:: A Model of Mizar Concepts -- Unification
:: by Grzegorz Bancerek
::
:: Received November 20, 2009
:: Copyright (c) 2009-2017 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 TARSKI, QC_LANG3, PBOOLE, MSUALG_1, CATALG_1, FINSEQ_1, XBOOLE_0,
ZFMISC_1, ARYTM_3, CARD_1, NAT_1, NUMBERS, XXREAL_0, ZF_LANG1, ORDINAL1,
TREES_A, ABIAN, CARD_3, MEMBER_1, FINSET_1, FUNCOP_1, FUNCT_1, TREES_4,
TREES_2, MSATERM, RELAT_1, MCART_1, MSAFREE, ZF_MODEL, AOFA_000,
FINSEQ_2, PARTFUN1, QC_LANG1, FUNCT_2, ORDINAL4, CAT_3, TREES_3,
ABCMIZ_0, ABCMIZ_1, ABCMIZ_A, STRUCT_0, FACIRC_1, INSTALG1, MSUALG_2,
COMPUT_1, BINTREE1, TREES_9, ARYTM_1, FUNCT_6, SUBSET_1, MARGREL1,
SETLIM_2;
notations TARSKI, XBOOLE_0, ZFMISC_1, XTUPLE_0, XFAMILY, SUBSET_1, DOMAIN_1,
RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCOP_1,
XCMPLX_0, XXREAL_0, ORDINAL1, NAT_D, MCART_1, FINSET_1, CARD_1, NUMBERS,
CARD_3, FINSEQ_1, FINSEQ_2, FINSEQ_4, FUNCT_6, TREES_1, TREES_2, TREES_3,
TREES_4, TREES_9, PBOOLE, STRUCT_0, MSUALG_1, MSUALG_2, MSAFREE,
EQUATION, MSATERM, INSTALG1, CATALG_1, MSAFREE3, AOFA_000, ABCMIZ_1;
constructors RELSET_1, DOMAIN_1, WELLORD2, TREES_9, EQUATION, NAT_D, FINSEQ_4,
CATALG_1, FACIRC_1, ABCMIZ_1, PRE_POLY, XTUPLE_0, XFAMILY;
registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, FUNCT_1, FINSET_1,
STRUCT_0, PBOOLE, MSUALG_2, FINSEQ_1, NAT_1, CARD_1, TREES_3, TREES_2,
FUNCOP_1, RELAT_1, INDEX_1, INSTALG1, MSAFREE3, WAYBEL26, FACIRC_1,
ABCMIZ_1, MSATERM, RELSET_1, XTUPLE_0;
requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL;
definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, PBOOLE, ABCMIZ_1;
equalities TARSKI, SUBSET_1, FINSEQ_1, MSAFREE, MSAFREE3, MSUALG_1, ABCMIZ_1,
ORDINAL1, CARD_1;
expansions TARSKI, XBOOLE_0, FUNCT_1, PBOOLE, ABCMIZ_1, ORDINAL1;
theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, XREAL_1, ZFMISC_1,
FUNCT_1, FUNCT_2, FINSEQ_1, FINSEQ_2, ENUMSET1, FUNCT_6, INSTALG1, NAT_1,
MCART_1, PBOOLE, RELAT_1, RELSET_1, CARD_1, CARD_5, ORDINAL1, MSUALG_2,
TREES_3, TREES_4, FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3,
YELLOW11, PARTFUN1, WELLORD2, ABCMIZ_1, TREES_9, XTUPLE_0, XREGULAR;
schemes FUNCT_1, NAT_1, RECDEF_1, CLASSES1, FINSEQ_1;
begin :: Preliminary
reserve i,j for Nat;
theorem
for x being pair object holds x = [x`1, x`2];
scheme MinimalElement{X() -> finite non empty set, R[set,set]}:
ex x being set st x in X() &
for y being set st y in X() holds not R[y,x]
provided
A1: for x,y being set st x in X() & y in X() & R[x,y] holds not R[y,x] and
A2: for x,y,z being set st x in X() & y in X() & z in X() & R[x,y] & R[y,z]
holds R[x,z]
proof
assume
A3: for x being set st x in X() ex y being set st y in X() & R[y,x];
set n = card X();
set x0 = the Element of X();
defpred P[Nat,set,set] means $2 in X() implies $3 in X() & R[ $3,$2];
A4: for m being Nat st 1 <= m & m < n+1
for x being set ex y being set st P[m,x,y]
proof
let m be Nat; assume 1 <= m & m < n+1;
let x be set;
per cases;
suppose
A5: x nin X();
set y = the set;
take y;
thus P[m,x,y] by A5;
end;
suppose
x in X(); then
consider y being set such that
A6: y in X() & R[y,x] by A3;
take y; thus thesis by A6;
end;
end;
consider p being FinSequence such that
A7: len p = n+1 and
A8: p.1 = x0 or n+1 = 0 and
A9: for i being Nat st 1 <= i & i < n+1 holds P[i, p.i, p.(i+1)]
from RECDEF_1:sch 3(A4);
defpred Q[Nat] means $1 in dom p implies p.$1 in X();
A10: Q[ 0] by FINSEQ_3:25;
A11: now let i be Nat; assume
A12: Q[i];
thus Q[i+1] proof assume
i+1 in dom p; then i+1 <= n+1 by A7,FINSEQ_3:25; then
A13: i < n+1 by NAT_1:13;
per cases;
suppose i = 0;
hence thesis by A8;
end;
suppose i > 0; then
i >= 0+1 & i is Element of NAT by NAT_1:13,ORDINAL1:def 12;
hence thesis by A12,A7,A9,A13,FINSEQ_3:25;
end;
end;
end;
A14: for i being Nat holds Q[i] from NAT_1:sch 2(A10,A11);
A15: rng p c= X()
proof
let x be object; assume x in rng p; then
ex i being object st i in dom p & x = p.i by FUNCT_1:def 3;
hence thesis by A14;
end;
A16: for i,j being Nat st 1 <= i & i < j & j <= n+1 holds R[p.j, p.i]
proof
let i,j be Nat; assume
A17: 1 <= i;
assume
A18: i < j; then
i+1 <= j by NAT_1:13; then
consider k being Nat such that
A19: j = i+1+k by NAT_1:10;
assume
A20: j <= n+1; then i <= n+1 by A18,XXREAL_0:2; then
A21: i in dom p by A17,A7,FINSEQ_3:25;
defpred S[Nat] means i+1+$1 <= n+1 implies R[p.(i+1+$1), p.i];
A22: S[ 0] proof assume i+1+0 <= n+1; then
A23: i < n+1 by NAT_1:13;
p.i in X() & i is Element of NAT by A14,A21;
hence R[p.(i+1+0), p.i] by A9,A17,A23;
end;
A24: now let k be Nat; assume
A25: S[k];
thus S[k+1] proof assume
A26: i+1+(k+1) <= n+1;
A27: i+1+(k+1) = i+1+k+1; then
A28: i+1+k < n+1 by A26,NAT_1:13;
A29: p.i in X() by A14,A21;
i+1+k = 1+(i+k); then
A30: 1 <= i+1+k by NAT_1:11; then
i+1+k in dom p by A7,A28,FINSEQ_3:25; then
A31: p.(i+1+k) in X() & i+1+k is Element of NAT by A14; then
p.(i+1+(k+1)) in X() & R[p.(i+1+(k+1)), p.(i+1+k)] by A9,A28,A27,A30;
hence R[p.(i+1+(k+1)), p.i] by A2,A28,A25,A31,A29;
end;
end;
for k being Nat holds S[k] from NAT_1:sch 2(A22,A24);
hence R[p.j, p.i] by A19,A20;
end;
A32: dom p = Seg(n+1) & card Seg(n+1) = n+1 by A7,FINSEQ_1:57,def 3;
Segm card rng p c= Segm card X() by A15,CARD_1:11; then
card rng p <= n & n < n+1 by NAT_1:19,39; then
not dom p, rng p are_equipotent by A32,CARD_1:5; then
p is not one-to-one by WELLORD2:def 4; then
consider i,j being object such that
A33: i in dom p & j in dom p & p.i = p.j & i <> j;
reconsider i,j as Nat by A33;
A34: 1 <= i & 1 <= j & i <= n+1 & j <= n+1 by A7,A33,FINSEQ_3:25;
p.i in rng p by A33,FUNCT_1:def 3; then
A35: p.i in X() by A15;
i < j or j < i by A33,XXREAL_0:1; then
R[p.i,p.i] by A16,A33,A34;
hence contradiction by A1,A35;
end;
scheme FiniteC{X() -> finite set, P[set]}:
P[X()]
provided
A1
: for A being Subset of X() st for B being set st B c< A holds P[B] holds P[A]
proof
defpred Q[Nat] means for A being Subset of X() st card A = $1 holds P[A];
A2: for n being Nat st for i being Nat st i < n holds Q[i] holds Q[n]
proof
let n be Nat such that
A3: for i being Nat st i < n holds Q[i];
let A be Subset of X() such that
A4: card A = n;
now
let B be set such that
A5: B c< A;
B c= A by A5; then
reconsider B9 = B as Subset of X() by XBOOLE_1:1;
card B9 < n by A4,A5,TREES_1:6;
hence P[B] by A3;
end;
hence thesis by A1;
end;
for n being Nat holds Q[n] from NAT_1:sch 4(A2); then
Q[card X()] & [#]X() = X();
hence thesis;
end;
scheme Numeration{X() -> finite set, R[set, set]}:
ex s being one-to-one FinSequence st rng s = X() &
for i,j st i in dom s & j in dom s & R[s.i, s.j] holds i < j
provided
A1: for x,y being set st x in X() & y in X() & R[x,y] holds not R[y,x] and
A2: for x,y,z being set st x in X() & y in X() & z in X() & R[x,y] & R[y,z]
holds R[x,z]
proof
defpred P[set] means
ex s being one-to-one FinSequence st rng s = $1 &
for i,j st i in dom s & j in dom s & R[s.i, s.j] holds i < j;
A3: P[{}]
proof
reconsider s = {} as one-to-one FinSequence;
take s; thus thesis;
end;
A4: for A being Subset of X() st for B being set st B c< A holds P[B]
holds P[A]
proof
let A be Subset of X() such that
A5: for B being set st B c< A holds P[B];
per cases;
suppose A is empty;
hence P[A] by A3;
end;
suppose A is non empty; then
reconsider A9 = A as non empty finite set;
A6: for x,y being set st x in A9 & y in A9 & R[x,y] holds not R[y,x] by A1;
A7: for x,y,z being set st x in A9 & y in A9 & z in A9 & R[x,y] & R[y,z]
holds R[x,z] by A2;
consider x being set such that
A8: x in A9 & for y being set st y in A9 holds not R[y,x]
from MinimalElement(A6,A7);
set B = A\{x};
A9: x nin B & B c= A by ZFMISC_1:56; then
B c< A by A8; then
consider s being one-to-one FinSequence such that
A10: rng s = B and
A11: for i,j st i in dom s & j in dom s & R[s.i, s.j] holds i < j by A5;
<*x*> is one-to-one & rng <*x*> = {x} & {x} misses B
by FINSEQ_1:39,FINSEQ_3:93,XBOOLE_1:79; then
reconsider s9 = <*x*>^s as one-to-one FinSequence by A10,FINSEQ_3:91;
A12: {x} c= A by A8,ZFMISC_1:31;
A13: len <*x*> = 1 by FINSEQ_1:40;
thus P[A]
proof
take s9;
thus
rng s9 = (rng <*x*>)\/ rng s by FINSEQ_1:31
.= {x} \/ B by A10,FINSEQ_1:38 .= A by A12,XBOOLE_1:45;
let i,j such that
A14: i in dom s9 & j in dom s9 & R[s9.i, s9.j];
A15: dom <*x*> = Seg 1 by FINSEQ_1:38;
per cases by A13,A14,FINSEQ_1:25;
suppose i in dom <*x*> & j in dom <*x*>; then
i = 1 & j = 1 by A15,FINSEQ_1:2,TARSKI:def 1; then
s9.i = x & s9.j = x by FINSEQ_1:41;
hence i < j by A8,A14;
end;
suppose
A16: i in dom <*x*> & ex n being Nat st n in dom s & j = 1 + n; then
A17: i = 1 by A15,FINSEQ_1:2,TARSKI:def 1;
consider n being Nat such that
A18: n in dom s & j = 1+n by A16;
1 <= n by A18,FINSEQ_3:25;
hence i < j by A17,A18,NAT_1:13;
end;
suppose
A19: j in dom <*x*> & ex n being Nat st n in dom s & i = 1 + n; then
j = 1 by A15,FINSEQ_1:2,TARSKI:def 1; then
A20: s9.j = x by FINSEQ_1:41;
consider n being Nat such that
A21: n in dom s & i = 1+n by A19;
s9.i = s.n by A13,A21,FINSEQ_1:def 7; then
s9.i in rng s by A21,FUNCT_1:def 3;
hence i < j by A8,A14,A20,A9,A10;
end;
suppose (ex n being Nat st n in dom s & i = 1 + n) &
ex n being Nat st n in dom s & j = 1 + n; then
consider ni,nj being Nat such that
A22: ni in dom s & i = 1+ni & nj in dom s & j = 1+nj;
s9.i = s.ni & s9.j = s.nj by A13,A22,FINSEQ_1:def 7; then
ni < nj by A11,A14,A22;
hence i < j by A22,XREAL_1:6;
end;
end;
end;
end;
thus P[X()] from FiniteC(A4);
end;
theorem Th2:
for x being variable holds varcl vars x = vars x
proof
let x be variable; x in Vars; then
consider A being Subset of Vars, j being Element of NAT such that
A1: x = [varcl A,j] & A is finite by ABCMIZ_1:18;
vars x = varcl A by A1;
hence thesis;
end;
theorem Th3:
for C being initialized ConstructorSignature for e being expression of C holds
e is compound iff not ex x being Element of Vars st e = x-term C
proof let C be initialized ConstructorSignature;
let e be expression of C;
(ex x being variable st e = x-term C) or
(ex c being constructor OperSymbol of C st
ex p being FinSequence of QuasiTerms C st
len p = len the_arity_of c & e = c-trm p) or
(ex a being expression of C, an_Adj C st e = (non_op C)term a) or
(ex a being expression of C, an_Adj C st
ex t being expression of C, a_Type C st
e = (ast C)term(a,t)) by ABCMIZ_1:53;
hence thesis;
end;
begin :: Standardized Constructor Signature
registration
cluster empty for quasi-loci;
existence by ABCMIZ_1:29;
end;
definition
let C be ConstructorSignature;
attr C is standardized means:
Def1:
for o being OperSymbol of C st o is constructor holds o in Constructors &
o`1 = the_result_sort_of o &
card o`2`1 = len the_arity_of o;
end;
theorem Th4:
for C being ConstructorSignature st C is standardized
for o being OperSymbol of C
holds o is constructor iff o in Constructors
proof
let C be ConstructorSignature such that
A1: C is standardized;
let o be OperSymbol of C;
thus o is constructor implies o in Constructors by A1;
assume o in Constructors; then
not o in {*, non_op} by ABCMIZ_1:39,XBOOLE_0:3;
hence o <> * & o <> non_op by TARSKI:def 2;
end;
registration
cluster MaxConstrSign -> standardized;
coherence
proof let o be OperSymbol of MaxConstrSign;
A1: the carrier' of MaxConstrSign = {*, non_op} \/ Constructors
by ABCMIZ_1:def 24;
assume
A2: o is constructor; then
A3: (the ResultSort of MaxConstrSign).o = o`1 &
card ((the Arity of MaxConstrSign).o) = card o`2`1
by ABCMIZ_1:def 24;
o <> * & o <> non_op by A2; then
not o in {*, non_op} by TARSKI:def 2;
hence thesis by A3,A1,XBOOLE_0:def 3;
end;
end;
registration
cluster initialized standardized strict for ConstructorSignature;
existence
proof take MaxConstrSign;
thus thesis;
end;
end;
definition
let C be initialized standardized ConstructorSignature;
let c be constructor OperSymbol of C;
func loci_of c -> quasi-loci equals c`2`1;
coherence
proof
reconsider c as Element of Constructors by Th4;
loci_of c is quasi-loci;
hence thesis;
end;
end;
registration
let C be ConstructorSignature;
cluster constructor for Subsignature of C;
existence
proof reconsider S = C as Subsignature of C by INSTALG1:15;
take S; thus thesis;
end;
end;
registration
let C be initialized ConstructorSignature;
cluster initialized for constructor Subsignature of C;
existence
proof reconsider S = C as constructor Subsignature of C by INSTALG1:15;
take S; thus thesis;
end;
end;
registration
let C be standardized ConstructorSignature;
cluster -> standardized for constructor Subsignature of C;
coherence
proof let S be constructor Subsignature of C;
let o be OperSymbol of S such that
A1: o <> * & o <> non_op;
A2: the carrier' of S c= the carrier' of C by INSTALG1:10;
reconsider c = o as OperSymbol of C by A2;
A3: c is constructor by A1;
the Arity of S = (the Arity of C)|the carrier' of S &
the ResultSort of S = (the ResultSort of C)|the carrier' of S
by INSTALG1:12; then
the_result_sort_of c = the_result_sort_of o &
the_arity_of c = the_arity_of o by FUNCT_1:49;
hence thesis by A3,Def1;
end;
end;
theorem
for S1,S2 being standardized ConstructorSignature
st the carrier' of S1 = the carrier' of S2
holds the ManySortedSign of S1 = the ManySortedSign of S2
proof let S1,S2 be standardized ConstructorSignature such that
A1: the carrier' of S1 = the carrier' of S2;
A2: the carrier of S1 = 3 & the carrier of S2 = 3 by ABCMIZ_1:def 9,YELLOW11:1;
now let o be OperSymbol of S1;
reconsider o2 = o as OperSymbol of S2 by A1;
per cases;
suppose o = * or o = non_op; then
(the Arity of S1).o = <*an_Adj*> & (the Arity of S2).o = <*an_Adj*> or
(the Arity of S1).o = <*an_Adj,a_Type*> &
(the Arity of S2).o = <*an_Adj,a_Type*> by ABCMIZ_1:def 9;
hence (the Arity of S1).o = (the Arity of S2).o;
end;
suppose o is constructor & o2 is constructor; then
card o`2`1 = len the_arity_of o & card o`2`1 = len the_arity_of o2 &
the_arity_of o = (len the_arity_of o) |-> a_Term &
the_arity_of o2 = (len the_arity_of o2) |-> a_Term
by Def1,ABCMIZ_1:37;
hence (the Arity of S1).o = the_arity_of o2 .= (the Arity of S2).o;
end;
end; then
A3: the Arity of S1 = the Arity of S2 by A1,A2,FUNCT_2:63;
now let o be OperSymbol of S1;
reconsider o2 = o as OperSymbol of S2 by A1;
per cases;
suppose o = * or o = non_op; then
(the ResultSort of S1).o = a_Type & (the ResultSort of S2).o = a_Type or
(the ResultSort of S1).o = an_Adj &
(the ResultSort of S2).o = an_Adj by ABCMIZ_1:def 9;
hence (the ResultSort of S1).o = (the ResultSort of S2).o;
end;
suppose o is constructor & o2 is constructor; then
the_result_sort_of o = o`1 & the_result_sort_of o2 = o`1
by Def1;
hence (the ResultSort of S1).o = the_result_sort_of o2
.= (the ResultSort of S2).o;
end;
end;
hence thesis by A1,A2,A3,FUNCT_2:63;
end;
theorem
for C being ConstructorSignature holds
C is standardized iff C is Subsignature of MaxConstrSign
proof let C be ConstructorSignature;
A1: the carrier' of MaxConstrSign = {*, non_op} \/ Constructors
by ABCMIZ_1:def 24;
A2: dom the Arity of MaxConstrSign = the carrier' of MaxConstrSign
by FUNCT_2:def 1;
A3: dom the ResultSort of MaxConstrSign = the carrier' of MaxConstrSign
by FUNCT_2:def 1;
thus C is standardized implies C is Subsignature of MaxConstrSign
proof assume
A4: for o being OperSymbol of C st o is constructor
holds o in Constructors &
o`1 = the_result_sort_of o &
card o`2`1 = len the_arity_of o;
A5: the carrier of C = 3 & the carrier of MaxConstrSign = 3
by ABCMIZ_1:def 9,YELLOW11:1;
A6: the Arity of C c= the Arity of MaxConstrSign
proof let x,y be object; assume
A7: [x,y] in the Arity of C; then
reconsider x as OperSymbol of C by ZFMISC_1:87;
x = * or x = non_op or x is constructor; then
x in {*, non_op} or x in Constructors by A4,TARSKI:def 2; then
reconsider c = x as OperSymbol of MaxConstrSign by A1,XBOOLE_0:def 3;
A8: y = (the Arity of C).x by A7,FUNCT_1:1;
per cases;
suppose x = * or x = non_op; then
c = * & y = <*an_Adj,a_Type*> or c = non_op & y = <*an_Adj*>
by A8,ABCMIZ_1:def 9; then
y = (the Arity of MaxConstrSign).c by ABCMIZ_1:def 9;
hence thesis by A2,FUNCT_1:def 2;
end;
suppose
A9: x is constructor; then
A10: x <> * & x <> non_op; then
A11: c is constructor;
reconsider y as set by TARSKI:1;
card x`2`1 = len the_arity_of x by A4,A9
.= card y by A7,FUNCT_1:1; then
A12: card y = card ((the Arity of MaxConstrSign).c)
by A11,ABCMIZ_1:def 24;
y in {a_Term}* & (the Arity of MaxConstrSign).c in {a_Term}*
by A8,A10,ABCMIZ_1:def 9; then
y = (the Arity of MaxConstrSign).c by A12,ABCMIZ_1:6;
hence thesis by A2,FUNCT_1:def 2;
end;
end;
the ResultSort of C c= the ResultSort of MaxConstrSign
proof let x,y be object; assume
A13: [x,y] in the ResultSort of C; then
reconsider x as OperSymbol of C by ZFMISC_1:87;
x is constructor or x = * or x = non_op; then
x in {*, non_op} or x in Constructors by A4,TARSKI:def 2; then
reconsider c = x as OperSymbol of MaxConstrSign by A1,XBOOLE_0:def 3;
A14: y = (the ResultSort of C).x by A13,FUNCT_1:1;
per cases;
suppose x = * or x = non_op; then
c = * & y = a_Type or c = non_op & y = an_Adj
by A14,ABCMIZ_1:def 9; then
y = (the ResultSort of MaxConstrSign).c by ABCMIZ_1:def 9;
hence thesis by A3,FUNCT_1:def 2;
end;
suppose
A15: x is constructor & c is constructor; then
x`1 = the_result_sort_of x by A4
.= y by A13,FUNCT_1:1; then
y = the_result_sort_of c by A15,Def1
.= (the ResultSort of MaxConstrSign).c;
hence thesis by A3,FUNCT_1:def 2;
end;
end;
hence thesis by A5,A6,INSTALG1:13;
end;
assume
A16: C is Subsignature of MaxConstrSign;
let o be OperSymbol of C such that
A17: o <> * & o <> non_op;
the carrier' of C c= the carrier' of MaxConstrSign &
o in the carrier' of C by A16,INSTALG1:10; then
reconsider c = o as OperSymbol of MaxConstrSign;
A18: c is constructor by A17;
not c in {*, non_op} by A17,TARSKI:def 2;
hence o in Constructors by A1,XBOOLE_0:def 3;
thus o`1 = (the ResultSort of MaxConstrSign).c by A18,ABCMIZ_1:def 24
.= ((the ResultSort of MaxConstrSign)|the carrier' of C).o
by FUNCT_1:49
.= (the ResultSort of C).o by A16,INSTALG1:12
.= the_result_sort_of o;
thus card o`2`1 = card ((the Arity of MaxConstrSign).c)
by A18,ABCMIZ_1:def 24
.= card (((the Arity of MaxConstrSign)|the carrier' of C).o)
by FUNCT_1:49
.= card ((the Arity of C).o) by A16,INSTALG1:12
.= len the_arity_of o;
end;
registration
let C be initialized ConstructorSignature;
cluster non compound for quasi-term of C;
existence
proof set x = the Element of Vars;
take x-term C; thus thesis;
end;
end;
registration
cluster -> pair for Element of Vars;
coherence
proof let x be Element of Vars;
Vars = {[varcl A, j] where A is Subset of Vars, j is Element of NAT:
A is finite} & x in Vars by ABCMIZ_1:18; then
ex A being Subset of Vars, j being Element of NAT st
x = [varcl A, j] & A is finite;
hence thesis;
end;
end;
theorem Th7:
for x being Element of Vars st vars x is natural holds vars x = 0
proof let x be Element of Vars;
assume x`1 is natural; then
reconsider n = x`1 as Element of NAT;
Vars = {[varcl A, j] where A is Subset of Vars, j is Element of NAT:
A is finite} & x in Vars by ABCMIZ_1:18; then
consider A being Subset of Vars, j being Element of NAT such that
A1: x = [varcl A, j] & A is finite;
set i = the Element of n;
assume A2: x`1 <> 0; then
A3: i in n;
reconsider i as Element of NAT by A2,ORDINAL1:10;
n = varcl A & vars x c= Vars by A1; then
i in Vars by A3;
hence thesis;
end;
theorem Th8:
Vars misses Constructors proof assume Vars meets Constructors; then
consider x being object such that
A1: x in Vars & x in Constructors by XBOOLE_0:3;
reconsider x as Element of Vars by A1;
consider A being Subset of Vars, j being Element of NAT such that
A2: x = [varcl A, j] & A is finite by A1,ABCMIZ_1:18;
x in Modes \/ Attrs or x in Funcs by A1,XBOOLE_0:def 3; then
x in Modes or x in Attrs or x in Funcs by XBOOLE_0:def 3; then
x`2 in [:QuasiLoci,NAT:] & x`2 = j by A2,MCART_1:10; then
ex a,b being object st a in QuasiLoci & b in NAT & [a,b] = j
by ZFMISC_1:def 2;
hence thesis;
end;
theorem
for x being Element of Vars holds x <> * & x <> non_op;
theorem Th10:
for C being standardized ConstructorSignature holds
Vars misses the carrier' of C
proof let C be standardized ConstructorSignature;
assume Vars meets the carrier' of C; then
consider x being object such that
A1: x in Vars & x in the carrier' of C by XBOOLE_0:3;
reconsider x as Element of Vars by A1;
reconsider c = x as OperSymbol of C by A1;
x = * or x = non_op or c is constructor; then
x = * or x = non_op or x in Constructors & Vars misses Constructors
by Th8,Def1;
hence thesis by XBOOLE_0:3;
end;
theorem Th11: :: see ABCMIZ_1:51
for C being initialized standardized ConstructorSignature
for e being expression of C
holds
(ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]) or
(ex o being OperSymbol of C st e.{} = [o, the carrier of C] &
( o in Constructors or o = * or o = non_op ))
proof let C be initialized standardized ConstructorSignature;
let e be expression of C;
set X = MSVars C;
set Y = X (\/) ((the carrier of C) --> {0});
reconsider q = e as Term of C,Y by MSAFREE3:8;
per cases by MSATERM:2;
suppose
ex s being SortSymbol of C, v being Element of Y.s st q.{} = [v,s]; then
consider s being SortSymbol of C, v being Element of Y.s such that
A1: q.{} = [v,s];
consider z being object such that
A2: z in dom the Sorts of Free(C,X) & e in (the Sorts of Free(C, X)).z
by CARD_5:2;
reconsider z as SortSymbol of C by A2;
the carrier of C = {a_Type, an_Adj, a_Term} by ABCMIZ_1:def 9; then
A3: z = a_Type or z = an_Adj or z = a_Term by ENUMSET1:def 1;
A4: q = root-tree [v,s] by A1,MSATERM:5; then
A5: the_sort_of q = s by MSATERM:14;
A6: the Sorts of Free(C, X) = C-Terms(X,Y) by MSAFREE3:24; then
the Sorts of Free(C, X) c= the Sorts of FreeMSA Y by PBOOLE:def 18; then
(the Sorts of Free(C, X)).z c= (the Sorts of FreeMSA Y).z &
FreeMSA Y = MSAlgebra(#FreeSort Y, FreeOper Y#); then
q in (the Sorts of FreeMSA Y).z &
(the Sorts of FreeMSA Y).z = FreeSort(Y, z) by A2,MSAFREE:def 11; then
A7: s = z by A5,MSATERM:def 5; then
v in (MSVars C).z by A4,A2,A6,MSAFREE3:18; then
A8: v in Vars & z = a_Term by A3,ABCMIZ_1:def 25; then
reconsider x = v as Element of Vars;
e = x-term C by A1,A7,A8,MSATERM:5;
hence thesis by A1,A7,A8;
end;
suppose
q.{} in [:the carrier' of C,{the carrier of C}:]; then
consider o,s being object such that
A9: o in the carrier' of C & s in {the carrier of C} & q.{} = [o,s]
by ZFMISC_1:def 2;
reconsider o as OperSymbol of C by A9;
o is constructor iff o <> * & o <> non_op; then
s = the carrier of C & (o in Constructors or o = * or o = non_op)
by A9,Def1,TARSKI:def 1;
hence thesis by A9;
end;
end;
registration
let C be initialized standardized ConstructorSignature;
let e be expression of C;
cluster e.{} -> pair;
coherence
proof
(ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]) or
(ex o being OperSymbol of C st e.{} = [o, the carrier of C] &
( o in Constructors or o = * or o = non_op )) by Th11;
hence thesis;
end;
end;
theorem Th12:
for C being initialized ConstructorSignature for e being expression of C
for o being OperSymbol of C st e.{} = [o, the carrier of C]
holds
e is expression of C, the_result_sort_of o
proof let C be initialized ConstructorSignature;
let e be expression of C;
let o be OperSymbol of C such that
A1: e.{} = [o, the carrier of C];
set X = MSVars C, Y = X (\/) ((the carrier of C) --> {0});
reconsider t = e as Term of C, Y by MSAFREE3:8;
variables_in t c= X by MSAFREE3:27; then
e in {t1 where t1 is Term of C, Y:
the_sort_of t1 = the_sort_of t & variables_in t1 c= X}; then
e in C-Terms(X,Y).the_sort_of t by MSAFREE3:def 5; then
A2: e in (the Sorts of Free(C, X)).the_sort_of t by MSAFREE3:24;
the_sort_of t = the_result_sort_of o by A1,MSATERM:17;
hence thesis by A2,ABCMIZ_1:def 28;
end;
theorem Th13:
for C being initialized standardized ConstructorSignature
for e being expression of C
holds
( (e.{})`1 = * implies e is expression of C, a_Type C ) &
( (e.{})`1 = non_op implies e is expression of C, an_Adj C )
proof let C be initialized standardized ConstructorSignature;
let e be expression of C;
per cases by Th11;
suppose
ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]; then
consider x being Element of Vars such that
A1: e = x-term C & e.{} = [x, a_Term];
thus thesis by A1;
end;
suppose
(ex o being OperSymbol of C st e.{} = [o, the carrier of C] &
( o in Constructors or o = * or o = non_op )); then
consider o being OperSymbol of C such that
A2: e.{} = [o, the carrier of C];
set X = MSVars C, Y = X (\/) ((the carrier of C) --> {0});
reconsider t = e as Term of C, Y by MSAFREE3:8;
variables_in t c= X by MSAFREE3:27; then
e in {t1 where t1 is Term of C, Y:
the_sort_of t1 = the_sort_of t & variables_in t1 c= X}; then
e in C-Terms(X,Y).the_sort_of t by MSAFREE3:def 5; then
A3: e in (the Sorts of Free(C, X)).the_sort_of t by MSAFREE3:24;
A4: the_result_sort_of non_op C = an_Adj C &
the_result_sort_of ast C = a_Type C by ABCMIZ_1:38;
A5: (e.{})`1 = o & non_op C = non_op & ast C = * by A2;
the_sort_of t = the_result_sort_of o by A2,MSATERM:17;
hence thesis by A3,A4,A5,ABCMIZ_1:def 28;
end;
end;
theorem Th14:
for C being initialized standardized ConstructorSignature
for e being expression of C
holds
(e.{})`1 in Vars & (e.{})`2 = a_Term & e is quasi-term of C or
(e.{})`2 = the carrier of C &
( (e.{})`1 in Constructors & (e.{})`1 in the carrier' of C or
(e.{})`1 = * or (e.{})`1 = non_op )
proof let C be initialized standardized ConstructorSignature;
let e be expression of C;
per cases by Th11;
suppose ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term];
then consider x being Element of Vars such that
A1: e = x-term C & e.{} = [x, a_Term];
thus thesis by A1;
end;
suppose ex o being OperSymbol of C st e.{} = [o, the carrier of C] &
( o in Constructors or o = * or o = non_op ); then
consider o being OperSymbol of C such that
A2: e.{} = [o, the carrier of C] &
( o in Constructors or o = * or o = non_op );
thus thesis by A2;
end;
end;
theorem
for C being initialized standardized ConstructorSignature
for e being expression of C
st (e.{})`1 in Constructors
holds e in (the Sorts of Free(C, MSVars C)).(e.{})`1`1
proof let C be initialized standardized ConstructorSignature;
let e be expression of C;
assume
A1: (e.{})`1 in Constructors;
per cases by Th11;
suppose
ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]; then
consider x being Element of Vars such that
A2: e = x-term C & e.{} = [x, a_Term];
(e.{})`1 = x by A2;
hence thesis by A1,Th8,XBOOLE_0:3;
end;
suppose
ex o being OperSymbol of C st e.{} = [o, the carrier of C] &
( o in Constructors or o = * or o = non_op ); then
consider o being OperSymbol of C such that
A3: e.{} = [o, the carrier of C];
A4: (e.{})`1 = o by A3;
* in {*, non_op} & non_op in {*, non_op} by TARSKI:def 2; then
o <> * & o <> non_op by A1,A4,ABCMIZ_1:39,XBOOLE_0:3; then
A5: o is constructor;
set X = MSVars C;
reconsider t = e as Term of C, X (\/) ((the carrier of C) --> {0})
by MSAFREE3:8;
A6: the_sort_of t = the_result_sort_of o by A3,MSATERM:17
.= o`1 by A5,Def1;
variables_in t c= X by MSAFREE3:27; then
e in {t1 where t1 is Term of C, X (\/) ((the carrier of C) --> {0}):
the_sort_of t1 = the_sort_of t & variables_in t1 c= X}; then
e in C-Terms(X, X (\/) ((the carrier of C)-->{0})).the_sort_of t
by MSAFREE3:def 5;
hence e in (the Sorts of Free(C, MSVars C)).(e.{})`1`1
by A4,A6,MSAFREE3:23;
end;
end;
theorem
for C being initialized standardized ConstructorSignature
for e being expression of C
holds not (e.{})`1 in Vars iff (e.{})`1 is OperSymbol of C
proof let C be initialized standardized ConstructorSignature;
let e be expression of C;
A1: (e.{})`1 in Vars or (e.{})`1 in the carrier' of C or
(e.{})`1 = ast C or (e.{})`1 = non_op C by Th14;
Vars misses the carrier' of C by Th10;
hence not (e.{})`1 in Vars iff (e.{})`1 is OperSymbol of C
by A1,XBOOLE_0:3;
end;
theorem Th17:
for C being initialized standardized ConstructorSignature
for e being expression of C
st (e.{})`1 in Vars
ex x being Element of Vars st x = (e.{})`1 & e = x-term C
proof let C be initialized standardized ConstructorSignature;
let t be expression of C such that
A1: (t.{})`1 in Vars;
set X = MSVars C;
set V = X (\/) ((the carrier of C) --> {0});
reconsider q = t as Term of C, V by MSAFREE3:8;
per cases by MSATERM:2;
suppose q.{} in [:the carrier' of C, {the carrier of C}:]; then
(q.{})`1 in the carrier' of C &
the carrier' of C misses Vars by Th10,MCART_1:10;
hence thesis by A1,XBOOLE_0:3;
end;
suppose
ex s being SortSymbol of C, v being Element of V.s st q.{} = [v,s]; then
consider s being SortSymbol of C, v being Element of V.s such that
A2: t.{} = [v,s];
A3: q = root-tree [v,s] by A2,MSATERM:5;
reconsider x = v as Element of Vars by A1,A2;
take x;
the carrier of C = {a_Type, an_Adj, a_Term} by ABCMIZ_1:def 9; then
A4: s = a_Term or s = a_Type or s = an_Adj by ENUMSET1:def 1;
((the carrier of C) --> {0}).s = {0} by FUNCOP_1:7; then
V.s = X.s \/ {0} by PBOOLE:def 4; then
A5: s = a_Term or V.s = {} \/ {0} by A4,ABCMIZ_1:def 25;
v in V.s & x <> 0;
hence thesis by A2,A3,A5;
end;
end;
theorem Th18:
for C being initialized standardized ConstructorSignature
for e being expression of C
st (e.{})`1 = *
ex a being expression of C, an_Adj C, q being expression of C, a_Type C st
e = [*,3]-tree(a,q)
proof let C be initialized standardized ConstructorSignature;
let e be expression of C such that
A1: (e.{})`1 = *;
not ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]
by A1;
then consider o being OperSymbol of C such that
A2: e.{} = [o, the carrier of C] and
o in Constructors or o = * or o = non_op by Th11;
set Y = (MSVars C) (\/) ((the carrier of C) --> {0});
reconsider t = e as Term of C, (MSVars C) (\/) ((the carrier of C) --> {0})
by MSAFREE3:8;
consider aa being ArgumentSeq of
Sym(o, (MSVars C) (\/) ((the carrier of C) --> {0})) such that
A3: t = [o, the carrier of C]-tree aa by A2,MSATERM:10;
A4: * = [o, the carrier of C]`1 by A1,A3,TREES_4:def 4 .= o;
A5: the_arity_of ast C = <*an_Adj C, a_Type C*> by ABCMIZ_1:38;
A6: len aa = len the_arity_of o by MSATERM:22
.= 2 by A4,A5,FINSEQ_1:44; then
dom aa = Seg 2 by FINSEQ_1:def 3; then
A7: 1 in dom aa & 2 in dom aa; then
reconsider t1 = aa.1, t2 = aa.2 as Term of C,
(MSVars C) (\/) ((the carrier of C) --> {0}) by MSATERM:22;
A8: len doms aa = len aa by TREES_3:38;
(doms aa).1 = dom t1 & (doms aa).2 = dom t2 by A7,FUNCT_6:22; then
A9: 0 < 2 & 0+1 = 1 & 1 < 2 & 1+1 = 2 & {} in (doms aa).1 & {} in (doms aa).2 &
<* 0*>^<*>NAT = <* 0*> & <* 1*>^<*>NAT = <* 1*>
by FINSEQ_1:34,TREES_1:22;
dom t = tree doms aa by A3,TREES_4:10; then
reconsider 00 = <* 0*>, 01 = <* 1*> as Element of dom t
by A6,A8,A9,TREES_3:def 15;
0 < 2 & 1 = 0+1 & 1 < 2 & 2 = 1+1 & aa is DTree-yielding; then
t1 = t|00 & t2 = t|01 by A3,A6,TREES_4:def 4; then
A10: t1 is expression of C & t2 is expression of C &
variables_in t1 c= variables_in t & variables_in t2 c= variables_in t
by MSAFREE3:32,33; then
A11: variables_in t1 c= MSVars C & variables_in t2 c= MSVars C by MSAFREE3:27;
the_sort_of t1 = (the_arity_of o).1 by A7,MSATERM:23
.= an_Adj C by A4,A5,FINSEQ_1:44; then
t1 in {s where s is Term of C,Y: the_sort_of s = an_Adj C &
variables_in s c= MSVars C} by A11; then
t1 in (C-Terms(MSVars C, Y)).an_Adj C by MSAFREE3:def 5; then
t1 in (the Sorts of Free(C, MSVars C)).an_Adj C by MSAFREE3:24; then
reconsider a = t1 as expression of C, an_Adj C by A10,ABCMIZ_1:def 28;
the_sort_of t2 = (the_arity_of o).2 by A7,MSATERM:23
.= a_Type C by A4,A5,FINSEQ_1:44; then
t2 in {s where s is Term of C,Y: the_sort_of s = a_Type C &
variables_in s c= MSVars C} by A11; then
t2 in (C-Terms(MSVars C, Y)).a_Type C by MSAFREE3:def 5; then
t2 in (the Sorts of Free(C, MSVars C)).a_Type C by MSAFREE3:24; then
reconsider q = t2 as expression of C, a_Type C by A10,ABCMIZ_1:def 28;
take a,q;
A12: the carrier of C = 3 by ABCMIZ_1:def 9,YELLOW11:1;
aa = <*a,q*> by A6,FINSEQ_1:44;
hence thesis by A3,A4,A12,TREES_4:def 6;
end;
theorem Th19:
for C being initialized standardized ConstructorSignature
for e being expression of C
st (e.{})`1 = non_op
ex a being expression of C, an_Adj C st e = [non_op,3]-tree a
proof let C be initialized standardized ConstructorSignature;
let e be expression of C such that
A1: (e.{})`1 = non_op;
not ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]
by A1;
then consider o being OperSymbol of C such that
A2: e.{} = [o, the carrier of C] and
o in Constructors or o = * or o = non_op by Th11;
set Y = (MSVars C) (\/) ((the carrier of C) --> {0});
reconsider t = e as Term of C, (MSVars C) (\/) ((the carrier of C) --> {0})
by MSAFREE3:8;
consider aa being ArgumentSeq of
Sym(o, (MSVars C) (\/) ((the carrier of C) --> {0})) such that
A3: t = [o, the carrier of C]-tree aa by A2,MSATERM:10;
A4: non_op = [o, the carrier of C]`1 by A1,A3,TREES_4:def 4 .= o;
A5: the_arity_of non_op C = <*an_Adj C*> by ABCMIZ_1:38;
A6: len aa = len the_arity_of o by MSATERM:22
.= 1 by A4,A5,FINSEQ_1:40; then
dom aa = Seg 1 by FINSEQ_1:def 3; then
A7: 1 in dom aa; then
reconsider t1 = aa.1 as Term of C,
(MSVars C) (\/) ((the carrier of C) --> {0}) by MSATERM:22;
A8: len doms aa = len aa by TREES_3:38;
(doms aa).1 = dom t1 by A7,FUNCT_6:22; then
A9: 0 < 1 & 0+1 = 1 & {} in (doms aa).1 & <* 0*>^<*>NAT = <* 0*>
by FINSEQ_1:34,TREES_1:22;
dom t = tree doms aa by A3,TREES_4:10; then
reconsider 00 = <* 0*> as Element of dom t by A6,A8,A9,TREES_3:def 15;
t1 = t|00 by A3,A6,A9,TREES_4:def 4; then
A10: t1 is expression of C & variables_in t1 c= variables_in t
by MSAFREE3:32,33; then
A11: variables_in t1 c= MSVars C by MSAFREE3:27;
the_sort_of t1 = (the_arity_of o).1 by A7,MSATERM:23
.= an_Adj C by A4,A5,FINSEQ_1:40; then
t1 in {s where s is Term of C,Y: the_sort_of s = an_Adj C &
variables_in s c= MSVars C} by A11; then
t1 in (C-Terms(MSVars C, Y)).an_Adj C by MSAFREE3:def 5; then
t1 in (the Sorts of Free(C, MSVars C)).an_Adj C by MSAFREE3:24; then
reconsider a = t1 as expression of C, an_Adj C by A10,ABCMIZ_1:def 28;
take a;
A12: the carrier of C = 3 by ABCMIZ_1:def 9,YELLOW11:1;
aa = <*a*> by A6,FINSEQ_1:40;
hence thesis by A3,A4,A12,TREES_4:def 5;
end;
theorem Th20:
for C being initialized standardized ConstructorSignature
for e being expression of C
st (e.{})`1 in Constructors
ex o being OperSymbol of C st o = (e.{})`1 & the_result_sort_of o = o`1 &
e is expression of C, the_result_sort_of o
proof let C be initialized standardized ConstructorSignature;
let e be expression of C such that
A1: (e.{})`1 in Constructors;
per cases by Th11;
suppose ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term];
then consider x being Element of Vars such that
A2: e = x-term C & e.{} = [x, a_Term];
(e.{})`1 = x & x in Vars by A2;
hence thesis by A1,Th8,XBOOLE_0:3;
end;
suppose ex o being OperSymbol of C st e.{} = [o, the carrier of C] &
( o in Constructors or o = * or o = non_op ); then
consider o being OperSymbol of C such that
A3: e.{} = [o, the carrier of C] &
( o in Constructors or o = * or o = non_op );
take o;
A4: (e.{})`1 = o & (e.{})`2 = the carrier of C by A3;
* in {*, non_op} & non_op in {*, non_op} by TARSKI:def 2; then
o <> * & o <> non_op by A1,A4,ABCMIZ_1:39,XBOOLE_0:3; then
o is constructor;
hence o = (e.{})`1 & the_result_sort_of o = o`1
by A3,Def1;
set X = MSVars C;
set V = X (\/) ((the carrier of C) --> {0});
reconsider q = e as Term of C,V by MSAFREE3:8;
A5: variables_in q c= X by MSAFREE3:27;
A6: the_sort_of q = the_result_sort_of o by A3,MSATERM:17;
(the Sorts of Free(C, MSVars C)).the_result_sort_of o
= C-Terms(X,V).the_result_sort_of o by MSAFREE3:24
.= {a where a is Term of C,V: the_sort_of a = the_result_sort_of o &
variables_in a c= X} by MSAFREE3:def 5;
hence
e in (the Sorts of Free(C, MSVars C)).the_result_sort_of o by A5,A6;
end;
end;
theorem Th21:
for C being initialized standardized ConstructorSignature
for t being quasi-term of C holds
t is compound iff (t.{})`1 in Constructors & (t.{})`1`1 = a_Term
proof let C be initialized standardized ConstructorSignature;
set X = MSVars C;
set V = X (\/) ((the carrier of C) --> {0});
let t be quasi-term of C;
C-Terms(X, V) c= the Sorts of FreeMSA V &
the Sorts of Free(C, X) = C-Terms(X, V) by MSAFREE3:24,PBOOLE:def 18; then
A1: FreeMSA V = MSAlgebra(#FreeSort V, FreeOper V#) &
(C-Terms(X, V)).a_Term C c= (the Sorts of FreeMSA V).a_Term C &
t in C-Terms(X,V).a_Term C
by ABCMIZ_1:def 28; then
t in (FreeSort V).a_Term C; then
A2: t in FreeSort(V,a_Term C) by MSAFREE:def 11;
A3: (MSVars C).a_Term = Vars & a_Term C = a_Term & a_Term = 2
by ABCMIZ_1:def 25;
reconsider q = t as Term of C, V by MSAFREE3:8;
per cases by MSATERM:2;
suppose
ex s being SortSymbol of C, v being Element of V.s st q.{} = [v,s]; then
consider s being SortSymbol of C, v being Element of V.s such that
A4: t.{} = [v,s];
A5: q = root-tree [v,s] & the_sort_of q = a_Term C
by A2,A4,MSATERM:5,def 5; then
A6: a_Term C = s & (t.{})`1 = v by A4,MSATERM:14; then
reconsider x = v as Element of Vars by A3,A5,A1,MSAFREE3:18;
q = x-term C & vars x <> 2 by A5,A6,Th7;
hence thesis by A6;
end;
suppose
q.{} in [:the carrier' of C,{the carrier of C}:]; then
consider o, k being object such that
A7: o in the carrier' of C & k in {the carrier of C} & q.{} = [o,k]
by ZFMISC_1:def 2;
reconsider o as OperSymbol of C by A7;
k = the carrier of C by A7,TARSKI:def 1; then
A8: the_result_sort_of o = the_sort_of q by A7,MSATERM:17
.= a_Term C by A1,MSAFREE3:17; then
o <> ast C & o <> non_op C by ABCMIZ_1:38; then
A9: o is constructor; then
A10: a_Term C = o`1 by A8,Def1 .= (q.{})`1`1 by A7;
A11: (q.{})`1 = o by A7;
now given x being Element of Vars such that
A12: q = x-term C;
q.{} = [x,a_Term] by A12,TREES_4:3; then
k = a_Term by A7,XTUPLE_0:1; then
2 = the carrier of C by A7,TARSKI:def 1;
hence contradiction by ABCMIZ_1:def 9,YELLOW11:1;
end;
hence thesis by A9,A10,A11,Th3,Def1;
end;
end;
theorem Th22:
for C being initialized standardized ConstructorSignature
for t being expression of C holds
t is non compound quasi-term of C iff (t.{})`1 in Vars
proof let C be initialized standardized ConstructorSignature;
let t be expression of C;
thus
now assume t is non compound quasi-term of C; then
consider x being Element of Vars such that
A1: t = x-term C by Th3;
t.{} = [x,a_Term] & x in Vars by A1,TREES_4:3;
hence (t.{})`1 in Vars;
end;
assume (t.{})`1 in Vars; then
ex x being Element of Vars st x = (t.{})`1 & t = x-term C by Th17;
hence thesis;
end;
theorem
for C being initialized standardized ConstructorSignature
for t being expression of C holds
t is quasi-term of C iff
(t.{})`1 in Constructors & (t.{})`1`1 = a_Term or (t.{})`1 in Vars
proof let C be initialized standardized ConstructorSignature;
let t be expression of C;
hereby assume t is quasi-term of C; then
reconsider tr = t as quasi-term of C;
tr is compound or tr is non compound;
hence (t.{})`1 in Constructors & (t.{})`1`1 = a_Term or
(t.{})`1 in Vars by Th21,Th22;
end;
assume that
A1: (t.{})`1 in Constructors & (t.{})`1`1 = a_Term or (t.{})`1 in Vars and
A2: t is not quasi-term of C;
A3: (t.{})`1 in Vars implies
ex x being Element of Vars st x = (t.{})`1 & t = x-term C by Th17; then
(t.{})`1 in Constructors & (t.{})`1`1 = a_Term by A1,A2; then
ex o being OperSymbol of C st o = (t.{})`1 & the_result_sort_of o = o`1 &
t is expression of C, the_result_sort_of o by Th20;
hence thesis by A2,A3,A1;
end;
theorem Th24:
for C being initialized standardized ConstructorSignature
for a being expression of C holds
a is positive quasi-adjective of C iff
(a .{})`1 in Constructors & (a .{})`1`1 = an_Adj
proof let C be initialized standardized ConstructorSignature;
set X = MSVars C;
set V = X (\/) ((the carrier of C) --> {0});
let t be expression of C;
consider A being MSSubset of FreeMSA V such that
A1: Free(C, X) = GenMSAlg A & A = (Reverse V)""X by MSAFREE3:def 1;
the Sorts of Free(C, X) is MSSubset of FreeMSA V
by A1,MSUALG_2:def 9; then
the Sorts of Free(C, X) c= the Sorts of FreeMSA V by PBOOLE:def 18; then
A2: (the Sorts of Free(C, MSVars C)).an_Adj C c=
(the Sorts of FreeMSA V).an_Adj C;
per cases by Th14;
suppose (t.{})`1 in Vars & (t.{})`2 = a_Term & t is quasi-term of C;
hence thesis by Th8,ABCMIZ_1:77,XBOOLE_0:3;
end;
suppose that
A3: (t.{})`2 = the carrier of C and
A4: (t.{})`1 in Constructors and
A5: (t.{})`1 in the carrier' of C;
reconsider o = (t.{})`1 as OperSymbol of C by A5;
reconsider tt = t as Term of C, V by MSAFREE3:8;
not o in {*, non_op} by A4,ABCMIZ_1:39,XBOOLE_0:3; then
o <> * & o <> non_op by TARSKI:def 2; then
o is constructor; then
A6: o`1 = the_result_sort_of o by Def1;
A7: t.{} = [o, (t.{})`2]; then
A8: the_sort_of tt = the_result_sort_of o by A3,MSATERM:17;
hereby assume t is positive quasi-adjective of C; then
A9: t in (the Sorts of Free(C, MSVars C)).an_Adj C by ABCMIZ_1:def 28;
thus (t.{})`1 in Constructors by A4;
assume (t.{})`1`1 <> an_Adj;
hence contradiction by A2,A9,A6,A8,MSAFREE3:7;
end;
assume (t.{})`1 in Constructors;
assume (t.{})`1`1 = an_Adj; then
reconsider t as expression of C, an_Adj C by A3,A6,A7,Th12;
t is positive proof given a being expression of C, an_Adj C such that
A10: t = (non_op C)term a;
t = [non_op, the carrier of C]-tree <*a*> by A10,ABCMIZ_1:43; then
t.{} = [non_op, the carrier of C] by TREES_4:def 4; then
(t.{})`1 = non_op; then
(t.{})`1 in {*, non_op} by TARSKI:def 2;
hence thesis by A4,ABCMIZ_1:39,XBOOLE_0:3;
end;
hence thesis;
end;
suppose
A11: (t.{})`1 = *; then
(t.{})`1 in {*, non_op} by TARSKI:def 2; then
A12: not (t.{})`1 in Constructors by ABCMIZ_1:39,XBOOLE_0:3;
consider a being expression of C, an_Adj C,
q being expression of C, a_Type C such that
A13: t = [*,3]-tree(a,q) by A11,Th18;
t = [*,3]-tree<*a,q*> by A13,TREES_4:def 6; then
t.{} = [*, 3] by TREES_4:def 4; then
(t.{})`1 = *; then
t is expression of C, a_Type C & a_Type C = a_Type & a_Type = 0 &
an_Adj C = an_Adj & an_Adj = 1 by Th13;
hence thesis by A12,ABCMIZ_1:48;
end;
suppose
A14: (t.{})`1 = non_op; then
(t.{})`1 in {*, non_op} by TARSKI:def 2; then
A15: not (t.{})`1 in Constructors by ABCMIZ_1:39,XBOOLE_0:3;
consider a being expression of C, an_Adj C such that
A16: t = [non_op,3]-tree a by A14,Th19;
t = [non_op,3]-tree <*a*> by A16,TREES_4:def 5
.= [non_op, the carrier of C]-tree <*a*> by ABCMIZ_1:def 9,YELLOW11:1
.= (non_op C)term a by ABCMIZ_1:43;
hence thesis by A15,ABCMIZ_1:def 37;
end;
end;
theorem
for C being initialized standardized ConstructorSignature
for a being quasi-adjective of C holds
a is negative iff (a .{})`1 = non_op
proof let C be initialized standardized ConstructorSignature;
let t be quasi-adjective of C;
per cases;
suppose
A1: t is positive expression of C, an_Adj C; then
(t.{})`1 in Constructors & non_op in {*, non_op} by Th24,TARSKI:def 2;
hence thesis by A1,ABCMIZ_1:39,XBOOLE_0:3;
end;
suppose
A2: t is negative expression of C, an_Adj C; then
consider a being expression of C, an_Adj C such that
A3: a is positive & t = (non_op C)term a by ABCMIZ_1:def 38;
t = [non_op, the carrier of C]-tree <*a*> by A3,ABCMIZ_1:43; then
t.{} = [non_op, the carrier of C] by TREES_4:def 4;
hence thesis by A2;
end;
end;
theorem
for C being initialized standardized ConstructorSignature
for t being expression of C holds
t is pure expression of C, a_Type C iff
(t.{})`1 in Constructors & (t.{})`1`1 = a_Type
proof let C be initialized standardized ConstructorSignature;
set X = MSVars C;
set V = X (\/) ((the carrier of C) --> {0});
let t be expression of C;
consider A being MSSubset of FreeMSA V such that
A1: Free(C, X) = GenMSAlg A & A = (Reverse V)""X by MSAFREE3:def 1;
the Sorts of Free(C, X) is MSSubset of FreeMSA V
by A1,MSUALG_2:def 9; then
the Sorts of Free(C, X) c= the Sorts of FreeMSA V by PBOOLE:def 18; then
A2: (the Sorts of Free(C, MSVars C)).a_Type C c=
(the Sorts of FreeMSA V).a_Type C;
per cases by Th14;
suppose (t.{})`1 in Vars & (t.{})`2 = a_Term & t is quasi-term of C;
hence thesis by Th8,ABCMIZ_1:48,XBOOLE_0:3;
end;
suppose that
A3: (t.{})`2 = the carrier of C and
A4: (t.{})`1 in Constructors and
A5: (t.{})`1 in the carrier' of C;
reconsider o = (t.{})`1 as OperSymbol of C by A5;
reconsider tt = t as Term of C, V by MSAFREE3:8;
not o in {*, non_op} by A4,ABCMIZ_1:39,XBOOLE_0:3; then
o <> * & o <> non_op by TARSKI:def 2; then
o is constructor; then
A6: o`1 = the_result_sort_of o by Def1;
A7: t.{} = [o, (t.{})`2]; then
A8: the_sort_of tt = the_result_sort_of o by A3,MSATERM:17;
thus
now assume t is pure expression of C, a_Type C; then
A9: t in (the Sorts of Free(C, MSVars C)).a_Type C by ABCMIZ_1:def 28;
thus (t.{})`1 in Constructors by A4;
assume (t.{})`1`1 <> a_Type;
hence contradiction by A2,A9,A6,A8,MSAFREE3:7;
end;
assume (t.{})`1 in Constructors;
assume (t.{})`1`1 = a_Type; then
reconsider t as expression of C, a_Type C by A3,A7,Th12,A6;
t is pure proof given a being expression of C, an_Adj C,
q being expression of C, a_Type C such that
A10: t = (ast C)term(a,q);
t = [*, the carrier of C]-tree <*a,q*> by A10,ABCMIZ_1:46; then
t.{} = [*, the carrier of C] by TREES_4:def 4; then
(t.{})`1 = *; then
(t.{})`1 in {*, non_op} by TARSKI:def 2;
hence thesis by A4,ABCMIZ_1:39,XBOOLE_0:3;
end;
hence thesis;
end;
suppose
A11: (t.{})`1 = *; then
(t.{})`1 in {*, non_op} by TARSKI:def 2; then
A12: not (t.{})`1 in Constructors by ABCMIZ_1:39,XBOOLE_0:3;
consider a being expression of C, an_Adj C,
q being expression of C, a_Type C
such that
A13: t = [*,3]-tree(a,q) by A11,Th18;
t = [*,3]-tree<*a,q*> by A13,TREES_4:def 6
.= [*, the carrier of C]-tree <*a,q*> by ABCMIZ_1:def 9,YELLOW11:1
.= (ast C)term(a,q) by ABCMIZ_1:46;
hence thesis by A12,ABCMIZ_1:def 41;
end;
suppose
A14: (t.{})`1 = non_op; then
(t.{})`1 in {*, non_op} by TARSKI:def 2; then
A15: not (t.{})`1 in Constructors by ABCMIZ_1:39,XBOOLE_0:3;
consider a being expression of C, an_Adj C such that
A16: t = [non_op,3]-tree a by A14,Th19;
t = [non_op,3]-tree <*a*> by A16,TREES_4:def 5; then
t.{} = [non_op,3] by TREES_4:def 4; then
(t.{})`1 = non_op; then
t is expression of C, an_Adj C by Th13;
hence thesis by A15,ABCMIZ_1:48;
end;
end;
begin :: Expressions
reserve i,j for Nat,
x for variable,
l for quasi-loci;
set MC = MaxConstrSign;
definition
mode expression is expression of MaxConstrSign;
mode valuation is valuation of MaxConstrSign;
mode quasi-adjective is quasi-adjective of MaxConstrSign;
func QuasiAdjs -> Subset of Free(MaxConstrSign, MSVars MaxConstrSign) equals
QuasiAdjs MaxConstrSign; coherence;
mode quasi-term is quasi-term of MaxConstrSign;
func QuasiTerms -> Subset of Free(MaxConstrSign, MSVars MaxConstrSign) equals
QuasiTerms MaxConstrSign; coherence;
mode quasi-type is quasi-type of MaxConstrSign;
func QuasiTypes -> set equals QuasiTypes MaxConstrSign; coherence;
end;
registration
cluster QuasiAdjs -> non empty; coherence;
cluster QuasiTerms -> non empty; coherence;
cluster QuasiTypes -> non empty; coherence;
end;
definition
redefine func Modes -> non empty Subset of Constructors;
coherence
proof
Modes c= Modes \/ Attrs & Modes \/ Attrs c= Constructors by XBOOLE_1:7;
hence thesis by XBOOLE_1:1;
end;
redefine func Attrs -> non empty Subset of Constructors;
coherence
proof
Attrs c= Modes \/ Attrs & Modes \/ Attrs c= Constructors by XBOOLE_1:7;
hence thesis by XBOOLE_1:1;
end;
redefine func Funcs -> non empty Subset of Constructors;
coherence by XBOOLE_1:7;
end;
reserve C for initialized ConstructorSignature,
c for constructor OperSymbol of C;
definition
func set-constr -> Element of Modes equals [a_Type,[{},0]];
coherence
proof
a_Type in {a_Type} & [{},0] in [:QuasiLoci,NAT:]
by ABCMIZ_1:29,TARSKI:def 1,ZFMISC_1:def 2;
hence thesis by ZFMISC_1:def 2;
end;
end;
theorem
kind_of set-constr = a_Type & loci_of set-constr = {} &
index_of set-constr = 0;
theorem Th28:
Constructors = [:{a_Type, an_Adj, a_Term}, [:QuasiLoci, NAT:]:]
proof
thus Constructors = [:{a_Type},[:QuasiLoci,NAT:]:] \/
[:{an_Adj},[:QuasiLoci,NAT:]:] \/ Funcs
.= [:{a_Type} \/ {an_Adj}, [:QuasiLoci,NAT:]:] \/ Funcs by ZFMISC_1:97
.= [:{a_Type, an_Adj}, [:QuasiLoci,NAT:]:] \/ Funcs by ENUMSET1:1
.= [:{a_Type, an_Adj} \/ {a_Term}, [:QuasiLoci,NAT:]:] by ZFMISC_1:97
.= [:{a_Type, an_Adj, a_Term}, [:QuasiLoci, NAT:]:] by ENUMSET1:3;
end;
theorem Th29:
[rng l, i] in Vars & l^<*[rng l,i]*> is quasi-loci
proof
varcl rng l = rng l by ABCMIZ_1:33;
hence [rng l, i] in Vars by ABCMIZ_1:17; then
reconsider x = [rng l, i] as variable;
rng l in {rng l, i} & {rng l, i} in x by TARSKI:def 2; then
vars x = rng l & x nin rng l by XREGULAR:7;
hence thesis by ABCMIZ_1:31;
end;
theorem Th30:
ex l st len l = i
proof
defpred P[Nat] means ex l st len l = $1;
<*>Vars is quasi-loci & len <*>Vars = 0 by ABCMIZ_1:29; then
A1: P[ 0 ];
A2: P[j] implies P[j+1] proof given l such that
A3: len l = j;
reconsider l1 = l^<*[rng l, 1]*> as quasi-loci by Th29;
take l1; thus thesis by A3,FINSEQ_2:16;
end;
P[j] from NAT_1:sch 2(A1,A2);
hence thesis;
end;
theorem Th31:
for X being finite Subset of Vars ex l st rng l = varcl X
proof
let X be finite Subset of Vars;
reconsider Y = varcl X as finite Subset of Vars by ABCMIZ_1:24;
defpred R[set, set] means $1 in $2`1;
A1: for x,y being set st x in Y & y in Y & R[x,y] holds not R[y,x]
proof
let x,y be set such that
A2: x in Y & y in Y & R[x,y] & R[y,x];
x in Vars by A2; then
consider A being Subset of Vars, j being Element of NAT such that
A3: x = [varcl A, j] & A is finite by ABCMIZ_1:18;
y in Vars by A2; then
consider B being Subset of Vars, k being Element of NAT such that
A4: y = [varcl B, k] & B is finite by ABCMIZ_1:18;
A5: y in varcl A & x in varcl B by A2,A3,A4;
A6: varcl A in {varcl A} & varcl B in {varcl B} by TARSKI:def 1;
{varcl A} in x & {varcl B} in y by A4,A3,TARSKI:def 2;
hence thesis by A5,A6,XREGULAR:10;
end;
A7: for x,y,z being set st x in Y & y in Y & z in Y & R[x,y] & R[y,z]
holds R[x,z]
proof
let x,y,z be set such that
A8: x in Y & y in Y & z in Y & R[x,y] & R[y,z];
y in Vars by A8; then
consider B being Subset of Vars, k being Element of NAT such that
A9: y = [varcl B, k] & B is finite by ABCMIZ_1:18;
z in Vars by A8; then
consider C being Subset of Vars, j being Element of NAT such that
A10: z = [varcl C, j] & C is finite by ABCMIZ_1:18;
A11: z`1 = varcl C & y`1 = varcl B by A10,A9; then
varcl B c= varcl C by A8,A9,ABCMIZ_1:def 1;
hence R[x,z] by A11,A8;
end;
consider l being one-to-one FinSequence such that
A12: rng l = Y and
A13: for i,j st i in dom l & j in dom l & R[l.i, l.j] holds i < j
from Numeration(A1,A7);
reconsider l as one-to-one FinSequence of Vars by A12,FINSEQ_1:def 4;
now let i be Nat, x be variable; assume
A14: i in dom l & x = l.i;
let y be variable; assume
A15: y in vars x;
x in Vars; then
consider A being Subset of Vars, j being Element of NAT such that
A16: x = [varcl A, j] & A is finite by ABCMIZ_1:18;
x in rng l & vars x = varcl A by A14,A16,FUNCT_1:def 3; then
vars x c= Y by A12,A16,ABCMIZ_1:def 1; then
consider a being object such that
A17: a in dom l & y = l.a by A12,A15,FUNCT_1:def 3;
reconsider a as Nat by A17;
take a;
thus a in dom l & a < i & y = l.a by A13,A14,A15,A17;
end; then
reconsider l as quasi-loci by ABCMIZ_1:30;
take l;
thus rng l = varcl X by A12;
end;
theorem Th32: :: to mozna uogolnic na X zamkniety na poddrzewa
:: (troche dodatkowych pojec i twierdzen)
for S being non empty non void ManySortedSign
for Y being non empty-yielding ManySortedSet of the carrier of S
for X,o being set, p being DTree-yielding FinSequence
st ex C st X = Union the Sorts of Free(S, Y)
holds o-tree p in X implies p is FinSequence of X
proof
let S be non empty non void ManySortedSign;
let Y be non empty-yielding ManySortedSet of the carrier of S;
let X,o be set;
let p be DTree-yielding FinSequence;
given C such that
A1: X = Union the Sorts of Free(S, Y);
assume o-tree p in X; then
reconsider e = o-tree p as Element of Free(S, Y) by A1;
rng p c= X
proof
let z be object; assume z in rng p; then
consider i being object such that
A2: i in dom p & z = p.i by FUNCT_1:def 3;
reconsider i as Nat by A2;
reconsider ppi = p.i as DecoratedTree by A2,TREES_3:24;
A3: 1 <= i & i <= len p by A2,FINSEQ_3:25; then
A4: (i-'1)+1 = i by XREAL_1:235;
then A5: i-'1 < len p by A3,NAT_1:13;
A6: len doms p = len p by TREES_3:38;
A7: (doms p).i = dom ppi by A2,FUNCT_6:22;
A8: dom e = tree doms p by TREES_4:10;
<*i-'1*>^<*>NAT = <*i-'1*> & <*>NAT in dom ppi
by FINSEQ_1:34,TREES_1:22; then
reconsider q = <*i-'1*> as Element of dom e
by A4,A5,A6,A7,A8,TREES_3:def 15;
e|q = z by A2,A4,A5,TREES_4:def 4; then
z is Element of Free(S, Y) by MSAFREE3:33;
hence thesis by A1;
end;
hence p is FinSequence of X by FINSEQ_1:def 4;
end;
definition
let C;
let e be expression of C;
mode subexpression of e -> expression of C means
it in Subtrees e;
existence by TREES_9:11;
func constrs e -> set equals
(proj1 rng e)/\the set of all o where o is constructor OperSymbol of C;
coherence;
end;
definition
let S be non empty non void ManySortedSign;
let X be non empty-yielding ManySortedSet of the carrier of S;
let e be Element of Free(S,X);
func main-constr e -> set
equals: :: dobre dla zestandaryzowanych (nie ma def)
Def9:
(e.{})`1 if e is compound otherwise {};
:: x-term C = [x, a_Term]-tree {}
:: (ast C)term(a,t) = [*, the carrier of C]-tree<*a,t*>
:: (non_op C)term a = [non_op, the carrier of C]-tree<*a*>
:: c-trm p = [c, the carrier of C]-tree p
:: problem gdy '{}' moze byc 'c'
correctness;
func args e -> DTree-yielding FinSequence means: ARGS:
e = (e.{})-tree it;
existence
proof
consider v being set, p being DTree-yielding FinSequence such that
A1: e = v-tree p by TREES_9:8;
A2: v = e.{} by A1,TREES_4:def 4;
thus thesis by A1,A2;
end;
uniqueness by TREES_4:15;
end;
definition
let S be non empty non void ManySortedSign;
let X be non empty-yielding ManySortedSet of the carrier of S;
let e be Element of Free(S,X);
redefine func args e -> FinSequence of Free(S, X);
coherence
proof
A1: e = (e.{})-tree args e by ARGS;
args e is FinSequence of Free(S, X) by A1,Th32;
hence thesis;
end;
end;
theorem Th33:
for C for e being expression of C holds e is subexpression of e
proof
let C be initialized ConstructorSignature;
let e be expression of C;
thus e in Subtrees e by TREES_9:11;
end;
theorem
main-constr (x -term C) = {} by Def9;
theorem Th35:
for c being constructor OperSymbol of C
for p being FinSequence of QuasiTerms C st len p = len the_arity_of c
holds main-constr (c-trm p) = c
proof
let c be constructor OperSymbol of C;
let p be FinSequence of QuasiTerms C;
assume len p = len the_arity_of c; then
c-trm p = [c, the carrier of C]-tree p by ABCMIZ_1:def 35; then
(c-trm p).{} = [c, the carrier of C] by TREES_4:def 4;
hence main-constr (c-trm p) = [c, the carrier of C]`1 by Def9
.= c;
end;
definition
let C;
let e be expression of C;
attr e is constructor means:
Def11:
e is compound & main-constr e is constructor OperSymbol of C;
end;
registration
let C;
cluster constructor -> compound for expression of C;
coherence;
end;
registration
let C;
cluster constructor for expression of C;
existence
proof
consider m, a being OperSymbol of C such that
A1: the_result_sort_of m = a_Type & the_arity_of m = {} &
the_result_sort_of a = an_Adj & the_arity_of a = {} by ABCMIZ_1:def 12;
the_arity_of ast C = <*an_Adj C, a_Type C*> &
the_arity_of non_op C = <*an_Adj C*> by ABCMIZ_1:38; then
reconsider m as constructor OperSymbol of C by A1,ABCMIZ_1:def 11;
set p = <*>QuasiTerms C;
take e = m-trm p; thus e is compound;
len p = len the_arity_of m by A1;
hence thesis by Th35;
end;
end;
registration
let C;
let e be constructor expression of C;
cluster constructor for subexpression of e;
existence
proof
e is subexpression of e by Th33;
hence thesis;
end;
end;
registration
let S be non void Signature;
let X be non empty-yielding ManySortedSet of S;
let t be Element of Free(S,X);
cluster rng t -> Relation-like;
coherence
proof
set Z = (the carrier of S)-->{0};
set Y = X (\/) Z;
t is Term of S,Y by MSAFREE3:8; then
rng t c= the carrier of DTConMSA Y by RELAT_1:def 19;
hence thesis;
end;
end;
theorem
for e being constructor expression of C holds main-constr e in constrs e
proof
let e be constructor expression of C;
A1: main-constr e = (e.{})`1 by Def9;
{} in dom e by TREES_1:22; then
e.{} in rng e by FUNCT_1:def 3; then
A2: main-constr e in proj1 rng e by A1,MCART_1:86;
main-constr e is constructor OperSymbol of C by Def11; then
main-constr e in the set of all c;
hence main-constr e in constrs e by A2,XBOOLE_0:def 4;
end;
begin :: Arity
reserve a,a9 for quasi-adjective,
t,t1,t2 for quasi-term,
T for quasi-type,
c for Element of Constructors;
definition
let C be non void Signature;
attr C is arity-rich means: Def12:
for n being Nat, s being SortSymbol of C holds
{o where o is OperSymbol of C: the_result_sort_of o = s &
len the_arity_of o = n} is infinite;
let o be OperSymbol of C;
attr o is nullary means: Def13: the_arity_of o = {};
attr o is unary means: Def14: len the_arity_of o = 1;
attr o is binary means: Def15: len the_arity_of o = 2;
end;
theorem Th37:
for C being non void Signature for o being OperSymbol of C holds
(o is nullary implies o is not unary) &
(o is nullary implies o is not binary) &
(o is unary implies o is not binary);
registration
let C be ConstructorSignature;
cluster non_op C -> unary;
coherence
proof
the_arity_of non_op C = <*an_Adj C*> by ABCMIZ_1:38;
hence len the_arity_of non_op C = 1 by FINSEQ_1:39;
end;
cluster ast C -> binary;
coherence
proof
the_arity_of ast C = <*an_Adj C, a_Type C*> by ABCMIZ_1:38;
hence len the_arity_of ast C = 2 by FINSEQ_1:44;
end;
end;
registration
let C be ConstructorSignature;
cluster nullary -> constructor for OperSymbol of C;
coherence
proof
let o be OperSymbol of C such that
A1: the_arity_of o = {};
the_arity_of ast C = <*an_Adj C, a_Type C*> &
the_arity_of non_op C = <*an_Adj C*> by ABCMIZ_1:38;
hence o <> * & o <> non_op by A1;
end;
end;
theorem Th38:
for C being ConstructorSignature holds C is initialized iff
ex m being OperSymbol of a_Type C, a being OperSymbol of an_Adj C st
m is nullary & a is nullary
proof
let C be ConstructorSignature;
hereby assume C is initialized; then
consider m, a being OperSymbol of C such that
A1: the_result_sort_of m = a_Type & the_arity_of m = {} &
the_result_sort_of a = an_Adj & the_arity_of a = {};
reconsider m as OperSymbol of a_Type C by A1,ABCMIZ_1:def 32;
reconsider a as OperSymbol of an_Adj C by A1,ABCMIZ_1:def 32;
take m, a;
thus m is nullary by A1;
thus a is nullary by A1;
end;
given m being OperSymbol of a_Type C, a being OperSymbol of an_Adj C
such that
A2: m is nullary & a is nullary;
take m,a;
the_result_sort_of non_op C = an_Adj C &
the_result_sort_of ast C = a_Type C by ABCMIZ_1:38;
hence thesis by A2,ABCMIZ_1:def 32;
end;
registration
let C be initialized ConstructorSignature;
cluster nullary constructor for OperSymbol of a_Type C;
existence
proof
consider m being OperSymbol of a_Type C, a being OperSymbol of an_Adj C
such that
A1: m is nullary & a is nullary by Th38;
take m; thus thesis by A1;
end;
cluster nullary constructor for OperSymbol of an_Adj C;
existence
proof
consider m being OperSymbol of a_Type C, a being OperSymbol of an_Adj C
such that
A2: m is nullary & a is nullary by Th38;
take a; thus thesis by A2;
end;
end;
registration
let C be initialized ConstructorSignature;
cluster nullary constructor for OperSymbol of C;
existence
proof
set o = the nullary constructor OperSymbol of a_Type C;
take o; thus thesis;
end;
end;
registration
cluster arity-rich -> with_an_operation_for_each_sort for
non void Signature;
coherence
proof
let S be non void Signature such that
A1: for n being Nat, s being SortSymbol of S holds
{o where o is OperSymbol of S: the_result_sort_of o = s &
len the_arity_of o = n} is infinite;
let v be object;
assume v in the carrier of S; then
reconsider v as SortSymbol of S;
set A = {o where o is OperSymbol of S: the_result_sort_of o = v &
len the_arity_of o = 0};
reconsider A as infinite set by A1;
set a = the Element of A;
a in A; then
consider o being OperSymbol of S such that
A2: a = o & the_result_sort_of o = v & len the_arity_of o = 0;
thus thesis by A2,FUNCT_2:4;
end;
cluster arity-rich -> initialized for ConstructorSignature;
coherence
proof
let C be ConstructorSignature such that
A3: C is arity-rich;
set Xt = {o where o is OperSymbol of C: the_result_sort_of o = a_Type C &
len the_arity_of o = 0};
set x = the Element of Xt;
Xt is infinite set by A3; then
x in Xt; then
consider m being OperSymbol of C such that
A4: x = m & the_result_sort_of m = a_Type C & len the_arity_of m = 0;
set Xa = {o where o is OperSymbol of C: the_result_sort_of o = an_Adj C &
len the_arity_of o = 0};
set x = the Element of Xa;
Xa is infinite set by A3; then
x in Xa; then
consider a being OperSymbol of C such that
A5: x = a & the_result_sort_of a = an_Adj C & len the_arity_of a = 0;
take m, a; thus thesis by A4,A5;
end;
end;
registration
cluster MaxConstrSign -> arity-rich;
coherence
proof set C = MaxConstrSign;
let n be Nat, s be SortSymbol of C;
A1: the carrier of C = {0,1,2} by ABCMIZ_1:def 9;
set X = {o where o is OperSymbol of C: the_result_sort_of o = s &
len the_arity_of o = n};
consider l being quasi-loci such that
A2: len l = n by Th30;
deffunc F(object) = [s,[l,$1]];
consider f being Function such that
A3: dom f = NAT & for i being object st i in NAT holds f.i = F(i)
from FUNCT_1:sch 3;
f is one-to-one
proof
let i,j be object; assume i in dom f & j in dom f; then
reconsider i,j as Element of NAT by A3;
f.i = F(i) & f.j = F(j) by A3; then
f.i = f.j implies [l,i] = [l,j] by XTUPLE_0:1;
hence thesis by XTUPLE_0:1;
end; then
A4: rng f is infinite by A3,CARD_1:59;
rng f c= X
proof
let y be object; assume y in rng f; then
consider x being object such that
A5: x in dom f & y = f.x by FUNCT_1:def 3;
reconsider x as Element of NAT by A3,A5;
A6: [l,x] in [:QuasiLoci, NAT:] & y = F(x) by A3,A5; then
y in Constructors by A1,Th28,ZFMISC_1:def 2; then
y in {*,non_op}\/Constructors by XBOOLE_0:def 3; then
reconsider y as OperSymbol of C by ABCMIZ_1:def 24;
A7: y is constructor by A6; then
A8: the_result_sort_of y = y`1 by ABCMIZ_1:def 24 .= s by A6;
len the_arity_of y = card y`2`1 by A7,ABCMIZ_1:def 24
.= card [l,x]`1 by A6
.= len l;
hence thesis by A2,A8;
end;
hence X is infinite by A4;
end;
end;
registration
cluster arity-rich initialized for ConstructorSignature;
existence
proof
take MaxConstrSign; thus thesis;
end;
end;
registration
let C be arity-rich ConstructorSignature;
let s be SortSymbol of C;
cluster nullary constructor for OperSymbol of s;
existence
proof
set X = {o where o is OperSymbol of C: the_result_sort_of o = s &
len the_arity_of o = 0};
X is infinite by Def12; then
consider m1,m2 being object such that
A1: m1 in X & m2 in X & m1 <> m2 by ZFMISC_1:def 10;
consider o1 being OperSymbol of C such that
A2: m1 = o1 & the_result_sort_of o1 = s & len the_arity_of o1 = 0 by A1;
reconsider m1 as OperSymbol of s by A2,ABCMIZ_1:def 32;
the_arity_of m1 = {} by A2; then
m1 is nullary;
hence thesis;
end;
cluster unary constructor for OperSymbol of s;
existence
proof
set X = {o where o is OperSymbol of C: the_result_sort_of o = s &
len the_arity_of o = 1};
X is infinite by Def12; then
consider m1,m2 being object such that
A3: m1 in X & m2 in X & m1 <> m2 by ZFMISC_1:def 10;
consider o1 being OperSymbol of C such that
A4: m1 = o1 & the_result_sort_of o1 = s & len the_arity_of o1 = 1 by A3;
consider o2 being OperSymbol of C such that
A5: m2 = o2 & the_result_sort_of o2 = s & len the_arity_of o2 = 1 by A3;
reconsider m1,m2 as OperSymbol of s by A4,A5,ABCMIZ_1:def 32;
A6: m1 is unary & m2 is unary by A4,A5; then
A7: m1 <> ast C & m2 <> ast C by Th37;
m1 <> non_op or m2 <> non_op by A3; then
m1 is constructor or m2 is constructor by A7;
hence thesis by A6;
end;
cluster binary constructor for OperSymbol of s;
existence
proof
set X = {o where o is OperSymbol of C: the_result_sort_of o = s &
len the_arity_of o = 2};
X is infinite by Def12; then
consider m1,m2 being object such that
A8: m1 in X & m2 in X & m1 <> m2 by ZFMISC_1:def 10;
consider o1 being OperSymbol of C such that
A9: m1 = o1 & the_result_sort_of o1 = s & len the_arity_of o1 = 2 by A8;
consider o2 being OperSymbol of C such that
A10: m2 = o2 & the_result_sort_of o2 = s & len the_arity_of o2 = 2 by A8;
reconsider m1,m2 as OperSymbol of s by A9,A10,ABCMIZ_1:def 32;
A11: m1 is binary & m2 is binary by A9,A10; then
A12: m1 <> non_op C & m2 <> non_op C by Th37;
m1 <> * or m2 <> * by A8; then
m1 is constructor or m2 is constructor by A12;
hence thesis by A11;
end;
end;
registration
let C be arity-rich ConstructorSignature;
cluster unary constructor for OperSymbol of C;
existence
proof
set o = the unary constructor OperSymbol of a_Type C;
take o; thus thesis;
end;
cluster binary constructor for OperSymbol of C;
existence
proof
set o = the binary constructor OperSymbol of a_Type C;
take o; thus thesis;
end;
end;
theorem Th39:
for o being nullary OperSymbol of C holds
[o, the carrier of C]-tree {} is expression of C, the_result_sort_of o
proof
let o be nullary OperSymbol of C;
set X = MSVars C;
set Z = (the carrier of C)-->{0};
set Y = X (\/) Z;
A1: the_arity_of o = {} by Def13;
A2: the Sorts of Free(C, X) = C-Terms(X, Y) by MSAFREE3:24;
for i being Nat st i in dom {} ex t being Term of C,Y st t = {}.i &
the_sort_of t = (the_arity_of o).i; then
reconsider p = {} as ArgumentSeq of Sym(o, Y) by A1,MSATERM:24;
A3: variables_in (Sym(o, Y)-tree p) c= X
proof let s be object; assume s in the carrier of C; then
reconsider s9 = s as SortSymbol of C;
let x be object; assume x in (variables_in (Sym(o, Y)-tree p)).s; then
ex t being DecoratedTree st t in rng p & x in (C variables_in t).s9
by MSAFREE3:11;
hence thesis;
end;
set s9 = the_result_sort_of o;
A4: the_sort_of (Sym(o, Y)-tree p) = the_result_sort_of o by MSATERM:20;
(the Sorts of Free(C, X)).s9 =
{t where t is Term of C,Y: the_sort_of t = s9 & variables_in t c= X}
by A2,MSAFREE3:def 5; then
[o, the carrier of C]-tree {} in (the Sorts of Free(C, X)).s9 by A3,A4;
hence thesis by ABCMIZ_1:41;
end;
definition
let C be initialized ConstructorSignature;
let m be nullary constructor OperSymbol of a_Type C;
redefine func m term -> pure expression of C, a_Type C;
coherence
proof
set T = m term;
the_arity_of m = 0 by Def13; then
len the_arity_of m = 0; then
A1: T = [m, the carrier of C]-tree {} by ABCMIZ_1:def 29;
ex m, a being OperSymbol of C st
the_result_sort_of m = a_Type & the_arity_of m = {} &
the_result_sort_of a = an_Adj & the_arity_of a = {}
by ABCMIZ_1:def 12; then
the_result_sort_of m = a_Type C by ABCMIZ_1:def 32; then
reconsider T as expression of C, a_Type C by A1,Th39;
T is pure
proof
given a being expression of C, an_Adj C,
t being expression of C, a_Type C such that
A2: T = (ast C)term(a,t);
T = [ *, the carrier of C]-tree <*a,t*> by A2,ABCMIZ_1:46;
hence thesis by A1,TREES_4:15;
end;
hence thesis;
end;
end;
definition
let c be Element of Constructors;
func @c -> constructor OperSymbol of MaxConstrSign equals c;
coherence
proof
* in {*, non_op} & non_op in {*, non_op} &
the carrier' of MC = {*, non_op} \/ Constructors
by ABCMIZ_1:def 24,TARSKI:def 2; then
c <> * & c <> non_op & c in the carrier' of MC
by ABCMIZ_1:39,XBOOLE_0:3,def 3;
hence thesis by ABCMIZ_1:def 11;
end;
end;
definition
let m be Element of Modes;
redefine func @m -> constructor OperSymbol of a_Type MaxConstrSign;
coherence
proof
A1: m`1 in {a_Type} by MCART_1:10;
the_result_sort_of @m = m`1 by ABCMIZ_1:def 24
.= a_Type by A1,TARSKI:def 1;
hence thesis by ABCMIZ_1:def 32;
end;
end;
registration
cluster @set-constr -> nullary;
coherence
proof
len the_arity_of @set-constr = card set-constr`2`1 by ABCMIZ_1:def 24
.= card [{},0]`1 .= card 0;
hence the_arity_of @set-constr = {};
end;
end;
theorem
the_arity_of @set-constr = {} by Def13;
definition
func set-type -> quasi-type equals
({}QuasiAdjs MaxConstrSign) ast ((@set-constr) term);
coherence;
end;
theorem
adjs set-type = {} & the_base_of set-type = (@set-constr) term;
definition
let l be FinSequence of Vars;
func args l -> FinSequence of QuasiTerms MaxConstrSign means:
Def18:
len it = len l & for i st i in dom l holds it.i = (l/.i)-term MaxConstrSign;
existence
proof
deffunc F(Nat) = (l/.$1)-term MaxConstrSign;
consider g being FinSequence such that
A1: len g = len l and
A2: for i st i in dom g holds g.i = F(i) from FINSEQ_1:sch 2;
A3: dom g = dom l by A1,FINSEQ_3:29;
rng g c= QuasiTerms MaxConstrSign
proof
let j be object; assume j in rng g; then
consider z being object such that
A4: z in dom g & j = g.z by FUNCT_1:def 3;
reconsider z as Nat by A4;
j = F(z) by A2,A4;
hence thesis by ABCMIZ_1:49;
end; then
reconsider g as FinSequence of QuasiTerms MaxConstrSign by FINSEQ_1:def 4;
take g;
thus thesis by A1,A2,A3;
end;
uniqueness
proof
let a1,a2 be FinSequence of QuasiTerms MaxConstrSign such that
A5: len a1 = len l and
A6: for i st i in dom l holds a1.i = (l/.i)-term MaxConstrSign and
A7: len a2 = len l and
A8: for i st i in dom l holds a2.i = (l/.i)-term MaxConstrSign;
A9: dom a1 = dom l & dom a2 = dom l by A5,A7,FINSEQ_3:29;
now let i;
assume i in dom a1; then
a1.i = (l/.i)-term MaxConstrSign & a2.i = (l/.i)-term MaxConstrSign
by A6,A8,A9;
hence a1.i = a2.i;
end;
hence thesis by A9;
end;
end;
definition
let c;
func base_exp_of c -> expression equals (@c)-trm args loci_of c;
coherence;
end;
theorem Th42:
for o being OperSymbol of MaxConstrSign holds
o is constructor iff o in Constructors
proof
let o be OperSymbol of MaxConstrSign;
A1: the carrier' of MaxConstrSign = {*, non_op} \/ Constructors
by ABCMIZ_1:def 24;
o is constructor iff o nin {*,non_op} by TARSKI:def 2;
hence thesis by A1,ABCMIZ_1:39,XBOOLE_0:3,def 3;
end;
theorem
for m being nullary OperSymbol of MaxConstrSign holds
main-constr (m term) = m
proof set C = MaxConstrSign;
let m be nullary OperSymbol of C;
the_arity_of m = 0 by Def13; then
len the_arity_of m = 0 & len {} = 0; then
A1: m term = [m, the carrier of C]-tree {} &
m-trm(<*>QuasiTerms C) = [m, the carrier of C]-tree {}
by ABCMIZ_1:def 29,def 35;
hence main-constr (m term) = ((m term).{})`1 by Def9
.= [m, the carrier of C]`1 by A1,TREES_4:def 4
.= m;
end;
theorem
for m being unary constructor OperSymbol of MaxConstrSign
for t holds main-constr (m term t) = m
proof set C = MaxConstrSign;
let m be unary constructor OperSymbol of C;
let t;
reconsider w = t as Element of QuasiTerms C by ABCMIZ_1:49;
reconsider p = <*w*> as FinSequence of QuasiTerms C;
A1: len the_arity_of m = 1 by Def14; then
the_arity_of m = 1 |-> a_Term by ABCMIZ_1:37
.= <*a_Term*> by FINSEQ_2:59; then
len p = 1 & (the_arity_of m).1 = a_Term C by FINSEQ_1:40; then
A2: m term t = [m, the carrier of C]-tree <*t*> &
m-trm p = [m, the carrier of C]-tree <*t*> by A1,ABCMIZ_1:def 30,def 35;
hence main-constr (m term t) = ((m term t).{})`1 by Def9
.= [m, the carrier of C]`1 by A2,TREES_4:def 4
.= m;
end;
theorem
for a holds main-constr ((non_op MaxConstrSign)term a) = non_op
proof set C = MaxConstrSign;
set m = non_op C;
let a;
A1: len the_arity_of m = 1 by Def14;
the_arity_of m = <*an_Adj*> by ABCMIZ_1:38; then
(the_arity_of m).1 = an_Adj C by FINSEQ_1:40; then
A2: m term a = [m, the carrier of C]-tree <*a*> by A1,ABCMIZ_1:def 30;
thus main-constr (m term a) = ((m term a).{})`1 by Def9
.= [m, the carrier of C]`1 by A2,TREES_4:def 4
.= non_op;
end;
theorem
for m being binary constructor OperSymbol of MaxConstrSign
for t1,t2 holds main-constr (m term(t1,t2)) = m
proof set C = MaxConstrSign;
let m be binary constructor OperSymbol of C;
let t1,t2;
reconsider w1 = t1, w2 = t2 as Element of QuasiTerms C by ABCMIZ_1:49;
reconsider p = <*w1,w2*> as FinSequence of QuasiTerms C;
A1: len the_arity_of m = 2 by Def15; then
the_arity_of m = 2 |-> a_Term by ABCMIZ_1:37
.= <*a_Term,a_Term*> by FINSEQ_2:61; then
(the_arity_of m).1 = a_Term C & (the_arity_of m).2 = a_Term C & len p = 2
by FINSEQ_1:44; then
A2: m term(t1,t2) = [m, the carrier of C]-tree <*t1,t2*> &
m-trm p = [m, the carrier of C]-tree p by A1,ABCMIZ_1:def 31,def 35;
hence main-constr (m term(t1,t2)) = ((m term(t1,t2)).{})`1 by Def9
.= [m, the carrier of C]`1 by A2,TREES_4:def 4
.= m;
end;
theorem
for q being expression of MaxConstrSign, a_Type MaxConstrSign
for a holds main-constr ((ast MaxConstrSign)term(a,q)) = *
proof set C = MaxConstrSign;
set m = ast C;
let q be expression of MaxConstrSign, a_Type MaxConstrSign;
let a;
A1: len the_arity_of m = 2 by Def15;
the_arity_of m = <*an_Adj C,a_Type C*> by ABCMIZ_1:38; then
(the_arity_of m).1 = an_Adj C & (the_arity_of m).2 = a_Type C
by FINSEQ_1:44; then
A2: m term(a,q) = [m, the carrier of C]-tree <*a,q*> by A1,ABCMIZ_1:def 31;
thus main-constr (m term(a,q)) = ((m term(a,q)).{})`1 by Def9
.= [m, the carrier of C]`1 by A2,TREES_4:def 4
.= *;
end;
definition
let T be quasi-type;
func constrs T -> set equals
(constrs the_base_of T) \/ union {constrs a: a in adjs T};
coherence;
end;
theorem
for q being pure expression of MaxConstrSign, a_Type MaxConstrSign
for A being finite Subset of QuasiAdjs MaxConstrSign holds
constrs(A ast q) = (constrs q) \/ union {constrs a: a in A};
theorem
constrs(a ast T) = (constrs a) \/ (constrs T)
proof
set A = {constrs a9: a9 in {a} \/ adjs T};
set B = {constrs a9: a9 in adjs T};
A1: A = B \/{constrs a}
proof
thus A c= B \/{constrs a} proof let z be object;
assume z in A; then
consider a9 such that
A2: z = constrs a9 & a9 in {a} \/ adjs T;
a9 in {a} or a9 in adjs T by A2,XBOOLE_0:def 3; then
a9 = a or a9 in adjs T by TARSKI:def 1; then
z in {constrs a} or z in B by A2,TARSKI:def 1;
hence thesis by XBOOLE_0:def 3;
end;
let z be object; assume
A3: z in B\/{constrs a};
A4: a in {a} by TARSKI:def 1;
per cases by A3,XBOOLE_0:def 3;
suppose z in B; then
consider a9 such that
A5: z = constrs a9 & a9 in adjs T;
a9 in {a} \/ adjs T by A5,XBOOLE_0:def 3;
hence thesis by A5;
end;
suppose z in {constrs a}; then
z = constrs a & a in {a} \/ adjs T by A4,TARSKI:def 1,XBOOLE_0:def 3;
hence thesis;
end;
end;
thus constrs(a ast T) = (constrs the_base_of (a ast T)) \/ union A
.= (constrs the_base_of T) \/ union A
.= (constrs the_base_of T) \/ ((union {constrs a}) \/ union B)
by A1,ZFMISC_1:78
.= (constrs the_base_of T) \/ ((constrs a) \/ union B) by ZFMISC_1:25
.= (constrs the_base_of T) \/ (constrs a) \/ union B by XBOOLE_1:4
.= (constrs a) \/ ((constrs the_base_of T) \/ union B) by XBOOLE_1:4
.= (constrs a) \/ (constrs T);
end;
begin :: Unification
definition
let C be initialized ConstructorSignature;
let t,p be expression of C;
pred t matches_with p means
ex f being valuation of C st t = p at f;
reflexivity
proof let t be expression of C;
take f = the empty valuation of C;
thus t at f = t by ABCMIZ_1:139;
end;
end;
theorem
for t1,t2,t3 being expression of C st t1 matches_with t2 & t2 matches_with t3
holds t1 matches_with t3
proof
let t1,t2,t3 be expression of C;
given f1 being valuation of C such that
A1: t1 = t2 at f1;
given f2 being valuation of C such that
A2: t2 = t3 at f2;
take f2 at f1;
thus thesis by A1,A2,ABCMIZ_1:149;
end;
definition
let C be initialized ConstructorSignature;
let A,B be Subset of QuasiAdjs C;
pred A matches_with B means
ex f being valuation of C st B at f c= A;
reflexivity
proof let t be Subset of QuasiAdjs C;
take f = the empty valuation of C;
let x be object; assume x in t at f; then
ex a being quasi-adjective of C st x = a at f & a in t;
hence x in t by ABCMIZ_1:139;
end;
end;
theorem
for A1,A2,A3 being Subset of QuasiAdjs C
st A1 matches_with A2 & A2 matches_with A3
holds A1 matches_with A3
proof
let t1,t2,t3 be Subset of QuasiAdjs C;
given f1 being valuation of C such that
A1: t2 at f1 c= t1;
given f2 being valuation of C such that
A2: t3 at f2 c= t2;
take f2 at f1;
(t3 at f2) at f1 c= t2 at f1 by A2,ABCMIZ_1:146; then
(t3 at f2) at f1 c= t1 by A1;
hence thesis by ABCMIZ_1:150;
end;
definition
let C be initialized ConstructorSignature;
let T,P be quasi-type of C;
pred T matches_with P means
ex f being valuation of C st
(adjs P) at f c= adjs T & (the_base_of P) at f = the_base_of T;
reflexivity
proof let t be quasi-type of C;
take f = the empty valuation of C;
thus (adjs t) at f c= adjs t
proof let x be object; assume x in (adjs t) at f; then
ex a being quasi-adjective of C st x = a at f & a in adjs t;
hence x in adjs t by ABCMIZ_1:139;
end;
thus thesis by ABCMIZ_1:139;
end;
end;
theorem
for T1,T2,T3 being quasi-type of C st T1 matches_with T2 & T2 matches_with T3
holds T1 matches_with T3
proof
let t1,t2,t3 be quasi-type of C;
given f1 being valuation of C such that
A1: (adjs t2) at f1 c= adjs t1 & (the_base_of t2) at f1 = the_base_of t1;
given f2 being valuation of C such that
A2: (adjs t3) at f2 c= adjs t2 & (the_base_of t3) at f2 = the_base_of t2;
take f2 at f1;
((adjs t3) at f2) at f1 c= (adjs t2) at f1 by A2,ABCMIZ_1:146; then
((adjs t3) at f2) at f1 c= adjs t1 by A1;
hence thesis by A1,A2,ABCMIZ_1:149,150;
end;
definition
let C be initialized ConstructorSignature;
let t1,t2 be expression of C;
let f be valuation of C;
::$N Unification of Mizar terms
pred f unifies t1,t2 means
t1 at f = t2 at f;
end;
theorem
for t1,t2 being expression of C for f being valuation of C st f unifies t1,t2
holds f unifies t2,t1;
definition
let C be initialized ConstructorSignature;
let t1,t2 be expression of C;
pred t1,t2 are_unifiable means
ex f being valuation of C st f unifies t1,t2;
reflexivity
proof
let t be expression of C;
set f = the valuation of C;
take f; thus t at f = t at f;
end;
symmetry
proof
let t1,t2 be expression of C;
given f being valuation of C such that
A1: f unifies t1,t2;
take f;
thus t2 at f = t1 at f by A1;
end;
end;
definition
let C be initialized ConstructorSignature;
let t1,t2 be expression of C;
pred t1,t2 are_weakly-unifiable means
ex g being irrelevant one-to-one valuation of C
st variables_in t2 c= dom g & t1,t2 at g are_unifiable;
reflexivity
proof
let t be expression of C;
take C idval variables_in t;
thus thesis by ABCMIZ_1:131,137;
end;
:: symmetry;
end;
:: theorem
:: for t1,t2 being expression of C st t1 matches_with t2
:: holds t1,t2 are_weakly-unifiable;
theorem
for t1,t2 being expression of C st t1,t2 are_unifiable
holds t1,t2 are_weakly-unifiable
proof
let t1,t2 be expression of C;
given f being valuation of C such that
A1: f unifies t1,t2;
take g = C idval variables_in t2;
thus variables_in t2 c= dom g by ABCMIZ_1:131;
take f; thus f unifies t1,t2 at g by A1,ABCMIZ_1:137;
end;
definition
let C be initialized ConstructorSignature;
let t,t1,t2 be expression of C;
pred t is_a_unification_of t1,t2 means
ex f being valuation of C st f unifies t1,t2 & t = t1 at f;
end;
theorem
for t1,t2,t being expression of C st t is_a_unification_of t1,t2
holds t is_a_unification_of t2,t1
proof
let t1,t2,t be expression of C;
given f being valuation of C such that
A1: f unifies t1,t2 & t = t1 at f;
take f;
thus f unifies t2,t1 & t = t2 at f by A1;
end;
theorem
for t1,t2,t being expression of C st t is_a_unification_of t1,t2
holds t matches_with t1 & t matches_with t2
proof
let t1,t2,t be expression of C;
given f being valuation of C such that
A1: f unifies t1,t2 & t = t1 at f;
thus ex f being valuation of C st t = t1 at f by A1;
take f; thus t = t2 at f by A1;
end;
definition
let C be initialized ConstructorSignature;
let t,t1,t2 be expression of C;
pred t is_a_general-unification_of t1,t2 means
t is_a_unification_of t1,t2 &
for u being expression of C st u is_a_unification_of t1,t2
holds u matches_with t;
end;
:: theorem
:: for t1,t2 being expression of C st t1,t2 are_unifiable
:: ex t being expression of C st t is_a_general-unification_of t1,t2;
begin :: Type distribution
theorem Th57:
for n being Nat for s being SortSymbol of MaxConstrSign
ex m being constructor OperSymbol of s
st len the_arity_of m = n
proof set C = MaxConstrSign;
let n be Nat;
let s be SortSymbol of C;
deffunc F(Nat) = [{},$1];
consider l being FinSequence such that
A1: len l = n and
A2: for i st i in dom l holds l.i = F(i) from FINSEQ_1:sch 2;
A3: l is one-to-one
proof
let i,j be object such that
A4: i in dom l & j in dom l & l.i = l.j;
reconsider i,j as Nat by A4;
l.i = F(i) & l.i = F(j) by A2,A4; then
i = F(j)`2;
hence thesis;
end;
rng l c= Vars
proof
let z be object; assume z in rng l; then
consider a being object such that
A5: a in dom l & z = l.a by FUNCT_1:def 3;
reconsider a as Nat by A5;
z = F(a) by A2,A5;
hence thesis by ABCMIZ_1:25;
end; then
reconsider l as one-to-one FinSequence of Vars by A3,FINSEQ_1:def 4;
for i being Nat, x being variable st i in dom l & x = l.i
for y being variable st y in vars x
ex j being Nat st j in dom l & j < i & y = l.j
proof
let i,x; assume i in dom l & x = l.i; then
x = F(i) by A2;
hence thesis;
end; then
reconsider l as quasi-loci by ABCMIZ_1:30;
set m = [s,[l,0]];
the carrier of C = {a_Type, an_Adj, a_Term} by ABCMIZ_1:def 9;
then A6: m in Constructors by Th28; then
m in {*,non_op}\/Constructors by XBOOLE_0:def 3; then
reconsider m as constructor OperSymbol of C by A6,Th42,ABCMIZ_1:def 24;
the_result_sort_of m = m`1 by ABCMIZ_1:def 24 .= s; then
reconsider m as constructor OperSymbol of s by ABCMIZ_1:def 32;
take m;
thus len the_arity_of m = card m`2`1 by ABCMIZ_1:def 24
.= card [l,0]`1
.= card l
.= n by A1;
end;
theorem Th58:
for l for s being SortSymbol of MaxConstrSign
for m being constructor OperSymbol of s st len the_arity_of m = len l
holds variables_in (m-trm args l) = rng l
proof
let l; set X = rng l;
set n = len l;
set C = MaxConstrSign;
let s be SortSymbol of C;
let m be constructor OperSymbol of s such that
A1: len the_arity_of m = n;
set p = args l;
set Y = {variables_in t where t is quasi-term of C: t in rng p};
A2: len p = len the_arity_of m by A1,Def18; then
A3: variables_in (m-trm p) = union Y by ABCMIZ_1:90;
A4: dom p = dom l by A1,A2,FINSEQ_3:29;
thus variables_in (m-trm p) c= X
proof
let s be object; assume s in variables_in (m-trm p); then
consider A being set such that
A5: s in A & A in Y by A3,TARSKI:def 4;
consider t being quasi-term of C such that
A6: A = variables_in t & t in rng p by A5;
consider z being object such that
A7: z in dom p & t = p.z by A6,FUNCT_1:def 3;
reconsider z as Element of NAT by A7;
l.z = l/.z by A4,A7,PARTFUN1:def 6; then
A8: l/.z in X by A4,A7,FUNCT_1:def 3;
p.z = (l/.z)-term C by A4,A7,Def18; then
A = {l/.z} by A6,A7,ABCMIZ_1:86;
hence thesis by A5,A8,TARSKI:def 1;
end;
let s be object; assume s in X; then
consider z being object such that
A9: z in dom l & s = l.z by FUNCT_1:def 3;
reconsider z as Element of NAT by A9;
set t = (l/.z)-term C;
p.z = t & l.z = l/.z by A9,Def18,PARTFUN1:def 6; then
variables_in t = {s} & t in rng p by A4,A9,ABCMIZ_1:86,FUNCT_1:def 3; then
s in {s} & {s} in Y by TARSKI:def 1;
hence thesis by A3,TARSKI:def 4;
end;
theorem Th59:
for X being finite Subset of Vars st varcl X = X
for s being SortSymbol of MaxConstrSign
ex m being constructor OperSymbol of s st ::a_Type MaxConstrSign
ex p being FinSequence of QuasiTerms MaxConstrSign
st len p = len the_arity_of m & vars (m-trm p) = X
proof
let X be finite Subset of Vars;
assume
A1: varcl X = X; then
consider l such that
A2: rng l = X by Th31;
set n = len l;
set C = MaxConstrSign;
let s be SortSymbol of C;
consider m being constructor OperSymbol of s such that
A3: len the_arity_of m = n by Th57;
take m;
set p = args l;
take p;
thus len p = len the_arity_of m by A3,Def18;
thus thesis by A1,A2,A3,Th58;
end;
definition
let d be PartFunc of Vars, QuasiTypes;
attr d is even means
for x,T st x in dom d & T = d.x holds vars T = vars x;
end;
definition
let l be quasi-loci;
mode type-distribution of l -> PartFunc of Vars, QuasiTypes means:
Def30:
dom it = rng l & it is even;
existence
proof
defpred P[object,object] means
ex x,T st $1 = x & $2 = T & vars T = vars x;
A1: for z being object st z in rng l ex y being object st P[z,y]
proof
set C = MaxConstrSign;
let z be object; assume z in rng l; then
reconsider x = z as variable;
varcl vars x = vars x by Th2; then
consider m being constructor OperSymbol of a_Type C,
p being FinSequence of QuasiTerms C such that
A2: len p = len the_arity_of m & vars (m-trm p) = vars x by Th59;
a_Type C in the carrier of C &
the carrier of C c= rng the ResultSort of C by ABCMIZ_1:def 54; then
consider o being object such that
A3: o in dom the ResultSort of C & a_Type C = (the ResultSort of C).o
by FUNCT_1:def 3;
reconsider o as OperSymbol of C by A3;
the_result_sort_of o = a_Type C by A3; then
the_result_sort_of m = a_Type C by ABCMIZ_1:def 32; then
reconsider q = m-trm p as pure expression of C, a_Type C
by A2,ABCMIZ_1:75;
set B = {} QuasiAdjs C;
reconsider T = B ast q as quasi-type;
take T, x, T;
thus thesis by A2,ABCMIZ_1:106;
end;
consider f being Function such that
A4: dom f = rng l and
A5: for z being object st z in rng l holds P[z,f.z] from CLASSES1:sch 1(A1);
rng f c= QuasiTypes
proof
let y be object; assume y in rng f; then
consider z being object such that
A6: z in dom f & y = f.z by FUNCT_1:def 3;
P[z,y] by A4,A5,A6;
hence thesis by ABCMIZ_1:def 43;
end; then
reconsider f as PartFunc of Vars, QuasiTypes by A4,RELSET_1:4;
take f; thus dom f = rng l by A4;
let x,T; assume x in dom f & T = f.x; then
P[x,T] by A4,A5;
hence thesis;
end;
end;
theorem
for l being empty quasi-loci holds {} is type-distribution of l
proof
let l be empty quasi-loci;
reconsider d = {} as PartFunc of Vars, QuasiTypes by RELSET_1:12;
dom d = rng l & d is even;
hence thesis by Def30;
end;