4:def 4;
then
A11: [da,db] in CompClass(R, CComp(ro)) by Def9;
A12: da in (the Sorts of A)-carrier_of CComp(ro) by Th5;
y = (OSQuotRes(R,o)).((Den(o,A)).a) by A7,FUNCT_1:12
.= OSClass(R, da) by Def14
.= Class( CompClass(R, CComp(ro)), da);
hence thesis by A12,A8,A11,EQREL_1:35;
end;
consider f be Function such that
A13: dom f = Ca & rng f c= Cr & for x be object st x in Ca holds P[x,f.x]
from FUNCT_1:sch 6(A1);
reconsider f as Function of ((OSClass R)# * the Arity of S).o, ((OSClass R
) * the ResultSort of S).o by A13,FUNCT_2:2;
take f;
thus thesis by A13;
end;
uniqueness
proof
set ao = the_arity_of o;
let F,G be Function of ((OSClass R)# * the Arity of S).o, ((OSClass R) *
the ResultSort of S).o;
assume that
A14: for a be Element of Args(o,A) st R #_os a in ((OSClass R)# * the
Arity of S).o holds F.(R #_os a) = ((OSQuotRes(R,o)) * (Den(o,A))).a and
A15: for a be Element of Args(o,A) st R #_os a in ((OSClass R)# * the
Arity of S).o holds G.(R #_os a) = ((OSQuotRes(R,o)) * (Den(o,A))).a;
A16: dom (the Arity of S) = the carrier' of S by FUNCT_2:def 1;
then dom ((OSClass R)# * the Arity of S) = dom (the Arity of S) by
PARTFUN1:def 2;
then
A17: ((OSClass R)# * the Arity of S).o = (OSClass R)#.((the Arity of S).o)
by A16,FUNCT_1:12
.= (OSClass R)#.ao by MSUALG_1:def 1;
A18: now
let x be object;
assume
A19: x in (OSClass R)#.ao;
then consider a be Element of Args(o,A) such that
A20: x = R #_os a by A17,Th14;
F.x = (OSQuotRes(R,o) * Den(o,A)).a by A14,A17,A19,A20;
hence F.x = G.x by A15,A17,A19,A20;
end;
dom F = (OSClass R)#.ao & dom G = (OSClass R)#.ao by A17,FUNCT_2:def 1;
hence thesis by A18,FUNCT_1:2;
end;
end;
definition
let S;
let A be non-empty OSAlgebra of S;
let R be OSCongruence of A;
func OSQuotCharact R -> ManySortedFunction of (OSClass R)# * the Arity of S,
(OSClass R) * the ResultSort of S means
:Def19:
for o be OperSymbol of S holds it.o = OSQuotCharact(R,o);
existence
proof
defpred P[object,object] means
for o be OperSymbol of S st o = $1 holds $2 =
OSQuotCharact(R,o);
set O = the carrier' of S;
A1: for x be object st x in O ex y be object st P[x,y]
proof
let x be object;
assume x in O;
then reconsider x1 = x as OperSymbol of S;
take OSQuotCharact(R,x1);
thus thesis;
end;
consider f be Function such that
A2: dom f = O & for x be object st x in O holds P[x,f.x] from CLASSES1:
sch 1( A1);
reconsider f as ManySortedSet of O by A2,PARTFUN1:def 2,RELAT_1:def 18;
for x be object st x in dom f holds f.x is Function
proof
let x be object;
assume x in dom f;
then reconsider x1 = x as OperSymbol of S;
f.x1 = OSQuotCharact(R,x1) by A2;
hence thesis;
end;
then reconsider f as ManySortedFunction of O by FUNCOP_1:def 6;
for i be object st i in O holds f.i is Function of ((OSClass R)# * the
Arity of S).i, ((OSClass R) * the ResultSort of S).i
proof
let i be object;
assume i in O;
then reconsider i1 = i as OperSymbol of S;
f.i1 = OSQuotCharact(R,i1) by A2;
hence thesis;
end;
then reconsider
f as ManySortedFunction of ((OSClass R)# * the Arity of S), ((
OSClass R) * the ResultSort of S) by PBOOLE:def 15;
take f;
thus thesis by A2;
end;
uniqueness
proof
let f,g be ManySortedFunction of ((OSClass R)# * the Arity of S), ((
OSClass R) * the ResultSort of S);
assume that
A3: for o being OperSymbol of S holds f.o = OSQuotCharact(R,o) and
A4: for o being OperSymbol of S holds g.o = OSQuotCharact(R,o);
now
let i be object;
assume i in the carrier' of S;
then reconsider i1 = i as OperSymbol of S;
f.i1 = OSQuotCharact(R,i1) by A3;
hence f.i = g.i by A4;
end;
hence thesis by PBOOLE:3;
end;
end;
definition
let S;
let U1 be non-empty OSAlgebra of S;
let R be OSCongruence of U1;
func QuotOSAlg(U1,R) -> OSAlgebra of S equals
MSAlgebra(# OSClass R,
OSQuotCharact R #);
coherence by OSALG_1:17;
end;
:: we could note that for discrete the QuotOSAlg and QuotMsAlg are equal
registration
let S;
let U1 be non-empty OSAlgebra of S;
let R be OSCongruence of U1;
cluster QuotOSAlg (U1,R) -> strict non-empty;
coherence by MSUALG_1:def 3;
end;
definition
let S;
let U1 be non-empty OSAlgebra of S;
let R be OSCongruence of U1;
let s be Element of S;
func OSNat_Hom(U1,R,s) -> Function of (the Sorts of U1).s,OSClass(R,s) means
:Def21:
for x being Element of (the Sorts of U1).s holds it.x = OSClass(R,x);
existence
proof
deffunc F(Element of (the Sorts of U1).s) = OSClass(R,$1);
thus ex F being Function of (the Sorts of U1).s,OSClass(R,s) st for x
being Element of (the Sorts of U1).s holds F.x = F(x) from FUNCT_2:sch 4;
end;
uniqueness
proof
let f,g be Function of (the Sorts of U1).s,OSClass (R,s);
assume that
A1: for x being Element of (the Sorts of U1).s holds f.x = OSClass(R,x ) and
A2: for x being Element of (the Sorts of U1).s holds g.x = OSClass(R,x );
A3: now
let x be object;
assume x in (the Sorts of U1).s;
then reconsider x1 = x as Element of (the Sorts of U1).s;
f.x = OSClass(R,x1) by A1;
hence f.x = g.x by A2;
end;
dom f = (the Sorts of U1).s & dom g = (the Sorts of U1).s by FUNCT_2:def 1;
hence thesis by A3,FUNCT_1:2;
end;
end;
definition
let S;
let U1 be non-empty OSAlgebra of S;
let R be OSCongruence of U1;
func OSNat_Hom(U1,R) -> ManySortedFunction of U1, QuotOSAlg (U1,R) means
:
Def22: for s be Element of S holds it.s = OSNat_Hom(U1,R,s);
existence
proof
deffunc F(Element of S) = OSNat_Hom(U1,R,$1);
consider f be Function such that
A1: dom f = the carrier of S & for d be Element of S holds f.d = F(d)
from FUNCT_1:sch 4;
reconsider f as ManySortedSet of the carrier of S by A1,PARTFUN1:def 2
,RELAT_1:def 18;
for x be object st x in dom f holds f.x is Function
proof
let x be object;
assume x in dom f;
then reconsider y = x as Element of S;
f.y = OSNat_Hom(U1,R,y) by A1;
hence thesis;
end;
then reconsider f as ManySortedFunction of the carrier of S by
FUNCOP_1:def 6;
for i be object st i in the carrier of S holds f.i is Function of (the
Sorts of U1).i, (OSClass R).i
proof
let i be object;
assume i in the carrier of S;
then reconsider s = i as Element of S;
OSClass(R,s) = (OSClass R).i & f.s = OSNat_Hom(U1,R,s) by A1,Def11;
hence thesis;
end;
then reconsider f as ManySortedFunction of the Sorts of U1,OSClass R by
PBOOLE:def 15;
reconsider f as ManySortedFunction of U1,QuotOSAlg (U1,R);
take f;
thus thesis by A1;
end;
uniqueness
proof
let F,G be ManySortedFunction of U1, QuotOSAlg (U1,R);
assume that
A2: for s be Element of S holds F.s = OSNat_Hom(U1,R,s) and
A3: for s be Element of S holds G.s = OSNat_Hom(U1,R,s);
now
let i be object;
assume i in the carrier of S;
then reconsider s = i as SortSymbol of S;
F.s = OSNat_Hom(U1,R,s) by A2;
hence F.i = G.i by A3;
end;
hence thesis by PBOOLE:3;
end;
end;
theorem
for U1 be non-empty OSAlgebra of S, R be OSCongruence of U1 holds
OSNat_Hom(U1,R) is_epimorphism U1, QuotOSAlg (U1,R) & OSNat_Hom(U1,R) is
order-sorted
proof
let U1 be non-empty OSAlgebra of S, R be OSCongruence of U1;
set F = OSNat_Hom(U1,R), QA = QuotOSAlg (U1,R), S1 = the Sorts of U1;
for o be Element of the carrier' of S st Args (o,U1) <> {} for x be
Element of Args(o,U1) holds (F.(the_result_sort_of o)).(Den(o,U1).x) = Den(o,QA
).(F#x)
proof
let o be Element of the carrier' of S such that
Args (o,U1) <> {};
let x be Element of Args(o,U1);
set ro = the_result_sort_of o, ar = the_arity_of o;
(the Arity of S).o = ar by MSUALG_1:def 1;
then
A1: ((OSClass R)# * the Arity of S).o = product ((OSClass R) * ar) by MSAFREE:1
;
A2: dom x = dom ar by MSUALG_3:6;
A3: for a be object st a in dom ar holds (F#x).a = (R #_os x).a
proof
let a be object;
assume
A4: a in dom ar;
then reconsider n = a as Nat;
set Fo = OSNat_Hom(U1,R,((ar)/.n)), s = ((ar)/.n);
A5: ex z being Element of S1.s st z = x.n & (R #_os x).n = OSClass(R,z)
by A4,Def13;
A6: n in dom ((the Sorts of U1) * ar) by A4,PARTFUN1:def 2;
then ((the Sorts of U1) * ar).n = (the Sorts of U1).(ar.n) by FUNCT_1:12
.= S1.s by A4,PARTFUN1:def 6;
then reconsider xn = x.n as Element of S1.s by A6,MSUALG_3:6;
thus (F#x).a = (F.((ar)/.n)).(x.n) by A2,A4,MSUALG_3:def 6
.= Fo.xn by Def22
.= (R #_os x).a by A5,Def21;
end;
dom (the Sorts of U1) = the carrier of S by PARTFUN1:def 2;
then rng (the ResultSort of S) c= dom (the Sorts of U1);
then
dom (the ResultSort of S) = the carrier' of S & dom ((the Sorts of U1
) * the ResultSort of S) = dom (the ResultSort of S) by FUNCT_2:def 1
,RELAT_1:27;
then
A7: ((the Sorts of U1) * the ResultSort of S).o = (the Sorts of U1).((the
ResultSort of S).o) by FUNCT_1:12
.= (the Sorts of U1).ro by MSUALG_1:def 2;
then reconsider dx = (Den(o,U1)).x as Element of S1.ro by MSUALG_1:def 5;
rng Den(o,U1) c= Result(o,U1) & Result(o,U1) = S1.ro by A7,MSUALG_1:def 5;
then rng Den(o,U1) c= dom OSQuotRes(R,o) by A7,FUNCT_2:def 1;
then
A8: dom Den(o,U1) = Args(o,U1) & dom ((OSQuotRes(R,o)) * Den(o,U1)) = dom
Den(o, U1) by FUNCT_2:def 1,RELAT_1:27;
dom (OSClass R) = the carrier of S by PARTFUN1:def 2;
then
dom (R #_os x) = dom ((OSClass R) * (the_arity_of o)) & rng ar c= dom
( OSClass R) by CARD_3:9;
then dom (F#x) = dom ar & dom (R #_os x) = dom ar by MSUALG_3:6,RELAT_1:27;
then
A9: F # x = R #_os x by A3,FUNCT_1:2;
Den(o,QA) = (OSQuotCharact R).o by MSUALG_1:def 6
.= OSQuotCharact(R,o) by Def19;
then Den(o,QA).(F#x) = ((OSQuotRes(R,o)) * (Den(o,U1))).x by A1,A9,Def18
.= (OSQuotRes(R,o)) . dx by A8,FUNCT_1:12
.= OSClass (R, dx) by Def14
.= (OSNat_Hom(U1,R,ro)).dx by Def21
.= (F.ro).((Den(o,U1)).x) by Def22;
hence thesis;
end;
hence F is_homomorphism U1,QA;
for i be set st i in the carrier of S holds rng (F.i) = (the Sorts of QA).i
proof
let i be set;
assume i in the carrier of S;
then reconsider s = i as Element of S;
reconsider f = F.i as Function of S1.s, (the Sorts of QA).s by
PBOOLE:def 15;
A10: dom f = S1.s by FUNCT_2:def 1;
A11: (the Sorts of QA).s = OSClass (R,s) by Def11;
for x be object st x in (the Sorts of QA).i holds x in rng f
proof
let x be object;
A12: f = OSNat_Hom(U1,R,s) by Def22;
assume x in (the Sorts of QA).i;
then consider a1 being set such that
A13: a1 in S1.s and
A14: x = Class(CompClass(R,CComp(s)),a1) by A11,Def10;
reconsider a2 = a1 as Element of S1.s by A13;
OSClass(R,a2) = x & f.a1 in rng f by A10,A13,A14,FUNCT_1:def 3;
hence thesis by A12,Def21;
end;
then (the Sorts of QA).i c= rng f;
hence thesis;
end;
hence F is "onto";
thus F is order-sorted
proof
reconsider S2 = S1 as OrderSortedSet of S by OSALG_1:17;
let s1,s2 be Element of S such that
A15: s1 <= s2;
A16: S2.s1 c= S2.s2 by A15,OSALG_1:def 16;
let a1 be set such that
A17: a1 in dom (F.s1);
A18: a1 in S1.s1 by A17;
then reconsider b2 = a1 as Element of S1.s2 by A16;
dom (F.s2) = S1.s2 by FUNCT_2:def 1;
hence a1 in dom (F.s2) by A16,A18;
reconsider b1 = a1 as Element of S1.s1 by A17;
thus (F.s1).a1 = OSNat_Hom(U1,R,s1).b1 by Def22
.= OSClass(R,b1) by Def21
.= OSClass(R,b2) by A15,Th4
.= OSNat_Hom(U1,R,s2).b2 by Def21
.= (F.s2).a1 by Def22;
end;
end;
theorem Th16:
for U1,U2 being non-empty OSAlgebra of S, F being
ManySortedFunction of U1,U2 st F is_homomorphism U1,U2 & F is order-sorted
holds MSCng(F) is OSCongruence of U1
proof
let U1,U2 be non-empty OSAlgebra of S, F be ManySortedFunction of U1,U2 such
that
A1: F is_homomorphism U1,U2 and
A2: F is order-sorted;
reconsider S1 = the Sorts of U1 as OrderSortedSet of S by OSALG_1:17;
MSCng(F) is os-compatible
proof
let s1,s2 be Element of S such that
A3: s1 <= s2;
reconsider s3 = s1, s4 = s2 as SortSymbol of S;
let x,y be set such that
A4: x in (the Sorts of U1).s1 & y in (the Sorts of U1).s1;
reconsider x1 = x, y1 = y as Element of (the Sorts of U1).s1 by A4;
S1.s3 c= S1.s4 by A3,OSALG_1:def 16;
then reconsider x2 = x, y2 = y as Element of (the Sorts of U1).s2 by A4;
A5: [x2,y2] in MSCng(F,s2) iff (F.s2).x2 = (F.s2).y2 by MSUALG_4:def 17;
dom (F.s3) = S1.s3 by FUNCT_2:def 1;
then
A6: (F.s3).x1 = (F.s4).x1 & (F.s3).y1 = (F.s4).y1 by A2,A3;
(MSCng(F)).s1 = MSCng(F,s1) by A1,MSUALG_4:def 18;
hence [x,y] in (MSCng(F)).s1 iff [x,y] in (MSCng(F)).s2 by A1,A5,A6,
MSUALG_4:def 17,def 18;
end;
hence thesis by Def2;
end;
:: just a casting func, currently no other way how to employ the type system
definition
let S;
let U1,U2 be non-empty OSAlgebra of S;
let F be ManySortedFunction of U1,U2;
assume
A1: F is_homomorphism U1,U2 & F is order-sorted;
func OSCng(F) -> OSCongruence of U1 equals
:Def23:
MSCng(F);
correctness by A1,Th16;
end;
definition
let S;
let U1,U2 be non-empty OSAlgebra of S;
let F be ManySortedFunction of U1,U2;
let s be Element of S;
assume that
A1: F is_homomorphism U1,U2 and
A2: F is order-sorted;
func OSHomQuot(F,s) -> Function of (the Sorts of (QuotOSAlg (U1,OSCng F))).s
,(the Sorts of U2).s means
:Def24:
for x be Element of (the Sorts of U1).s
holds it.(OSClass(OSCng(F),x)) = (F.s).x;
existence
proof
set qa = QuotOSAlg (U1,OSCng F), cqa = the Sorts of qa, S1 = the Sorts of
U1, S2 = the Sorts of U2;
defpred P[object,object] means
for a be Element of S1.s st $1 = OSClass(OSCng(F)
,a) holds $2 = (F.s).a;
A3: cqa.s = OSClass (OSCng(F),s) by Def11;
A4: for x be object st x in cqa.s ex y be object st y in S2.s & P[x,y]
proof
let x be object;
assume x in cqa.s;
then consider a being set such that
A5: a in (the Sorts of U1).s and
A6: x = Class( CompClass(OSCng(F),CComp(s)), a) by A3,Def10;
reconsider a as Element of S1.s by A5;
take y = (F.s).a;
thus y in S2.s;
let b be Element of S1.s;
assume
A7: x = OSClass(OSCng(F),b);
x = OSClass(OSCng(F),a) by A6;
then [b,a] in (OSCng(F)).s by A7,Th12;
then [b,a] in (MSCng(F)).s by A1,A2,Def23;
then [b,a] in MSCng(F,s) by A1,MSUALG_4:def 18;
hence thesis by MSUALG_4:def 17;
end;
consider G being Function such that
A8: dom G = cqa.s & rng G c= S2.s & for x be object st x in cqa.s holds
P[x,G.x] from FUNCT_1:sch 6(A4);
reconsider G as Function of cqa.s,S2.s by A8,FUNCT_2:def 1,RELSET_1:4;
take G;
let a be Element of S1.s;
thus thesis by A3,A8;
end;
uniqueness
proof
set qa = QuotOSAlg (U1, OSCng F), cqa = the Sorts of qa, u1 = the Sorts of
U1, u2 = the Sorts of U2;
let H,G be Function of cqa.s,u2.s;
assume that
A9: for a be Element of u1.s holds H.(OSClass(OSCng(F),a)) = (F.s).a and
A10: for a be Element of u1.s holds G.(OSClass(OSCng(F),a)) = (F.s).a;
A11: cqa.s = OSClass (OSCng(F),s) by Def11;
for x be object st x in cqa.s holds H.x = G.x
proof
let x be object;
assume x in cqa.s;
then consider y being set such that
A12: y in (the Sorts of U1).s and
A13: x = Class( CompClass(OSCng(F),CComp(s)), y) by A11,Def10;
reconsider y1 = y as Element of u1.s by A12;
A14: OSClass(OSCng(F),y1) = x by A13;
hence H.x = (F.s).y1 by A9
.= G.x by A10,A14;
end;
hence thesis by FUNCT_2:12;
end;
end;
:: this seems a bit too permissive, but same as the original
:: we should assume F order-sorted probably
definition
let S;
let U1,U2 be non-empty OSAlgebra of S;
let F be ManySortedFunction of U1,U2;
func OSHomQuot(F) -> ManySortedFunction of (QuotOSAlg (U1, OSCng F)),U2
means
:Def25:
for s be Element of S holds it.s = OSHomQuot(F,s);
existence
proof
deffunc F(Element of S) = OSHomQuot(F,$1);
consider f be Function such that
A1: dom f = the carrier of S & for d be Element of S holds f.d = F(d)
from FUNCT_1:sch 4;
reconsider f as ManySortedSet of the carrier of S by A1,PARTFUN1:def 2
,RELAT_1:def 18;
for x be object st x in dom f holds f.x is Function
proof
let x be object;
assume x in dom f;
then reconsider y = x as Element of S;
f.y = OSHomQuot(F,y) by A1;
hence thesis;
end;
then reconsider f as ManySortedFunction of the carrier of S by
FUNCOP_1:def 6;
for i be object st i in the carrier of S holds f.i is Function of (the
Sorts of QuotOSAlg (U1, OSCng F)).i, (the Sorts of U2).i
proof
let i be object;
assume i in the carrier of S;
then reconsider s = i as Element of S;
f.s = OSHomQuot(F,s) by A1;
hence thesis;
end;
then reconsider
f as ManySortedFunction of the Sorts of QuotOSAlg (U1,OSCng F),
the Sorts of U2 by PBOOLE:def 15;
reconsider f as ManySortedFunction of QuotOSAlg (U1,OSCng F),U2;
take f;
thus thesis by A1;
end;
uniqueness
proof
let H,G be ManySortedFunction of QuotOSAlg (U1,OSCng F),U2;
assume that
A2: for s be Element of S holds H.s = OSHomQuot(F,s) and
A3: for s be Element of S holds G.s = OSHomQuot(F,s);
now
let i be object;
assume i in the carrier of S;
then reconsider s = i as SortSymbol of S;
H.s = OSHomQuot(F,s) by A2;
hence H.i = G.i by A3;
end;
hence thesis by PBOOLE:3;
end;
end;
theorem Th17:
for U1,U2 be non-empty OSAlgebra of S, F be ManySortedFunction
of U1,U2 st F is_homomorphism U1,U2 & F is order-sorted holds OSHomQuot(F)
is_monomorphism QuotOSAlg (U1,OSCng F),U2 & OSHomQuot(F) is order-sorted
proof
let U1,U2 be non-empty OSAlgebra of S, F be ManySortedFunction of U1,U2;
set mc = OSCng(F), qa = QuotOSAlg (U1,mc), qh = OSHomQuot(F), S1 = the Sorts
of U1;
assume that
A1: F is_homomorphism U1,U2 and
A2: F is order-sorted;
for o be Element of the carrier' of S st Args (o,qa) <> {} for x be
Element of Args(o,qa) holds (qh.(the_result_sort_of o)).(Den(o,qa).x) = Den(o,
U2).(qh#x)
proof
let o be Element of the carrier' of S such that
Args (o,qa) <> {};
let x be Element of Args(o,qa);
reconsider o1 = o as OperSymbol of S;
set ro = the_result_sort_of o, ar = the_arity_of o;
A3: dom x = dom ar by MSUALG_3:6;
Args(o,qa) = ((OSClass mc)# * (the Arity of S)).o by MSUALG_1:def 4;
then consider a be Element of Args(o,U1) such that
A4: x = mc #_os a by Th14;
A5: dom a = dom ar by MSUALG_3:6;
A6: now
let y be object;
assume
A7: y in dom ar;
then reconsider n = y as Nat;
ar.n in rng ar by A7,FUNCT_1:def 3;
then reconsider s = ar.n as SortSymbol of S;
A8: ar/.n = ar.n by A7,PARTFUN1:def 6;
then consider an being Element of S1.s such that
A9: an = a.n and
A10: x.n = OSClass(mc,an) by A4,A7,Def13;
(qh # x).n = (qh.s).(x.n) by A3,A7,A8,MSUALG_3:def 6
.= OSHomQuot(F,s).OSClass(mc,an) by A10,Def25
.= (F.s).an by A1,A2,Def24
.= (F # a).n by A5,A7,A8,A9,MSUALG_3:def 6;
hence (qh # x).y = (F # a).y;
end;
o in the carrier' of S;
then o in dom (S1 * the ResultSort of S) by PARTFUN1:def 2;
then
A11: ((the Sorts of U1) * the ResultSort of S).o = (the Sorts of U1).((the
ResultSort of S).o) by FUNCT_1:12
.= (the Sorts of U1).ro by MSUALG_1:def 2;
then rng Den(o,U1) c= Result(o,U1) & Result(o,U1) = S1.ro by MSUALG_1:def 5
;
then rng Den(o,U1) c= dom OSQuotRes(mc,o) by A11,FUNCT_2:def 1;
then
A12: dom Den(o,U1) = Args(o,U1) & dom ((OSQuotRes(mc,o)) * Den(o,U1)) =
dom Den(o,U1) by FUNCT_2:def 1,RELAT_1:27;
ar = (the Arity of S).o by MSUALG_1:def 1;
then
A13: product((OSClass mc) * ar) = ((OSClass mc)# * the Arity of S).o by
MSAFREE:1;
reconsider da = (Den(o,U1)).a as Element of S1.ro by A11,MSUALG_1:def 5;
A14: qh.ro = OSHomQuot(F,ro) by Def25;
Den(o,qa) = (OSQuotCharact mc).o by MSUALG_1:def 6
.= OSQuotCharact(mc,o1) by Def19;
then Den(o,qa).x = (OSQuotRes(mc,o) * Den(o,U1)).a by A4,A13,Def18
.= (OSQuotRes(mc,o)) . da by A12,FUNCT_1:12
.= OSClass (OSCng(F),da) by Def14;
then
A15: (qh.ro).(Den(o,qa).x) = (F.ro).((Den(o,U1)).a) by A1,A2,A14,Def24
.= Den(o,U2).(F#a) by A1;
dom (qh # x) = dom ar & dom (F # a) = dom ar by MSUALG_3:6;
hence thesis by A6,A15,FUNCT_1:2;
end;
hence qh is_homomorphism qa,U2;
for i be set st i in the carrier of S holds (qh.i) is one-to-one
proof
let i be set;
set f = qh.i;
assume i in the carrier of S;
then reconsider s = i as SortSymbol of S;
A16: f = OSHomQuot(F,s) by Def25;
for x1,x2 be object st x1 in dom f & x2 in dom f & f.x1 = f.x2
holds x1 = x2
proof
let x1,x2 be object;
assume that
A17: x1 in dom f and
A18: x2 in dom f and
A19: f.x1 = f.x2;
x2 in (OSClass mc).s by A16,A18,FUNCT_2:def 1;
then x2 in OSClass (mc,s) by Def11;
then consider a2 being set such that
A20: a2 in S1.s and
A21: x2 = Class( CompClass(OSCng(F),CComp(s)), a2) by Def10;
reconsider a2 as Element of S1.s by A20;
A22: x2 = OSClass(OSCng(F),a2) by A21;
x1 in (OSClass mc).s by A16,A17,FUNCT_2:def 1;
then x1 in OSClass (mc,s) by Def11;
then consider a1 being set such that
A23: a1 in S1.s and
A24: x1 = Class( CompClass(OSCng(F),CComp(s)), a1) by Def10;
reconsider a1 as Element of S1.s by A23;
(F.s).a1 = f.(OSClass(OSCng(F),a1)) & (F.s).a2 = f.(OSClass(OSCng(F
),a2)) by A1,A2,A16,Def24;
then [a1,a2] in MSCng(F,s) by A19,A24,A21,MSUALG_4:def 17;
then [a1,a2] in (MSCng(F)).s by A1,MSUALG_4:def 18;
then
A25: [a1,a2] in (OSCng(F)).s by A1,A2,Def23;
x1 = OSClass(OSCng(F),a1) by A24;
hence thesis by A22,A25,Th12;
end;
hence thesis by FUNCT_1:def 4;
end;
hence qh is "1-1" by MSUALG_3:1;
thus OSHomQuot(F) is order-sorted
proof
reconsider S1O = S1 as OrderSortedSet of S by OSALG_1:17;
reconsider sqa = the Sorts of qa as OrderSortedSet of S;
let s1,s2 being Element of S such that
A26: s1 <= s2;
let a1 be set such that
A27: a1 in dom (qh.s1);
a1 in (OSClass OSCng(F)).s1 by A27;
then a1 in OSClass (OSCng(F),s1) by Def11;
then consider x being set such that
A28: x in S1.s1 and
A29: a1 = Class( CompClass(OSCng(F),CComp(s1)), x) by Def10;
S1O.s1 c= S1O.s2 by A26,OSALG_1:def 16;
then reconsider x2 = x as Element of S1.s2 by A28;
A30: a1 = OSClass(OSCng(F),x2) by A26,A29,Th4;
reconsider s3 = s1, s4 = s2 as Element of S;
A31: dom (qh.s1) = (the Sorts of qa).s1 & dom (qh.s2) = (the Sorts of qa).
s2 by FUNCT_2:def 1;
reconsider x1 = x as Element of S1.s1 by A28;
x1 in dom (F.s3) by A28,FUNCT_2:def 1;
then
A32: (F.s3).x1 = (F.s4).x1 by A2,A26;
sqa.s1 c= sqa.s2 by A26,OSALG_1:def 16;
hence a1 in dom (qh.s2) by A27,A31;
thus (qh.s1).a1 = OSHomQuot(F,s1).(OSClass(OSCng(F),x1)) by A29,Def25
.= (F.s2).x1 by A1,A2,A32,Def24
.= OSHomQuot(F,s2).(OSClass(OSCng(F),x2)) by A1,A2,Def24
.= (qh.s2).a1 by A30,Def25;
end;
end;
theorem Th18:
for U1,U2 be non-empty OSAlgebra of S, F be ManySortedFunction
of U1,U2 st F is_epimorphism U1,U2 & F is order-sorted holds OSHomQuot(F)
is_isomorphism QuotOSAlg (U1,OSCng F),U2
proof
let U1,U2 be non-empty OSAlgebra of S, F be ManySortedFunction of U1,U2;
set mc = OSCng(F), qa = QuotOSAlg (U1,mc), qh = OSHomQuot(F);
assume that
A1: F is_epimorphism U1,U2 and
A2: F is order-sorted;
set Sq = the Sorts of qa, S1 = the Sorts of U1, S2 = the Sorts of U2;
A3: F is_homomorphism U1,U2 by A1;
A4: F is "onto" by A1;
for i be set st i in the carrier of S holds rng (qh.i) = S2.i
proof
let i be set;
set f = qh.i;
assume i in the carrier of S;
then reconsider s = i as SortSymbol of S;
A5: rng (F.s) = S2.s by A4;
A6: qh.i = OSHomQuot(F,s) by Def25;
then
A7: dom f = Sq.s by FUNCT_2:def 1;
thus rng f c= S2.i by A6,RELAT_1:def 19;
let x be object;
assume x in S2.i;
then consider a be object such that
A8: a in dom (F.s) and
A9: (F.s).a = x by A5,FUNCT_1:def 3;
A10: Sq.s = OSClass (OSCng(F),s) by Def11;
reconsider a as Element of S1.s by A8;
f.(OSClass(OSCng(F),a)) = x by A2,A3,A6,A9,Def24;
hence thesis by A7,A10,FUNCT_1:def 3;
end;
then
A11: qh is "onto";
qh is_monomorphism qa,U2 by A2,A3,Th17;
then qh is_homomorphism qa,U2 & qh is "1-1";
hence thesis by A11,MSUALG_3:13;
end;
theorem
for U1,U2 be non-empty OSAlgebra of S, F be ManySortedFunction of U1,
U2 st F is_epimorphism U1,U2 & F is order-sorted holds QuotOSAlg (U1,OSCng F),
U2 are_isomorphic
proof
let U1,U2 be non-empty OSAlgebra of S, F be ManySortedFunction of U1,U2;
assume F is_epimorphism U1,U2 & F is order-sorted;
then OSHomQuot(F) is_isomorphism QuotOSAlg (U1,OSCng F),U2 by Th18;
hence thesis;
end;
:: monotone OSCongruence ... monotonicity is properly stronger
:: than MSCongruence, so we define it more broadly and prove the
:: ccluster then, however if used for other things than OSCongruences
:: the name of the attribute should be changed
:: this condition is nontrivial only for nonmonotone osas (see further),
:: where Den(o1,U1).x1 can differ from Den(o2,U2).x1
:: is OK for constants ... their Args is always {{}}, so o1 <= o2
:: implies for them [Den(o1,U1).{},Den(o2,U1).{}] in R
definition
let S be OrderSortedSign, U1 be non-empty OSAlgebra of S, R be
MSEquivalence-like OrderSortedRelation of U1;
attr R is monotone means
:Def26:
for o1,o2 being OperSymbol of S st o1 <= o2
for x1 being Element of Args(o1,U1) for x2 being Element of Args(o2,U1) st (
for y being Nat st y in dom x1 holds [x1.y,x2.y] in R.((the_arity_of o2)/.y) )
holds [Den(o1,U1).x1,Den(o2,U1).x2] in R.(the_result_sort_of o2);
end;
theorem Th20:
for S being OrderSortedSign, U1 being non-empty OSAlgebra of S
holds [| the Sorts of U1, the Sorts of U1 |] is OSCongruence of U1
proof
let S be OrderSortedSign, U1 be non-empty OSAlgebra of S;
reconsider C = [| the Sorts of U1, the Sorts of U1 |] as MSCongruence of U1
by MSUALG_5:18;
C is os-compatible
proof
reconsider O1 = (the Sorts of U1) as OrderSortedSet of S by OSALG_1:17;
let s1,s2 be Element of S such that
A1: s1 <= s2;
reconsider s3 = s1, s4 = s2 as Element of S;
A2: O1.s3 c= O1.s4 by A1,OSALG_1:def 16;
A3: C.s1 = [:(the Sorts of U1).s1,(the Sorts of U1).s1:] & C.s2 = [:(the
Sorts of U1).s2,(the Sorts of U1).s2:] by PBOOLE:def 16;
let x,y be set;
assume x in (the Sorts of U1).s1 & y in (the Sorts of U1).s1;
hence [x,y] in C.s1 iff [x,y] in C.s2 by A2,A3,ZFMISC_1:87;
end;
hence thesis by Def2;
end;
theorem Th21:
for S being OrderSortedSign, U1 being non-empty OSAlgebra of S,
R being OSCongruence of U1 st R = [| (the Sorts of U1), (the Sorts of U1) |]
holds R is monotone
proof
let S be OrderSortedSign, U1 be non-empty OSAlgebra of S, R be OSCongruence
of U1 such that
A1: R = [| (the Sorts of U1), (the Sorts of U1) |];
reconsider O1 = the Sorts of U1 as OrderSortedSet of S by OSALG_1:17;
let o1,o2 be OperSymbol of S such that
A2: o1 <= o2;
set s2 = the_result_sort_of o2, s1 = the_result_sort_of o1;
s1 <= s2 by A2;
then
A3: O1.s1 c= O1.s2 by OSALG_1:def 16;
let x1 be Element of Args(o1,U1);
let x2 be Element of Args(o2,U1) such that
for y being Nat st y in dom x1 holds [x1.y,x2.y] in R.((the_arity_of o2)
/.y);
A4: Den(o1,U1).x1 in (the Sorts of U1).s1 & Den(o2,U1).x2 in (the Sorts of
U1). s2 by MSUALG_9:18;
R.s2 = [:(the Sorts of U1).s2,(the Sorts of U1).s2:] by A1,PBOOLE:def 16;
hence thesis by A3,A4,ZFMISC_1:87;
end;
registration
let S be OrderSortedSign, U1 be non-empty OSAlgebra of S;
cluster monotone for OSCongruence of U1;
existence
proof
reconsider R = [| the Sorts of U1, the Sorts of U1 |] as OSCongruence of
U1 by Th20;
take R;
thus thesis by Th21;
end;
end;
registration
let S be OrderSortedSign, U1 be non-empty OSAlgebra of S;
cluster monotone for MSEquivalence-like OrderSortedRelation of U1;
existence
proof
set R = the monotone OSCongruence of U1;
take R;
thus thesis;
end;
end;
theorem Th22:
for S being OrderSortedSign, U1 being non-empty OSAlgebra of S,
R being monotone MSEquivalence-like OrderSortedRelation of U1 holds R is
MSCongruence-like
proof
let S be OrderSortedSign, U1 be non-empty OSAlgebra of S, R be monotone
MSEquivalence-like OrderSortedRelation of U1;
for o be (Element of the carrier' of S), x,y be Element of Args(o,U1) st
(for n be Nat st n in dom x holds [x.n,y.n] in R.((the_arity_of o)/.n)) holds [
Den(o,U1).x,Den(o,U1).y] in R.(the_result_sort_of o) by Def26;
hence thesis by MSUALG_4:def 4;
end;
registration
let S be OrderSortedSign, U1 be non-empty OSAlgebra of S;
cluster monotone -> MSCongruence-like for MSEquivalence-like
OrderSortedRelation of U1;
coherence by Th22;
end;
theorem Th23:
for S being OrderSortedSign, U1 being monotone non-empty
OSAlgebra of S, R being OSCongruence of U1 holds R is monotone
proof
let S be OrderSortedSign, U1 be monotone non-empty OSAlgebra of S, R be
OSCongruence of U1;
let o1,o2 be OperSymbol of S such that
A1: o1 <= o2;
let x1 be Element of Args(o1,U1);
Args(o1,U1) c= Args(o2,U1) by A1,OSALG_1:26;
then reconsider x3 = x1 as Element of Args(o2,U1);
let x2 be Element of Args(o2,U1);
assume
for y being Nat st y in dom x1 holds [x1.y,x2.y] in R.((the_arity_of o2)/.y);
then
A2: [Den(o2,U1).x3,Den(o2,U1).x2] in R.(the_result_sort_of o2) by
MSUALG_4:def 4;
x1 in Args(o1,U1);
then
A3: x1 in dom Den(o1,U1) by FUNCT_2:def 1;
Den(o1,U1) c= Den(o2,U1) by A1,OSALG_1:27;
hence thesis by A2,A3,GRFUNC_1:2;
end;
registration
let S be OrderSortedSign, U1 be monotone non-empty OSAlgebra of S;
cluster -> monotone for OSCongruence of U1;
coherence by Th23;
end;
:: monotonicity of quotient by monotone oscongruence
registration
let S;
let U1 be non-empty OSAlgebra of S;
let R be monotone OSCongruence of U1;
cluster QuotOSAlg(U1,R) -> monotone;
coherence
proof
reconsider O1 = the Sorts of U1 as OrderSortedSet of S by OSALG_1:17;
set A = QuotOSAlg(U1,R);
let o1,o2 be OperSymbol of S such that
A1: o1 <= o2;
A2: Args(o1,A) c= Args(o2,A) by A1,OSALG_1:26;
Den(o2,A) = (OSQuotCharact R).o2 by MSUALG_1:def 6;
then
A3: Den(o2,A) = OSQuotCharact( R,o2) by Def19;
o2 in the carrier' of S;
then
A4: o2 in dom (the ResultSort of S) by FUNCT_2:def 1;
Den(o1,A) = (OSQuotCharact R).o1 by MSUALG_1:def 6;
then
A5: Den(o1,A) = OSQuotCharact( R,o1) by Def19;
o1 in the carrier' of S;
then
A6: o1 in dom (the ResultSort of S) by FUNCT_2:def 1;
A7: (the_arity_of o1) <= (the_arity_of o2) by A1;
then len (the_arity_of o1) = len (the_arity_of o2);
then
A8: dom (the_arity_of o1) = dom (the_arity_of o2) by FINSEQ_3:29;
A9: the_result_sort_of o1 <= the_result_sort_of o2 by A1;
then
A10: O1.(the_result_sort_of o1) c= O1.(the_result_sort_of o2) by OSALG_1:def 17
;
A11: for x being object st x in dom Den(o1,A)
holds Den(o1,A).x = Den(o2,A).x
proof
let x be object;
assume x in dom Den(o1,A);
then
A12: x in Args(o1,A);
then
A13: x in ((OSClass R)# * the Arity of S).o1 by MSUALG_1:def 4;
then consider a1 being Element of Args(o1,U1) such that
A14: x = R #_os a1 by Th14;
Result(o1,U1) = ((the Sorts of U1) * the ResultSort of S).o1 by
MSUALG_1:def 5
.= (the Sorts of U1).((the ResultSort of S).o1) by A6,FUNCT_1:13
.= (the Sorts of U1).(the_result_sort_of o1) by MSUALG_1:def 2;
then reconsider da1 = Den(o1,U1).a1 as Element of (the Sorts of U1).(
the_result_sort_of o1);
reconsider da12 = da1 as Element of (the Sorts of U1).(
the_result_sort_of o2) by A10;
a1 in Args(o1,U1);
then a1 in dom Den(o1,U1) by FUNCT_2:def 1;
then
A15: ((OSQuotRes(R,o1)) * (Den(o1,U1))).a1 = OSQuotRes(R,o1).da1 by FUNCT_1:13
.= OSClass(R,da1) by Def14;
A16: OSQuotCharact(R,o1).(R #_os a1) = ((OSQuotRes(R,o1)) * (Den(o1,U1))
).a1 by A13,A14,Def18;
x in Args(o2,A) by A2,A12;
then
A17: x in ((OSClass R)# * the Arity of S).o2 by MSUALG_1:def 4;
then consider a2 being Element of Args(o2,U1) such that
A18: x = R #_os a2 by Th14;
for y being Nat st y in dom a1 holds [a1.y,a2.y] in R.((
the_arity_of o2)/.y)
proof
let y be Nat such that
A19: y in dom a1;
A20: y in dom (the_arity_of o1) by A19,MSUALG_6:2;
then consider
z1 being Element of (the Sorts of U1).((the_arity_of o1)/.y )
such that
A21: z1 = a1.y and
A22: (R #_os a1).y = OSClass(R, z1) by Def13;
reconsider s1 = (the_arity_of o1).y, s2 = (the_arity_of o2).y as
SortSymbol of S by A8,A20,PARTFUN1:4;
A23: y in dom (the_arity_of o2) by A8,A19,MSUALG_6:2;
then
A24: (the_arity_of o2)/.y = (the_arity_of o2).y by PARTFUN1:def 6;
A25: (the_arity_of o1)/.y = (the_arity_of o1).y & s1 <= s2 by A7,A20,
PARTFUN1:def 6;
then (the Sorts of U1).((the_arity_of o1)/.y) c= (the Sorts of U1).((
the_arity_of o2)/.y) by A24,OSALG_1:def 17;
then reconsider
z12 = z1 as Element of (the Sorts of U1).((the_arity_of o2)
/.y);
consider z2 being Element of (the Sorts of U1).((the_arity_of o2)/.y)
such that
A26: z2 = a2.y and
A27: (R #_os a2).y = OSClass(R, z2) by A23,Def13;
OSClass(R, z2) = OSClass(R, z12) by A14,A18,A22,A27,A24,A25,Th4;
hence thesis by A21,A26,Th12;
end;
then
A28: [Den(o1,U1).a1,Den(o2,U1).a2] in R.(the_result_sort_of o2) by A1,Def26;
Result(o2,U1) = ((the Sorts of U1) * the ResultSort of S).o2 by
MSUALG_1:def 5
.= (the Sorts of U1).((the ResultSort of S).o2) by A4,FUNCT_1:13
.= (the Sorts of U1).(the_result_sort_of o2) by MSUALG_1:def 2;
then reconsider da2 = Den(o2,U1).a2 as Element of (the Sorts of U1).(
the_result_sort_of o2);
a2 in Args(o2,U1);
then a2 in dom Den(o2,U1) by FUNCT_2:def 1;
then
A29: ((OSQuotRes(R,o2)) * (Den(o2,U1))).a2 = OSQuotRes(R,o2).da2 by FUNCT_1:13
.= OSClass(R,da2) by Def14;
OSClass(R,da1) = OSClass(R,da12) by A9,Th4
.= OSClass(R,da2) by A28,Th12;
hence thesis by A5,A3,A17,A14,A18,A16,A15,A29,Def18;
end;
dom Den(o2,A) = Args(o2,A) & dom Den(o1,A) = Args(o1,A) by FUNCT_2:def 1;
then dom Den(o1,A) = (dom Den(o2,A)) /\ Args(o1,A) by A2,XBOOLE_1:28;
hence thesis by A11,FUNCT_1:46;
end;
end;
theorem
for U1 being non-empty OSAlgebra of S, U2 being monotone non-empty
OSAlgebra of S, F being ManySortedFunction of U1,U2 st F is_homomorphism U1,U2
& F is order-sorted holds OSCng(F) is monotone
proof
let U1 be non-empty OSAlgebra of S, U2 be monotone non-empty OSAlgebra of S,
F be ManySortedFunction of U1,U2 such that
A1: F is_homomorphism U1,U2 and
A2: F is order-sorted;
reconsider S1 = the Sorts of U1 as OrderSortedSet of S by OSALG_1:17;
set O1 = the Sorts of U1;
let o1,o2 being OperSymbol of S such that
A3: o1 <= o2;
A4: Den(o2,U2)|Args(o1,U2) = Den(o1,U2) by A3,OSALG_1:def 21;
set R = OSCng(F), rs1 = the_result_sort_of o1, rs2 = the_result_sort_of o2;
let x1 being (Element of Args(o1,U1)), x2 being Element of Args(o2,U1) such
that
A5: for y being Nat st y in dom x1 holds [x1.y,x2.y] in R.((the_arity_of
o2)/.y);
Args(o1,U1) c= Args(o2,U1) by A3,OSALG_1:26;
then reconsider x12=x1 as Element of Args(o2,U1);
set D1 = Den(o1,U1).x1, D2 = Den(o2,U1).x2, M = MSCng(F);
A6: D1 in S1.rs1 by MSUALG_9:18;
F#x1 in Args(o1,U2);
then
A7: F#x1 in dom Den(o1,U2) by FUNCT_2:def 1;
A8: rs1 <= rs2 by A3;
then S1.rs1 c= S1.rs2 by OSALG_1:def 16;
then reconsider D11=D1,D12=Den(o2,U1).x12 as Element of O1.rs2 by
MSUALG_9:18;
D1 in dom (F.rs1) by A6,FUNCT_2:def 1;
then (F.rs2).(Den(o1,U1).x1) = (F.rs1).(Den(o1,U1).x1) by A2,A8
.= Den(o1,U2).(F#x1) by A1
.= Den(o2,U2).(F#x1) by A7,A4,FUNCT_1:47
.= Den(o2,U2).(F#x12) by A2,A3,OSALG_3:12
.= (F.rs2).(Den(o2,U1).x12) by A1;
then
A9: D2 in O1.rs2 & [D11,D12] in MSCng(F,rs2) by MSUALG_4:def 17,MSUALG_9:18;
field(R.rs2) = (O1.rs2) by ORDERS_1:12;
then
A10: (R.rs2) is_transitive_in (O1.rs2) by RELAT_2:def 16;
A11: [Den(o2,U1).x12,Den(o2,U1).x2] in R.rs2 by A5,MSUALG_4:def 4;
M.rs2 = MSCng(F,rs2) & M = R by A1,A2,Def23,MSUALG_4:def 18;
hence thesis by A11,A9,A10,RELAT_2:def 8;
end;
:: these are a bit more general versions of OSHomQuot, that
:: I need for OSAFREE; more proper way how to do this is restating
:: the MSUALG_9 quotient theorems for OSAs, but that's more work
definition
let S;
let U1,U2 be non-empty OSAlgebra of S;
let F be ManySortedFunction of U1,U2;
let R be OSCongruence of U1;
let s be Element of S;
assume that
A1: F is_homomorphism U1,U2 and
A2: F is order-sorted and
A3: R c= OSCng F;
func OSHomQuot(F,R,s) -> Function of (the Sorts of (QuotOSAlg (U1,R))).s,(
the Sorts of U2).s means
:Def27:
for x be Element of (the Sorts of U1).s holds it.(OSClass(R,x)) = (F.s).x;
existence
proof
set qa = QuotOSAlg (U1,R), cqa = the Sorts of qa, S1 = the Sorts of U1, S2
= the Sorts of U2;
defpred P[object,object] means
for a be Element of S1.s st $1 = OSClass(R,a)
holds $2 = (F.s).a;
A4: cqa.s = OSClass (R,s) by Def11;
A5: for x be object st x in cqa.s ex y be object st y in S2.s & P[x,y]
proof
let x be object;
A6: R.s c= (OSCng F).s by A3,PBOOLE:def 2;
assume x in cqa.s;
then consider a being set such that
A7: a in (the Sorts of U1).s and
A8: x = Class( CompClass(R,CComp(s)), a) by A4,Def10;
reconsider a as Element of S1.s by A7;
take y = (F.s).a;
thus y in S2.s;
let b be Element of S1.s;
assume
A9: x = OSClass(R,b);
x = OSClass(R,a) by A8;
then [b,a] in R.s by A9,Th12;
then [b,a] in (OSCng(F)).s by A6;
then [b,a] in (MSCng(F)).s by A1,A2,Def23;
then [b,a] in MSCng(F,s) by A1,MSUALG_4:def 18;
hence thesis by MSUALG_4:def 17;
end;
consider G being Function such that
A10: dom G = cqa.s & rng G c= S2.s & for x be object st x in cqa.s holds
P[x,G.x] from FUNCT_1:sch 6(A5);
reconsider G as Function of cqa.s,S2.s by A10,FUNCT_2:def 1,RELSET_1:4;
take G;
let a be Element of S1.s;
thus thesis by A4,A10;
end;
uniqueness
proof
set qa = QuotOSAlg (U1, R), cqa = the Sorts of qa, u1 = the Sorts of U1,
u2 = the Sorts of U2;
let H,G be Function of cqa.s,u2.s;
assume that
A11: for a be Element of u1.s holds H.(OSClass(R,a)) = (F.s).a and
A12: for a be Element of u1.s holds G.(OSClass(R,a)) = (F.s).a;
A13: cqa.s = OSClass (R,s) by Def11;
for x be object st x in cqa.s holds H.x = G.x
proof
let x be object;
assume x in cqa.s;
then consider y being set such that
A14: y in (the Sorts of U1).s and
A15: x = Class( CompClass(R,CComp(s)), y) by A13,Def10;
reconsider y1 = y as Element of u1.s by A14;
A16: OSClass(R,y1) = x by A15;
hence H.x = (F.s).y1 by A11
.= G.x by A12,A16;
end;
hence thesis by FUNCT_2:12;
end;
end;
:: this seems a bit too permissive, but same as the original
:: we should assume F order-sorted probably
definition
let S;
let U1,U2 be non-empty OSAlgebra of S;
let F be ManySortedFunction of U1,U2;
let R be OSCongruence of U1;
func OSHomQuot(F,R) -> ManySortedFunction of (QuotOSAlg (U1, R)),U2 means
:
Def28: for s be Element of S holds it.s = OSHomQuot(F,R,s);
existence
proof
deffunc F(Element of S) = OSHomQuot(F,R,$1);
consider f be Function such that
A1: dom f = the carrier of S & for d be Element of S holds f.d = F(d)
from FUNCT_1:sch 4;
reconsider f as ManySortedSet of the carrier of S by A1,PARTFUN1:def 2
,RELAT_1:def 18;
for x be object st x in dom f holds f.x is Function
proof
let x be object;
assume x in dom f;
then reconsider y = x as Element of S;
f.y = OSHomQuot(F,R,y) by A1;
hence thesis;
end;
then reconsider f as ManySortedFunction of the carrier of S by
FUNCOP_1:def 6;
for i be object st i in the carrier of S holds f.i is Function of (the
Sorts of QuotOSAlg (U1, R)).i, (the Sorts of U2).i
proof
let i be object;
assume i in the carrier of S;
then reconsider s = i as Element of S;
f.s = OSHomQuot(F,R,s) by A1;
hence thesis;
end;
then reconsider
f as ManySortedFunction of the Sorts of QuotOSAlg (U1,R),the
Sorts of U2 by PBOOLE:def 15;
reconsider f as ManySortedFunction of QuotOSAlg (U1,R),U2;
take f;
thus thesis by A1;
end;
uniqueness
proof
let H,G be ManySortedFunction of QuotOSAlg (U1,R),U2;
assume that
A2: for s be Element of S holds H.s = OSHomQuot(F,R,s) and
A3: for s be Element of S holds G.s = OSHomQuot(F,R,s);
now
let i be object;
assume i in the carrier of S;
then reconsider s = i as SortSymbol of S;
H.s = OSHomQuot(F,R,s) by A2;
hence H.i = G . i by A3;
end;
hence thesis by PBOOLE:3;
end;
end;
theorem
for U1,U2 be non-empty OSAlgebra of S, F be ManySortedFunction of U1,
U2, R be OSCongruence of U1 st F is_homomorphism U1,U2 & F is order-sorted & R
c= OSCng F holds OSHomQuot(F,R) is_homomorphism QuotOSAlg (U1,R),U2 & OSHomQuot
(F,R) is order-sorted
proof
let U1,U2 be non-empty OSAlgebra of S, F be ManySortedFunction of U1,U2, R
be OSCongruence of U1;
set mc = R, qa = QuotOSAlg (U1,mc), qh = OSHomQuot(F,R), S1 = the Sorts of
U1;
assume that
A1: F is_homomorphism U1,U2 and
A2: F is order-sorted and
A3: R c= OSCng F;
for o be Element of the carrier' of S st Args (o,qa) <> {} for x be
Element of Args(o,qa) holds (qh.(the_result_sort_of o)).(Den(o,qa).x) = Den(o,
U2).(qh#x)
proof
let o be Element of the carrier' of S such that
Args (o,qa) <> {};
let x be Element of Args(o,qa);
reconsider o1 = o as OperSymbol of S;
set ro = the_result_sort_of o, ar = the_arity_of o;
A4: dom x = dom ar by MSUALG_3:6;
Args(o,qa) = ((OSClass mc)# * (the Arity of S)).o by MSUALG_1:def 4;
then consider a be Element of Args(o,U1) such that
A5: x = mc #_os a by Th14;
A6: dom a = dom ar by MSUALG_3:6;
A7: now
let y be object;
assume
A8: y in dom ar;
then reconsider n = y as Nat;
ar.n in rng ar by A8,FUNCT_1:def 3;
then reconsider s = ar.n as SortSymbol of S;
A9: ar/.n = ar.n by A8,PARTFUN1:def 6;
then consider an being Element of S1.s such that
A10: an = a.n and
A11: x.n = OSClass(mc,an) by A5,A8,Def13;
(qh # x).n = (qh.s).(x.n) by A4,A8,A9,MSUALG_3:def 6
.= OSHomQuot(F,R,s).OSClass(mc,an) by A11,Def28
.= (F.s).an by A1,A2,A3,Def27
.= (F # a).n by A6,A8,A9,A10,MSUALG_3:def 6;
hence (qh # x).y = (F # a).y;
end;
o in the carrier' of S;
then o in dom (S1 * the ResultSort of S) by PARTFUN1:def 2;
then
A12: ((the Sorts of U1) * the ResultSort of S).o = (the Sorts of U1).((the
ResultSort of S).o) by FUNCT_1:12
.= (the Sorts of U1).ro by MSUALG_1:def 2;
then rng Den(o,U1) c= Result(o,U1) & Result(o,U1) = S1.ro by MSUALG_1:def 5
;
then rng Den(o,U1) c= dom OSQuotRes(mc,o) by A12,FUNCT_2:def 1;
then
A13: dom Den(o,U1) = Args(o,U1) & dom ((OSQuotRes(mc,o)) * Den(o,U1)) =
dom Den(o,U1) by FUNCT_2:def 1,RELAT_1:27;
ar = (the Arity of S).o by MSUALG_1:def 1;
then
A14: product((OSClass mc) * ar) = ((OSClass mc)# * the Arity of S).o by
MSAFREE:1;
reconsider da = (Den(o,U1)).a as Element of S1.ro by A12,MSUALG_1:def 5;
A15: qh.ro = OSHomQuot(F,R,ro) by Def28;
Den(o,qa) = (OSQuotCharact mc).o by MSUALG_1:def 6
.= OSQuotCharact(mc,o1) by Def19;
then Den(o,qa).x = (OSQuotRes(mc,o) * Den(o,U1)).a by A5,A14,Def18
.= (OSQuotRes(mc,o)) . da by A13,FUNCT_1:12
.= OSClass (R,da) by Def14;
then
A16: (qh.ro).(Den(o,qa).x) = (F.ro).((Den(o,U1)).a) by A1,A2,A3,A15,Def27
.= Den(o,U2).(F#a) by A1;
dom (qh # x) = dom ar & dom (F # a) = dom ar by MSUALG_3:6;
hence thesis by A7,A16,FUNCT_1:2;
end;
hence qh is_homomorphism qa,U2;
thus OSHomQuot(F,R) is order-sorted
proof
reconsider S1O = S1 as OrderSortedSet of S by OSALG_1:17;
reconsider sqa = the Sorts of qa as OrderSortedSet of S;
let s1,s2 being Element of S such that
A17: s1 <= s2;
let a1 be set such that
A18: a1 in dom (qh.s1);
a1 in (OSClass R).s1 by A18;
then a1 in OSClass (R,s1) by Def11;
then consider x being set such that
A19: x in S1.s1 and
A20: a1 = Class( CompClass(R,CComp(s1)), x) by Def10;
S1O.s1 c= S1O.s2 by A17,OSALG_1:def 16;
then reconsider x2 = x as Element of S1.s2 by A19;
A21: a1 = OSClass(R,x2) by A17,A20,Th4;
reconsider s3 = s1, s4 = s2 as Element of S;
A22: dom (qh.s1) = (the Sorts of qa).s1 & dom (qh.s2) = (the Sorts of qa).
s2 by FUNCT_2:def 1;
reconsider x1 = x as Element of S1.s1 by A19;
x1 in dom (F.s3) by A19,FUNCT_2:def 1;
then
A23: (F.s3).x1 = (F.s4).x1 by A2,A17;
sqa.s1 c= sqa.s2 by A17,OSALG_1:def 16;
hence a1 in dom (qh.s2) by A18,A22;
thus (qh.s1).a1 = OSHomQuot(F,R,s1).(OSClass(R,x1)) by A20,Def28
.= (F.s2).x1 by A1,A2,A3,A23,Def27
.= OSHomQuot(F,R,s2).(OSClass(R,x2)) by A1,A2,A3,Def27
.= (qh.s2).a1 by A21,Def28;
end;
end;