:: Reduction Relations
:: by Grzegorz Bancerek
::
:: Received November 14, 1995
:: Copyright (c) 1995-2018 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, FINSEQ_1, ORDINAL4, XBOOLE_0, SUBSET_1, ARYTM_3,
RELAT_1, XXREAL_0, CARD_1, NAT_1, FUNCT_1, FINSEQ_5, ARYTM_1, TARSKI,
WELLORD1, PBOOLE, RELAT_2, FUNCOP_1, EQREL_1, REWRITE1;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XXREAL_0,
NAT_1, FUNCT_1, RELAT_1, RELAT_2, WELLORD1, EQREL_1, FINSEQ_1, FUNCOP_1,
PBOOLE, FINSEQ_5;
constructors SETFAM_1, WELLORD1, XXREAL_0, XREAL_0, NAT_1, EQREL_1, PBOOLE,
FINSEQ_5, RELSET_1, NUMBERS;
registrations XBOOLE_0, RELAT_1, FUNCT_1, ORDINAL1, PARTFUN1, NAT_1, XXREAL_0,
XREAL_0, FINSEQ_1, CARD_1;
requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM;
begin :: Forgetting concatenation and reduction sequence
definition
let p,q be FinSequence;
func p$^q -> FinSequence means
:: REWRITE1:def 1
it = p^q if p = {} or q = {} otherwise
ex i being Nat, r being FinSequence st len p = i+1 & r = p|Seg i &
it = r^q;
end;
reserve p,q,r for FinSequence,
x,y for object;
theorem :: REWRITE1:1
{}$^p = p & p$^{} = p;
theorem :: REWRITE1:2
q <> {} implies (p^<*x*>)$^q = p^q;
theorem :: REWRITE1:3
(p^<*x*>)$^(<*y*>^q) = p^<*y*>^q;
theorem :: REWRITE1:4
q <> {} implies <*x*>$^q = q;
theorem :: REWRITE1:5
p <> {} implies ex x,q st p = <*x*>^q & len p = len q+1;
scheme :: REWRITE1:sch 1
PathCatenation {P[set,set], p,q() -> FinSequence}:
for i being Nat st i in dom (p()$^q()) & i+1 in dom (p()$^q())
for x,y being set st x = (p( )$^q()).i & y = (p()$^q()).(i+1) holds P[x,y]
provided
for i being Nat st i in dom p() & i+1 in dom p() holds P[
p().i, p().(i+1)] and
for i being Nat st i in dom q() & i+1 in dom q() holds P[
q().i, q().(i+1)] and
len p() > 0 & len q() > 0 & p().len p() = q().1;
definition
let R be Relation;
mode RedSequence of R -> FinSequence means
:: REWRITE1:def 2
len it > 0 & for i being
Nat st i in dom it & i+1 in dom it holds [it.i, it.(i+1)] in R;
end;
registration
let R be Relation;
cluster -> non empty for RedSequence of R;
end;
theorem :: REWRITE1:6
for R being Relation, a being object holds <*a*> is RedSequence of R;
theorem :: REWRITE1:7
for R being Relation, a,b being object st [a,b] in R holds <*a,b*>
is RedSequence of R;
theorem :: REWRITE1:8
for R being Relation, p,q being RedSequence of R st p.len p = q.1
holds p$^q is RedSequence of R;
theorem :: REWRITE1:9
for R being Relation, p being RedSequence of R holds Rev p is
RedSequence of R~;
theorem :: REWRITE1:10
for R,Q being Relation st R c= Q for p being RedSequence of R
holds p is RedSequence of Q;
begin :: Reducibility, convertibility, and normal forms
definition
let R be Relation;
let a,b be object;
pred R reduces a,b means
:: REWRITE1:def 3
ex p being RedSequence of R st p.1 = a & p. len p = b;
end;
definition
let R be Relation;
let a,b be object;
pred a,b are_convertible_wrt R means
:: REWRITE1:def 4
R \/ R~ reduces a,b;
end;
theorem :: REWRITE1:11
for R being Relation, a,b being object holds R reduces a,b iff ex p
being FinSequence st len p > 0 & p.1 = a & p.len p = b &
for i being Nat st i in dom p & i+1 in dom p holds [p.i, p.(i+1)] in R;
theorem :: REWRITE1:12
for R being Relation, a being object holds R reduces a,a;
theorem :: REWRITE1:13
for a,b being object st {} reduces a,b holds a = b;
theorem :: REWRITE1:14
for R being Relation, a,b being object st R reduces a,b & not a in
field R holds a = b;
theorem :: REWRITE1:15
for R being Relation, a,b being object st [a,b] in R holds R reduces a,b;
theorem :: REWRITE1:16
for R being Relation, a,b,c being object st R reduces a,b & R
reduces b,c holds R reduces a,c;
theorem :: REWRITE1:17
for R being Relation, p being RedSequence of R, i,j being
Nat st i in dom p & j in dom p & i <= j holds R reduces p.i,p.j;
theorem :: REWRITE1:18
for R being Relation, a,b being object st R reduces a,b & a <> b
holds a in field R & b in field R;
theorem :: REWRITE1:19
for R being Relation, a,b being object st R reduces a,b holds a in field
R iff b in field R;
theorem :: REWRITE1:20
for R being Relation, a,b being object holds R reduces a,b iff a =
b or [a,b] in R[*];
theorem :: REWRITE1:21
for R being Relation, a,b being object
holds R reduces a,b iff R[*] reduces a,b;
theorem :: REWRITE1:22
for R,Q being Relation st R c= Q for a,b being object st R reduces
a,b holds Q reduces a,b;
theorem :: REWRITE1:23
for R being Relation, X being set, a,b being object holds R reduces a,b
iff R \/ id X reduces a,b;
theorem :: REWRITE1:24
for R being Relation, a,b being object st R reduces a,b
holds R~ reduces b,a;
theorem :: REWRITE1:25
for R being Relation, a,b being object st R reduces a,b holds a,b
are_convertible_wrt R & b,a are_convertible_wrt R;
theorem :: REWRITE1:26
for R being Relation, a being object holds a,a are_convertible_wrt R;
theorem :: REWRITE1:27
for a,b being object st a,b are_convertible_wrt {} holds a = b;
theorem :: REWRITE1:28
for R being Relation, a,b being object st a,b are_convertible_wrt R & not
a in field R holds a = b;
theorem :: REWRITE1:29
for R being Relation, a,b being object st [a,b] in R holds a,b
are_convertible_wrt R;
theorem :: REWRITE1:30
for R being Relation, a,b,c being object st a,b are_convertible_wrt
R & b,c are_convertible_wrt R holds a,c are_convertible_wrt R;
theorem :: REWRITE1:31
for R being Relation, a,b being object st a,b are_convertible_wrt R holds
b,a are_convertible_wrt R;
theorem :: REWRITE1:32
for R being Relation, a,b being object st a,b are_convertible_wrt R
& a <> b holds a in field R & b in field R;
definition
let R be Relation;
let a be object;
pred a is_a_normal_form_wrt R means
:: REWRITE1:def 5
not ex b being object st [a,b] in R;
end;
theorem :: REWRITE1:33
for R being Relation, a,b being object st a is_a_normal_form_wrt R
& R reduces a,b holds a = b;
theorem :: REWRITE1:34
for R being Relation, a being object st not a in field R holds a
is_a_normal_form_wrt R;
definition
let R be Relation;
let a,b be object;
pred b is_a_normal_form_of a,R means
:: REWRITE1:def 6
b is_a_normal_form_wrt R & R reduces a,b;
pred a,b are_convergent_wrt R means
:: REWRITE1:def 7
ex c being object st R reduces a,c & R reduces b,c;
pred a,b are_divergent_wrt R means
:: REWRITE1:def 8
ex c being object st R reduces c,a & R reduces c,b;
pred a,b are_convergent<=1_wrt R means
:: REWRITE1:def 9
ex c being object st ([a,c] in R or a = c) & ([b,c] in R or b = c);
pred a,b are_divergent<=1_wrt R means
:: REWRITE1:def 10
ex c being object st ([c,a] in R or a = c) & ([c,b] in R or b = c);
end;
theorem :: REWRITE1:35
for R being Relation, a being object st not a in field R holds a
is_a_normal_form_of a,R;
theorem :: REWRITE1:36
for R being Relation, a,b being object st R reduces a,b holds a,b
are_convergent_wrt R & a,b are_divergent_wrt R & b,a are_convergent_wrt R & b,a
are_divergent_wrt R;
theorem :: REWRITE1:37
for R being Relation, a,b being object st a,b are_convergent_wrt R
or a,b are_divergent_wrt R holds a,b are_convertible_wrt R;
theorem :: REWRITE1:38
for R being Relation, a being object holds a,a are_convergent_wrt R
& a,a are_divergent_wrt R;
theorem :: REWRITE1:39
for a,b being object st a,b are_convergent_wrt {} or a,b
are_divergent_wrt {} holds a = b;
theorem :: REWRITE1:40
for R being Relation, a,b being object st a,b are_convergent_wrt R holds
b,a are_convergent_wrt R;
theorem :: REWRITE1:41
for R being Relation, a,b being object st a,b are_divergent_wrt R holds b
,a are_divergent_wrt R;
theorem :: REWRITE1:42
for R being Relation, a,b,c being object st R reduces a,b & b,c
are_convergent_wrt R or a,b are_convergent_wrt R & R reduces c,b holds a,c
are_convergent_wrt R;
theorem :: REWRITE1:43
for R being Relation, a,b,c being object st R reduces b,a & b,c
are_divergent_wrt R or a,b are_divergent_wrt R & R reduces b,c holds a,c
are_divergent_wrt R;
theorem :: REWRITE1:44
for R being Relation, a,b being object st a,b are_convergent<=1_wrt
R holds a,b are_convergent_wrt R;
theorem :: REWRITE1:45
for R being Relation, a,b being object st a,b are_divergent<=1_wrt
R holds a,b are_divergent_wrt R;
definition
let R be Relation;
let a be object;
pred a has_a_normal_form_wrt R means
:: REWRITE1:def 11
ex b being object st b is_a_normal_form_of a,R;
end;
theorem :: REWRITE1:46
for R being Relation, a being object st not a in field R holds a
has_a_normal_form_wrt R;
definition
let R be Relation, a be object;
assume that
a has_a_normal_form_wrt R and
for b,c being object st b is_a_normal_form_of a,R & c
is_a_normal_form_of a,R holds b = c;
func nf(a,R) -> set means
:: REWRITE1:def 12
it is_a_normal_form_of a,R;
end;
begin :: Terminating reductions
definition
let R be Relation;
attr R is co-well_founded means
:: REWRITE1:def 13
R~ is well_founded;
attr R is weakly-normalizing means
:: REWRITE1:def 14
for a being object st a in field R holds a has_a_normal_form_wrt R;
attr R is strongly-normalizing means
:: REWRITE1:def 15 :: terminating, Noetherian
for f being ManySortedSet of NAT ex i
being Nat st not [f.i,f.(i+1)] in R;
end;
definition
let R be Relation;
redefine attr R is co-well_founded means
:: REWRITE1:def 16
for Y being set st Y c= field R & Y <> {}
ex a being object st a in Y & for b being object st b in Y & a <> b
holds not [a,b] in R;
end;
scheme :: REWRITE1:sch 2
coNoetherianInduction{R() -> Relation, P[object]}:
for a being object st a in field R() holds P[a]
provided
R() is co-well_founded and
for a being object
st for b being object st [a,b] in R() & a <> b holds P[b]
holds P[a];
registration
cluster strongly-normalizing -> irreflexive co-well_founded for Relation;
cluster co-well_founded irreflexive -> strongly-normalizing for Relation;
end;
registration
cluster empty -> weakly-normalizing strongly-normalizing for Relation;
end;
theorem :: REWRITE1:47
for Q being co-well_founded Relation, R being Relation st R c= Q holds
R is co-well_founded;
registration
cluster strongly-normalizing -> weakly-normalizing for Relation;
end;
begin :: Church-Rosser property
definition
let R,Q be Relation;
pred R commutes-weakly_with Q means
:: REWRITE1:def 17
for a,b,c being object st [a,b] in R & [a,c
] in Q ex d being object st Q reduces b,d & R reduces c,d;
symmetry;
pred R commutes_with Q means
:: REWRITE1:def 18
for a,b,c being object st R reduces a,b &
Q reduces a,c ex d being object st Q reduces b,d & R reduces c,d;
symmetry;
end;
theorem :: REWRITE1:48
for R,Q being Relation st R commutes_with Q holds R commutes-weakly_with Q;
definition
let R be Relation;
attr R is with_UN_property means
:: REWRITE1:def 19
for a,b being object st a
is_a_normal_form_wrt R & b is_a_normal_form_wrt R & a,b are_convertible_wrt R
holds a = b;
attr R is with_NF_property means
:: REWRITE1:def 20
for a,b being object st a is_a_normal_form_wrt
R & a,b are_convertible_wrt R holds R reduces b,a;
attr R is subcommutative means
:: REWRITE1:def 21
for a,b,c being object st [a,b] in R &
[a,c] in R holds b,c are_convergent<=1_wrt R;
attr R is confluent means
:: REWRITE1:def 22
for a,b being object st a,b are_divergent_wrt R
holds a,b are_convergent_wrt R;
attr R is with_Church-Rosser_property means
:: REWRITE1:def 23
for a,b being object st a,b are_convertible_wrt R
holds a,b are_convergent_wrt R;
attr R is locally-confluent means
:: REWRITE1:def 24
for a,b,c being object st [a,b] in R
& [a,c] in R holds b,c are_convergent_wrt R;
end;
theorem :: REWRITE1:49
for R being Relation st R is subcommutative for a,b,c being object
st R reduces a,b & [a,c] in R holds b,c are_convergent_wrt R;
theorem :: REWRITE1:50
for R being Relation holds R is confluent iff R commutes_with R;
theorem :: REWRITE1:51
for R being Relation holds R is confluent iff for a,b,c being
object st R reduces a,b & [a,c] in R holds b,c are_convergent_wrt R;
theorem :: REWRITE1:52
for R being Relation holds R is locally-confluent iff R
commutes-weakly_with R;
registration
cluster with_Church-Rosser_property -> confluent for Relation;
cluster confluent -> locally-confluent with_Church-Rosser_property for
Relation;
cluster subcommutative -> confluent for Relation;
cluster with_Church-Rosser_property -> with_NF_property for Relation;
cluster with_NF_property -> with_UN_property for Relation;
cluster with_UN_property weakly-normalizing -> with_Church-Rosser_property
for Relation;
end;
registration
cluster empty -> subcommutative for Relation;
end;
theorem :: REWRITE1:53
for R being with_UN_property Relation for a,b,c being object st b
is_a_normal_form_of a,R & c is_a_normal_form_of a,R holds b = c;
theorem :: REWRITE1:54
for R being with_UN_property weakly-normalizing Relation for a
being object holds nf(a,R) is_a_normal_form_of a,R;
theorem :: REWRITE1:55
for R being with_UN_property weakly-normalizing Relation for a,b
being object st a,b are_convertible_wrt R holds nf(a,R) = nf(b,R);
registration
cluster strongly-normalizing locally-confluent -> confluent for Relation;
end;
definition
let R be Relation;
attr R is complete means
:: REWRITE1:def 25
R is confluent strongly-normalizing;
end;
registration
cluster complete -> confluent strongly-normalizing for Relation;
cluster confluent strongly-normalizing -> complete for Relation;
end;
registration
cluster complete for non empty Relation;
end;
theorem :: REWRITE1:56
for R,Q being with_Church-Rosser_property Relation st R commutes_with
Q holds R \/ Q is with_Church-Rosser_property;
theorem :: REWRITE1:57
for R being Relation holds R is confluent iff R[*] is locally-confluent;
theorem :: REWRITE1:58
for R being Relation holds R is confluent iff R[*] is subcommutative;
begin :: Completion method
definition
let R,Q be Relation;
pred R,Q are_equivalent means
:: REWRITE1:def 26
for a,b being object holds a,b
are_convertible_wrt R iff a,b are_convertible_wrt Q;
symmetry;
end;
definition
let R be Relation;
let a,b be object;
pred a,b are_critical_wrt R means
:: REWRITE1:def 27
a,b are_divergent<=1_wrt R & not a,b are_convergent_wrt R;
end;
theorem :: REWRITE1:59
for R being Relation, a,b being object st a,b are_critical_wrt R
holds a,b are_convertible_wrt R;
theorem :: REWRITE1:60
for R being Relation st not ex a,b being object st a,b are_critical_wrt R
holds R is locally-confluent;
theorem :: REWRITE1:61
for R,Q being Relation st for a,b being object st [a,b] in Q holds a,b
are_critical_wrt R holds R, R \/ Q are_equivalent;
theorem :: REWRITE1:62
for R being Relation ex Q being complete Relation st field Q c=
field R & for a,b being object holds a,b are_convertible_wrt R iff a,b
are_convergent_wrt Q;
definition
let R be Relation;
mode Completion of R -> complete Relation means
:: REWRITE1:def 28
for a,b being object
holds a,b are_convertible_wrt R iff a,b are_convergent_wrt it;
end;
theorem :: REWRITE1:63
for R being Relation, C being Completion of R holds R,C are_equivalent;
theorem :: REWRITE1:64
for R being Relation, Q being complete Relation st R,Q are_equivalent
holds Q is Completion of R;
theorem :: REWRITE1:65
for R being Relation, C being Completion of R, a,b being object holds a,b
are_convertible_wrt R iff nf(a,C) = nf(b,C);