:: Analysis of Algorithms: An Example of a Sort Algorithm
:: by Grzegorz Bancerek
::
:: Received November 9, 2012
:: Copyright (c) 2012-2016 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 INT_1, AOFA_A00, XBOOLE_0, STRUCT_0, RELAT_1, PBOOLE, MSAFREE4,
NAT_1, SUBSET_1, ORDINAL1, MSUALG_1, AOFA_000, ZF_MODEL, FUNCT_1,
INCPROJ, FUNCT_2, CARD_1, GRAPHSP, AOFA_I00, QC_LANG1, MSAFREE, AOFA_A01,
ARYTM_3, NEWTON, ABCMIZ_1, ZFMISC_1, XREAL_0, MSUALG_3, GROUP_6, REAL_1,
MSUALG_2, TARSKI, MARGREL1, XBOOLEAN, PARTFUN1, FINSEQ_1, XXREAL_0,
CARD_3, NUMBERS, FUNCT_7, ARYTM_1, PRELAMB, REALSET1, TREES_4, POWER,
PROB_2, FUNCT_6, PZFMISC1, ALGSTR_4, ORDINAL4, MEMBERED, FINSET_1,
XXREAL_2, AFINSQ_1, EXCHSORT, FUNCT_4, FUNCOP_1, FINSEQ_4, MODELC_3,
UNIALG_1, WELLORD1, ORDERS_2, LFUZZY_0, REARRAN1, SETLIM_2, MCART_1,
EQUATION, FOMODEL2, TREES_2;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, FINSET_1, WELLORD2, MCART_1,
RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, MARGREL1, XTUPLE_0, FUNCT_4,
FUNCT_6, FUNCT_7, FUNCOP_1, FINSEQ_1, FINSEQ_2, FINSEQ_4, AFINSQ_1,
CARD_1, CARD_3, POWER, MEMBERED, STRUCT_0, ORDINAL1, NAT_1, BINOP_1,
PBOOLE, TREES_3, TREES_4, PZFMISC1, TREES_2, NUMBERS, XCMPLX_0, XXREAL_0,
XREAL_0, XXREAL_2, INT_1, NAT_D, NEWTON, TREES_1, PROB_2, ORDERS_2,
WAYBEL_0, LFUZZY_0, UNIALG_1, FREEALG, COMPUT_1, PUA2MSS1, MSUALG_1,
MSUALG_2, MSUALG_3, MSAFREE, MSAFREE1, MSUALG_6, MSAFREE3, MSATERM,
AOFA_000, EXCHSORT, MSAFREE4, AOFA_A00;
constructors XBOOLE_0, AOFA_000, AOFA_A00, SUBSET_1, BINOP_1, ORDINAL1, NAT_1,
INSTALG1, MSAFREE3, ZFMISC_1, NEWTON, XCMPLX_0, MSAFREE4, CATALG_1,
AUTALG_1, POWER, PUA2MSS1, MSUALG_3, FINSEQ_4, NUMBERS, RELSET_1, REAL_1,
MSAFREE1, PZFMISC1, PRALG_2, SQUARE_1, NAT_D, XXREAL_0, XXREAL_2,
EXCHSORT, AFINSQ_1, WELLORD2, FUNCT_4, PARTFUN1, COMPUT_1, LFUZZY_0,
MEMBERED, WAYBEL_0, MCART_1, MSATERM, TREES_9, TREES_2;
registrations AOFA_000, AOFA_A00, ORDINAL1, NUMBERS, PBOOLE, MSUALG_1,
INSTALG1, MSAFREE4, FUNCOP_1, RELSET_1, STRUCT_0, INT_1, XREAL_0,
MSUALG_2, FUNCT_1, FINSEQ_1, XBOOLEAN, MARGREL1, XBOOLE_0, NAT_1,
RELAT_1, MSUALG_9, POWER, MEMBERED, XXREAL_2, EXCHSORT, FINSET_1,
AFINSQ_1, CARD_1, UNIALG_1, LFUZZY_0, TREES_3, TREES_2, VALUED_0,
XTUPLE_0;
requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL;
definitions TARSKI, MEMBERED, XBOOLE_0, RELAT_1, FUNCT_2, XXREAL_2, STRUCT_0,
UNIALG_1, PBOOLE, MSUALG_3, AOFA_000, AOFA_A00;
equalities FINSEQ_1, XBOOLEAN, MARGREL1, PUA2MSS1, MSUALG_1, AOFA_000,
AOFA_A00, CARD_1, ORDINAL1;
expansions TARSKI, FUNCT_2, FINSEQ_1, UNIALG_1, PBOOLE, MSUALG_3, AOFA_000,
AOFA_A00;
theorems TARSKI, XBOOLE_0, XBOOLE_1, ZFMISC_1, INT_1, RELAT_1, XXREAL_0,
PARTFUN1, FUNCT_1, FUNCT_2, FINSEQ_1, FINSEQ_2, FINSEQ_3, FINSEQ_4,
FINSET_1, ORDINAL1, AFINSQ_1, FUNCOP_1, CARD_5, XBOOLEAN, MARGREL1,
POWER, NEWTON, FIB_NUM2, NAT_1, NAT_D, PRE_FF, FUNCT_4, INT_2, XXREAL_2,
TREES_4, TREES_1, CARD_3, FUNCT_7, PBOOLE, PRE_CIRC, MSAFREE, MSSUBFAM,
PRALG_2, EXTENS_1, MSAFREE3, MSAFREE4, EQUATION, MSUALG_2, MSUALG_3,
AOFA_000, AOFA_A00, EXCHSORT, MEMBERED, CARD_1, XREAL_1, LFUZZY_0,
MSAFREE1, INSTALG1, TREES_9, SUBSET_1;
schemes FUNCT_2, NAT_1, FINSEQ_1, RECDEF_1, AOFA_000, CLASSES1;
begin :: Exponentiation by squaring revisited
theorem Th1:
1 mod 2 = 1 & 2 mod 2 = 0 by NAT_D:24,NAT_D:25;
theorem Th2:
for S being non empty non void ManySortedSign
for A being MSAlgebra over S
for B being MSSubAlgebra of A
for s being SortSymbol of S
for a being set st a in (the Sorts of B).s
holds a in (the Sorts of A).s
proof
let S be non empty non void ManySortedSign;
let A be MSAlgebra over S;
let B be MSSubAlgebra of A;
let s be SortSymbol of S;
the Sorts of B is MSSubset of A by MSUALG_2:def 9;
then (the Sorts of B).s c= (the Sorts of A).s by PBOOLE:def 18,def 2;
hence thesis;
end;
theorem Th3:
for I being non empty set, a,b,c being set, i being Element of I
holds c in (i-singleton a).b iff b = i & c = a
proof
let I be non empty set;
let a,b,c be set;
let i be Element of I;
A1: (i-singleton a).i = {a} & for b being set st b in I & b <> i holds
(i-singleton a).b = {} by AOFA_A00:6;
dom (i-singleton a) = I by PARTFUN1:def 2;
then
A2: for b being set st b nin I holds (i-singleton a).b = {} by FUNCT_1:def 2;
hereby
assume A3: c in (i-singleton a).b;
thus b = i by A3,A1,A2;
hence c = a by A1,A3,TARSKI:def 1;
end;
thus thesis by A1,TARSKI:def 1;
end;
theorem Th4:
for I being non empty set, a,b,c,d being set, i,j being Element of I
holds c in ((i-singleton a)(\/)(j-singleton d)).b iff
b = i & c = a or b = j & c = d
proof
let I be non empty set;
let a,b,c,d be set;
let i,j be Element of I;
hereby
assume A1: c in ((i-singleton a)(\/)(j-singleton d)).b;
assume A2: not (b = i & c = a);
b in dom((i-singleton a)(\/)(j-singleton d)) by A1,FUNCT_1:def 2;
then b in I by PARTFUN1:def 2;
then c in ((i-singleton a).b)\/((j-singleton d).b) by A1,PBOOLE:def 4;
then c in (i-singleton a).b or c in (j-singleton d).b by XBOOLE_0:def 3;
hence b = j & c = d by A2,Th3;
end;
assume A3: b = i & c = a or b = j & c = d;
then c in (i-singleton a).b or c in (j-singleton d).b by Th3;
then c in ((i-singleton a).b)\/((j-singleton d).b) by XBOOLE_0:def 3;
hence c in ((i-singleton a)(\/)(j-singleton d)).b by A3,PBOOLE:def 4;
end;
definition
let S be (4,1) integer bool-correct non empty non void BoolSignature;
let A be non-empty MSAlgebra over S;
attr A is integer means: Def1:
ex C being image of A st C is (4,1) integer bool-correct MSAlgebra over S;
end;
theorem Th5:
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
holds Image id the Sorts of A = the MSAlgebra of A
proof
let S be non empty non void ManySortedSign;
let A be non-empty MSAlgebra over S;
the MSAlgebra of A is strict non-empty MSSubAlgebra of A &
id the Sorts of A is_homomorphism A,A &
(id the Sorts of A).:.:the Sorts of A = the Sorts of A
by EQUATION:15,MSUALG_2:5,MSUALG_3:9;
hence Image id the Sorts of A = the MSAlgebra of A by MSUALG_3:def 12;
end;
theorem Th6:
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
holds A is image of A
proof
let S be non empty non void ManySortedSign;
let A be non-empty MSAlgebra over S;
A is A-Image
proof
now
take B = A;
reconsider h = id the Sorts of A as ManySortedFunction of
the Sorts of A, the Sorts of B;
take h;
thus h is_homomorphism A,B by MSUALG_3:9;
thus the MSAlgebra of A = Image h by Th5;
end;
hence thesis by MSAFREE4:def 4;
end;
hence thesis;
end;
registration
let S be (4,1) integer bool-correct non empty non void BoolSignature;
cluster integer for non-empty MSAlgebra over S;
existence
proof
set C = the (4,1) integer bool-correct non-empty MSAlgebra over S;
reconsider C1 = C as image of C by Th6;
take C,C1; thus thesis;
end;
end;
registration
let S be (4,1) integer bool-correct non empty non void BoolSignature;
let A be integer non-empty MSAlgebra over S;
cluster bool-correct for image of A;
existence
proof
consider C being image of A such that
A1: C is (4,1) integer bool-correct MSAlgebra over S by Def1;
take C; thus thesis by A1;
end;
end;
registration
let S be (4,1) integer bool-correct non empty non void BoolSignature;
let A be integer non-empty MSAlgebra over S;
cluster (4,1) integer for bool-correct image of A;
existence
proof
ex C being image of A st
C is (4,1) integer bool-correct MSAlgebra over S by Def1;
hence thesis;
end;
end;
theorem Th7:
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for o being OperSymbol of S, a being set, r being SortSymbol of S
st o is_of_type a,r
holds Den(o,A) is Function of (the Sorts of A)#.a, (the Sorts of A).r &
Args(o,A) = (the Sorts of A)#.a & Result(o,A) = (the Sorts of A).r
proof
let S be non empty non void ManySortedSign;
let A be non-empty MSAlgebra over S;
let o be OperSymbol of S;
let a be set;
let r be SortSymbol of S;
assume A1: (the Arity of S).o = a & (the ResultSort of S).o = r;
then
A2: ((the Sorts of A)#*the Arity of S).o = (the Sorts of A)#.a by FUNCT_2:15;
((the Sorts of A)*the ResultSort of S).o = (the Sorts of A).r
by A1,FUNCT_2:15;
hence Den(o,A) is Function of (the Sorts of A)#.a, (the Sorts of A).r
by A2;
thus thesis by A1,FUNCT_2:15;
end;
registration
let S be bool-correct non empty non void BoolSignature;
let A be bool-correct non-empty MSAlgebra over S;
cluster -> bool-correct for non-empty MSSubAlgebra of A;
coherence
proof
A1: (the Sorts of A).the bool-sort of S = BOOLEAN &
Den(In((the connectives of S).1, the carrier' of S), A).{} = TRUE &
for x,y be boolean object holds
Den(In((the connectives of S).2, the carrier' of S), A).<*x*> = 'not' x &
Den(In((the connectives of S).3, the carrier' of S), A).<*x,y*> = x '&' y
by AOFA_A00:def 32;
let B be non-empty MSSubAlgebra of A;
the Sorts of B is MSSubset of A by MSUALG_2:def 9;
hence (the Sorts of B).the bool-sort of S c= BOOLEAN
by A1,PBOOLE:def 18,def 2;
set o1 = In((the connectives of S).1, the carrier' of S),
o2 = In((the connectives of S).2, the carrier' of S);
set b = the bool-sort of S;
3 <= len the connectives of S by AOFA_A00:def 31;
then 1 <= len the connectives of S by XXREAL_0:2;
then 1 in dom the connectives of S by FINSEQ_3:25;
then o1 = (the connectives of S).1 by FUNCT_1:102,SUBSET_1:def 8;
then o1 is_of_type {},b by AOFA_A00:def 31;
then
A2: Den(o1,B) is Function of (the Sorts of B)#.{}, (the Sorts of B).b &
Args(o1,B) = (the Sorts of B)#.{} by Th7;
(the Sorts of B)#.<*>the carrier of S = {{}} by PRE_CIRC:2;
then
A3: {} in (the Sorts of B)#.{} by TARSKI:def 1;
A4: Den(o1,B).{} = Den(o1,A).{} by A2,A3,EQUATION:19
.= TRUE by AOFA_A00:def 32;
then
A5: TRUE in (the Sorts of B).the bool-sort of S by A2,A3,FUNCT_2:5;
A6: <*b*> in (the carrier of S)* by FINSEQ_1:def 11;
A7: dom the Sorts of B = the carrier of S by PARTFUN1:def 2;
3 <= len the connectives of S by AOFA_A00:def 31;
then 2 <= len the connectives of S by XXREAL_0:2;
then 2 in dom the connectives of S by FINSEQ_3:25;
then o2 = (the connectives of S).2 by FUNCT_1:102,SUBSET_1:def 8;
then o2 is_of_type <*b*>,b by AOFA_A00:def 31;
then
A8: Den(o2,B) is Function of (the Sorts of B)#.<*b*>, (the Sorts of B).b &
Args(o2,B) = (the Sorts of B)#.<*b*> by Th7;
then
A9: Args(o2,B) = product ((the Sorts of B)*<*b*>) by A6,FINSEQ_2:def 5
.= product <*(the Sorts of B).b*> by A7,FINSEQ_2:34;
then
A10: <*TRUE*> in Args(o2,B) by A5,FINSEQ_3:123;
Den(o2,B).<*TRUE*>
= Den(o2,A).<*TRUE*> by A9,A5,FINSEQ_3:123,EQUATION:19
.= 'not' TRUE by AOFA_A00:def 32 .= FALSE;
then FALSE in (the Sorts of B).b by A8,A10,FUNCT_2:5;
hence
A11: BOOLEAN c= (the Sorts of B).b by A5,ZFMISC_1:32;
thus Den(In((the connectives of S).1, the carrier' of S), B).{}
= TRUE by A4;
let x,y be boolean object;
A12: <*b,b*> in (the carrier of S)* by FINSEQ_1:def 11;
A13: x in BOOLEAN & y in BOOLEAN by MARGREL1:def 12;
thus Den(o2, B).<*x*> = Den(o2,A).<*x*>
by A13,A11,A9,FINSEQ_3:123,EQUATION:19
.= 'not' x by AOFA_A00:def 32;
set o3 = In((the connectives of S).3, the carrier' of S);
3 <= len the connectives of S by AOFA_A00:def 31;
then 3 in dom the connectives of S by FINSEQ_3:25;
then o3 = (the connectives of S).3 by FUNCT_1:102,SUBSET_1:def 8;
then o3 is_of_type <*b,b*>,b by AOFA_A00:def 31;
then
Den(o3,B) is Function of (the Sorts of B)#.<*b,b*>, (the Sorts of B).b &
Args(o3,B) = (the Sorts of B)#.<*b,b*> by Th7;
then
Args(o3,B) = product ((the Sorts of B)*<*b,b*>) by A12,FINSEQ_2:def 5
.= product <*(the Sorts of B).b,(the Sorts of B).b*> by A7,FINSEQ_2:125;
hence Den(o3, B).<*x,y*> = Den(o3,A).<*x,y*>
by A11,A13,FINSEQ_3:124,EQUATION:19
.= x '&' y by AOFA_A00:def 32;
end;
end;
registration
let S be (4,1) integer bool-correct non empty non void BoolSignature;
let A be (4,1) integer bool-correct non-empty MSAlgebra over S;
cluster -> (4,1) integer for non-empty MSSubAlgebra of A;
coherence
proof
let B be non-empty MSSubAlgebra of A;
set n = 4;
consider I being SortSymbol of S such that
A1: I = 1 & (the connectives of S).n is_of_type {},I &
(the Sorts of A).I = INT &
Den(In((the connectives of S).n, the carrier' of S), A).{} = 0 &
Den(In((the connectives of S).(n+1), the carrier' of S), A).{} = 1 &
for i,j being Integer holds
Den(In((the connectives of S).(n+2), the carrier' of S), A).<*i*> = -i &
Den(In((the connectives of S).(n+3), the carrier' of S), A).<*i,j*> = i+j &
Den(In((the connectives of S).(n+4), the carrier' of S), A).<*i,j*> = i*j &
(j <> 0 implies
Den(In((the connectives of S).(n+5), the carrier' of S), A).<*i,j*>
= i div j) &
Den(In((the connectives of S).(n+6), the carrier' of S), A).<*i,j*>
= IFGT(i,j,FALSE,TRUE) by AOFA_A00:def 50;
reconsider I as integer SortSymbol of S by A1,AOFA_A00:def 40;
take I;
thus I = 1 & (the connectives of S).n is_of_type {},I by A1;
the Sorts of B is MSSubset of A by MSUALG_2:def 9;
hence (the Sorts of B).I c= INT by A1,PBOOLE:def 18,def 2;
n+6 <= len the connectives of S & 4 <= 10 by AOFA_A00:def 39;
then n <= len the connectives of S & n+1 <= len the connectives of S &
1 <= 5 by XXREAL_0:2;
then n in dom the connectives of S & n+1 in dom the connectives of S
by FINSEQ_3:25;
then reconsider o1 = (the connectives of S).n,
o2 = (the connectives of S).(n+1) as OperSymbol of S by FUNCT_1:102;
o1 is_of_type {},I & o2 is_of_type {},I by AOFA_A00:53;
then
A2: Den(o1,B) is Function of (the Sorts of B)#.{}, (the Sorts of B).I &
Den(o2,B) is Function of (the Sorts of B)#.{}, (the Sorts of B).I &
Args(o1,B) = (the Sorts of B)#.{} & Args(o2,B) = (the Sorts of B)#.{}
by Th7;
(the Sorts of B)#.<*>the carrier of S = {{}} by PRE_CIRC:2;
then
A3: {} in (the Sorts of B)#.{} by TARSKI:def 1;
then
A4: Den(o1,B).{} in (the Sorts of B).I & Den(o2,B).{} in (the Sorts of B).I
by A2,FUNCT_2:5;
A5: Den(o1,B).{} = Den(o1,A).{} by A2,A3,EQUATION:19
.= 0 by A1,SUBSET_1:def 8;
A6: Den(o2,B).{} = Den(o2,A).{} by A2,A3,EQUATION:19
.= 1 by A1,SUBSET_1:def 8;
defpred P[Nat] means $1 in (the Sorts of B).I & -$1 in (the Sorts of B).I;
A7: P[0] by A2,A3,A5,FUNCT_2:5;
n+6 <= len the connectives of S & 4 <= 10 by AOFA_A00:def 39;
then n+2 <= len the connectives of S & n+3 <= len the connectives of S &
1 <= 6 & 1 <= 7 by XXREAL_0:2;
then n+2 in dom the connectives of S & n+3 in dom the connectives of S
by FINSEQ_3:25;
then reconsider o3 = (the connectives of S).(n+2),
o4 = (the connectives of S).(n+3) as OperSymbol of S by FUNCT_1:102;
o3 is_of_type <*I*>,I & o4 is_of_type <*I,I*>,I by AOFA_A00:53;
then
A8: Den(o3,B) is Function of (the Sorts of B)#.<*I*>, (the Sorts of B).I &
Den(o4,B) is Function of (the Sorts of B)#.<*I,I*>, (the Sorts of B).I &
Args(o3,B) = (the Sorts of B)#.<*I*> &
Args(o4,B) = (the Sorts of B)#.<*I,I*> by Th7;
A9: dom the Sorts of B = the carrier of S by PARTFUN1:def 2;
<*I*> is Element of (the carrier of S)* by FINSEQ_1:def 11;
then
A10: Args(o3,B) = product ((the Sorts of B)*<*I*>) by A8,FINSEQ_2:def 5
.= product <*(the Sorts of B).I*> by A9,FINSEQ_2:34;
<*I,I*> is Element of (the carrier of S)* by FINSEQ_1:def 11;
then
A11: Args(o4,B) = product ((the Sorts of B)*<*I,I*>) by A8,FINSEQ_2:def 5
.= product <*(the Sorts of B).I,(the Sorts of B).I*> by A9,FINSEQ_2:125;
A12: for i being Nat st P[i] holds P[i+1]
proof let i be Nat;
assume
A13: P[i];
then
A14: <*i,1*> in Args(o4,B) by A6,A4,A11,FINSEQ_3:124;
Den(o4,B).<*i,1*> = Den(o4,A).<*i,1*>
by A13,A6,A4,A11,FINSEQ_3:124,EQUATION:19
.= Den(In(o4,the carrier' of S),A).<*i,1*> by SUBSET_1:def 8
.= i+1 by A1;
hence
A15: i+1 in (the Sorts of B).I by A14,A8,FUNCT_2:5;
then
A16: <*i+1*> in Args(o3,B) by A10,FINSEQ_3:123;
Den(o3,B).<*i+1*> = Den(o3,A).<*i+1*> by A15,A10,FINSEQ_3:123,EQUATION:19
.= Den(In(o3,the carrier' of S),A).<*i+1*> by SUBSET_1:def 8
.= -(i+1) by A1;
hence -(i+1) in (the Sorts of B).I by A16,A8,FUNCT_2:5;
end;
A17: for i being Nat holds P[i] from NAT_1:sch 2(A7,A12);
thus
A18: INT c= (the Sorts of B).I
proof let x be Integer;
x in INT by INT_1:def 2;
then consider n being Nat such that
A19: x = n or x = -n by INT_1:def 1;
thus thesis by A17,A19;
end;
thus Den(In((the connectives of S).n, the carrier' of S), B).{} = 0 &
Den(In((the connectives of S).(n+1), the carrier' of S), B).{} = 1
by A5,A6,SUBSET_1:def 8;
let i,j be Integer;
A20: i in INT & j in INT by INT_1:def 2;
<*i*> in Args(o3,B) by A20,A18,A10,FINSEQ_3:123;
then <*i*> in Args(In(o3,the carrier' of S),B) by SUBSET_1:def 8;
hence
Den(In((the connectives of S).(n+2), the carrier' of S), B).<*i*>
= Den(In((the connectives of S).(n+2), the carrier' of S), A).<*i*>
by EQUATION:19 .= -i by A1;
<*i,j*> in Args(o4,B) by A11,A20,A18,FINSEQ_3:124;
then <*i,j*> in Args(In(o4,the carrier' of S),B) by SUBSET_1:def 8;
hence
Den(In((the connectives of S).(n+3), the carrier' of S), B).<*i,j*>
= Den(In((the connectives of S).(n+3), the carrier' of S), A).<*i,j*>
by EQUATION:19 .= i+j by A1;
n+6 <= len the connectives of S & 4 <= 10 by AOFA_A00:def 39;
then n+4 <= len the connectives of S & n+5 <= len the connectives of S &
1 <= 8 & 1 <= 9 by XXREAL_0:2;
then n+4 in dom the connectives of S & n+5 in dom the connectives of S
by FINSEQ_3:25;
then reconsider o5 = (the connectives of S).(n+4),
o6 = (the connectives of S).(n+5) as OperSymbol of S by FUNCT_1:102;
o5 is_of_type <*I,I*>,I & o6 is_of_type <*I,I*>,I by AOFA_A00:53;
then
A21: Args(o5,B) = (the Sorts of B)#.<*I,I*> &
Args(o6,B) = (the Sorts of B)#.<*I,I*> by Th7;
then <*i,j*> in Args(o5,B) by A8,A11,A20,A18,FINSEQ_3:124;
then <*i,j*> in Args(In(o5,the carrier' of S),B) by SUBSET_1:def 8;
hence
Den(In((the connectives of S).(n+4), the carrier' of S), B).<*i,j*>
= Den(In((the connectives of S).(n+4), the carrier' of S), A).<*i,j*>
by EQUATION:19 .= i*j by A1;
hereby assume
A22: j <> 0;
<*i,j*> in Args(o6,B) by A8,A11,A20,A18,A21,FINSEQ_3:124;
then <*i,j*> in Args(In(o6,the carrier' of S),B) by SUBSET_1:def 8;
hence
Den(In((the connectives of S).(n+5), the carrier' of S), B).<*i,j*>
= Den(In((the connectives of S).(n+5), the carrier' of S), A).<*i,j*>
by EQUATION:19 .= i div j by A22,A1;
end;
n+6 <= len the connectives of S & 4 <= 10 by AOFA_A00:def 39;
then n+6 in dom the connectives of S by FINSEQ_3:25;
then reconsider o7 = (the connectives of S).(n+6) as OperSymbol of S
by FUNCT_1:102;
o7 is_of_type <*I,I*>,the bool-sort of S by AOFA_A00:53;
then
Args(o7,B) = (the Sorts of B)#.<*I,I*> by Th7;
then <*i,j*> in Args(o7,B) by A8,A11,A20,A18,FINSEQ_3:124;
then <*i,j*> in Args(In(o7,the carrier' of S),B) by SUBSET_1:def 8;
hence
Den(In((the connectives of S).(n+6), the carrier' of S), B).<*i,j*>
= Den(In((the connectives of S).(n+6), the carrier' of S), A).<*i,j*>
by EQUATION:19 .= IFGT(i,j,FALSE,TRUE) by A1;
end;
end;
registration
let S be (4,1) integer bool-correct non empty non void BoolSignature;
let X be non-empty ManySortedSet of the carrier of S;
cluster Free(S,X) -> integer for non-empty MSAlgebra over S;
coherence
proof let F be non-empty MSAlgebra over S;
assume
A1: F = Free(S,X);
set A = the (4,1) integer bool-correct non-empty MSAlgebra over S;
reconsider G = FreeGen X as GeneratorSet of F by A1,MSAFREE3:31;
set f = the ManySortedFunction of G, the Sorts of A;
FreeGen X is free & F = FreeMSA X by A1,MSAFREE3:31;
then consider h being ManySortedFunction of F,A such that
A2: h is_homomorphism F,A & h||G = f by MSAFREE:def 5;
reconsider C = Image h as image of F by A2,MSAFREE4:def 4;
take C; thus C is (4,1) integer bool-correct MSAlgebra over S;
end;
end;
theorem Th8:
for S being non empty non void ManySortedSign
for A1,A2,B1 being MSAlgebra over S, B2 being non-empty MSAlgebra over S
st the MSAlgebra of A1 = the MSAlgebra of A2 &
the MSAlgebra of B1 = the MSAlgebra of B2
for h1 being ManySortedFunction of A1,B1
for h2 being ManySortedFunction of A2,B2 st h1 = h2 &
h1 is_epimorphism A1,B1 holds h2 is_epimorphism A2,B2
by MSAFREE4:30;
registration
let S be (4,1) integer bool-correct non empty non void BoolSignature;
let X be non-empty ManySortedSet of the carrier of S;
cluster vf-free integer for all_vars_including inheriting_operations
free_in_itself (X,S)-terms non-empty VarMSAlgebra over S;
existence
proof
set A = Free(S,X);
consider V being ManySortedMSSet of the Sorts of A,the Sorts of A,
B being all_vars_including inheriting_operations free_in_itself
(X,S)-terms VarMSAlgebra over S such that
A1: B = VarMSAlgebra(#the Sorts of A, the Charact of A, V#) &
B is vf-free by AOFA_A00:39;
take B; thus B is vf-free by A1;
consider C being image of A such that
A2: C is (4,1) integer bool-correct MSAlgebra over S by Def1;
consider h being ManySortedFunction of A,C such that
A3: h is_epimorphism A,C by MSAFREE4:def 5;
reconsider g = h as ManySortedFunction of B,C by A1;
the MSAlgebra of C = the MSAlgebra of C;
then reconsider D = C as image of B by A1,A3,Th8,MSAFREE4:def 5;
take D; thus thesis by A2;
end;
end;
definition
let S be non empty non void ManySortedSign;
let X be non-empty ManySortedSet of the carrier of S;
let T be all_vars_including inheriting_operations (X,S)-terms
MSAlgebra over S;
func FreeGen T -> non-empty GeneratorSet of T equals FreeGen X;
coherence by MSAFREE4:45;
end;
registration
let S be non empty non void ManySortedSign;
let X0 be countable non-empty ManySortedSet of the carrier of S;
let T be all_vars_including inheriting_operations free_in_itself
(X0,S)-terms MSAlgebra over S;
cluster FreeGen T -> (Equations(S, T))-free non-empty;
coherence by MSAFREE4:75;
end;
definition
let S be non empty non void ManySortedSign;
let X be non-empty ManySortedSet of the carrier of S;
let T be all_vars_including inheriting_operations (X,S)-terms
MSAlgebra over S;
let G be GeneratorSet of T;
attr G is basic means: Def3: FreeGen T c= G;
let s be SortSymbol of S;
let x be Element of G.s;
attr x is pure means: Def4: x in (FreeGen T).s;
end;
registration
let S be non empty non void ManySortedSign;
let X be non-empty ManySortedSet of the carrier of S;
let T be all_vars_including inheriting_operations (X,S)-terms
MSAlgebra over S;
cluster FreeGen T -> basic;
coherence;
end;
registration
let S be non empty non void ManySortedSign;
let X be non-empty ManySortedSet of the carrier of S;
let T be all_vars_including inheriting_operations (X,S)-terms
MSAlgebra over S;
cluster basic for non-empty GeneratorSet of T;
existence
proof
reconsider G = FreeGen X as non-empty GeneratorSet of T by MSAFREE4:45;
take G; thus FreeGen T c= G;
end;
end;
registration
let S be non empty non void ManySortedSign;
let X be non-empty ManySortedSet of the carrier of S;
let T be all_vars_including inheriting_operations (X,S)-terms
MSAlgebra over S;
let G be basic GeneratorSet of T;
let s be SortSymbol of S;
cluster pure for Element of G.s;
existence
proof
set x = the Element of (FreeGen T).s;
x in (FreeGen T).s & (FreeGen T).s c= G.s by Def3,PBOOLE:def 2;
then reconsider x as Element of G.s;
take x; thus x in (FreeGen T).s;
end;
end;
theorem
for S being non empty non void ManySortedSign
for X being non-empty ManySortedSet of the carrier of S
for T being all_vars_including inheriting_operations (X,S)-terms
MSAlgebra over S
for G being basic GeneratorSet of T
for s being SortSymbol of S
for a being set holds a is pure Element of G.s iff a in (FreeGen T).s
proof
let S be non empty non void ManySortedSign;
let X be non-empty ManySortedSet of the carrier of S;
let T be all_vars_including inheriting_operations (X,S)-terms
MSAlgebra over S;
let G be basic GeneratorSet of T;
let s be SortSymbol of S;
let a be set;
(FreeGen T).s c= G.s by Def3,PBOOLE:def 2;
hence thesis by Def4;
end;
definition
let S be non empty non void ManySortedSign;
let X be non-empty ManySortedSet of the carrier of S;
let T be all_vars_including inheriting_operations free_in_itself (X,S)-terms
MSAlgebra over S;
let G be GeneratorSystem over S,X,T;
attr G is basic means: Def5: the generators of G is basic;
end;
registration
let S be non empty non void ManySortedSign;
let X be non-empty ManySortedSet of the carrier of S;
let T be all_vars_including inheriting_operations free_in_itself (X,S)-terms
MSAlgebra over S;
cluster basic for GeneratorSystem over S,X,T;
existence
proof
set Y = the basic non-empty GeneratorSet of T;
set SV = the ManySortedFunction of Y, FreeGen X;
set ST = the ManySortedMSSet of Y, the carrier of S;
take G = GeneratorSystem(#Y,SV,ST#);
thus the generators of G is basic;
end;
end;
registration
let S be non empty non void ManySortedSign;
let X be non-empty ManySortedSet of the carrier of S;
let T be all_vars_including inheriting_operations free_in_itself (X,S)-terms
MSAlgebra over S;
let G be basic GeneratorSystem over S,X,T;
cluster the generators of G -> basic;
coherence by Def5;
end;
reserve
S for (4,1) integer bool-correct non empty non void BoolSignature,
X for non-empty ManySortedSet of the carrier of S,
T for vf-free integer all_vars_including inheriting_operations free_in_itself
(X,S)-terms VarMSAlgebra over S,
C for (4,1) integer bool-correct non-empty image of T,
G for basic GeneratorSystem over S,X,T,
A for IfWhileAlgebra of the generators of G,
I for integer SortSymbol of S,
x,y,z,m for pure (Element of (the generators of G).I),
b for pure (Element of (the generators of G).the bool-sort of S),
t,t1,t2 for Element of T,I,
P for Algorithm of A,
s,s1,s2 for Element of C-States(the generators of G);
definition
let S be bool-correct non empty non void BoolSignature;
let A be non-empty MSAlgebra over S;
func \falseA -> Element of A, the bool-sort of S equals \not\trueA;
coherence;
end;
reserve
f for ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G, b);
theorem Th10:
\falseC = FALSE
proof
\trueC = TRUE by AOFA_A00:def 32;
hence \falseC = 'not' TRUE by AOFA_A00:def 32 .= FALSE;
end;
definition
let S be bool-correct non empty non void BoolSignature;
let X be non-empty ManySortedSet of the carrier of S;
let T be all_vars_including inheriting_operations free_in_itself (X,S)-terms
MSAlgebra over S;
let G be GeneratorSystem over S,X,T;
let b be Element of (the generators of G).the bool-sort of S;
let C be image of T;
let A be preIfWhileAlgebra;
let f be ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G, b);
let s be Element of C-States(the generators of G);
let P be Algorithm of A;
redefine func f.(s,P) -> Element of C-States(the generators of G);
coherence
proof
thus f.(s,P) is Element of C-States(the generators of G);
end;
end;
definition
let S be non empty non void ManySortedSign;
let T be non-empty MSAlgebra over S;
let G be non-empty GeneratorSet of T;
let s be SortSymbol of S;
let x be Element of G.s;
func @x -> Element of T,s equals x;
coherence
proof
G.s c= (the Sorts of T).s & x in G.s by PBOOLE:def 18,def 2;
hence x is Element of (the Sorts of T).s;
end;
end;
definition
let S,X,T,G,A,b,I,t1,t2;
func b leq(t1, t2, A) -> Algorithm of A equals b:=(leq(t1,t2),A);
coherence;
func b gt(t1, t2, A) -> Algorithm of A equals b:=(\not(leq(t1,t2)),A);
coherence;
end;
definition
let S,X,T,I;
func \2(T,I) -> Element of T,I equals \1(T,I)+\1(T,I);
coherence;
end;
definition
let S,X,T,G,A,b,I,t;
func t is_odd(b,A) -> Algorithm of A equals b gt(t mod \2(T,I),\0(T,I),A);
coherence;
func t is_even(b,A) -> Algorithm of A equals b leq(t mod \2(T,I),\0(T,I),A);
coherence;
end;
registration
let S,X,T,G,C,I,s;
let x be Element of (the generators of G).I;
cluster s.I.x -> integer;
coherence
proof
(the Sorts of C).I = INT & s is ManySortedFunction of the generators of G,
the Sorts of C by AOFA_A00:48,55;
then s.I is Function of (the generators of G).I, INT by PBOOLE:def 15;
then s.I.x in INT by FUNCT_2:5;
hence thesis;
end;
end;
registration
let S,X,T,G,C,I,s,t;
cluster t value_at(C,s) -> integer;
coherence
proof
(the Sorts of C).I = INT by AOFA_A00:55;
hence thesis;
end;
end;
reserve u for ManySortedFunction of FreeGen T, the Sorts of C;
registration
let S,X,T,C,I,u,t;
cluster t value_at(C,u) -> integer;
coherence
proof
(the Sorts of C).I = INT by AOFA_A00:55;
hence thesis;
end;
end;
registration
let S,X,T,G,C,s;
let t be Element of T, the bool-sort of S;
cluster t value_at(C,s) -> boolean;
coherence
proof
(the Sorts of C).the bool-sort of S = BOOLEAN by AOFA_A00:def 32;
hence thesis;
end;
end;
registration
let S,X,T,C,u;
let t be Element of T, the bool-sort of S;
cluster t value_at(C,u) -> boolean;
coherence
proof
(the Sorts of C).the bool-sort of S = BOOLEAN by AOFA_A00:def 32;
hence thesis;
end;
end;
theorem Th11:
for o being OperSymbol of S st
o = In((the connectives of S).1, the carrier' of S)
holds o = (the connectives of S).1 &
the_arity_of o = {} & the_result_sort_of o = the bool-sort of S
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).1, the carrier' of S);
4+6 <= len the connectives of S by AOFA_A00:def 39;
then 1 <= len the connectives of S by XXREAL_0:2;
then 1 in dom the connectives of S by FINSEQ_3:25;
hence o = (the connectives of S).1 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type {}, the bool-sort of S by AOFA_A00:def 31;
hence the_arity_of o = {} & the_result_sort_of o = the bool-sort of S;
end;
theorem Th12:
for o being OperSymbol of S st
o = In((the connectives of S).2, the carrier' of S)
holds o = (the connectives of S).2 &
the_arity_of o = <*the bool-sort of S*> &
the_result_sort_of o = the bool-sort of S
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).2, the carrier' of S);
4+6 <= len the connectives of S by AOFA_A00:def 39;
then 2 <= len the connectives of S by XXREAL_0:2;
then 2 in dom the connectives of S by FINSEQ_3:25;
hence o = (the connectives of S).2 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type <*the bool-sort of S*>, the bool-sort of S
by AOFA_A00:def 31;
hence the_arity_of o = <*the bool-sort of S*> &
the_result_sort_of o = the bool-sort of S;
end;
theorem Th13:
for o being OperSymbol of S st
o = In((the connectives of S).3, the carrier' of S)
holds o = (the connectives of S).3 &
the_arity_of o = <*the bool-sort of S, the bool-sort of S*> &
the_result_sort_of o = the bool-sort of S
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).3, the carrier' of S);
4+6 <= len the connectives of S by AOFA_A00:def 39;
then 3 <= len the connectives of S by XXREAL_0:2;
then 3 in dom the connectives of S by FINSEQ_3:25;
hence o = (the connectives of S).3 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type <*the bool-sort of S, the bool-sort of S*>,
the bool-sort of S by AOFA_A00:def 31;
hence the_arity_of o = <*the bool-sort of S, the bool-sort of S*> &
the_result_sort_of o = the bool-sort of S;
end;
theorem Th14:
for o being OperSymbol of S st
o = In((the connectives of S).4, the carrier' of S)
holds the_arity_of o = {} & the_result_sort_of o = I
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).4, the carrier' of S);
4+6 <= len the connectives of S by AOFA_A00:def 39;
then 4 <= len the connectives of S by XXREAL_0:2;
then 4 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S).4 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type {}, I by AOFA_A00:53;
hence the_arity_of o = {} & the_result_sort_of o = I;
end;
theorem Th15:
for o being OperSymbol of S st
o = In((the connectives of S).5, the carrier' of S)
holds the_arity_of o = {} & the_result_sort_of o = I
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).5, the carrier' of S);
4+6 <= len the connectives of S by AOFA_A00:def 39;
then 5 <= len the connectives of S by XXREAL_0:2;
then 5 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S).5 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type {}, I by AOFA_A00:53;
hence the_arity_of o = {} & the_result_sort_of o = I;
end;
theorem Th16:
for o being OperSymbol of S st
o = In((the connectives of S).6, the carrier' of S)
holds the_arity_of o = <*I*> & the_result_sort_of o = I
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).6, the carrier' of S);
4+6 <= len the connectives of S by AOFA_A00:def 39;
then 6 <= len the connectives of S by XXREAL_0:2;
then 6 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S).6 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type <*I*>, I by AOFA_A00:53;
hence the_arity_of o = <*I*> & the_result_sort_of o = I;
end;
theorem Th17:
for o being OperSymbol of S st
o = In((the connectives of S).7, the carrier' of S)
holds the_arity_of o = <*I,I*> & the_result_sort_of o = I
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).7, the carrier' of S);
4+6 <= len the connectives of S by AOFA_A00:def 39;
then 7 <= len the connectives of S by XXREAL_0:2;
then 7 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S).7 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type <*I,I*>, I by AOFA_A00:53;
hence the_arity_of o = <*I,I*> & the_result_sort_of o = I;
end;
theorem Th18:
for o being OperSymbol of S st
o = In((the connectives of S).8, the carrier' of S)
holds the_arity_of o = <*I,I*> & the_result_sort_of o = I
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).8, the carrier' of S);
4+6 <= len the connectives of S by AOFA_A00:def 39;
then 8 <= len the connectives of S by XXREAL_0:2;
then 8 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S).8 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type <*I,I*>, I by AOFA_A00:53;
hence the_arity_of o = <*I,I*> & the_result_sort_of o = I;
end;
theorem Th19:
for o being OperSymbol of S st
o = In((the connectives of S).9, the carrier' of S)
holds the_arity_of o = <*I,I*> & the_result_sort_of o = I
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).9, the carrier' of S);
4+6 <= len the connectives of S by AOFA_A00:def 39;
then 9 <= len the connectives of S by XXREAL_0:2;
then 9 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S).9 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type <*I,I*>, I by AOFA_A00:53;
hence the_arity_of o = <*I,I*> & the_result_sort_of o = I;
end;
theorem Th20:
for o being OperSymbol of S st
o = In((the connectives of S).10, the carrier' of S)
holds the_arity_of o = <*I,I*> & the_result_sort_of o = the bool-sort of S
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).10, the carrier' of S);
4+6 <= len the connectives of S by AOFA_A00:def 39;
then 10 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S).10 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type <*I,I*>, the bool-sort of S by AOFA_A00:53;
hence the_arity_of o = <*I,I*> & the_result_sort_of o = the bool-sort of S;
end;
theorem Th21:
for S being non empty non void ManySortedSign
for o being OperSymbol of S st the_arity_of o = {}
for A being MSAlgebra over S holds Args(o,A) = {{}}
proof
let S be non empty non void ManySortedSign;
let o be OperSymbol of S;
assume A1: the_arity_of o = {};
let A be MSAlgebra over S;
thus Args(o,A) = product ((the Sorts of A)*the_arity_of o) by PRALG_2:3
.= {{}} by A1,CARD_3:10;
end;
theorem Th22:
for S being non empty non void ManySortedSign
for a being SortSymbol of S
for o being OperSymbol of S st the_arity_of o = <*a*>
for A being MSAlgebra over S holds
Args(o,A) = product <*(the Sorts of A).a*>
proof
let S be non empty non void ManySortedSign;
let a be SortSymbol of S;
let o be OperSymbol of S;
assume A1: the_arity_of o = <*a*>;
let A be MSAlgebra over S;
A2: dom the Sorts of A = the carrier of S by PARTFUN1:def 2;
thus Args(o,A) = product ((the Sorts of A)*the_arity_of o) by PRALG_2:3
.= product <*(the Sorts of A).a*> by A1,A2,FINSEQ_2:34;
end;
theorem Th23:
for S being non empty non void ManySortedSign
for a,b being SortSymbol of S
for o being OperSymbol of S st the_arity_of o = <*a,b*>
for A being MSAlgebra over S holds
Args(o,A) = product <*(the Sorts of A).a, (the Sorts of A).b*>
proof
let S be non empty non void ManySortedSign;
let a,b be SortSymbol of S;
let o be OperSymbol of S;
assume A1: the_arity_of o = <*a,b*>;
let A be MSAlgebra over S;
A2: dom the Sorts of A = the carrier of S by PARTFUN1:def 2;
thus Args(o,A) = product ((the Sorts of A)*the_arity_of o) by PRALG_2:3
.= product <*(the Sorts of A).a, (the Sorts of A).b*>
by A1,A2,FINSEQ_2:125;
end;
theorem Th24:
for S being non empty non void ManySortedSign
for a,b,c being SortSymbol of S
for o being OperSymbol of S st the_arity_of o = <*a,b,c*>
for A being MSAlgebra over S holds
Args(o,A) = product <*(the Sorts of A).a, (the Sorts of A).b,
(the Sorts of A).c*>
proof
let S be non empty non void ManySortedSign;
let a,b,c be SortSymbol of S;
let o be OperSymbol of S;
assume A1: the_arity_of o = <*a,b,c*>;
let A be MSAlgebra over S;
A2: dom the Sorts of A = the carrier of S by PARTFUN1:def 2;
thus Args(o,A) = product ((the Sorts of A)*the_arity_of o) by PRALG_2:3
.= product <*(the Sorts of A).a, (the Sorts of A).b, (the Sorts of A).c*>
by A1,A2,FINSEQ_2:126;
end;
theorem Th25:
for S being non empty non void ManySortedSign
for A,B being non-empty MSAlgebra over S
for s being SortSymbol of S
for a being Element of A,s
for h being ManySortedFunction of A,B
for o being OperSymbol of S st the_arity_of o = <*s*>
for p being Element of Args(o,A)
st p = <*a*> holds h#p = <*h.s.a*>
proof
let S be non empty non void ManySortedSign;
let A,B be non-empty MSAlgebra over S;
let s be SortSymbol of S;
let a be Element of A,s;
let h be ManySortedFunction of A,B;
let o be OperSymbol of S such that
A1: the_arity_of o = <*s*>;
let p be Element of Args(o,A);
assume A2: p = <*a*>;
A3: dom p = dom the_arity_of o & dom(h#p) = dom the_arity_of o by MSUALG_3:6;
then
A4: dom(h#p) = Seg 1 by A2,FINSEQ_1:38;
then
A5: len p = 1 & len (h#p) = 1 by A3,FINSEQ_1:def 3;
1 in Seg 1;
then (h#p).1 = h.((the_arity_of o)/.1).(p.1) by A3,A4,MSUALG_3:def 6
.= h.s.(p.1) by A1,FINSEQ_4:16 .= h.s.a by A2,FINSEQ_1:40;
hence h#p = <*h.s.a*> by A5,FINSEQ_1:40;
end;
theorem Th26:
for S being non empty non void ManySortedSign
for A,B being non-empty MSAlgebra over S
for s1,s2 being SortSymbol of S
for a being Element of A,s1, b being Element of A,s2
for h being ManySortedFunction of A,B
for o being OperSymbol of S st the_arity_of o = <*s1,s2*>
for p being Element of Args(o,A)
st p = <*a,b*> holds h#p = <*h.s1.a, h.s2.b*>
proof
let S be non empty non void ManySortedSign;
let A,B be non-empty MSAlgebra over S;
let s1,s2 be SortSymbol of S;
let a be Element of A,s1, b be Element of A,s2;
let h be ManySortedFunction of A,B;
let o be OperSymbol of S such that
A1: the_arity_of o = <*s1,s2*>;
let p be Element of Args(o,A);
assume A2: p = <*a,b*>;
A3: dom p = dom the_arity_of o & dom(h#p) = dom the_arity_of o by MSUALG_3:6;
then
A4: dom(h#p) = Seg 2 by A2,FINSEQ_1:89;
then
A5: len <*a,b*> = 2 & len (h#p) = 2 by A2,A3,FINSEQ_1:def 3;
1 in Seg 2;
then
A6: (h#p).1 = h.((the_arity_of o)/.1).(p.1) by A3,A4,MSUALG_3:def 6
.= h.s1.(p.1) by A1,FINSEQ_4:17 .= h.s1.a by A2,FINSEQ_1:44;
2 in Seg 2;
then (h#p).2 = h.((the_arity_of o)/.2).(p.2) by A3,A4,MSUALG_3:def 6
.= h.s2.(p.2) by A1,FINSEQ_4:17 .= h.s2.b by A2,FINSEQ_1:44;
hence h#p = <*h.s1.a, h.s2.b*> by A5,A6,FINSEQ_1:44;
end;
theorem Th27:
for S being non empty non void ManySortedSign
for A,B being non-empty MSAlgebra over S
for s1,s2,s3 being SortSymbol of S
for a being Element of A,s1, b being Element of A,s2, c being Element of A,s3
for h being ManySortedFunction of A,B
for o being OperSymbol of S st the_arity_of o = <*s1,s2,s3*>
for p being Element of Args(o,A)
st p = <*a,b,c*> holds h#p = <*h.s1.a,h.s2.b,h.s3.c*>
proof
let S be non empty non void ManySortedSign;
let A,B be non-empty MSAlgebra over S;
let s1,s2,s3 be SortSymbol of S;
let a be Element of A,s1;
let b be Element of A,s2;
let c be Element of A,s3;
let h be ManySortedFunction of A,B;
let o be OperSymbol of S such that
A1: the_arity_of o = <*s1,s2,s3*>;
let p be Element of Args(o,A);
assume A2: p = <*a,b,c*>;
A3: dom p = dom the_arity_of o & dom(h#p) = dom the_arity_of o by MSUALG_3:6;
then
A4: dom(h#p) = Seg 3 by A2,FINSEQ_1:89;
then
A5: len p = 3 & len (h#p) = 3 by A3,FINSEQ_1:def 3;
1 in Seg 3;
then
A6: (h#p).1 = h.((the_arity_of o)/.1).(p.1) by A3,A4,MSUALG_3:def 6
.= h.s1.(p.1) by A1,FINSEQ_4:18 .= h.s1.a by A2,FINSEQ_1:45;
2 in Seg 3;
then
A7: (h#p).2 = h.((the_arity_of o)/.2).(p.2) by A3,A4,MSUALG_3:def 6
.= h.s2.(p.2) by A1,FINSEQ_4:18 .= h.s2.b by A2,FINSEQ_1:45;
3 in Seg 3;
then (h#p).3 = h.((the_arity_of o)/.3).(p.3) by A3,A4,MSUALG_3:def 6
.= h.s3.(p.3) by A1,FINSEQ_4:18 .= h.s3.c by A2,FINSEQ_1:45;
hence h#p = <*h.s1.a,h.s2.b,h.s3.c*> by A5,A6,A7,FINSEQ_1:45;
end;
theorem Th28:
for h being ManySortedFunction of T,C st h is_homomorphism T,C
for a being SortSymbol of S
for t being Element of T,a
holds t value_at(C,h||FreeGen T) = h.a.t
proof
let h be ManySortedFunction of T,C;
assume A1: h is_homomorphism T,C;
set s = h||FreeGen T;
let a be SortSymbol of S;
let t be Element of T,a;
FreeGen T is_transformable_to the Sorts of C by MSAFREE4:21;
then
A2: doms s = FreeGen T by MSSUBFAM:17;
thus t value_at(C,s) = h.a.t by A2,A1,AOFA_A00:def 21;
end;
theorem Th29:
for h being ManySortedFunction of T,C
st h is_homomorphism T,C & s = h||the generators of G
for a being SortSymbol of S
for t being Element of T,a
holds t value_at(C,s) = h.a.t
proof
let h be ManySortedFunction of T,C;
assume A1: h is_homomorphism T,C;
assume A2: s = h||the generators of G;
let a be SortSymbol of S;
let t be Element of T,a;
A3: s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
the generators of G is_transformable_to the Sorts of C by MSAFREE4:21;
then
A4: doms s = the generators of G by A3,MSSUBFAM:17;
thus t value_at(C,s) = h.a.t by A4,A1,A2,AOFA_A00:def 21;
end;
theorem Th30:
\trueT value_at(C,s) = TRUE
proof
A1: s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A2: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
the generators of G is_transformable_to the Sorts of C
by MSAFREE4:21;
then
doms s = the generators of G by A1,MSSUBFAM:17;
then consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A3: f is_homomorphism T,C & Q = doms s & s = f||Q &
\trueT value_at(C,s) = f.(the bool-sort of S).\trueT by A2,AOFA_A00:def 21;
set o = In((the connectives of S).1, the carrier' of S);
A4: o = (the connectives of S).1 &
the_arity_of o = {} & the_result_sort_of o = the bool-sort of S by Th11;
then
Args(o,T) = {{}} by Th21;
then reconsider p = {} as Element of Args(o,T) by TARSKI:def 1;
dom(f#p) = {} & dom p = {} by A4,MSUALG_3:6;
then
A5: p = f#p;
f.(the bool-sort of S).\trueT
= \trueC by A5,A3,A4
.= TRUE by AOFA_A00:def 32;
hence thesis by A3;
end;
theorem Th31:
for t being Element of T, the bool-sort of S holds
\nott value_at(C,s) = \not(t value_at(C,s))
proof
let t be Element of T, the bool-sort of S;
s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
A2: (\nott) value_at(C,s) = f.(the bool-sort of S).(\nott) by A1,Th29;
set o = In((the connectives of S).2, the carrier' of S);
A3: the_arity_of o = <*the bool-sort of S*> &
the_result_sort_of o = the bool-sort of S by Th12;
then Args(o,T) = product <*(the Sorts of T).the bool-sort of S*> by Th22;
then reconsider p = <*t*> as Element of Args(o,T) by FINSEQ_3:123;
thus (\nott) value_at(C,s) = Den(o,C).(f#p) by A1,A2,A3
.= Den(o,C).<*f.(the bool-sort of S).t*> by A3,Th25
.= \not(t value_at(C,s)) by A1,Th29;
end;
theorem
for a being boolean object
for t being Element of T, the bool-sort of S holds
\nott value_at(C,s) = 'not' a iff t value_at(C,s) = a
proof
let a be boolean object;
let t be Element of T, the bool-sort of S;
hereby
assume \nott value_at(C,s) = 'not' a;
then \not(t value_at(C,s)) = 'not' a by Th31;
then 'not' (t value_at(C,s)) = 'not' a by AOFA_A00:def 32;
hence t value_at(C,s) = a;
end;
assume A1: t value_at(C,s) = a;
\not(t value_at(C,s)) = \nott value_at(C,s) by Th31;
hence \nott value_at(C,s) = 'not' a by A1,AOFA_A00:def 32;
end;
theorem Th33:
for a being Element of C, the bool-sort of S
for x being boolean object holds
\nota = 'not' x iff a = x
proof
let a be Element of C, the bool-sort of S;
a in (the Sorts of C).the bool-sort of S;
then a in BOOLEAN by AOFA_A00:def 32;
then reconsider b = a as boolean object;
let x be boolean object;
hereby
assume \nota = 'not' x;
then 'not' b = 'not' x by AOFA_A00:def 32;
hence a = x;
end;
assume a = x;
hence \nota = 'not' x by AOFA_A00:def 32;
end;
theorem
(\falseT) value_at(C,s) = FALSE
proof
thus (\falseT) value_at(C,s) = \not((\trueT) value_at(C,s)) by Th31
.= 'not' ((\trueT) value_at(C,s)) by AOFA_A00:def 32 .= 'not' TRUE by Th30
.= FALSE;
end;
theorem
for t1,t2 being Element of T, the bool-sort of S holds
t1\andt2 value_at(C,s) = (t1 value_at(C,s))\and(t2 value_at(C,s))
proof
let t1,t2 be Element of T, the bool-sort of S;
s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
A2: t1 value_at(C,s) = f.(the bool-sort of S).t1 by A1,Th29;
A3: t1\andt2 value_at(C,s) = f.(the bool-sort of S).(t1\andt2) by A1,Th29;
set o = In((the connectives of S).3, the carrier' of S);
A4: the_arity_of o = <*the bool-sort of S,the bool-sort of S*> &
the_result_sort_of o = the bool-sort of S by Th13;
then Args(o,T) = product <*(the Sorts of T).the bool-sort of S,
(the Sorts of T).the bool-sort of S*> by Th23;
then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
thus (t1\andt2) value_at(C,s) = Den(o,C).(f#p)
by A1,A3,A4
.= Den(o,C).<*f.(the bool-sort of S).t1,f.(the bool-sort of S).t2*>
by A4,Th26
.= (t1 value_at(C,s))\and(t2 value_at(C,s)) by A2,A1,Th29;
end;
theorem Th36:
\0(T,I) value_at(C,s) = 0
proof
A1: s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A2: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
the generators of G is_transformable_to the Sorts of C
by MSAFREE4:21;
then
doms s = the generators of G by A1,MSSUBFAM:17;
then consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A3: f is_homomorphism T,C & Q = doms s & s = f||Q &
\0(T,I) value_at(C,s) = f.I.\0(T,I) by A2,AOFA_A00:def 21;
set o = In((the connectives of S).4, the carrier' of S);
A4: the_arity_of o = {} & the_result_sort_of o = I by Th14;
then
Args(o,T) = {{}} by Th21;
then reconsider p = {} as Element of Args(o,T) by TARSKI:def 1;
dom(f#p) = {} & dom p = {} by A4,MSUALG_3:6;
then
A5: p = f#p;
f.I.\0(T,I)
= \0(C,I) by A5,A3,A4 .= 0 by AOFA_A00:55;
hence thesis by A3;
end;
theorem Th37:
\1(T,I) value_at(C,s) = 1
proof
A1: s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A2: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
the generators of G is_transformable_to the Sorts of C
by MSAFREE4:21;
then
doms s = the generators of G by A1,MSSUBFAM:17;
then consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A3: f is_homomorphism T,C & Q = doms s & s = f||Q &
\1(T,I) value_at(C,s) = f.I.\1(T,I) by A2,AOFA_A00:def 21;
set o = In((the connectives of S).5, the carrier' of S);
A4: the_arity_of o = {} & the_result_sort_of o = I by Th15;
then
Args(o,T) = {{}} by Th21;
then reconsider p = {} as Element of Args(o,T) by TARSKI:def 1;
dom(f#p) = {} & dom p = {} by A4,MSUALG_3:6;
then
A5: p = f#p;
f.I.\1(T,I)
= \1(C,I) by A5,A3,A4 .= 1 by AOFA_A00:55;
hence thesis by A3;
end;
theorem Th38:
(-t) value_at(C,s) = -(t value_at(C,s))
proof
A1: s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A2: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
the generators of G is_transformable_to the Sorts of C
by MSAFREE4:21;
then
A3: doms s = the generators of G by A1,MSSUBFAM:17;
then consider f1 being ManySortedFunction of T,C,
Q1 being GeneratorSet of T such that
A4: f1 is_homomorphism T,C & Q1 = doms s & s = f1||Q1 &
t value_at(C,s) = f1.I.t by A2,AOFA_A00:def 21;
consider f2 being ManySortedFunction of T,C,
Q2 being GeneratorSet of T such that
A5: f2 is_homomorphism T,C & Q2 = doms s & s = f2||Q2 &
(-t) value_at(C,s) = f2.I.(-t) by A2,A3,AOFA_A00:def 21;
set o = In((the connectives of S).6, the carrier' of S);
A6: the_arity_of o = <*I*> & the_result_sort_of o = I by Th16;
then Args(o,T) = product <*(the Sorts of T).I*> by Th22;
then reconsider p = <*t*> as Element of Args(o,T) by FINSEQ_3:123;
thus (-t) value_at(C,s) = Den(o,C).(f2#p) by A5,A6
.= Den(o,C).<*f2.I.t*> by A6,Th25
.= -(t value_at(C,s)) by A4,A5,EXTENS_1:19;
end;
theorem Th39:
(t1+t2) value_at(C,s) = (t1 value_at(C,s))+(t2 value_at(C,s))
proof
A1: s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A2: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
the generators of G is_transformable_to the Sorts of C
by MSAFREE4:21;
then
A3: doms s = the generators of G by A1,MSSUBFAM:17;
then consider f1 being ManySortedFunction of T,C,
Q1 being GeneratorSet of T such that
A4: f1 is_homomorphism T,C & Q1 = doms s & s = f1||Q1 &
t1 value_at(C,s) = f1.I.t1 by A2,AOFA_A00:def 21;
consider f2 being ManySortedFunction of T,C,
Q2 being GeneratorSet of T such that
A5: f2 is_homomorphism T,C & Q2 = doms s & s = f2||Q2 &
t2 value_at(C,s) = f2.I.t2 by A2,A3,AOFA_A00:def 21;
consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A6: f is_homomorphism T,C & Q = doms s & s = f||Q &
(t1+t2) value_at(C,s) = f.I.(t1+t2) by A2,A3,AOFA_A00:def 21;
A7: f = f1 & f = f2 by A4,A5,A6,EXTENS_1:19;
set o = In((the connectives of S).7, the carrier' of S);
A8: the_arity_of o = <*I,I*> & the_result_sort_of o = I by Th17;
then Args(o,T) = product <*(the Sorts of T).I, (the Sorts of T).I*>
by Th23;
then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
thus (t1+t2) value_at(C,s) = Den(o,C).(f#p) by A6,A8
.= (t1 value_at(C,s))+(t2 value_at(C,s)) by A4,A5,A7,A8,Th26;
end;
theorem Th40:
\2(T,I) value_at (C,s) = 2
proof
A1: \1(T,I) value_at(C,s) = 1 by Th37;
thus \2(T,I) value_at(C,s)
= (\1(T,I) value_at(C,s))+(\1(T,I) value_at(C,s)) by Th39
.= 2 by A1,AOFA_A00:55;
end;
theorem Th41:
(t1-t2) value_at(C,s) = (t1 value_at(C,s))-(t2 value_at(C,s))
proof
thus (t1-t2) value_at(C,s) = (t1 value_at(C,s))+((-t2) value_at(C,s))
by Th39
.= (t1 value_at(C,s))-(t2 value_at(C,s)) by Th38;
end;
theorem Th42:
(t1*t2) value_at(C,s) = (t1 value_at(C,s))*(t2 value_at(C,s))
proof
s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
A2: t2 value_at(C,s) = f.I.t2 by A1,Th29;
A3: (t1*t2) value_at(C,s) = f.I.(t1*t2) by A1,Th29;
set o = In((the connectives of S).8, the carrier' of S);
A4: the_arity_of o = <*I,I*> & the_result_sort_of o = I by Th18;
then Args(o,T) = product <*(the Sorts of T).I, (the Sorts of T).I*>
by Th23;
then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
thus (t1*t2) value_at(C,s) = Den(o,C).(f#p) by A1,A3,A4
.= Den(o,C).<*f.I.t1,f.I.t2*> by A4,Th26
.= (t1 value_at(C,s))*(t2 value_at(C,s)) by A1,A2,Th29;
end;
theorem Th43:
(t1 div t2) value_at(C,s) = (t1 value_at(C,s))div(t2 value_at(C,s))
proof
A1: s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A2: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
the generators of G is_transformable_to the Sorts of C
by MSAFREE4:21;
then
A3: doms s = the generators of G by A1,MSSUBFAM:17;
then consider f1 being ManySortedFunction of T,C,
Q1 being GeneratorSet of T such that
A4: f1 is_homomorphism T,C & Q1 = doms s & s = f1||Q1 &
t1 value_at(C,s) = f1.I.t1 by A2,AOFA_A00:def 21;
consider f2 being ManySortedFunction of T,C,
Q2 being GeneratorSet of T such that
A5: f2 is_homomorphism T,C & Q2 = doms s & s = f2||Q2 &
t2 value_at(C,s) = f2.I.t2 by A2,A3,AOFA_A00:def 21;
consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A6: f is_homomorphism T,C & Q = doms s & s = f||Q &
(t1 div t2) value_at(C,s) = f.I.(t1 div t2) by A2,A3,AOFA_A00:def 21;
A7: f = f1 & f = f2 by A4,A5,A6,EXTENS_1:19;
set o = In((the connectives of S).9, the carrier' of S);
A8: the_arity_of o = <*I,I*> & the_result_sort_of o = I by Th19;
then Args(o,T) = product <*(the Sorts of T).I, (the Sorts of T).I*>
by Th23;
then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
thus (t1 div t2) value_at(C,s) = Den(o,C).(f#p) by A6,A8
.= (t1 value_at(C,s))div(t2 value_at(C,s)) by A4,A5,A7,A8,Th26;
end;
theorem Th44:
(t1 mod t2) value_at(C,s) = (t1 value_at(C,s)) mod (t2 value_at(C,s))
proof
thus (t1 mod t2) value_at(C,s) = (t1-(t1 div t2)*t2) value_at(C,s)
.= (t1 value_at(C,s))-(((t1 div t2)*t2) value_at(C,s)) by Th41
.= (t1 value_at(C,s))-(((t1 div t2) value_at(C,s))*(t2 value_at(C,s)))
by Th42
.= (t1 value_at(C,s)) mod (t2 value_at(C,s)) by Th43;
end;
theorem Th45:
leq(t1,t2) value_at(C,s) = leq(t1 value_at(C,s), t2 value_at(C,s))
proof
A1: s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A2: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
the generators of G is_transformable_to the Sorts of C
by MSAFREE4:21;
then
A3: doms s = the generators of G by A1,MSSUBFAM:17;
then consider f1 being ManySortedFunction of T,C,
Q1 being GeneratorSet of T such that
A4: f1 is_homomorphism T,C & Q1 = doms s & s = f1||Q1 &
t1 value_at(C,s) = f1.I.t1 by A2,AOFA_A00:def 21;
consider f2 being ManySortedFunction of T,C,
Q2 being GeneratorSet of T such that
A5: f2 is_homomorphism T,C & Q2 = doms s & s = f2||Q2 &
t2 value_at(C,s) = f2.I.t2 by A2,A3,AOFA_A00:def 21;
consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A6: f is_homomorphism T,C & Q = doms s & s = f||Q &
leq(t1,t2) value_at(C,s) = f.(the bool-sort of S).leq(t1,t2)
by A2,A3,AOFA_A00:def 21;
A7: f = f1 & f = f2 by A4,A5,A6,EXTENS_1:19;
set o = In((the connectives of S).10, the carrier' of S);
A8: the_arity_of o = <*I,I*> & the_result_sort_of o = the bool-sort of S
by Th20;
then Args(o,T) = product <*(the Sorts of T).I, (the Sorts of T).I*>
by Th23;
then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
thus leq(t1,t2) value_at(C,s) = Den(o,C).(f#p) by A6,A8
.= leq(t1 value_at(C,s), t2 value_at(C,s)) by A4,A5,A7,A8,Th26;
end;
theorem Th46:
\trueT value_at(C,u) = TRUE
proof
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
FreeGen T is_transformable_to the Sorts of C
by MSAFREE4:21;
then
doms u = FreeGen T by MSSUBFAM:17;
then consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A2: f is_homomorphism T,C & Q = doms u & u = f||Q &
\trueT value_at(C,u) = f.(the bool-sort of S).\trueT by A1,AOFA_A00:def 21;
set o = In((the connectives of S).1, the carrier' of S);
A3: o = (the connectives of S).1 &
the_arity_of o = {} & the_result_sort_of o = the bool-sort of S by Th11;
then
Args(o,T) = {{}} by Th21;
then reconsider p = {} as Element of Args(o,T) by TARSKI:def 1;
dom(f#p) = {} & dom p = {} by A3,MSUALG_3:6;
then
A4: p = f#p;
f.(the bool-sort of S).\trueT = \trueC by A4,A2,A3
.= TRUE by AOFA_A00:def 32;
hence thesis by A2;
end;
theorem Th47:
for t being Element of T, the bool-sort of S holds
\nott value_at(C,u) = \not(t value_at(C,u))
proof
let t be Element of T, the bool-sort of S;
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
A2: (\nott) value_at(C,u) = f.(the bool-sort of S).(\nott) by A1,Th28;
set o = In((the connectives of S).2, the carrier' of S);
A3: the_arity_of o = <*the bool-sort of S*> &
the_result_sort_of o = the bool-sort of S by Th12;
then Args(o,T) = product <*(the Sorts of T).the bool-sort of S*> by Th22;
then reconsider p = <*t*> as Element of Args(o,T) by FINSEQ_3:123;
thus (\nott) value_at(C,u) = Den(o,C).(f#p) by A1,A2,A3
.= Den(o,C).<*f.(the bool-sort of S).t*> by A3,Th25
.= \not(t value_at(C,u)) by A1,Th28;
end;
theorem
for a being boolean object
for t being Element of T, the bool-sort of S holds
\nott value_at(C,u) = 'not' a iff t value_at(C,u) = a
proof
let a be boolean object;
let t be Element of T, the bool-sort of S;
hereby
assume \nott value_at(C,u) = 'not' a;
then \not(t value_at(C,u)) = 'not' a by Th47;
then 'not' (t value_at(C,u)) = 'not' a by AOFA_A00:def 32;
hence t value_at(C,u) = a;
end;
assume A1: t value_at(C,u) = a;
\not(t value_at(C,u)) = \nott value_at(C,u) by Th47;
hence \nott value_at(C,u) = 'not' a by A1,AOFA_A00:def 32;
end;
theorem
(\falseT) value_at(C,u) = FALSE
proof
thus (\falseT) value_at(C,u) = \not((\trueT) value_at(C,u)) by Th47
.= 'not' ((\trueT) value_at(C,u)) by AOFA_A00:def 32 .= 'not' TRUE by Th46
.= FALSE;
end;
theorem
for t1,t2 being Element of T, the bool-sort of S holds
t1\andt2 value_at(C,u) = (t1 value_at(C,u))\and(t2 value_at(C,u))
proof
let t1,t2 be Element of T, the bool-sort of S;
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
A2: t1 value_at(C,u) = f.(the bool-sort of S).t1 by A1,Th28;
A3: t1\andt2 value_at(C,u) = f.(the bool-sort of S).(t1\andt2) by A1,Th28;
set o = In((the connectives of S).3, the carrier' of S);
A4: the_arity_of o = <*the bool-sort of S,the bool-sort of S*> &
the_result_sort_of o = the bool-sort of S by Th13;
then Args(o,T) = product <*(the Sorts of T).the bool-sort of S,
(the Sorts of T).the bool-sort of S*> by Th23;
then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
thus (t1\andt2) value_at(C,u) = Den(o,C).(f#p)
by A1,A3,A4
.= Den(o,C).<*f.(the bool-sort of S).t1,f.(the bool-sort of S).t2*>
by A4,Th26
.= (t1 value_at(C,u))\and(t2 value_at(C,u)) by A2,A1,Th28;
end;
theorem
\0(T,I) value_at(C,u) = 0
proof
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
FreeGen T is_transformable_to the Sorts of C
by MSAFREE4:21;
then
doms u = FreeGen T by MSSUBFAM:17;
then consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A2: f is_homomorphism T,C & Q = doms u & u = f||Q &
\0(T,I) value_at(C,u) = f.I.\0(T,I) by A1,AOFA_A00:def 21;
set o = In((the connectives of S).4, the carrier' of S);
A3: the_arity_of o = {} & the_result_sort_of o = I by Th14;
then
Args(o,T) = {{}} by Th21;
then reconsider p = {} as Element of Args(o,T) by TARSKI:def 1;
dom(f#p) = {} & dom p = {} by A3,MSUALG_3:6;
then
A4: p = f#p;
f.I.\0(T,I) = \0(C,I) by A4,A2,A3 .= 0 by AOFA_A00:55;
hence thesis by A2;
end;
theorem Th52:
\1(T,I) value_at(C,u) = 1
proof
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
FreeGen T is_transformable_to the Sorts of C
by MSAFREE4:21;
then
doms u = FreeGen T by MSSUBFAM:17;
then consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A2: f is_homomorphism T,C & Q = doms u & u = f||Q &
\1(T,I) value_at(C,u) = f.I.\1(T,I) by A1,AOFA_A00:def 21;
set o = In((the connectives of S).5, the carrier' of S);
A3: the_arity_of o = {} & the_result_sort_of o = I by Th15;
then
Args(o,T) = {{}} by Th21;
then reconsider p = {} as Element of Args(o,T) by TARSKI:def 1;
dom(f#p) = {} & dom p = {} by A3,MSUALG_3:6;
then
A4: p = f#p;
f.I.\1(T,I) = \1(C,I) by A4,A2,A3 .= 1 by AOFA_A00:55;
hence thesis by A2;
end;
theorem Th53:
(-t) value_at(C,u) = -(t value_at(C,u))
proof
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
FreeGen T is_transformable_to the Sorts of C
by MSAFREE4:21;
then
A2: doms u = FreeGen T by MSSUBFAM:17;
then consider f1 being ManySortedFunction of T,C,
Q1 being GeneratorSet of T such that
A3: f1 is_homomorphism T,C & Q1 = doms u & u = f1||Q1 &
t value_at(C,u) = f1.I.t by A1,AOFA_A00:def 21;
consider f2 being ManySortedFunction of T,C,
Q2 being GeneratorSet of T such that
A4: f2 is_homomorphism T,C & Q2 = doms u & u = f2||Q2 &
(-t) value_at(C,u) = f2.I.(-t) by A1,A2,AOFA_A00:def 21;
set o = In((the connectives of S).6, the carrier' of S);
A5: the_arity_of o = <*I*> & the_result_sort_of o = I by Th16;
then Args(o,T) = product <*(the Sorts of T).I*> by Th22;
then reconsider p = <*t*> as Element of Args(o,T) by FINSEQ_3:123;
thus (-t) value_at(C,u) = Den(o,C).(f2#p) by A4,A5
.= Den(o,C).<*f2.I.t*> by A5,Th25
.= -(t value_at(C,u)) by A3,A4,EXTENS_1:19;
end;
theorem Th54:
(t1+t2) value_at(C,u) = (t1 value_at(C,u))+(t2 value_at(C,u))
proof
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
FreeGen T is_transformable_to the Sorts of C
by MSAFREE4:21;
then
A2: doms u = FreeGen T by MSSUBFAM:17;
then consider f1 being ManySortedFunction of T,C,
Q1 being GeneratorSet of T such that
A3: f1 is_homomorphism T,C & Q1 = doms u & u = f1||Q1 &
t1 value_at(C,u) = f1.I.t1 by A1,AOFA_A00:def 21;
consider f2 being ManySortedFunction of T,C,
Q2 being GeneratorSet of T such that
A4: f2 is_homomorphism T,C & Q2 = doms u & u = f2||Q2 &
t2 value_at(C,u) = f2.I.t2 by A1,A2,AOFA_A00:def 21;
consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A5: f is_homomorphism T,C & Q = doms u & u = f||Q &
(t1+t2) value_at(C,u) = f.I.(t1+t2) by A1,A2,AOFA_A00:def 21;
A6: f = f1 & f = f2 by A3,A4,A5,EXTENS_1:19;
set o = In((the connectives of S).7, the carrier' of S);
A7: the_arity_of o = <*I,I*> & the_result_sort_of o = I by Th17;
then Args(o,T) = product <*(the Sorts of T).I, (the Sorts of T).I*>
by Th23;
then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
thus (t1+t2) value_at(C,u) = Den(o,C).(f#p) by A5,A7
.= Den(o,C).<*f.I.t1,f.I.t2*> by A7,Th26
.= (t1 value_at(C,u))+(t2 value_at(C,u)) by A3,A4,A6;
end;
theorem
\2(T,I) value_at (C,u) = 2
proof
A1: \1(T,I) value_at(C,u) = 1 by Th52;
thus \2(T,I) value_at(C,u)
= (\1(T,I) value_at(C,u))+(\1(T,I) value_at(C,u)) by Th54
.= 2 by A1,AOFA_A00:55;
end;
theorem Th56:
(t1-t2) value_at(C,u) = (t1 value_at(C,u))-(t2 value_at(C,u))
proof
thus (t1-t2) value_at(C,u) = (t1 value_at(C,u))+((-t2) value_at(C,u))
by Th54
.= (t1 value_at(C,u))-(t2 value_at(C,u)) by Th53;
end;
theorem Th57:
(t1*t2) value_at(C,u) = (t1 value_at(C,u))*(t2 value_at(C,u))
proof
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
A2: t2 value_at(C,u) = f.I.t2 by A1,Th28;
A3: (t1*t2) value_at(C,u) = f.I.(t1*t2) by A1,Th28;
set o = In((the connectives of S).8, the carrier' of S);
A4: the_arity_of o = <*I,I*> & the_result_sort_of o = I by Th18;
then Args(o,T) = product <*(the Sorts of T).I, (the Sorts of T).I*>
by Th23;
then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
thus (t1*t2) value_at(C,u) = Den(o,C).(f#p) by A1,A3,A4
.= Den(o,C).<*f.I.t1,f.I.t2*> by A4,Th26
.= (t1 value_at(C,u))*(t2 value_at(C,u)) by A1,A2,Th28;
end;
theorem Th58:
(t1 div t2) value_at(C,u) = (t1 value_at(C,u))div(t2 value_at(C,u))
proof
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
FreeGen T is_transformable_to the Sorts of C
by MSAFREE4:21;
then
A2: doms u = FreeGen T by MSSUBFAM:17;
then consider f1 being ManySortedFunction of T,C,
Q1 being GeneratorSet of T such that
A3: f1 is_homomorphism T,C & Q1 = doms u & u = f1||Q1 &
t1 value_at(C,u) = f1.I.t1 by A1,AOFA_A00:def 21;
consider f2 being ManySortedFunction of T,C,
Q2 being GeneratorSet of T such that
A4: f2 is_homomorphism T,C & Q2 = doms u & u = f2||Q2 &
t2 value_at(C,u) = f2.I.t2 by A1,A2,AOFA_A00:def 21;
consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A5: f is_homomorphism T,C & Q = doms u & u = f||Q &
(t1 div t2) value_at(C,u) = f.I.(t1 div t2) by A1,A2,AOFA_A00:def 21;
A6: f = f1 & f = f2 by A3,A4,A5,EXTENS_1:19;
set o = In((the connectives of S).9, the carrier' of S);
A7: the_arity_of o = <*I,I*> & the_result_sort_of o = I by Th19;
then Args(o,T) = product <*(the Sorts of T).I, (the Sorts of T).I*>
by Th23;
then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
thus (t1 div t2) value_at(C,u) = Den(o,C).(f#p) by A5,A7
.= (t1 value_at(C,u))div(t2 value_at(C,u)) by A3,A4,A6,A7,Th26;
end;
theorem
(t1 mod t2) value_at(C,u) = (t1 value_at(C,u)) mod (t2 value_at(C,u))
proof
thus (t1 mod t2) value_at(C,u) = (t1-(t1 div t2)*t2) value_at(C,u)
.= (t1 value_at(C,u))-(((t1 div t2)*t2) value_at(C,u)) by Th56
.= (t1 value_at(C,u))-(((t1 div t2) value_at(C,u))*(t2 value_at(C,u)))
by Th57
.= (t1 value_at(C,u)) mod (t2 value_at(C,u)) by Th58;
end;
theorem
leq(t1,t2) value_at(C,u) = leq(t1 value_at(C,u), t2 value_at(C,u))
proof
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
FreeGen T is_transformable_to the Sorts of C
by MSAFREE4:21;
then
A2: doms u = FreeGen T by MSSUBFAM:17;
then consider f1 being ManySortedFunction of T,C,
Q1 being GeneratorSet of T such that
A3: f1 is_homomorphism T,C & Q1 = doms u & u = f1||Q1 &
t1 value_at(C,u) = f1.I.t1 by A1,AOFA_A00:def 21;
consider f2 being ManySortedFunction of T,C,
Q2 being GeneratorSet of T such that
A4: f2 is_homomorphism T,C & Q2 = doms u & u = f2||Q2 &
t2 value_at(C,u) = f2.I.t2 by A1,A2,AOFA_A00:def 21;
consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A5: f is_homomorphism T,C & Q = doms u & u = f||Q &
leq(t1,t2) value_at(C,u) = f.(the bool-sort of S).leq(t1,t2)
by A1,A2,AOFA_A00:def 21;
A6: f = f1 & f = f2 by A3,A4,A5,EXTENS_1:19;
set o = In((the connectives of S).10, the carrier' of S);
A7: the_arity_of o = <*I,I*> & the_result_sort_of o = the bool-sort of S
by Th20;
then Args(o,T) = product <*(the Sorts of T).I, (the Sorts of T).I*>
by Th23;
then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
thus leq(t1,t2) value_at(C,u) = Den(o,C).(f#p) by A5,A7
.= leq(t1 value_at(C,u), t2 value_at(C,u)) by A3,A4,A6,A7,Th26;
end;
theorem Th61:
for a being SortSymbol of S
for x being Element of (the generators of G).a holds
@x value_at(C,s) = s.a.x
proof
let a be SortSymbol of S;
let x be Element of (the generators of G).a;
s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider h being ManySortedFunction of T,C such that
A1: h is_homomorphism T,C & s = h||the generators of G by AOFA_A00:def 19;
@x value_at(C,s) = h.a.x by A1,Th29 .= ((h.a)|((the generators of G).a)).x
by FUNCT_1:49;
hence @x value_at(C,s) = s.a.x by A1,MSAFREE:def 1;
end;
theorem Th62:
for a being SortSymbol of S
for x being pure Element of (the generators of G).a
for u being ManySortedFunction of FreeGen T, the Sorts of C holds
@x value_at(C,u) = u.a.x
proof
let a be SortSymbol of S;
let x be pure Element of (the generators of G).a;
let u be ManySortedFunction of FreeGen T, the Sorts of C;
consider h being ManySortedFunction of T,C such that
A1: h is_homomorphism T,C & u = h||FreeGen T by MSAFREE4:46;
FreeGen T is_transformable_to the Sorts of C by MSAFREE4:21;
then
A2: doms u = FreeGen T by MSSUBFAM:17;
then consider f being ManySortedFunction of T,C,
Q being GeneratorSet of T such that
A3: f is_homomorphism T,C & Q = doms u & u = f||Q & @x value_at(C,u) = f.a.@x
by A1,AOFA_A00:def 21;
@x value_at(C,u) = h.a.x by A1,A2,A3,EXTENS_1:19
.= ((h.a)|((FreeGen T).a)).x by Def4,FUNCT_1:49;
hence @x value_at(C,u) = u.a.x by A1,MSAFREE:def 1;
end;
theorem Th63:
for i,j being Integer, a,b being Element of C,I st a = i & b = j
holds a-b = i-j
proof
let i,j be Integer;
let a,b be Element of C,I;
assume A1: a = i;
assume b = j;
then -b = -j by AOFA_A00:55;
then a+-b = i+-j by A1,AOFA_A00:55;
hence a-b = i-j;
end;
theorem Th64:
for i,j being Integer, a,b being Element of C,I st a = i & b = j & j <> 0
holds a mod b = i mod j
proof
let i,j be Integer;
let a,b be Element of C,I;
assume A1: a = i;
assume A2: b = j;
assume A3: j <> 0;
then a div b = i div j by A1,A2,AOFA_A00:55;
then (a div b)*b = (i div j)*j by A2,AOFA_A00:55;
then a-(a div b)*b = i-(i div j)*j by A1,Th63;
hence a mod b = i mod j by A3,INT_1:def 10;
end;
theorem Th65:
G is C-supported & f in C-Execution(A,b,\falseC) implies
for a being SortSymbol of S, x being pure Element of (the generators of G).a
for t being Element of T,a
holds
f.(s,x:=(t,A)).a.x = t value_at(C,s) &
(for z being pure Element of (the generators of G).a st z <> x
holds f.(s, x:=(t,A)).a.z = s.a.z) &
for b being SortSymbol of S st a <> b holds
(for z being pure Element of (the generators of G).b holds
f.(s, x:=(t,A)).b.z = s.b.z)
proof
assume
A1: G is C-supported;
assume
A2: f in C-Execution(A,b,\falseC);
let a be SortSymbol of S;
let x be pure Element of (the generators of G).a;
let t be Element of T,a;
reconsider x0 = @x as Element of G,a by AOFA_A00:def 22;
thus
f.(s, x:=(t,A)).a.x = succ(s,x0,t value_at(C,s)).a.x by A2,AOFA_A00:def 28
.= t value_at(C,s) by A1,AOFA_A00:def 27;
hereby
let z be pure Element of (the generators of G).a; assume
A3: z <> x;
A4: x in (FreeGen T).a & z in (FreeGen T).a &
FreeGen X is ManySortedSubset of the generators of G
by A1,Def4;
then vf @x = a-singleton(x) & FreeGen X c= the generators of G
by AOFA_A00:41,PBOOLE:def 18;
then (vf @x).a = {x} & (FreeGen X).a c= (the generators of G).a
by AOFA_A00:6;
then
A5: z nin (vf @x).a & @x is Element of G,a
by A3,TARSKI:def 1,AOFA_A00:def 22;
thus f.(s, x:=(t,A)).a.z = succ(s,x0,t value_at(C,s)).a.z
by A2,AOFA_A00:def 28
.= s.a.z by A1,A3,A5,A4,AOFA_A00:def 27;
end;
let b be SortSymbol of S; assume
A6: a <> b;
hereby
let z be pure Element of (the generators of G).b;
A7: x in (FreeGen T).a & z in (FreeGen T).b &
FreeGen X is ManySortedSubset of the generators of G
by A1,Def4;
then vf @x = a-singleton(x) & FreeGen X c= the generators of G
by AOFA_A00:41,PBOOLE:def 18;
then
A8: z nin (vf @x).b & @x is Element of G,a
by A6,AOFA_A00:6,AOFA_A00:def 22;
thus f.(s, x:=(t,A)).b.z = succ(s,x0,t value_at(C,s)).b.z
by A2,AOFA_A00:def 28
.= s.b.z by A1,A6,A8,A7,AOFA_A00:def 27;
end;
end;
theorem Th66:
G is C-supported & f in C-Execution(A,b,\falseC) implies
(t1 value_at(C,s) < t2 value_at(C,s) iff
f.(s, b gt(t2,t1,A)) in (\falseC)-States(the generators of G,b)) &
(t1 value_at(C,s) <= t2 value_at(C,s) iff
f.(s, b leq(t1,t2,A)) in (\falseC)-States(the generators of G,b)) &
(for x holds f.(s, b gt(t1,t2,A)).I.x = s.I.x &
f.(s, b leq(t1,t2,A)).I.x = s.I.x) &
for c being pure Element of (the generators of G).the bool-sort of S
st c <> b holds
f.(s, b gt(t1,t2,A)).(the bool-sort of S).c = s.(the bool-sort of S).c &
f.(s, b leq(t1,t2,A)).(the bool-sort of S).c = s.(the bool-sort of S).c
proof assume
A1: G is C-supported & f in C-Execution(A,b,\falseC);
A2: f.(s, b gt(t2,t1,A)) is ManySortedFunction of the generators of G,
the Sorts of C by AOFA_A00:48;
reconsider b0 = @b as Element of G, the bool-sort of S
by AOFA_A00:def 22;
A3: \not(leq(t2,t1)) value_at (C,s)
= \not(leq(t2,t1) value_at(C,s)) by Th31
.= \not(leq(t2 value_at(C,s), t1 value_at(C,s))) by Th45;
then
A4: f.(s, b gt(t2,t1,A)).(the bool-sort of S).b
= succ(s, b0, \not(leq(t2 value_at(C,s), t1 value_at(C,s)))).
(the bool-sort of S).b by A1,AOFA_A00:def 28
.= \not(leq(t2 value_at(C,s), t1 value_at(C,s))) by A1,AOFA_A00:def 27;
A5: 'not' FALSE = TRUE & TRUE <> FALSE &
for x being boolean object holds x <> FALSE iff x = TRUE
by XBOOLEAN:def 3;
\trueC = TRUE by AOFA_A00:def 32;
then
A6: \falseC = 'not' TRUE by AOFA_A00:def 32 .= FALSE;
t1 value_at(C,s) < t2 value_at(C,s) iff
leq(t2 value_at(C,s), t1 value_at(C,s)) = FALSE by AOFA_A00:55;
then t1 value_at(C,s) < t2 value_at(C,s) iff
\notleq(t2 value_at(C,s), t1 value_at(C,s)) <> \falseC by A6,A3,A5,Th33;
hence (t1 value_at(C,s) < t2 value_at(C,s) iff
f.(s, b gt(t2,t1,A)) in (\falseC)-States(the generators of G,b))
by A2,A4,AOFA_A00:def 20;
A7: f.(s, b leq(t1,t2,A)) is ManySortedFunction of the generators of G,
the Sorts of C by AOFA_A00:48;
leq(t1,t2) value_at (C,s)
= leq(t1 value_at(C,s), t2 value_at(C,s)) by Th45;
then
A8: f.(s, b leq(t1,t2,A)).(the bool-sort of S).b
= succ(s, b0, leq(t1 value_at(C,s), t2 value_at(C,s))).
(the bool-sort of S).b by A1,AOFA_A00:def 28
.= leq(t1 value_at(C,s), t2 value_at(C,s)) by A1,AOFA_A00:def 27;
\trueC = TRUE by AOFA_A00:def 32;
then
A9: \falseC = 'not' TRUE by AOFA_A00:def 32 .= FALSE;
t1 value_at(C,s) <= t2 value_at(C,s) iff
leq(t1 value_at(C,s), t2 value_at(C,s)) <> \falseC
by A9,AOFA_A00:55;
hence t1 value_at(C,s) <= t2 value_at(C,s) iff
f.(s, b leq(t1,t2,A)) in (\falseC)-States(the generators of G,b)
by A7,A8,AOFA_A00:def 20;
b in (FreeGen T).the bool-sort of S by Def4;
then
A10: vf b0 = (the bool-sort of S)-singleton(b) by AOFA_A00:41;
hereby let x;
A11: I <> the bool-sort of S by AOFA_A00:53;
A12: x in (FreeGen T).I by Def4;
A13: x nin (vf b0).I by A10,A11,AOFA_A00:6;
thus f.(s, b gt(t1,t2,A)).I.x
= succ(s,b0,\notleq(t1,t2)value_at(C,s)).I.x by A1,AOFA_A00:def 28
.= s.I.x by A11,A12,A13,A1,AOFA_A00:def 27;
thus f.(s, b leq(t1,t2,A)).I.x
= succ(s,b0,leq(t1,t2)value_at(C,s)).I.x by A1,AOFA_A00:def 28
.= s.I.x by A11,A12,A13,A1,AOFA_A00:def 27;
end;
let c be pure Element of (the generators of G).the bool-sort of S;
assume A14: c <> b;
(vf b0).the bool-sort of S = {b} by A10,AOFA_A00:6;
then
A15: c nin (vf b0).the bool-sort of S by A14,TARSKI:def 1;
A16: c in (FreeGen T).the bool-sort of S by Def4;
thus f.(s, b gt(t1,t2,A)).(the bool-sort of S).c
= succ(s,b0,\notleq(t1,t2)value_at(C,s)).(the bool-sort of S).c
by A1,AOFA_A00:def 28
.= s.(the bool-sort of S).c by A14,A15,A16,A1,AOFA_A00:def 27;
thus f.(s, b leq(t1,t2,A)).(the bool-sort of S).c
= succ(s,b0,leq(t1,t2)value_at(C,s)).(the bool-sort of S).c
by A1,AOFA_A00:def 28
.= s.(the bool-sort of S).c by A14,A15,A16,A1,AOFA_A00:def 27;
end;
registration
let i,j be Real;
let a,b be boolean object;
cluster IFGT(i,j,a,b) -> boolean;
coherence by XXREAL_0:def 11;
end;
theorem Th67:
G is C-supported & f in C-Execution(A,b,\falseC) implies
f.(s, t is_odd(b,A)).(the bool-sort of S).b = (t value_at(C,s)) mod 2 &
f.(s, t is_even(b,A)).(the bool-sort of S).b = ((t value_at(C,s))+1) mod 2 &
for z holds f.(s, t is_odd(b,A)).I.z = s.I.z &
f.(s, t is_even(b,A)).I.z = s.I.z
proof assume
A1: G is C-supported & f in C-Execution(A,b,\falseC);
reconsider b0 = @b as Element of G,the bool-sort of S by AOFA_A00:def 22;
A2: \2(T,I) value_at(C,s) = 2 & \0(T,I) value_at(C,s) = 0 by Th36,Th40;
then
A3: (t value_at(C,s)) mod (\2(T,I) value_at(C,s))
= (t value_at(C,s)) mod 2 by Th64;
then
A4: leq((t value_at(C,s)) mod (\2(T,I) value_at(C,s)),
\0(T,I) value_at(C,s))
= IFGT((t value_at(C,s)) mod 2, 0, FALSE, TRUE) by A2,AOFA_A00:55;
reconsider Z = IFGT((t value_at(C,s)) mod 2, 0, FALSE, TRUE) as boolean
object;
A5: f.(s, t is_odd(b,A)).(the bool-sort of S).b
= succ(s,b0,\notleq(t mod \2(T,I),\0(T,I)) value_at(C,s))
.(the bool-sort of S).b by A1,AOFA_A00:def 28
.= \notleq(t mod \2(T,I),\0(T,I)) value_at(C,s) by A1,AOFA_A00:def 27
.= \not(leq(t mod \2(T,I),\0(T,I)) value_at(C,s)) by Th31
.= \notleq((t mod \2(T,I)) value_at(C,s),\0(T,I) value_at(C,s)) by Th45
.= \notleq((t value_at(C,s)) mod (\2(T,I) value_at(C,s)),
\0(T,I) value_at(C,s)) by Th44
.= 'not' Z by A4,AOFA_A00:def 32;
hereby
per cases by PRE_FF:6;
suppose
A6: (t value_at(C,s)) mod 2 = 0;
hence f.(s, t is_odd(b,A)).(the bool-sort of S).b
= 'not' TRUE by A5,XXREAL_0:def 11
.= (t value_at(C,s)) mod 2 by A6;
end;
suppose
A7: (t value_at(C,s)) mod 2 = 1;
hence f.(s, t is_odd(b,A)).(the bool-sort of S).b
= 'not' FALSE by A5,XXREAL_0:def 11
.= (t value_at(C,s)) mod 2 by A7;
end;
end;
A8: f.(s, t is_even(b,A)).(the bool-sort of S).b
= succ(s,b0,leq(t mod \2(T,I),\0(T,I)) value_at(C,s))
.(the bool-sort of S).b by A1,AOFA_A00:def 28
.= leq(t mod \2(T,I),\0(T,I)) value_at(C,s) by A1,AOFA_A00:def 27
.= leq((t mod \2(T,I)) value_at(C,s),\0(T,I) value_at(C,s)) by Th45
.= leq((t value_at(C,s)) mod (\2(T,I) value_at(C,s)),
\0(T,I) value_at(C,s)) by Th44
.= IFGT((t value_at(C,s)) mod 2, 0, FALSE, TRUE) by A3,A2,AOFA_A00:55;
hereby
per cases by PRE_FF:6;
suppose
A9: (t value_at(C,s)) mod 2 = 0;
hence f.(s, t is_even(b,A)).(the bool-sort of S).b
= (0+1) mod 2 by Th1,A8,XXREAL_0:def 11
.= ((t value_at(C,s))+1) mod 2 by A9,Th1,NAT_D:66;
end;
suppose
A10: (t value_at(C,s)) mod 2 = 1;
hence f.(s, t is_even(b,A)).(the bool-sort of S).b
= 1+1 mod 2 by Th1,A8,XXREAL_0:def 11
.= ((t value_at(C,s))+1) mod 2 by A10,Th1,NAT_D:66;
end;
end;
let z;
A11: I <> the bool-sort of S by AOFA_A00:53;
b in (FreeGen T).the bool-sort of S by Def4;
then (vf @b) = (the bool-sort of S)-singleton(b) by AOFA_A00:41;
then
A12: z nin (vf @b).I by A11,AOFA_A00:6;
A13: z in (FreeGen T).I by Def4;
thus f.(s, t is_odd(b,A)).I.z
= succ(s,b0,\notleq(t mod \2(T,I),\0(T,I)) value_at(C,s)).I.z
by A1,AOFA_A00:def 28
.= s.I.z by A1,A11,A12,A13,AOFA_A00:def 27;
thus f.(s, t is_even(b,A)).I.z
= succ(s,b0,leq(t mod \2(T,I),\0(T,I)) value_at(C,s)).I.z
by A1,AOFA_A00:def 28
.= s.I.z by A1,A11,A12,A13,AOFA_A00:def 27;
end;
definition
let S,X,T,G,A;
attr A is elementary means
rng the assignments of A c= ElementaryInstructions A;
end;
theorem Th68:
A is elementary implies
for a being SortSymbol of S
for x being Element of (the generators of G).a
for t being Element of T,a holds
x:=(t,A) in ElementaryInstructions A
proof assume
A1: rng the assignments of A c= ElementaryInstructions A;
let a be SortSymbol of S;
let x be Element of (the generators of G).a;
let t be Element of T,a;
[x,t] in [:(the generators of G).a,(the Sorts of T).a:] by ZFMISC_1:87;
then [x,t] in [|the generators of G, the Sorts of T|].a &
dom [|the generators of G, the Sorts of T|] = the carrier of S
by PARTFUN1:def 2,PBOOLE:def 16;
then x:=(t,A) in rng the assignments of A by FUNCT_2:4,CARD_5:2;
hence thesis by A1;
end;
registration
let S,X,T,G;
cluster elementary for strict IfWhileAlgebra of the generators of G;
existence
proof
set W = the infinite IfWhileAlgebra of the generators of G;
set f = the Function of Union [|the generators of G, the Sorts of T|],
ElementaryInstructions W;
reconsider f as Function of Union [|the generators of G, the Sorts of T|],
the carrier of W by FUNCT_2:7;
set A = ProgramAlgStr(#the carrier of W, the charact of W, f#);
set X = the generators of G;
set J = S;
A is partial quasi_total non-empty non empty; then
reconsider A as partial quasi_total non-empty non empty strict
ProgramAlgStr over J,T,X;
A is with_empty-instruction with_catenation with_if-instruction
with_while-instruction
by AOFA_000:def 10,AOFA_000:def 11,AOFA_000:def 12,AOFA_000:def 13; then
reconsider A as with_empty-instruction with_catenation with_if-instruction
with_while-instruction
partial quasi_total non-empty non empty strict ProgramAlgStr over J,T,X;
A is non degenerated well_founded ECIW-strict infinite
proof
hereby let I1,I2 be Element of A;
reconsider J1 = I1, J2 = I2 as Element of W;
EmptyIns A = EmptyIns W & I1\;I2 = J1\;J2;
hence (I1 <> EmptyIns A implies I1\;I2 <> I2) &
(I2 <> EmptyIns A implies I1\;I2 <> I1) &
(I1 <> EmptyIns A or I2 <> EmptyIns A implies I1\;I2 <> EmptyIns A)
by AOFA_000:def 24;
end;
hereby let C,I1,I2 be Element of A;
reconsider C1 = C, J1 = I1, J2 = I2 as Element of W;
if-then-else(C,I1,I2) = if-then-else(C1,J1,J2) &
EmptyIns W = EmptyIns A;
hence if-then-else(C,I1,I2) <> EmptyIns A by AOFA_000:def 24;
end;
hereby let C,I be Element of A;
reconsider C1 = C, J = I as Element of W;
EmptyIns A = EmptyIns W & while(C,I) = while(C1,J);
hence while(C,I) <> EmptyIns A by AOFA_000:def 24;
end;
hereby let I1,I2,C,J1,J2 be Element of A;
reconsider C1 = C, K1 = I1, K2 = I2, L1 = J1, L2 = J2 as Element of W;
if-then-else(C,J1,J2) = if-then-else(C1,L1,L2) & I1\;I2 = K1\;K2 &
EmptyIns W = EmptyIns A;
hence I1 = EmptyIns A or I2 = EmptyIns A or
I1\;I2 <> if-then-else(C,J1,J2) by AOFA_000:def 24;
end;
hereby let I1,I2,C,J be Element of A;
reconsider C1 = C, K1 = I1, K2 = I2, L = J as Element of W;
EmptyIns W = EmptyIns A & I1\;I2 = K1\;K2 & while(C,J) = while(C1,L);
hence I1 <> EmptyIns A & I2 <> EmptyIns A implies I1\;I2 <> while(C,J)
by AOFA_000:def 24;
end;
hereby let C1,I1,I2,C2,J be Element of A;
reconsider C3 = C1, K1 = I1, K2 = I2, C4 = C2, L = J as Element of W;
while(C2,J) = while(C4,L) &
if-then-else(C1,I1,I2) = if-then-else(C3,K1,K2);
hence if-then-else(C1,I1,I2) <> while(C2,J) by AOFA_000:def 24;
end;
thus A is well_founded
proof
A1: the UAStr of W = the UAStr of A; then
ElementaryInstructions W = ElementaryInstructions A &
ElementaryInstructions W is GeneratorSet of W
by AOFA_A00:43,AOFA_000:def 25;
hence ElementaryInstructions A is GeneratorSet of A
by A1,AOFA_A00:46;
end;
the UAStr of A = the UAStr of W; then
signature A = signature W by AOFA_A00:47;
hence signature A = ECIW-signature by AOFA_000:def 27;
the UAStr of A = the UAStr of W;
hence ElementaryInstructions A is infinite by AOFA_A00:43;
end;
then reconsider A as infinite strict IfWhileAlgebra of X;
take A;
the UAStr of A = the UAStr of W; then
ElementaryInstructions A = ElementaryInstructions W by AOFA_A00:43;
hence rng the assignments of A c= ElementaryInstructions A
by RELAT_1:def 19;
end;
end;
registration
let S,X,T,G;
let A be elementary IfWhileAlgebra of the generators of G;
let a be SortSymbol of S;
let x be Element of (the generators of G).a;
let t be Element of T,a;
cluster x:=(t,A) -> absolutely-terminating;
coherence
proof
x:=(t,A) in ElementaryInstructions A by Th68;
hence thesis by AOFA_000:95;
end;
end;
theorem
for A being elementary IfWhileAlgebra of the generators of G
for f being ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G,b) holds
G is C-supported & f in C-Execution(A,b,\falseC) &
(ex d being Function st d.x = 1 & d.y = 2 & d.m = 3) implies
y:=(\1(T,I),A)\;
while(b gt(@m, \0(T,I), A),
if-then(@m is_odd(b,A), y:=(@y*@x,A))\;
m:=(@m div \2(T,I),A)\; x:=(@x*@x,A))
is_terminating_wrt f, {s: s.I.m >= 0}
proof
let A be elementary IfWhileAlgebra of the generators of G;
let f be ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G,b);
assume
A1: G is C-supported & f in C-Execution(A,b,\falseC);
set ST = C-States(the generators of G);
set TV = (\falseC)-States(the generators of G,b);
set P = {s: s.I.m >= 0};
given d being Function such that
A2: d.x = 1 and
A3: d.y = 2 and
A4: d.m = 3;
A5: I <> the bool-sort of S by AOFA_A00:53;
set W = b gt(@m, \0(T,I),A);
A6: y:=(\1(T,I),A) is_terminating_wrt f, P by AOFA_000:107;
deffunc F(Element of ST) = In($1.I.m, NAT);
defpred R[Element of ST] means $1.I.m > 0;
set K = if-then(@m is_odd(b,A), y:=(@y*@x,A));
set J = K\; m:=(@m div \2(T,I),A)\; x:=(@x*@x,A);
A7: f complies_with_if_wrt TV by AOFA_000:def 32;
A8: P is_invariant_wrt W,f
proof
let s be Element of ST;
assume s in P;
then
A9: ex s1 being Element of ST st s = s1 & s1.I.m >= 0;
f.(s, W).I.m = s.I.m by A1,Th66;
hence f.(s, W) in P by A9;
end;
A10: for s being Element of ST st s in P & f.(f.(s,J),W) in TV
holds f.(s,J) in P
proof
let s be Element of ST such that
s in P;
A11: \0(T,I) value_at(C,f.(s,J)) = 0 by Th36;
assume f.(f.(s,J),W) in TV;
then @m value_at(C,f.(s,J)) > \0(T,I) value_at(C,f.(s,J)) by A1,Th66;
then f.(s,J).I.m > 0 by A11,Th61;
hence thesis;
end;
A12: m <> y by A4,A3;
A13: P is_invariant_wrt y:=(\1(T,I),A), f
proof
let s;
assume s in P;
then
A14: ex s9 being Element of ST st s = s9 & s9.I.m >= 0;
f.(s, y:=(\1(T,I),A)).I.m = s.I.m by A12,A1,Th65;
hence f.(s, y:=(\1(T,I),A)) in P by A14;
end;
A15: m <> x by A4,A2;
A16: for s st f.(s,W) in P holds f iteration_terminates_for J\;W, f.(s,W)
proof
A17: for s being Element of ST st R[s] holds (R[f.(s,J\;W)] iff f.(s,J\;W)
in TV) & F(f.(s,J\;W)) < F(s)
proof
let s be Element of ST such that
A18: s.I.m > 0;
A19: F(s) = s.I.m by A18,SUBSET_1:def 8,INT_1:3;
set q1 = f.(s,K);
set q0 = f.(s, @m is_odd(b,A));
set sJ = f.(s,J);
set sC = f.(sJ,W);
A20: f.(s,J\;W) = sC by AOFA_000:def 29;
A21: sC.I.m = sJ.I.m & \0(T,I) value_at(C,sC) = 0 by Th36,A5,A1,Th65;
A22: @m value_at(C,sJ) = sJ.I.m & \0(T,I) value_at(C,sJ) = 0 by Th36,Th61;
thus R[f.(s,J\;W)] iff f.(s,J\;W) in TV by A21,A20,A22,A1,Th66;
set q2 = f.(q1,m:=(@m div \2(T,I),A));
set q3 = f.(q2,x:=(@x*@x,A));
A23: q1 = f.(q0, y:=(@y*@x,A)) or q1 = f.(q0, EmptyIns A)
by A7;
A24: @m value_at(C,q1) = q1.I.m by Th61;
A25: \2(T,I) value_at(C,q1) = 2 by Th40;
q2 = f.(s,K\;m:=(@m div \2(T,I),A)) by AOFA_000:def 29;
then q3 = f.(s,J) by AOFA_000:def 29;
then
A26: sJ.I.m = q2.I.m by A15,A1,Th65
.= (@m div \2(T,I)) value_at(C,q1) by A1,Th65
.= (@m value_at(C,q1)) div (\2(T,I) value_at(C,q1)) by Th43
.= (q1.I.m) div 2 by A24,A25,AOFA_A00:55
.= (q0.I.m) div 2 by A12,A23,A1,Th65,AOFA_000:def 28
.= (s.I.m) div 2 by A5,Th65,A1;
A27: sC.I.m = sJ.I.m by A1,Th66;
then sC.I.m in NAT by A18,A26,INT_1:3,61;
then F(sC) = sC.I.m by SUBSET_1:def 8;
hence thesis by A18,A27,A20,A26,A19,INT_1:56;
end;
let s0 be Element of ST such that
f.(s0,W) in P;
set s1 = f.(s0,W);
A28: s0.I.m = @m value_at(C,s0) & \0(T,I) value_at(C,s0) = 0 by Th36,Th61;
then
A29: s0.I.m <= 0 implies s1 nin TV by A1,Th66;
s0.I.m > 0 implies s1 in TV by A28,A1,Th66;
then
A30: f.(s0,W) in TV iff R[f.(s0,W)] by A29,A1,Th66;
thus f iteration_terminates_for J\;W, f.(s0,W) from AOFA_000:sch 3(A30,A17
);
end;
J is_terminating_wrt f,P by AOFA_000:107;
then while(b gt(@m, \0(T,I), A), if-then(@m is_odd(b,A), y:=(@y*@x,A))\;
m:=(@m div \2(T,I),A)\; x:=(@x*@x,A) )
is_terminating_wrt f, P by A8,A10,A16,AOFA_000:104,118;
hence thesis by A6,A13,AOFA_000:111;
end;
theorem
G is C-supported &
(ex d being Function st d.b = 0 & d.x = 1 & d.y = 2 & d.m = 3) implies
for s being Element of C-States(the generators of G)
for n being Nat st n = s.I.m holds
f in C-Execution(A,b,\falseC) implies
f.(s, y:=(\1(T,I),A)\;
while(b gt(@m, \0(T,I), A),
if-then(@m is_odd(b,A), y:=(@y*@x,A))\;
m:=(@m div \2(T,I),A)\; x:=(@x*@x,A)) ).I.y
= (s.I.x)|^n
proof
assume
A1: G is C-supported;
given d being Function such that
A2: d.b = 0 & d.x = 1 & d.y = 2 & d.m = 3;
let s;
let n be Nat;
assume that
A3: n = s.I.m and
A4: f in C-Execution(A,b,\falseC);
set Q = S;
set S = C-States(the generators of G);
set W = T; set g = f;
set T = (\falseC)-States(the generators of G, b);
set s0 = f.(s, y:=(\1(W,I),A));
A5: f complies_with_if_wrt T by AOFA_000:def 32;
defpred R[Element of S] means $1.I.m > 0;
set Z = C;
set C = b gt(@m, \0(W,I), A);
defpred P[Element of S] means
(s.I.x)|^n = ($1.I.y)*(($1.I.x)to_power($1.I.m)) &
$1.I.m >= 0;
deffunc F(Element of S) = In($1.I.m, NAT);
set Y = I;
set I = if-then(@m is_odd(b,A), y:=(@y*@x,A));
set J = I\; m:=(@m div \2(W,Y),A)\; x:=(@x*@x,A);
A6: m <> y by A2;
then
A7: s0.Y.m = s.Y.m by A1,A4,Th65;
A8: for s being Element of S st P[s] holds P[g.(s,C) qua Element of S] &
(g.(s,C) in T iff R[g.(s,C) qua Element of S])
proof
let s be Element of S such that
A9: P[s];
set s1 = g.(s, C);
A10: s1.Y.x = s.Y.x by A1,A4,Th66;
s1.Y.m = s.Y.m by A1,A4,Th66;
hence P[g.(s,C) qua Element of S] by A1,A4,A9,A10,Th66;
A11: \0(W,Y) value_at(Z,s) = 0 & @m value_at(Z,s) = s.Y.m by Th36,Th61;
then
A12: s.Y.m <= 0 implies s1 nin T by A1,A4,Th66;
s.Y.m > 0 implies s1 in T by A11,A1,A4,Th66;
hence thesis by A1,A4,A12,Th66;
end;
A13: s0.Y.y = \1(W,Y) value_at(Z,s) by A1,A4,Th65 .= 1 by Th37;
set fs = g.(s0, while(C,J));
set s1 = g.(s0,C);
A14: (fs.Y.x) to_power 0 = 1 by POWER:24;
A15: m <> x by A2;
A16: for s being Element of S st R[s] holds (R[g.(s,J\;C) qua Element of S] iff
g.(s,J\;C) in T) & F(g.(s,J\;C) qua Element of S) < F(s)
proof
let s be Element of S such that
A17: s.Y.m > 0;
A18: F(s) = s.Y.m by A17,SUBSET_1:def 8,INT_1:3;
set q1 = g.(s,I);
set q0 = g.(s, @m is_odd(b,A));
set sJ = g.(s,J);
set sC = g.(sJ,C);
A19: g.(s,J\;C) = sC by AOFA_000:def 29;
A20: \0(W,Y) value_at(Z,sJ) = 0 & @m value_at(Z,sJ) = sJ.Y.m by Th36,Th61;
then
A21: sJ.Y.m <= 0 implies sC nin T by A1,A4,Th66;
sJ.Y.m > 0 implies sC in T by A20,A1,A4,Th66;
hence R[g.(s,J\;C) qua Element of S] iff g.(s,J\;C) in T
by A21,A19,A1,A4,Th66;
set q2 = g.(q1,m:=(@m div \2(W,Y), A));
set q3 = g.(q2,x:=(@x*@x,A));
A22: q1 = g.(q0, y:=(@y*@x,A)) or q1 = g.(q0, EmptyIns A)
by A5;
A23: @m value_at(Z,q1) = q1.Y.m & \2(W,Y)value_at(Z,q1) = 2 by Th61,Th40;
q2 = g.(s,I\;m:=(@m div \2(W,Y), A)) by AOFA_000:def 29;
then q3 = g.(s,J) by AOFA_000:def 29;
then
A24: sJ.Y.m = q2.Y.m by A1,A4,A15,Th65
.= (@m div \2(W,Y))value_at(Z,q1) by A1,A4,Th65
.= (@m value_at(Z,q1)) div (\2(W,Y)value_at(Z,q1)) by Th43
.= (q1.Y.m) div 2 by A23,AOFA_A00:55
.= (q0.Y.m) div 2 by A1,A4,A6,A22,Th65,AOFA_000:def 28
.= (s.Y.m) div 2 by A1,A4,Th67;
A25: sC.Y.m = sJ.Y.m by A1,A4,Th66;
then sC.Y.m in NAT by A17,A24,INT_1:3,61;
then F(sC qua Element of S) = sC.Y.m by SUBSET_1:def 8;
hence thesis by A17,A25,A19,A24,A18,INT_1:56;
end;
set q = s;
A26: x <> y by A2;
A27: for s being Element of S st P[s] & s in T & R[s]
holds P[g.(s,J) qua Element of S]
proof
let s be Element of S such that
A28: P[s] and
s in T and
R[s];
reconsider sm = s.Y.m as Element of NAT by A28,INT_1:3;
s.Y.m = ((sm) div 2)*2+((sm) mod 2) by NEWTON:66;
then
A29: (q.Y.x)|^n = (s.Y.y)*(((s.Y.x)to_power((sm div 2)*2))*
((s.Y.x)to_power(sm mod 2))) by A28,FIB_NUM2:5
.= (s.Y.y)*((s.Y.x)to_power(sm mod 2))*((s.Y.x)to_power((sm div 2)*2))
.= (s.Y.y)*((s.Y.x)to_power(sm mod 2))*
(((s.Y.x)to_power 2) to_power (sm div 2)) by NEWTON:9
.= (s.Y.y)*((s.Y.x)to_power(sm mod 2))*
(((s.Y.x)*(s.Y.x)) to_power (sm div 2)) by NEWTON:81;
set q1 = g.(s,I);
set q0 = g.(s, @m is_odd(b,A));
set sJ = g.(s,J);
set q2 = g.(q1,m:=(@m div \2(W,Y), A));
set q3 = g.(q2,x:=(@x*@x,A));
A30: q1 = g.(q0, y:=(@y*@x,A)) or q1 = g.(q0, EmptyIns A)
by A5;
A31: q2.Y.x = q1.Y.x by A1,A4,A15,Th65
.= q0.Y.x by A1,A4,A26,A30,Th65,AOFA_000:def 28;
A32: q2.Y.y = q1.Y.y by A1,A4,A6,Th65;
A33: q0.Y.y = s.Y.y by A1,A4,Th67;
A34: q0.Y.x = s.Y.x by A1,A4,Th67;
q2 = g.(s,I\;m:=(@m div \2(W,Y), A)) by AOFA_000:def 29;
then
A35: q3 = g.(s,J) by AOFA_000:def 29;
then
A36: sJ.Y.y = q2.Y.y by A1,A4,A26,Th65;
A37: sm div 2 = s.Y.m div 2;
A38: now
A39: @m value_at(Z,s) = s.Y.m by Th61;
then
A40: q0.(the bool-sort of Q).b = (s.Y.m) mod 2 by A1,A4,Th67;
per cases by A37,A40,NAT_D:12;
suppose
A41: q0.(the bool-sort of Q).b = FALSE;
q0 is ManySortedFunction of the generators of G, the Sorts of Z &
\falseZ = FALSE by Th10,AOFA_A00:48;
then q0 nin T by A41,AOFA_A00:def 20;
then q1 = g.(q0, EmptyIns A) by A5;
then
A42: q1.Y.y = q0.Y.y by AOFA_000:def 28;
A43: (s.Y.y)*1 = s.Y.y;
(s.Y.x)to_power 0 = 1 by POWER:24;
hence (s.Y.y)*((s.Y.x)to_power(sm mod 2)) = sJ.Y.y
by A1,A4,A39,A36,A32,A33,A41,A42,A43,Th67;
end;
suppose
A44: q0.(the bool-sort of Q).b = TRUE;
A45: @y value_at(Z,q0) = q0.Y.y & @x value_at(Z,q0) = q0.Y.x by Th61;
q0 is ManySortedFunction of the generators of G, the Sorts of Z &
\falseZ = FALSE by Th10,AOFA_A00:48;
then q0 in T by A44,AOFA_A00:def 20;
then q1 = g.(q0, y:=(@y*@x,A)) by A5;
then
A46: q1.Y.y = (@y*@x) value_at(Z,q0) by A1,A4,Th65
.= (@y value_at(Z,q0))*(@x value_at(Z,q0)) by Th42
.= (q0.Y.y)*(q0.Y.x) by A45,AOFA_A00:55;
(s.Y.x)to_power 1 = s.Y.x & @m value_at(Z,s) = s.Y.m
by Th61,POWER:25;
hence (s.Y.y)*((s.Y.x)to_power(sm mod 2)) = sJ.Y.y
by A1,A4,A34,A36,A32,A33,A44,A46,Th67;
end;
end;
A47: @m value_at(Z,q1) = q1.Y.m & \2(W,Y) value_at(Z,q1) = 2 by Th61,Th40;
A48: sJ.Y.m = q2.Y.m by A1,A4,A15,A35,Th65
.= (@m div \2(W,Y)) value_at(Z,q1) by A1,A4,Th65
.= (@m value_at(Z,q1))div(\2(W,Y) value_at(Z,q1)) by Th43
.= (q1.Y.m) div 2 by A47,AOFA_A00:55
.= (q0.Y.m) div 2 by A1,A4,A6,A30,Th65,AOFA_000:def 28
.= (s.Y.m) div 2 by A1,A4,Th67;
A49: @x value_at(Z,q2) = q2.Y.x by Th61;
sJ.Y.x = (@x*@x) value_at(Z,q2) by A1,A4,A35,Th65
.= (@x value_at(Z,q2))*(@x value_at(Z,q2)) by Th42
.= (q2.Y.x)*(q2.Y.x) by A49,AOFA_A00:55;
hence thesis by A48,A29,A31,A34,A38;
end;
@m value_at(Z,s0) = s0.Y.m & \0(W,Y) value_at(Z,s0) = 0 &
s1.Y.m = s0.Y.m by A1,A4,Th36,Th61,Th66;
then
A50: g.(s0,C) in T iff R[g.(s0,C) qua Element of S] by A1,A4,Th66;
A51: g iteration_terminates_for J\;C, g.(s0,C) from AOFA_000:sch 3(A50, A16);
s0.Y.x = s.Y.x by A1,A4,A26,Th65;
then
A52: P[s0 qua Element of S] by A3,A7,A13,POWER:41;
A53: P[g.(s0,while(C,J)) qua Element of S] &
not R[g.(s0,while(C,J)) qua Element of S]
from AOFA_000:sch 5(A52,A51,A27,A8);
then fs.Y.m = 0;
hence thesis by A53,A14,AOFA_000:def 29;
end;
begin :: Calculation of maximum
registration
let X be non empty set;
let f be FinSequence of X^omega;
let x be Nat;
cluster f.x -> Sequence-like finite Function-like Relation-like;
coherence
proof
x in dom f or x nin dom f;
then f.x in X^omega or f.x = <%>X by FUNCT_1:def 2,102;
hence thesis;
end;
end;
registration
let X be non empty set;
cluster -> Function-yielding for FinSequence of X^omega;
coherence;
end;
registration
let i be Nat;
let f be i-based finite array;
let a,x be set;
cluster f+*(a,x) -> i-based finite segmental;
coherence
proof
A1: dom(f+*(a,x)) = dom f by FUNCT_7:30;
then
(for b being Ordinal st b in dom (f+*(a,x)) holds i in dom f & i c= b) &
ex c,b being Ordinal st dom (f+*(a,x)) = c\b by EXCHSORT:def 1,def 2;
hence thesis by A1,FINSET_1:10,EXCHSORT:def 1,def 2;
end;
end;
registration
let X be non empty set;
let f be X-valued Function;
let a be set;
let x be Element of X;
cluster f+*(a,x) -> X-valued;
coherence
proof
let y be object; assume y in rng (f+*(a,x));
then consider z being object such that
A1: z in dom (f+*(a,x)) & y = (f+*(a,x)).z by FUNCT_1:def 3;
A2: dom (f+*(a,x)) = dom f & (z = a or z <> a) by FUNCT_7:30;
then y = x or y = f.z by A1,FUNCT_7:31,32;
hence thesis by A1,A2,FUNCT_1:102;
end;
end;
scheme Sch1{X()->non empty set,
j()->Nat,
B()->set,
F(set,set,set)->set,
A(set)->set}:
ex f being FinSequence of X()^omega st
len f = j() & (f.1 = B() or j() = 0) &
for i being Nat st 1 <= i & i < j() holds f.(i+1) = F(f.i,i,A(i))
provided
A1: for a being 0-based finite array of X()
for i being Nat st 1 <= i & i < j()
for x being Element of X() holds
F(a,i,x) is 0-based finite array of X()
and
A2: B() is 0-based finite array of X()
and
A3: for i being Nat st i < j() holds A(i) in X()
proof
defpred P[set,set,set] means $3 = F($2,$1,A($1));
A4: for n being Nat st 1 <= n & n < j()
for x being set ex y being set st P[n,x,y];
consider f being FinSequence such that
A5: len f = j() & (f.1 = B() or j() = 0) &
for i being Nat st 1 <= i & i < j() holds P[i,f.i,f.(i+1)]
from RECDEF_1:sch 3(A4);
defpred Q[Nat] means 1 <= $1 & $1 <= j() implies f.$1 in X()^omega;
A6: Q[0];
A7: now let i be Nat;
assume
A8: Q[i];
thus Q[i+1]
proof
assume
A9: 1 <= i+1 & i+1 <= j();
reconsider x = A(i) as Element of X() by A3,A9,NAT_1:13;
per cases;
suppose i = 0;
hence f.(i+1) in X()^omega by A2,A5,A9,AFINSQ_1:def 7;
end;
suppose i > 0;
then
A10: i >= 1+0 & i in NAT by NAT_1:13,ORDINAL1:def 12;
f.(i+1) = F(f.i,i,x) & f.i is 0-based finite array of X()
by A8,A10,A5,A9,NAT_1:13;
then f.(i+1) is 0-based finite array of X() by A1,A9,A10,NAT_1:13;
hence f.(i+1) in X()^omega by AFINSQ_1:def 7;
end;
end;
end;
A11: for i being Nat holds Q[i] from NAT_1:sch 2(A6,A7);
rng f c= X()^omega
proof
let x be object;
assume x in rng f;
then consider y being object such that
A12: y in dom f & x = f.y by FUNCT_1:def 3;
reconsider y as Nat by A12;
1 <= y & y <= j() by A12,A5,FINSEQ_3:25;
hence x in X()^omega by A11,A12;
end;
then reconsider f as FinSequence of X()^omega by FINSEQ_1:def 4;
take f;
thus len f = j() & (f.1 = B() or j() = 0) by A5;
let i be Nat;
assume 1 <= i & i < j();
hence thesis by A5;
end;
theorem Th71:
for S being (11,1,1)-array non empty non void BoolSignature
for J,L being set, K being SortSymbol of S
st (the connectives of S).11 is_of_type <*J,L*>, K
holds J = the_array_sort_of S &
for I being integer SortSymbol of S holds the_array_sort_of S <> I
proof
let S be (11,1,1)-array non empty non void BoolSignature;
let J0,L0 be set, K0 be SortSymbol of S;
assume A1: (the connectives of S).11 is_of_type <*J0,L0*>, K0;
consider J,K,L being Element of S such that
A2: L = 1 & K = 1 & J <> L & J <> K &
(the connectives of S).11 is_of_type <*J,K*>, L &
(the connectives of S).(11+1) is_of_type <*J,K,L*>, J &
(the connectives of S).(11+2) is_of_type <*J*>, K &
(the connectives of S).(11+3) is_of_type <*K,L*>, J by AOFA_A00:def 51;
A3: the_array_sort_of S = J by A2;
thus J0 = <*J0,L0*>.1 by FINSEQ_1:44
.= (the Arity of S).((the connectives of S).11).1 by A1
.= <*J,K*>.1 by A2
.= the_array_sort_of S by A3,FINSEQ_1:44;
thus thesis by A2,AOFA_A00:def 40;
end;
theorem Th72:
for S being 1-1-connectives (4,1) integer (11,1,1)-array 11 array-correct
bool-correct non empty non void BoolSignature
for I being integer SortSymbol of S
for A being (4,1) integer (11,1,1)-array bool-correct non-empty
MSAlgebra over S
for a,b being Element of A,I st a = 0
holds init.array(a,b) = {}
proof
let S be 1-1-connectives (4,1) integer (11,1,1)-array 11 array-correct
bool-correct non empty non void BoolSignature;
let I be integer SortSymbol of S;
let A be (4,1) integer (11,1,1)-array bool-correct non-empty
MSAlgebra over S;
let a,b be Element of A,I;
assume A1: a = 0;
set o = (the connectives of S).14;
consider J,K being Element of S such that
A2: K = 1 & (the connectives of S).11 is_of_type <*J,1*>, K &
(the Sorts of A).J = ((the Sorts of A).K)^omega &
(the Sorts of A).1 = INT &
(for a being 0-based finite array of (the Sorts of A).K holds
(for i being Integer st i in dom a holds
Den((the connectives of S)/.11,A).<*a,i*> = a.i &
for x being Element of A,K holds
Den((the connectives of S)/.(11+1),A).<*a,i,x*> = a+*(i,x)) &
Den((the connectives of S)/.(11+2),A).<*a*> = card a) &
for i being Integer, x being Element of A,K st i >= 0 holds
Den((the connectives of S)/.(11+3),A).<*i,x*> = Segm(i)-->x
by AOFA_A00:def 52;
A3: I = 1 by AOFA_A00:def 40;
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 14 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S)/.14 & o in the carrier' of S
by FUNCT_1:102,PARTFUN1:def 6;
hence init.array(a,b) = Den((the connectives of S)/.14, A).<*a,b*>
by SUBSET_1:def 8
.= Segm(0 qua set)-->b by A1,A2,A3
.= {};
end;
theorem Th73:
for S being (11,1,1)-array 11 array-correct bool-correct non empty non void
BoolSignature
for I being integer SortSymbol of S holds
the_array_sort_of S <> I &
(the connectives of S).11 is_of_type <*the_array_sort_of S,I*>, I &
(the connectives of S).(11+1) is_of_type <*the_array_sort_of S,I,I*>,
the_array_sort_of S &
(the connectives of S).(11+2) is_of_type <*the_array_sort_of S*>, I &
(the connectives of S).(11+3) is_of_type <*I,I*>, the_array_sort_of S
proof
let S be (11,1,1)-array 11 array-correct bool-correct non empty
non void BoolSignature;
let I be integer SortSymbol of S;
consider J,K,L being Element of S such that
A1: L = 1 & K = 1 & J <> L & J <> K &
(the connectives of S).11 is_of_type <*J,K*>, L &
(the connectives of S).(11+1) is_of_type <*J,K,L*>, J &
(the connectives of S).(11+2) is_of_type <*J*>, K &
(the connectives of S).(11+3) is_of_type <*K,L*>, J by AOFA_A00:def 51;
the_array_sort_of S = J & I = 1 by A1,AOFA_A00:def 40;
hence thesis by A1;
end;
theorem Th74:
for S being 1-1-connectives (11,1,1)-array 11 array-correct (4,1) integer
bool-correct non empty non void BoolSignature
for I being integer SortSymbol of S
for A being (11,1,1)-array (4,1) integer bool-correct non-empty
MSAlgebra over S
holds (the Sorts of A).the_array_sort_of S = INT^omega &
(for i,j being Element of A,I st i is non negative Integer
holds init.array(i,j) = Segm(i)-->j) &
for a being Element of (the Sorts of A).the_array_sort_of S
holds length(a,I) = card a &
for i being Element of A,I
for f being Function st f = a & i in dom f
holds a.(i) = f.i &
for x being Element of A,I holds (a,i)<-x = f+*(i,x)
proof
let S be 1-1-connectives (11,1,1)-array 11 array-correct (4,1) integer
bool-correct non empty non void BoolSignature;
let I be integer SortSymbol of S;
let A be (11,1,1)-array (4,1) integer bool-correct non-empty
MSAlgebra over S;
consider J,K being Element of S such that
A1: K = 1 & (the connectives of S).11 is_of_type <*J,1*>, K &
(the Sorts of A).J = ((the Sorts of A).K)^omega &
(the Sorts of A).1 = INT &
(for a being 0-based finite array of (the Sorts of A).K holds
(for i being Integer st i in dom a holds
Den((the connectives of S)/.11,A).<*a,i*> = a.i &
for x being Element of A,K holds
Den((the connectives of S)/.(11+1),A).<*a,i,x*> = a+*(i,x)) &
Den((the connectives of S)/.(11+2),A).<*a*> = card a) &
for i being Integer, x being Element of A,K st i >= 0 holds
Den((the connectives of S)/.(11+3),A).<*i,x*> = Segm(i)-->x
by AOFA_A00:def 52;
thus (the Sorts of A).the_array_sort_of S = INT^omega by A1,Th71;
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 11 <= len the connectives of S &
12 <= len the connectives of S &
13 <= len the connectives of S &
14 <= len the connectives of S by XXREAL_0:2;
then 11 in dom the connectives of S &
12 in dom the connectives of S &
13 in dom the connectives of S &
14 in dom the connectives of S by FINSEQ_3:25;
then
A2: (the connectives of S).11 in the carrier' of S &
(the connectives of S).12 in the carrier' of S &
(the connectives of S).13 in the carrier' of S &
(the connectives of S).14 in the carrier' of S &
(the connectives of S).11 = (the connectives of S)/.11 &
(the connectives of S).12 = (the connectives of S)/.12 &
(the connectives of S).13 = (the connectives of S)/.13 &
(the connectives of S).14 = (the connectives of S)/.14
by FUNCT_1:102,PARTFUN1:def 6;
A3: I = 1 by AOFA_A00:def 40;
hereby
let i,j be Element of A,I;
reconsider ii = i as Integer by A3;
assume
i is non negative Integer;
then
A4: ii >= 0 & I = K by A1,AOFA_A00:def 40;
thus init.array(i,j) = Den((the connectives of S)/.14,A).<*ii,j*>
by A2,SUBSET_1:def 8
.= Segm(i)-->j by A1,A4;
end;
let a be Element of (the Sorts of A).the_array_sort_of S;
J = the_array_sort_of S by A1,Th71;
then reconsider b = a as XFinSequence of INT by A1;
thus length(a,I)
= Den((the connectives of S)/.13,A).<*b*> by A2,SUBSET_1:def 8
.= card a by A1;
let i be Element of A,I;
let f be Function;
assume A5: f = a & i in dom f;
thus a.(i) = Den((the connectives of S)/.11,A).<*a,i*> by A2,SUBSET_1:def 8
.= b.i by A1,A5,A3
.= f.i by A5;
let x be Element of A,I;
thus (a,i)<-x = Den((the connectives of S)/.12,A).<*b,i,x*>
by A2,SUBSET_1:def 8
.= f+*(i,x) by A1,A3,A5;
end;
registration
let a be 0-based finite array;
cluster len-a -> finite;
coherence
proof
len a = len-a by EXCHSORT:24;
hence thesis;
end;
end;
registration
let S be 1-1-connectives (4,1) integer (11,1,1)-array 11 array-correct
bool-correct non empty non void BoolSignature;
let A be (11,1,1)-array (4,1) integer bool-correct non-empty
MSAlgebra over S;
cluster -> (11,1,1)-array for non-empty MSSubAlgebra of A;
coherence
proof
let Q be non-empty MSSubAlgebra of A;
set I = the integer SortSymbol of S;
A1: (the Sorts of Q).I = INT by AOFA_A00:55;
then reconsider 00 = 0 as Element of Q,I by INT_1:def 2;
reconsider 0a = 00 as Element of A,I by Th2;
consider J,K being Element of S such that
A2: K = 1 & (the connectives of S).11 is_of_type <*J,1*>, K &
(the Sorts of A).J = ((the Sorts of A).K)^omega &
(the Sorts of A).1 = INT &
(for a being 0-based finite array of (the Sorts of A).K holds
(for i being Integer st i in dom a holds
Den((the connectives of S)/.11,A).<*a,i*> = a.i &
for x being Element of A,K holds
Den((the connectives of S)/.(11+1),A).<*a,i,x*> = a+*(i,x)) &
Den((the connectives of S)/.(11+2),A).<*a*> = card a) &
for i being Integer, x being Element of A,K st i >= 0 holds
Den((the connectives of S)/.(11+3),A).<*i,x*> = Segm(i)-->x
by AOFA_A00:def 52;
consider J1,K1,L1 being Element of S such that
A3: L1 = 1 & K1 = 1 & J1 <> L1 & J1 <> K1 &
(the connectives of S).11 is_of_type <*J1,K1*>, L1 &
(the connectives of S).(11+1) is_of_type <*J1,K1,L1*>, J1 &
(the connectives of S).(11+2) is_of_type <*J1*>, K1 &
(the connectives of S).(11+3) is_of_type <*K1,L1*>, J1 by AOFA_A00:def 51;
A4: I = 1 by AOFA_A00:def 40;
A5: the_array_sort_of S = J & the_array_sort_of S = J1 by A3,A2,Th71;
A6: for a being 0-based finite array of INT holds
a in (the Sorts of Q).the_array_sort_of S
proof
let a be 0-based finite array of INT;
set o = In((the connectives of S).14, the carrier' of S);
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 14 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S).14 by FUNCT_1:102,SUBSET_1:def 8;
then the_arity_of o = <*K,K*> by A3,A2;
then
A7: Args(o,Q) = product <*INT,INT*> & 0 in INT
by A2,A4,A1,Th23,INT_1:def 2;
per cases;
suppose len a = 0;
then a = <%>INT;
then a = init.array(0a,0a) by Th72
.= init.array(00,00) by A7,FINSEQ_3:124,EQUATION:19;
hence a in (the Sorts of Q).the_array_sort_of S;
end;
suppose
A8: len a <> 0;
deffunc F(array, Integer, set) = $1+*($2,$3);
deffunc G(Integer) = a.($1);
set j = len a;
A9: for a being 0-based finite array of INT
for i being Nat st 1 <= i & i < j
for x being Element of INT holds
F(a,i,x) is 0-based finite array of INT;
set B = Segm((len a qua set))-->a.0;
A10: B is 0-based finite array of INT;
A11: for i being Nat st i 0 implies len (f.$1) = j) &
for i being Nat st i < $1 holds a.i = f.$1.i;
A13: Q[0];
A14: now let i be Nat; assume
A15: Q[i];
thus Q[i+1]
proof assume
A16: i+1 <= len a;
per cases;
suppose
A17: i = 0;
thus i+1 <> 0 implies len(f.(i+1)) = j
by A12,A16,A17,FUNCOP_1:13;
let k be Nat;
assume k < i+1;
then k <= 0 & k >= 0 by A17,NAT_1:13;
then
A18: k = 0;
then k < len a by A16;
hence a.k = f.(i+1).k
by A17,A12,A18,FUNCOP_1:7,AFINSQ_1:86;
end;
suppose
A19: i > 0;
then
A20: i >= 0+1 & i < j by A16,NAT_1:13;
then
A21: f.(i+1) = F(f.i,i,G(i)) by A12;
thus i+1 <> 0 implies len(f.(i+1)) = j
by A15,A19,A21,A16,NAT_1:13,FUNCT_7:30;
let k be Nat;
assume k < i+1;
then
A22: k <= i & k < j by A16,XXREAL_0:2,NAT_1:13;
per cases by A22,XXREAL_0:1;
suppose k = i;
then f.(i+1) = f.i+*(k,a.k) & k in dom (f.i)
by A15,A12,A20,AFINSQ_1:86;
hence a.k = f.(i+1).k by FUNCT_7:31;
end;
suppose
A23: k < i;
A24: f.(i+1) = f.i+*(i,a.i) & k in dom (f.i)
by A20,A12,A15,A22,AFINSQ_1:86;
thus a.k = (f.i).k by A15,A23,A16,NAT_1:13 .= f.(i+1).k
by A23,A24,FUNCT_7:32;
end;
end;
end;
end;
A25: for i being Nat holds Q[i] from NAT_1:sch 2(A13,A14);
then
(len a <> 0 implies len (f.len a) = j)
& for i being Nat st i < len a holds a.i = (f.len a).i;
then
A26: f.len a = a by A8,AFINSQ_1:9;
defpred R[Nat] means 1 <= $1 & $1 <= len a implies
f.$1 in (the Sorts of Q).the_array_sort_of S;
A27: R[0];
A28: now let i be Nat;
assume
A29: R[i];
thus R[i+1]
proof
assume
A30: 1 <= i+1 & i+1 <= len a;
then
A31: i < len a by NAT_1:13;
per cases by NAT_1:14;
suppose
A32: i >= 1;
then reconsider fi = f.i as Element of Q, the_array_sort_of S
by A29,A30,NAT_1:13;
fi in (the Sorts of A).J by A5,Th2;
then reconsider ff = fi as finite Sequence of INT by A2;
reconsider ii = i as Element of (the Sorts of Q).I
by A1,INT_1:def 2;
reconsider a = G(i) as Element of (the Sorts of Q).I
by A1,A11,A30,NAT_1:13;
reconsider b = a, ia = ii as Element of (the Sorts of A).I
by Th2;
reconsider fj = fi as Element of A, the_array_sort_of S by Th2;
f.(i+1) = F(f.i,i,G(i)) & (1=0+1 implies i > 0) &
f.i in (the Sorts of Q).the_array_sort_of S
by A29,A32,A12,A30,NAT_1:13;
then
len (f.i) = j by A31,A25;
then
A33: i in dom ff by A31,AFINSQ_1:86;
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 12 <= len the connectives of S by XXREAL_0:2;
then 12 in dom the connectives of S by FINSEQ_3:25;
then
A34: (the connectives of S).12 = (the connectives of S)/.12 &
(the connectives of S).12 in the carrier' of S
by PARTFUN1:def 6,FUNCT_1:102;
then the_arity_of ((the connectives of S)/.12)
= <*the_array_sort_of S, I, I*> by A3,A4;
then
A35: Args((the connectives of S)/.12, Q) = product <*(the Sorts of Q)
.the_array_sort_of S, (the Sorts of Q).I, (the Sorts of Q).I*>
by Th24;
f.(i+1) = ff+*(i,a) by A32,A12,A30,NAT_1:13
.= Den((the connectives of S)/.12, A).<*fj,ia,b*>
by A4,A2,A33
.= Den((the connectives of S)/.12, Q).<*fi,ii,a*>
by A35,FINSEQ_3:125,EQUATION:19
.= (fi,ii)<-a by A34,SUBSET_1:def 8;
hence f.(i+1) in (the Sorts of Q).the_array_sort_of S;
end;
suppose
A36: i = 0;
reconsider a0 = a.0, ii = j as Element of Q, I
by A1,INT_1:def 2;
reconsider b0 = a0, ia = ii as Element of A, I by Th2;
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 14 in dom the connectives of S by FINSEQ_3:25;
then
A37: (the connectives of S).14 = (the connectives of S)/.14 &
(the connectives of S).14 in the carrier' of S
by PARTFUN1:def 6,FUNCT_1:102;
then the_arity_of ((the connectives of S)/.14) = <*I, I*>
by A3,A4;
then
A38: Args((the connectives of S)/.14, Q)
= product <*(the Sorts of Q).I, (the Sorts of Q).I*>
by Th23;
f.(i+1) = Den((the connectives of S)/.14, A).<*ia,b0*>
by A36,A12,A8,A4,A2
.= Den((the connectives of S)/.14, Q).<*ii,a0*>
by A38,FINSEQ_3:124,EQUATION:19
.= init.array(ii,a0) by A37,SUBSET_1:def 8;
hence thesis;
end;
end;
end;
A39: for i being Nat holds R[i] from NAT_1:sch 2(A27,A28);
0 < len a by A8;
then 0+1 <= len a by NAT_1:13;
hence a in (the Sorts of Q).the_array_sort_of S by A26,A39;
end;
end;
take J,K;
thus K = 1 by A2;
thus (the connectives of S).11 is_of_type <*J,1*>, K by A2;
the Sorts of Q is MSSubset of A by MSUALG_2:def 9;
hence (the Sorts of Q).J c= ((the Sorts of Q).K)^omega
by A1,A4,A2,PBOOLE:def 2,def 18;
thus
A40: ((the Sorts of Q).K)^omega c= (the Sorts of Q).J
by A5,A6,A1,A4,A2;
thus (the Sorts of Q).1 = INT by AOFA_A00:55,A4;
hereby
let a be 0-based finite array of (the Sorts of Q).K;
A41: a in ((the Sorts of Q).K)^omega by AFINSQ_1:def 7;
hereby let i be Integer; assume
A42: i in dom a;
A43: i in INT by INT_1:def 2;
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 11 <= len the connectives of S by XXREAL_0:2;
then 11 in dom the connectives of S by FINSEQ_3:25;
then
(the connectives of S).11 = (the connectives of S)/.11 &
(the connectives of S).11 in the carrier' of S
by PARTFUN1:def 6,FUNCT_1:102;
then the_arity_of ((the connectives of S)/.11)
= <*the_array_sort_of S, I*> by A3,A4;
then
A44: Args((the connectives of S)/.11, Q) = product <*(the Sorts of Q)
.the_array_sort_of S, (the Sorts of Q).I*> by Th23;
thus Den((the connectives of S)/.11,Q).<*a,i*>
= Den((the connectives of S)/.11,A).<*a,i*>
by A44,A41,A40,A43,A5,A1,FINSEQ_3:124,EQUATION:19
.= a.i by A42,A1,A4,A2;
let x be Element of Q,K;
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 12 <= len the connectives of S by XXREAL_0:2;
then 12 in dom the connectives of S by FINSEQ_3:25;
then
(the connectives of S).12 = (the connectives of S)/.12 &
(the connectives of S).12 in the carrier' of S
by PARTFUN1:def 6,FUNCT_1:102;
then the_arity_of ((the connectives of S)/.12)
= <*the_array_sort_of S, I, I*> by A3,A4;
then
A45: Args((the connectives of S)/.12, Q) = product <*(the Sorts of Q)
.the_array_sort_of S, (the Sorts of Q).I, (the Sorts of Q).I*>
by Th24;
thus Den((the connectives of S)/.(11+1),Q).<*a,i,x*>
= Den((the connectives of S)/.(11+1),A).<*a,i,x*>
by A45,A1,A41,A40,A2,A5,A4,A43,FINSEQ_3:125,EQUATION:19
.= a+*(i,x) by A1,A4,A2,A42;
end;
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 13 <= len the connectives of S by XXREAL_0:2;
then 13 in dom the connectives of S by FINSEQ_3:25;
then
(the connectives of S).13 = (the connectives of S)/.13 &
(the connectives of S).13 in the carrier' of S
by PARTFUN1:def 6,FUNCT_1:102;
then the_arity_of ((the connectives of S)/.13)
= <*the_array_sort_of S*> by A3;
then
A46: Args((the connectives of S)/.13, Q) = product <*(the Sorts of Q)
.the_array_sort_of S*> by Th22;
thus Den((the connectives of S)/.(11+2),Q).<*a*>
= Den((the connectives of S)/.(11+2),A).<*a*>
by A46,A41,A40,A5,FINSEQ_3:123,EQUATION:19
.= card a by A1,A4,A2;
end;
let i be Integer, x be Element of Q,K; assume
A47: i >= 0;
A48: i in INT by INT_1:def 2;
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 14 in dom the connectives of S by FINSEQ_3:25;
then
(the connectives of S).14 = (the connectives of S)/.14 &
(the connectives of S).14 in the carrier' of S
by PARTFUN1:def 6,FUNCT_1:102;
then the_arity_of ((the connectives of S)/.14)
= <*I, K*> by A3,A2,A4;
then
A49: Args((the connectives of S)/.14, Q) = product <*(the Sorts of Q).I,
(the Sorts of Q).K*> by Th23;
thus Den((the connectives of S)/.(11+3),Q).<*i,x*>
= Den((the connectives of S)/.(11+3),A).<*i,x*>
by A49,A48,A1,FINSEQ_3:124,EQUATION:19
.= Segm(i)-->x by A47,A1,A4,A2;
end;
end;
definition
let S be 1-1-connectives (4,1) integer (11,1,1)-array 11 array-correct
bool-correct non empty non void BoolSignature;
let A be non-empty MSAlgebra over S;
attr A is integer-array means: Def14:
ex C being image of A st
C is (4,1) integer (11,1,1)-array bool-correct MSAlgebra over S;
end;
registration
let S be 1-1-connectives (4,1) integer (11,1,1)-array 11 array-correct
bool-correct non empty non void BoolSignature;
let X be non-empty ManySortedSet of the carrier of S;
cluster Free(S,X) -> integer-array for non-empty MSAlgebra over S;
coherence
proof let F be non-empty MSAlgebra over S;
assume
A1: F = Free(S,X);
set A = the (4,1) integer (11,1,1)-array bool-correct non-empty
MSAlgebra over S;
reconsider G = FreeGen X as GeneratorSet of F by A1,MSAFREE3:31;
set f = the ManySortedFunction of G, the Sorts of A;
FreeGen X is free & F = FreeMSA X by A1,MSAFREE3:31;
then consider h being ManySortedFunction of F,A such that
A2: h is_homomorphism F,A & h||G = f by MSAFREE:def 5;
reconsider C = Image h as image of F by A2,MSAFREE4:def 4;
take C; thus thesis;
end;
end;
registration
let S be 1-1-connectives (4,1) integer (11,1,1)-array 11 array-correct
bool-correct non empty non void BoolSignature;
cluster integer-array -> integer for non-empty MSAlgebra over S;
coherence;
let X be non-empty ManySortedSet of the carrier of S;
cluster vf-free integer-array for all_vars_including inheriting_operations
free_in_itself (X,S)-terms non-empty strict
VarMSAlgebra over S;
existence
proof
set A = Free(S,X);
consider V being ManySortedMSSet of the Sorts of A,the Sorts of A,
B being all_vars_including inheriting_operations free_in_itself
(X,S)-terms VarMSAlgebra over S such that
A1: B = VarMSAlgebra(#the Sorts of A, the Charact of A, V#) &
B is vf-free by AOFA_A00:39;
reconsider B as all_vars_including inheriting_operations free_in_itself
(X,S)-terms strict VarMSAlgebra over S by A1;
take B; thus B is vf-free by A1;
consider C being image of A such that
A2: C is (4,1) integer (11,1,1)-array bool-correct MSAlgebra over S by Def14;
consider h being ManySortedFunction of A,C such that
A3: h is_epimorphism A,C by MSAFREE4:def 5;
reconsider g = h as ManySortedFunction of B,C by A1;
the MSAlgebra of C = the MSAlgebra of C;
then reconsider D = C as image of B by A1,A3,Th8,MSAFREE4:def 5;
take D; thus thesis by A2;
end;
end;
registration
let S be 1-1-connectives (4,1) integer (11,1,1)-array 11 array-correct
bool-correct non empty non void BoolSignature;
cluster integer-array for non-empty MSAlgebra over S;
existence
proof
set A = the integer-array non-empty VarMSAlgebra over S;
take A;
thus thesis;
end;
end;
registration
let S be 1-1-connectives (4,1) integer (11,1,1)-array 11 array-correct
bool-correct non empty non void BoolSignature;
let A be integer-array non-empty MSAlgebra over S;
cluster (4,1) integer (11,1,1)-array for bool-correct image of A;
existence
proof
consider C being image of A such that
A1: C is (4,1) integer (11,1,1)-array bool-correct MSAlgebra over S by Def14;
thus thesis by A1;
end;
end;
reserve
S for 1-1-connectives (4,1) integer (11,1,1)-array 11 array-correct
bool-correct non empty non void BoolSignature,
X for non-empty ManySortedSet of the carrier of S,
T for vf-free all_vars_including inheriting_operations free_in_itself
(X,S)-terms integer-array non-empty VarMSAlgebra over S,
C for (11,1,1)-array (4,1) integer bool-correct non-empty image of T,
G for basic GeneratorSystem over S,X,T,
A for IfWhileAlgebra of the generators of G,
I for integer SortSymbol of S,
x,y,m,i for pure (Element of (the generators of G).I),
M,N for pure (Element of (the generators of G).the_array_sort_of S),
b for pure (Element of (the generators of G).the bool-sort of S),
s,s1 for (Element of C-States(the generators of G));
registration
let S;
let A be (11,1,1)-array bool-correct non-empty MSAlgebra over S;
cluster -> Relation-like Function-like
for Element of (the Sorts of A).the_array_sort_of S;
coherence
proof
let M be Element of (the Sorts of A).the_array_sort_of S;
set I = the integer SortSymbol of S;
consider J,K being Element of S such that
A1: K = 1 & (the connectives of S).11 is_of_type <*J,1*>, K &
(the Sorts of A).J = ((the Sorts of A).K)^omega &
(the Sorts of A).1 = INT &
(for a being 0-based finite array of (the Sorts of A).K holds
(for i being Integer st i in dom a holds
Den((the connectives of S)/.11,A).<*a,i*> = a.i &
for x being Element of A,K holds
Den((the connectives of S)/.(11+1),A).<*a,i,x*> = a+*(i,x)) &
Den((the connectives of S)/.(11+2),A).<*a*> = card a) &
for i being Integer, x being Element of A,K st i >= 0 holds
Den((the connectives of S)/.(11+3),A).<*i,x*> = Segm(i)-->x
by AOFA_A00:def 52;
J = the_array_sort_of S by A1,Th71;
hence thesis by A1;
end;
end;
registration
let S;
let A be (11,1,1)-array bool-correct non-empty MSAlgebra over S;
cluster -> finite Sequence-like
for Element of (the Sorts of A).the_array_sort_of S;
coherence
proof
let M be Element of (the Sorts of A).the_array_sort_of S;
set I = the integer SortSymbol of S;
consider J,K being Element of S such that
A1: K = 1 & (the connectives of S).11 is_of_type <*J,1*>, K &
(the Sorts of A).J = ((the Sorts of A).K)^omega &
(the Sorts of A).1 = INT &
(for a being 0-based finite array of (the Sorts of A).K holds
(for i being Integer st i in dom a holds
Den((the connectives of S)/.11,A).<*a,i*> = a.i &
for x being Element of A,K holds
Den((the connectives of S)/.(11+1),A).<*a,i,x*> = a+*(i,x)) &
Den((the connectives of S)/.(11+2),A).<*a*> = card a) &
for i being Integer, x being Element of A,K st i >= 0 holds
Den((the connectives of S)/.(11+3),A).<*i,x*> = Segm(i)-->x
by AOFA_A00:def 52;
J = the_array_sort_of S by A1,Th71;
hence thesis by A1;
end;
end;
theorem Th75:
for o being OperSymbol of S st
o = In((the connectives of S).11, the carrier' of S)
holds the_arity_of o = <*the_array_sort_of S,I*> &
the_result_sort_of o = I
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).11, the carrier' of S);
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 11 <= len the connectives of S by XXREAL_0:2;
then 11 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S).11 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type <*the_array_sort_of S,I*>, I by Th73;
hence the_arity_of o = <*the_array_sort_of S,I*> &
the_result_sort_of o = I;
end;
theorem Th76:
for o being OperSymbol of S st
o = In((the connectives of S).12, the carrier' of S)
holds the_arity_of o = <*the_array_sort_of S,I,I*> &
the_result_sort_of o = the_array_sort_of S
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).12, the carrier' of S);
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 12 <= len the connectives of S by XXREAL_0:2;
then 12 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S).12 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type <*the_array_sort_of S,I,I*>, the_array_sort_of S
by Th73;
hence the_arity_of o = <*the_array_sort_of S,I,I*> &
the_result_sort_of o = the_array_sort_of S;
end;
theorem Th77:
for o being OperSymbol of S st
o = In((the connectives of S).13, the carrier' of S)
holds the_arity_of o = <*the_array_sort_of S*> &
the_result_sort_of o = I
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).13, the carrier' of S);
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 13 <= len the connectives of S by XXREAL_0:2;
then 13 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S).13 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type <*the_array_sort_of S*>, I by Th73;
hence the_arity_of o = <*the_array_sort_of S*> &
the_result_sort_of o = I;
end;
theorem Th78:
for o being OperSymbol of S st
o = In((the connectives of S).14, the carrier' of S)
holds the_arity_of o = <*I,I*> &
the_result_sort_of o = the_array_sort_of S
proof
let o be OperSymbol of S;
assume A1: o = In((the connectives of S).14, the carrier' of S);
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 14 in dom the connectives of S by FINSEQ_3:25;
then o = (the connectives of S).14 by A1,FUNCT_1:102,SUBSET_1:def 8;
then o is_of_type <*I,I*>, the_array_sort_of S by Th73;
hence the_arity_of o = <*I,I*> &
the_result_sort_of o = the_array_sort_of S;
end;
theorem Th79:
for t being Element of T, the_array_sort_of S holds
for t1 being Element of T, I holds
t.(t1) value_at(C,s) = (t value_at(C,s)).(t1 value_at(C,s))
proof
let t be Element of T, the_array_sort_of S;
let t1 be Element of T, I;
set o = In((the connectives of S).11, the carrier' of S);
s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
A2: t value_at(C,s) = f.(the_array_sort_of S).t by A1,Th29;
A3: (t.t1) value_at(C,s) = f.I.(t.t1) by A1,Th29;
A4: the_arity_of o = <*the_array_sort_of S,I*> &
the_result_sort_of o = I by Th75;
then Args(o,T) = product <*(the Sorts of T).the_array_sort_of S,
(the Sorts of T).I*> by Th23;
then reconsider p = <*t,t1*> as Element of Args(o,T) by FINSEQ_3:124;
thus (t.t1) value_at(C,s) = Den(o,C).(f#p) by A1,A3,A4
.= Den(o,C).<*f.(the_array_sort_of S).t,f.I.t1*> by A4,Th26
.= (t value_at(C,s)).(t1 value_at(C,s)) by A1,A2,Th29;
end;
theorem Th80:
for t being Element of T, the_array_sort_of S holds
for t1,t2 being Element of T, I holds
(t,t1)<-t2 value_at(C,s) =
(t value_at(C,s), t1 value_at(C,s))<-(t2 value_at(C,s))
proof
let t be Element of T, the_array_sort_of S;
let t1,t2 be Element of T, I;
set o = In((the connectives of S).12, the carrier' of S);
s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
A2: t2 value_at(C,s) = f.I.t2 by A1,Th29;
A3: t value_at(C,s) = f.(the_array_sort_of S).t by A1,Th29;
A4: (t,t1)<-t2 value_at(C,s) = f.(the_array_sort_of S).((t,t1)<-t2) by A1,Th29;
A5: the_arity_of o = <*the_array_sort_of S,I,I*> &
the_result_sort_of o = the_array_sort_of S by Th76;
then Args(o,T) = product <*(the Sorts of T).the_array_sort_of S,
(the Sorts of T).I, (the Sorts of T).I*> by Th24;
then reconsider p = <*t,t1,t2*> as Element of Args(o,T) by FINSEQ_3:125;
thus (t,t1)<-t2 value_at(C,s) = Den(o,C).(f#p) by A1,A4,A5
.= Den(o,C).<*f.(the_array_sort_of S).t,f.I.t1,f.I.t2*> by A5,Th27
.= (t value_at(C,s), t1 value_at(C,s))<-(t2 value_at(C,s)) by A1,A2,A3,Th29
;
end;
theorem Th81:
for t being Element of T, the_array_sort_of S holds
length(t,I) value_at(C,s) = length(t value_at(C,s), I)
proof
let t be Element of T, the_array_sort_of S;
set o = In((the connectives of S).13, the carrier' of S);
s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
A2: length(t,I) value_at(C,s) = f.I.(length(t,I)) by A1,Th29;
A3: the_arity_of o = <*the_array_sort_of S*> &
the_result_sort_of o = I by Th77;
then Args(o,T) = product <*(the Sorts of T).the_array_sort_of S*> by Th22;
then reconsider p = <*t*> as Element of Args(o,T) by FINSEQ_3:123;
thus (length(t,I)) value_at(C,s) = Den(o,C).(f#p)
by A1,A2,A3
.= Den(o,C).<*f.(the_array_sort_of S).t*> by A3,Th25
.= length(t value_at(C,s),I) by A1,Th29;
end;
theorem
for t1,t2 being Element of T, I holds
init.array(t1,t2) value_at(C,s) =
init.array(t1 value_at(C,s), t2 value_at(C,s))
proof
let t1,t2 be Element of T, I;
set o = In((the connectives of S).14, the carrier' of S);
s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
A2: t2 value_at(C,s) = f.I.t2 by A1,Th29;
A3: init.array(t1,t2) value_at(C,s) = f.(the_array_sort_of S)
.(init.array(t1,t2)) by A1,Th29;
A4: the_arity_of o = <*I,I*> &
the_result_sort_of o = the_array_sort_of S by Th78;
then Args(o,T) = product <*(the Sorts of T).I, (the Sorts of T).I*>
by Th23;
then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
thus (init.array(t1,t2)) value_at(C,s) = Den(o,C).(f#p)
by A1,A3,A4
.= Den(o,C).<*f.I.t1,f.I.t2*> by A4,Th26
.= init.array(t1 value_at(C,s),t2 value_at(C,s)) by A1,A2,Th29;
end;
reserve u for ManySortedFunction of FreeGen T, the Sorts of C;
theorem
for t being Element of T, the_array_sort_of S holds
for t1 being Element of T, I holds
t.(t1) value_at(C,u) = (t value_at(C,u)).(t1 value_at(C,u))
proof
let t be Element of T, the_array_sort_of S;
let t1 be Element of T, I;
set o = In((the connectives of S).11, the carrier' of S);
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
A2: t value_at(C,u) = f.(the_array_sort_of S).t by A1,Th28;
A3: (t.t1) value_at(C,u) = f.I.(t.t1) by A1,Th28;
A4: the_arity_of o = <*the_array_sort_of S,I*> &
the_result_sort_of o = I by Th75;
then Args(o,T) = product <*(the Sorts of T).the_array_sort_of S,
(the Sorts of T).I*> by Th23;
then reconsider p = <*t,t1*> as Element of Args(o,T) by FINSEQ_3:124;
thus (t.t1) value_at(C,u) = Den(o,C).(f#p) by A1,A3,A4
.= Den(o,C).<*f.(the_array_sort_of S).t,f.I.t1*> by A4,Th26
.= (t value_at(C,u)).(t1 value_at(C,u)) by A1,A2,Th28;
end;
theorem Th84:
for t being Element of T, the_array_sort_of S holds
for t1,t2 being Element of T, I holds
(t,t1)<-t2 value_at(C,u) =
(t value_at(C,u), t1 value_at(C,u))<-(t2 value_at(C,u))
proof
let t be Element of T, the_array_sort_of S;
let t1,t2 be Element of T, I;
set o = In((the connectives of S).12, the carrier' of S);
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
A2: t2 value_at(C,u) = f.I.t2 by A1,Th28;
A3: t value_at(C,u) = f.(the_array_sort_of S).t by A1,Th28;
A4: (t,t1)<-t2 value_at(C,u) = f.(the_array_sort_of S).((t,t1)<-t2) by A1,Th28;
A5: the_arity_of o = <*the_array_sort_of S,I,I*> &
the_result_sort_of o = the_array_sort_of S by Th76;
then Args(o,T) = product <*(the Sorts of T).the_array_sort_of S,
(the Sorts of T).I, (the Sorts of T).I*> by Th24;
then reconsider p = <*t,t1,t2*> as Element of Args(o,T) by FINSEQ_3:125;
thus (t,t1)<-t2 value_at(C,u) = Den(o,C).(f#p) by A1,A4,A5
.= Den(o,C).<*f.(the_array_sort_of S).t,f.I.t1,f.I.t2*> by A5,Th27
.= (t value_at(C,u), t1 value_at(C,u))<-(t2 value_at(C,u))
by A1,A2,A3,Th28;
end;
theorem
for t being Element of T, the_array_sort_of S holds
length(t,I) value_at(C,u) = length(t value_at(C,u), I)
proof
let t be Element of T, the_array_sort_of S;
set o = In((the connectives of S).13, the carrier' of S);
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
A2: length(t,I) value_at(C,u) = f.I.(length(t,I)) by A1,Th28;
A3: the_arity_of o = <*the_array_sort_of S*> &
the_result_sort_of o = I by Th77;
then Args(o,T) = product <*(the Sorts of T).the_array_sort_of S*> by Th22;
then reconsider p = <*t*> as Element of Args(o,T) by FINSEQ_3:123;
thus (length(t,I)) value_at(C,u) = Den(o,C).(f#p)
by A1,A2,A3
.= Den(o,C).<*f.(the_array_sort_of S).t*> by A3,Th25
.= length(t value_at(C,u),I) by A1,Th28;
end;
theorem
for t1,t2 being Element of T, I holds
init.array(t1,t2) value_at(C,u) =
init.array(t1 value_at(C,u), t2 value_at(C,u))
proof
let t1,t2 be Element of T, I;
set o = In((the connectives of S).14, the carrier' of S);
consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
A2: t2 value_at(C,u) = f.I.t2 by A1,Th28;
A3: init.array(t1,t2) value_at(C,u) = f.(the_array_sort_of S)
.(init.array(t1,t2)) by A1,Th28;
A4: the_arity_of o = <*I,I*> &
the_result_sort_of o = the_array_sort_of S by Th78;
then Args(o,T) = product <*(the Sorts of T).I, (the Sorts of T).I*>
by Th23;
then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
thus (init.array(t1,t2)) value_at(C,u) = Den(o,C).(f#p)
by A1,A3,A4
.= Den(o,C).<*f.I.t1,f.I.t2*> by A4,Th26
.= init.array(t1 value_at(C,u),t2 value_at(C,u)) by A1,A2,Th28;
end;
Lm1:
now
let S,X,T,I;
let i be Integer;
let f1 being Function of INT, (the Sorts of T).I such that
A1: f1.0 = \0(T,I) &
for j being Nat, t being Element of T,I st f1.j = t
holds f1.(j+1) = t+\1(T,I) & f1.(-(j+1)) = -(t+\1(T,I));
let f2 being Function of INT, (the Sorts of T).I such that
A2: f2.0 = \0(T,I) &
for j being Nat, t being Element of T,I st f2.j = t
holds f2.(j+1) = t+\1(T,I) & f2.(-(j+1)) = -(t+\1(T,I));
defpred P[Nat] means f1.$1 = f2.$1;
A3: P[0] by A1,A2;
A4: for i being Nat st P[i] holds P[i+1]
proof
let i be Nat;
assume
A5: P[i];
reconsider j = i as Element of INT by INT_1:def 2;
thus f1.(i+1) = (f1.j)+\1(T,I) by A1 .= f2.(i+1) by A2,A5;
end;
A6: for i being Nat holds P[i] from NAT_1:sch 2(A3,A4);
now let i be Element of INT;
consider n being Nat such that
A7: i = n or i = -n by INT_1:def 1;
per cases by A7;
suppose i = n or i = -n & n = 0;
hence f1.i = f2.i by A6;
end;
suppose
A8: i = -n & n <> 0;
then consider m being Nat such that
A9: n = m+1 by NAT_1:6;
reconsider m0 = m, m1 = m+1 as Element of INT by INT_1:def 2;
thus f1.i = -(f1.m0+\1(T,I)) by A1,A9,A8
.= -(f2.m0+\1(T,I)) by A6 .= f2.i by A2,A9,A8;
end;
end;
hence f1 = f2;
end;
definition
let S,X,T,I;
let i be Integer;
func ^(i,T,I) -> Element of T,I means: Def15:
ex f being Function of INT, (the Sorts of T).I st
it = f.i & f.0 = \0(T,I) &
for j being Nat, t being Element of T,I st f.j = t
holds f.(j+1) = t+\1(T,I) & f.(-(j+1)) = -(t+\1(T,I));
existence
proof
deffunc F(Nat,Element of T,I) = $2+\1(T,I);
consider f being Function of NAT,(the Sorts of T).I such that
A1: f.0 = \0(T,I) & for i being Nat holds f.(i+1) = F(i,f.i) from NAT_1:sch 12;
set X = {-j where j is Nat: j <> 0};
-1 in X;
then reconsider X as non empty set;
X is integer-membered
proof
let x be object;
assume x in X;
then ex j being Nat st x = -j & j <> 0;
hence thesis;
end;
then reconsider X as non empty integer-membered set;
deffunc G(Integer) = -(f.In(-$1,NAT));
consider g being Function of X,(the Sorts of T).I such that
A2: for i being Element of X holds g.i = G(i) from FUNCT_2:sch 4;
set h = f+*g;
A3: dom h = (dom f) \/ dom g by FUNCT_4:def 1
.= NAT \/ dom g by FUNCT_2:def 1 .= NAT \/ X by FUNCT_2:def 1;
A4: NAT \/ X = INT
proof
thus NAT \/ X c= INT by MEMBERED:5;
let x be Integer; assume x in INT;
then consider n being Nat such that
A5: x = n or x = -n by INT_1:def 1;
A6: n in NAT by ORDINAL1:def 12;
per cases by A5;
suppose x = n;
hence thesis by XBOOLE_0:def 3,A6;
end;
suppose x = -n & n = 0;
hence thesis by XBOOLE_0:def 3;
end;
suppose x = -n & n <> 0;
then x in X;
hence thesis by XBOOLE_0:def 3;
end;
end;
rng f c= (the Sorts of T).I & rng g c= (the Sorts of T).I
by RELAT_1:def 19;
then rng h c= (rng f)\/rng g & (rng f)\/rng g c= (the Sorts of T).I
by XBOOLE_1:8,FUNCT_4:17;
then reconsider h as Function of INT,(the Sorts of T).I
by A3,A4,FUNCT_2:2,XBOOLE_1:1;
reconsider j = i as Element of INT by INT_1:def 2;
reconsider t = h.j as Element of T,I;
take t, h; thus t = h.i;
-0 nin X
proof
assume -0 in X;
then ex n being Nat st -0 = -n & n <> 0;
hence contradiction;
end;
then 0 nin dom g by FUNCT_2:def 1;
hence h.0 = \0(T,I) by A1,FUNCT_4:11;
let j be Nat, t be Element of T,I; assume
A7: h.j = t;
j nin X
proof
assume j in X;
then consider n being Nat such that
A8: j = -n & n <> 0;
j in NAT by ORDINAL1:def 12;
hence contradiction by A8,INT_2:6;
end;
then j nin dom g by FUNCT_2:def 1;
then
A9: h.j = f.j by FUNCT_4:11;
j+1 nin X
proof
assume j+1 in X;
then consider n being Nat such that
A10: j+1 = -n & n <> 0;
thus contradiction by A10;
end;
then j+1 nin dom g by FUNCT_2:def 1;
hence h.(j+1) = f.(j+1) by FUNCT_4:11 .= t+\1(T,I) by A1,A7,A9;
A11: -(j+1) in X;
then -(j+1) in dom g by FUNCT_2:def 1;
hence h.(-(j+1)) = g.(-(j+1)) by FUNCT_4:13 .= G(-(j+1)) by A11,A2
.= -(f.(j+1))
.= -(t+\1(T,I)) by A1,A7,A9;
end;
uniqueness by Lm1;
end;
theorem Th87:
^(0,T,I) = \0(T,I)
proof
ex f being Function of INT, (the Sorts of T).I st
^(0,T,I) = f.0 & f.0 = \0(T,I) &
for j being Nat, t being Element of T,I st f.j = t
holds f.(j+1) = t+\1(T,I) & f.(-(j+1)) = -(t+\1(T,I)) by Def15;
hence thesis;
end;
theorem Th88:
for n being Nat holds ^(n+1,T,I) = ^(n,T,I)+\1(T,I) &
^(-(n+1),T,I) = - ^(n+1,T,I)
proof
let n be Nat;
consider f being Function of INT, (the Sorts of T).I such that
A1: ^(n+1,T,I) = f.(n+1) & f.0 = \0(T,I) &
for j being Nat, t being Element of T,I st f.j = t
holds f.(j+1) = t+\1(T,I) & f.(-(j+1)) = -(t+\1(T,I)) by Def15;
consider g being Function of INT, (the Sorts of T).I such that
A2: ^(n,T,I) = g.n & g.0 = \0(T,I) &
for j being Nat, t being Element of T,I st g.j = t
holds g.(j+1) = t+\1(T,I) & g.(-(j+1)) = -(t+\1(T,I)) by Def15;
consider h being Function of INT, (the Sorts of T).I such that
A3: ^(-(n+1),T,I) = h.(-(n+1)) & h.0 = \0(T,I) &
for j being Nat, t being Element of T,I st h.j = t
holds h.(j+1) = t+\1(T,I) & h.(-(j+1)) = -(t+\1(T,I)) by Def15;
A4: f = g by A1,A2,Lm1;
^(n,T,I) = f.n by A1,A2,Lm1;
hence
A5: ^(n+1,T,I) = ^(n,T,I)+\1(T,I) by A1;
f = h by A1,A3,Lm1;
hence ^(-(n+1),T,I) = - ^(n+1,T,I) by A3,A5,A4,A2;
end;
theorem
^(1,T,I) = \0(T,I)+\1(T,I)
proof 1 = 0+1;
hence ^(1,T,I) = ^(0,T,I)+\1(T,I) by Th88 .= \0(T,I)+\1(T,I) by Th87;
end;
theorem Th90:
for i being Integer holds ^(i,T,I) value_at(C,s) = i
proof
let i be Integer;
defpred P[Nat] means ^($1,T,I) value_at(C,s) = $1;
^(0,T,I) = \0(T,I) by Th87;
then
A1: P[0] by Th36;
A2: now let i be Nat;
assume
A3: P[i];
A4: \1(T,I) value_at(C,s) = 1 by Th37;
^(i+1,T,I) = ^(i,T,I)+\1(T,I) by Th88;
then ^(i+1,T,I) value_at(C,s)
= (^(i,T,I) value_at(C,s))+(\1(T,I) value_at(C,s)) by Th39
.= (^(i,T,I) value_at(C,s))+1 by A4,AOFA_A00:55;
hence P[i+1] by A3;
end;
A5: for i being Nat holds P[i] from NAT_1:sch 2(A1,A2);
i in INT by INT_1:def 2;
then consider n being Nat such that
A6: i = n or i = -n by INT_1:def 1;
per cases by A6;
suppose i = n or i = -n & n = 0;
hence thesis by A5;
end;
suppose
A7: i = -n & n <> 0;
then consider m being Nat such that
A8: n = m+1 by NAT_1:6;
\1(T,I) value_at(C,s) = 1 & ^(m,T,I) value_at(C,s) = m by A5,Th37;
then
A9: (^(m,T,I) value_at(C,s))+(\1(T,I) value_at(C,s)) = m+1 by AOFA_A00:55;
^(i,T,I) = - ^(m+1,T,I) by A7,A8,Th88;
hence ^(i,T,I) value_at(C,s)
= -(^(m+1,T,I) value_at(C,s)) by Th38
.= -((^(m,T,I)+\1(T,I)) value_at(C,s)) by Th88
.= -((^(m,T,I) value_at(C,s))+(\1(T,I) value_at(C,s))) by Th39
.= i by A7,A8,A9,AOFA_A00:55;
end;
end;
definition
let S,X,T,G,I,M;
let i be Integer;
func M.(i,I) -> Element of T,I equals @M.(^(i,T,I));
coherence;
end;
registration
let S,X,T,G,C,s,M;
cluster s.(the_array_sort_of S).M -> Function-like Relation-like;
coherence
proof
s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then s.(the_array_sort_of S) is Function of (the generators of G)
.the_array_sort_of S, (the Sorts of C).the_array_sort_of S
by PBOOLE:def 15;
then
A1: s.(the_array_sort_of S).M in (the Sorts of C).the_array_sort_of S
by FUNCT_2:5;
thus thesis by A1;
end;
end;
registration
let S,X,T,G,C,s,M;
cluster s.(the_array_sort_of S).M -> finite Sequence-like INT-valued;
coherence
proof
s is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then s.(the_array_sort_of S) is Function of (the generators of G)
.the_array_sort_of S, (the Sorts of C).the_array_sort_of S
by PBOOLE:def 15;
then
A1: s.(the_array_sort_of S).M in (the Sorts of C).the_array_sort_of S
by FUNCT_2:5;
(the Sorts of C).the_array_sort_of S = INT^omega by Th74;
hence thesis by A1;
end;
end;
registration
let S,X,T,G,C,s,M;
cluster rng (s.(the_array_sort_of S).M) -> finite integer-membered;
coherence;
end;
theorem
for j being Integer st j in dom (s.(the_array_sort_of S).M) &
M.(j,I) in (the generators of G).I
holds s.(the_array_sort_of S).M.j = s.I.(M.(j,I))
proof
let j be Integer;
assume
A1: j in dom (s.(the_array_sort_of S).M);
assume
A2: M.(j,I) in (the generators of G).I;
reconsider s1 = s as ManySortedFunction of the generators of G,
the Sorts of C by AOFA_A00:48;
consider h being ManySortedFunction of T,C such that
A3: h is_homomorphism T,C & s1 = h||the generators of G by AOFA_A00:def 19;
A4: ^(j,T,I) value_at(C,s) = j & @M value_at(C,s) = s.(the_array_sort_of S).M
by Th61,Th90;
s.I.(M.(j,I)) = (h.I)|((the generators of G).I).(M.(j,I))
by A3,MSAFREE:def 1
.= h.I.(@M.(^(j,T,I))) by A2,FUNCT_1:49
.= @M.(^(j,T,I)) value_at(C,s) by A3,Th29
.= (@M value_at(C,s)).(^(j,T,I) value_at(C,s)) by Th79
.= (s1.(the_array_sort_of S).M).j by A1,A4,Th74;
hence s.(the_array_sort_of S).M.j = s.I.(M.(j,I));
end;
theorem
for j being Integer st j in dom (s.(the_array_sort_of S).M) &
@M.(@i) in (the generators of G).I & j = @i value_at(C,s) holds
s.(the_array_sort_of S).M.(@i value_at(C,s)) = s.I.(@M.(@i))
proof
let j be Integer;
assume A1: j in dom (s.(the_array_sort_of S).M);
assume A2: @M.(@i) in (the generators of G).I;
assume A3: j = @i value_at(C,s);
reconsider s1 = s as ManySortedFunction of the generators of G,
the Sorts of C by AOFA_A00:48;
consider h being ManySortedFunction of T,C such that
A4: h is_homomorphism T,C & s1 = h||the generators of G by AOFA_A00:def 19;
s.(the_array_sort_of S).M = @M value_at(C,s) by Th61;
hence s.(the_array_sort_of S).M.(@i value_at(C,s))
= (@M value_at(C,s)).(@i value_at(C,s)) by A1,A3,Th74
.= @M.@i value_at(C,s) by Th79
.= h.I.(@M.@i) by A4,Th29
.= ((h.I)|((the generators of G).I)).(@M.@i) by A2,FUNCT_1:49
.= s.I.(@M.(@i)) by A4,MSAFREE:def 1;
end;
registration
let X be non empty set;
cluster X^omega -> infinite;
coherence
proof
set x = the Element of X;
set Y = the set of all n-->x where n is Nat;
A1: Y c= X^omega
proof
let a be object;
assume a in Y;
then ex n being Nat st a = n qua set-->x;
hence thesis by AFINSQ_1:def 7;
end;
defpred P[object,object] means
ex z being set st z = $1 & $2 = card z;
A2: for e being object st e in Y ex u being object st P[e,u]
proof
let e be object such that e in Y;
reconsider e as set by TARSKI:1;
take card e;
thus thesis;
end;
consider f being Function such that
A3: dom f = Y & for a being object st a in Y holds P[a,f.a]
from CLASSES1:sch 1(A2);
rng f = NAT
proof
thus rng f c= NAT
proof
let a be object; assume a in rng f;
then consider b being object such that
A4: b in dom f & a = f.b by FUNCT_1:def 3;
consider n being Nat such that
A5: b = n-->x by A3,A4;
ex z being set st z = b & f.b = card z by A3,A4;
then a = card dom(n-->x) by A4,A5,CARD_1:62
.= card n by FUNCOP_1:13 .= n;
hence a in NAT by ORDINAL1:def 12;
end;
let n be Nat;
assume n in NAT;
A6: n-->x in Y;
then ex z being set st z = n-->x & f.(n-->x) = card z by A3;
then f.(n-->x) = card dom (n-->x) by CARD_1:62
.= card n by FUNCOP_1:13 .= n;
hence n in rng f by A3,A6,FUNCT_1:def 3;
end;
hence thesis by A1,A3,FINSET_1:8;
end;
end;
theorem
for f being ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G, b) st
f in C-Execution(A,b,\falseC) & G is C-supported & i <> m &
s.(the_array_sort_of S).M <> {} holds
for n being Nat st
f.(s, m:=(\0(T,I),A)\;
for-do(i:=(\1(T,I),A), b gt(length(@M,I),@i,A), i:=(@i+\1(T,I),A),
if-then(b gt(@M.(@i), @M.(@m), A), m:=(@i,A)))).I.m = n
for X being non empty finite integer-membered set
st X = rng (s.(the_array_sort_of S).M)
holds (M.(n,I)) value_at(C,s) = max X
proof let f be ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G, b) such that
A1: f in C-Execution(A,b,\falseC) & G is C-supported & i <> m and
A2: s.(the_array_sort_of S).M <> {};
let n be Nat;
assume
A3: f.(s, m:=(\0(T,I),A)\;
for-do(i:=(\1(T,I),A), b gt(length(@M,I),@i,A), i:=(@i+\1(T,I),A),
if-then(b gt(@M.(@i), @M.(@m), A), m:=(@i,A)
)
)
).I.m = n;
let X be non empty finite integer-membered set;
assume
A4: X = rng (s.(the_array_sort_of S).M);
set ST = C-States(the generators of G);
set TV = (\falseC)-States(the generators of G, b);
defpred R[Element of ST] means
s.(the_array_sort_of S).M = $1.(the_array_sort_of S).M;
reconsider sm = s as ManySortedFunction of the generators of G,
the Sorts of C by AOFA_A00:48;
reconsider z = sm.(the_array_sort_of S).M as 0-based finite array of INT;
defpred P[Element of ST] means R[$1] &
$1.I.i in NAT & $1.I.m in NAT & $1.I.i <= len z & $1.I.m < $1.I.i &
$1.I.m < len z &
for mx being Integer st mx = $1.I.m
for j being Nat st j < $1.I.i
holds z.j <= z.mx;
defpred Q[Element of ST] means R[$1] &
$1.I.i < (length(@M,I)) value_at(C,s);
set s0 = s;
set s1 = f.(s,m:=(\0(T,I),A));
set s2 = f.(s1,i:=(\1(T,I),A));
set W = b gt(length(@M,I),@i,A);
set K = i:=(@i+\1(T,I),A);
set s3 = f.(s2,W);
set CJ = b gt(@M.(@i), @M.(@m), A);
set IJ = m:=(@i,A);
set J = if-then(CJ, IJ);
set a = the_array_sort_of S;
A5: I <> the bool-sort of S by AOFA_A00:53;
A6: f complies_with_if_wrt TV by AOFA_000:def 32;
A7: s1.I.m = \0(T,I) value_at(C,s) by A1,Th65;
A8:\0(T,I) value_at(C,s) = 0 by Th36;
A9: s2.I.m = s1.I.m by A1,Th65;
A10: s2.I.i = \1(T,I) value_at(C,s1) by A1,Th65 .= 1 by Th37;
A11: s3.I.i = s2.I.i by A1,A5,Th65;
consider J1,K1,L1 being Element of S such that
A12: L1 = 1 & K1 = 1 & J1 <> L1 & J1 <> K1 &
(the connectives of S).11 is_of_type <*J1,K1*>, L1 &
(the connectives of S).(11+1) is_of_type <*J1,K1,L1*>, J1 &
(the connectives of S).(11+2) is_of_type <*J1*>, K1 &
(the connectives of S).(11+3) is_of_type <*K1,L1*>, J1 by AOFA_A00:def 51;
A13: (the Sorts of C).the_array_sort_of S = INT^omega &
(the Sorts of C).the bool-sort of S = BOOLEAN by Th74,AOFA_A00:def 32;
A14: the bool-sort of S <> I by AOFA_A00:53;
A15: the_array_sort_of S <> I by A12,Th71;
then
A16: s1.(the_array_sort_of S).M = s.(the_array_sort_of S).M by A1,Th65;
A17: s3.(the_array_sort_of S).M = s2.(the_array_sort_of S).M by A13,A1,Th65;
A18: P[s2]
proof
thus R[s2] by A15,A1,Th65,A16;
thus s2.I.i in NAT & s2.I.m in NAT by A7,A8,A9,A10;
0 < len z & 0+1 = 1 by A2,NAT_1:3;
hence s2.I.i <= len z & s2.I.m < s2.I.i & s2.I.m < len z
by A7,A8,A9,A10,NAT_1:13;
let mx be Integer; assume
A19: mx = s2.I.m;
let j be Nat; assume
A20: j < s2.I.i;
1=0+1;
then j <= 0 & j >= 0 by A20,A10,NAT_1:13;
then
A21: j = 0;
thus z.j <= z.mx by A21,A19,A8,A9,A1,Th65;
end;
deffunc F(Element of ST)
= In((len(s0.(the_array_sort_of S).M))-$1.I.i,NAT);
A22: f.(s2,W) in TV iff Q[f.(s2,W)]
proof
A23: @i value_at(C,s2) < length(@M,I) value_at(C,s2) iff f.(s2, W) in TV
by A1,Th66;
length(@M,I) value_at(C,s2) = length(@M value_at(C,s2),I) by Th81
.= len (@M value_at(C,s2)) by Th74
.= len(s2.(the_array_sort_of S).M) by Th61
.= len(s0.(the_array_sort_of S).M) by A15,A1,Th65,A16
.= len(@M value_at(C,s0)) by Th61
.= length(@M value_at(C,s0), I) by Th74
.= length(@M,I) value_at(C,s0) by Th81;
hence thesis by A15,A1,Th65,A16,A17,A11,A23,Th61;
end;
A24: len(@M value_at(C,s0))
= length(@M value_at(C,s0),I) by Th74
.= length(@M,I) value_at(C,s0) by Th81;
A25: for s being Element of ST st Q[s]
holds (Q[f.(s,J\;K\;W)] iff f.(s,J\;K\;W) in TV) & F(f.(s,J\;K\;W)) < F(s)
proof
let s be Element of ST;
assume A26: Q[s];
A27: f.(s,J\;K\;W) = f.(f.(s, J\;K), W) & f.(s,J\;K) = f.(f.(s,J),K)
by AOFA_000:def 29;
hereby
A28: f.(s,J\;K\;W).I.i = f.(s,J\;K).I.i by A14,A27,A1,Th65;
A29: s.a.M = f.(s,CJ).a.M by A13,A1,Th65;
A30: s.I.i = f.(s,CJ).I.i by A14,A1,Th65;
A31: now
per cases;
suppose f.(s,CJ) in TV;
then f.(s,J) = f.(f.(s,CJ),IJ) by A6;
hence f.(s,J).a.M = s.a.M & f.(s,J).I.i = s.I.i
by A15,A1,A29,A30,Th65;
end;
suppose f.(s,CJ) nin TV;
then f.(s,J) = f.(f.(s,CJ),EmptyIns A) by A6;
hence f.(s,J).a.M = s.a.M & f.(s,J).I.i = s.I.i
by A29,A30,AOFA_000:def 28;
end;
end;
A32: (f.(s,J\;K).(the_array_sort_of S).M)
= (s.(the_array_sort_of S).M) by A31,A27,A15,A1,Th65;
length(@M,I) value_at(C,f.(s,J\;K))
= length(@M value_at(C,f.(s,J\;K)),I) by Th81
.= len(@M value_at(C,f.(s,J\;K))) by Th74
.= len(f.(s,J\;K).(the_array_sort_of S).M) by Th61
.= len(@M value_at(C,s0)) by A32,A26,Th61;
then
Q[f.(s,J\;K\;W)] iff @i value_at(C,f.(s,J\;K)) <
length(@M,I) value_at(C,f.(s,J\;K))
by A26,A28,A24,A32,A27,A13,A1,Th65,Th61;
hence Q[f.(s,J\;K\;W)] iff f.(s,J\;K\;W) in TV
by A1,A27,Th66;
end;
reconsider sJ = f.(s,J) as ManySortedFunction of the generators of G,
the Sorts of C by AOFA_A00:48;
reconsider a = sJ.I.i as Element of C,I;
A33: @i value_at(C,f.(s,J)) = f.(s,J).I.i &
\1(T,I) value_at(C,f.(s,J)) = 1 by Th37,Th61;
f.(s,J\;K\;W) = f.(f.(s,J\;K),W) by AOFA_000:def 29
.= f.(f.(f.(s,J),K),W) by AOFA_000:def 29;
then
A34: f.(s,J\;K\;W).I.i
= f.(f.(s,J),K).I.i by A1,Th65,A14
.= (@i+\1(T,I)) value_at(C,f.(s,J)) by A1,Th65
.= (@i value_at(C,f.(s,J)))+(\1(T,I) value_at(C,f.(s,J))) by Th39
.= f.(s,J).I.i + 1 by A33,AOFA_A00:55;
@M value_at(C,s0) = s0.(the_array_sort_of S).M &
s.I.i < (length(@M,I)) value_at(C,s0) &
(length(@M,I)) value_at(C,s0) = length(@M value_at(C,s0),I)
by A26,Th61,Th81;
then
A35: s.I.i < len(s0.(the_array_sort_of S).M) by Th74;
then
A36: (len(s0.(the_array_sort_of S).M))-s.I.i > 0 by XREAL_1:50;
(len(s0.(the_array_sort_of S).M))-s.I.i >= 0+1 by A35,XREAL_1:50,INT_1:7;
then
A37: (len(s0.(the_array_sort_of S).M))-s.I.i-1 >= 1-1 by XREAL_1:9;
per cases;
suppose @M.(@i) value_at(C,s) > @M.(@m) value_at(C,s);
then f.(s,CJ) in TV & f complies_with_if_wrt TV
by A1,Th66,AOFA_000:def 32;
then
A38: f.(s,J).I.i = f.(f.(s,CJ),IJ).I.i
.= f.(s,CJ).I.i by A1,Th65
.= s.I.i by A14,A1,Th65;
F(f.(s,J\;K\;W)) = (len(s0.(the_array_sort_of S).M))-s.I.i-1
by A34,A38,A37,INT_1:3,SUBSET_1:def 8;
then F(f.(s,J\;K\;W)) = F(s)-1 by A36,INT_1:3,SUBSET_1:def 8;
hence F(f.(s,J\;K\;W)) < F(s) by XREAL_1:44;
end;
suppose @M.(@i) value_at(C,s) <= @M.(@m) value_at(C,s);
then f.(s,CJ) nin TV & f complies_with_if_wrt TV
by A1,Th66,AOFA_000:def 32;
then f.(s,J) = f.(f.(s,CJ),EmptyIns A)
.= f.(s,CJ) by AOFA_000:def 28;
then f.(s,J).I.i = s.I.i by A1,Th65,A14;
then F(f.(s,J\;K\;W))
= len(s0.(the_array_sort_of S).M)-s.I.i-1
by A34,A37,INT_1:3,SUBSET_1:def 8
.= F(s)-1 by A36,INT_1:3,SUBSET_1:def 8;
hence F(f.(s,J\;K\;W)) < F(s) by XREAL_1:44;
end;
end;
A39: f iteration_terminates_for J\;K\;W, f.(s2,W) from AOFA_000:sch 3(A22,A25);
A40: for s being Element of ST st P[s] & s in TV & Q[s]
holds P[f.(s,J\;K)]
proof
let s be Element of ST;
assume A41: P[s];
assume s in TV;
assume A42: Q[s];
A43: s.a.M = f.(s,CJ).a.M by A13,A1,Th65;
thus R[f.(s,J\;K)]
proof
per cases;
suppose f.(s,CJ) in TV;
then f.(s,J) = f.(f.(s,CJ),IJ) by A6;
then
A44: f.(s,J).a.M = s0.a.M by A41,A15,A1,A43,Th65;
f.(s,J\;K) = f.(f.(s,J),K) by AOFA_000:def 29;
hence thesis by A44,A15,A1,Th65;
end;
suppose f.(s,CJ) nin TV;
then f.(s,J) = f.(f.(s,CJ),EmptyIns A) by A6;
then
A45: f.(s,J).a.M = s0.a.M by A41,A43,AOFA_000:def 28;
f.(s,J\;K) = f.(f.(s,J),K) by AOFA_000:def 29;
hence thesis by A45,A15,A1,Th65;
end;
end;
A46: @i value_at(C,f.(s,J)) = f.(s,J).I.i & @i value_at(C,s) = s.I.i &
@m value_at(C,s) = s.I.m & \1(T,I) value_at(C,f.(s,J)) = 1
by Th61,Th37;
A47: f.(s,J\;K) = f.(f.(s,J),K) by AOFA_000:def 29;
then
A48: f.(s,J\;K).I.i = (@i+\1(T,I)) value_at(C,f.(s,J)) by A1,Th65
.= (@i value_at(C,f.(s,J)))+(\1(T,I) value_at(C,f.(s,J))) by Th39
.= f.(s,J).I.i+1 by A46,AOFA_A00:55;
A49: f.(s,J\;K).I.m = f.(s,J).I.m by A47,A1,Th65;
A50: f.(s,CJ).I.i = s.I.i & f.(s,CJ).I.m = s.I.m &
f.(s,CJ).a.M = s.a.M by A13,A14,A1,Th65;
A51: s.I.i is Nat & @M value_at(C,s) = s.a.M & @M value_at(C,s0) = s0.a.M
by A41,Th61;
then
A52: @i value_at(C,s) in dom (@M value_at(C,s))
by A46,A24,A42,AFINSQ_1:86;
A53: @m value_at(C,s) in dom (@M value_at(C,s))
by A51,A46,A41,AFINSQ_1:86;
A54: z.(s.I.i) = (@M value_at(C,s)).(s.I.i) by A41,Th61
.= (@M value_at(C,s) qua Function).(@i value_at(C,s))
by Th61
.= (@M value_at(C,s)).(@i value_at(C,s)) by A52,Th74
.= @M.(@i) value_at(C,s) by Th79;
A55: z.(s.I.m) = (@M value_at(C,s)).(s.I.m) by A41,Th61
.= (@M value_at(C,s) qua Function).(@m value_at(C,s))
by Th61
.= (@M value_at(C,s)).(@m value_at(C,s)) by A53,Th74
.= @M.(@m) value_at(C,s) by Th79;
A56: now per cases;
case
z.(s.I.i) > z.(s.I.m);
then f.(s,CJ) in TV by A1,A54,A55,Th66;
then
A57: f.(s,J) = f.(f.(s,CJ),IJ) by A6;
hence f.(s,J).I.i = s.I.i by A50,A1,Th65;
thus f.(s,J).I.m = @i value_at(C,f.(s,CJ)) by A57,A1,Th65
.= s.I.i by A50,Th61;
thus f.(s,J).a.M = s.a.M by A57,A50,A1,A15,Th65;
end;
case
z.(s.I.i) <= z.(s.I.m);
then f.(s,CJ) nin TV by A1,A54,A55,Th66;
then
A58: f.(s,J) = f.(f.(s,CJ),EmptyIns A) by A6;
hence f.(s,J).I.i = s.I.i by A50,AOFA_000:def 28;
thus f.(s,J).I.m = s.I.m by A58,A50,AOFA_000:def 28;
thus f.(s,J).a.M = s.a.M by A58,A50,AOFA_000:def 28;
end;
end;
reconsider sIi = s.I.i as Element of NAT by A41;
A59: f.(s,J\;K).I.i = sIi+1 & sIi+1 in NAT by A48,A56,ORDINAL1:def 12;
thus f.(s,J\;K).I.i in NAT & f.(s,J\;K).I.m in NAT
by A56,A41,A47,A1,Th65,A48,ORDINAL1:def 12;
len z = length(@M,I) value_at(C,s0) by A24,Th61;
hence
f.(s,J\;K).I.i <= len z by A56,A48,A42,INT_1:7;
thus f.(s,J\;K).I.m < f.(s,J\;K).I.i by A56,A48,A49,A41,NAT_1:13;
thus f.(s,J\;K).I.m < len z by A24,Th61,A56,A49,A41,A42;
let mx be Integer;
assume
A60: mx = f.(s,J\;K).I.m;
let j be Nat; assume
A61: j < f.(s,J\;K).I.i;
per cases by A61,A59,NAT_1:22;
suppose
j < s.I.i & z.(s.I.i) <= z.(s.I.m);
hence z.j <= z.mx by A60,A41,A56,A49;
end;
suppose
j < s.I.i & z.(s.I.i) > z.(s.I.m);
then z.j <= z.(s.I.m) by A41;
hence z.j <= z.mx by A60,A56,A49,XXREAL_0:2;
end;
suppose
j = s.I.i & z.(s.I.i) <= z.(s.I.m);
hence z.j <= z.mx by A60,A56,A47,A1,Th65;
end;
suppose
j = s.I.i & z.(s.I.i) > z.(s.I.m);
hence z.j <= z.mx by A60,A56,A47,A1,Th65;
end;
end;
A62: for s being Element of ST st P[s] holds
P[f.(s,W)] &
(f.(s,W) in TV iff Q[f.(s,W)])
proof
let s be Element of ST;
assume A63: P[s];
thus R[f.(s,W)] by A63,A1,A13,Th65;
A64: f.(s,W).I.i = s.I.i & f.(s,W).I.m = s.I.m by A1,A14,Th65;
thus f.(s,W).I.i in NAT & f.(s,W).I.m in NAT by A63,A1,A14,Th65;
thus f.(s,W).I.i <= len z & f.(s,W).I.m < f.(s,W).I.i by A64,A63;
thus f.(s,W).I.m < len z by A1,A14,Th65,A63;
thus for mx being Integer st mx = f.(s,W).I.m
for j being Nat st j < f.(s,W).I.i holds z.j <= z.mx by A64,A63;
A65: length(@M,I) value_at(C,s)
= length(@M value_at(C,s),I) by Th81
.= len(@M value_at(C,s)) by Th74
.= len(s.(the_array_sort_of S).M) by Th61
.= len(@M value_at(C,s0)) by A63,Th61;
hereby
assume f.(s,W) in TV;
then @i value_at(C,s) < length(@M,I) value_at(C,s) &
s.I.i = @i value_at(C,s) by A1,Th66,Th61;
hence Q[f.(s,W)] by A63,A1,A14,Th65,A13,A24,A65;
end;
assume Q[f.(s,W)];
then @i value_at(C,s) < length(@M,I) value_at(C,s)
by A64,A24,A65,Th61;
hence f.(s,W) in TV by A1,Th66;
end;
A66: P[f.(s2, while(W, J\;K))] &
not Q[f.(s2, while(W, J\;K))]
from AOFA_000:sch 5(A18,A39,A40,A62);
A67: f.(s, m:=(\0(T,I),A)\;for-do(i:=(\1(T,I),A),W,K,J))
= f.(s1, for-do(i:=(\1(T,I),A),W,K,J)) by AOFA_000:def 29
.= f.(f.(s1, i:=(\1(T,I),A)), while(W, J\;K)) by AOFA_000:def 29;
then
A68: n in dom z by A66,A3,AFINSQ_1:86;
A69: ^(n,T,I) value_at(C,s) = n by Th90;
A70: z = @M value_at(C,s) by Th61;
A71: z.(f.(s, m:=(\0(T,I),A)\;for-do(i:=(\1(T,I),A),W,K,J)).I.m)
= (@M value_at(C,s)).(^(n,T,I) value_at(C,s)) by A68,A70,A69,Th74,A3
.= (@M. ^(n,T,I)) value_at(C,s) by Th79;
A72: (M.(n,I)) value_at(C,s) is UpperBound of X
proof
let x be ExtReal;
assume x in X;
then consider j being object such that
A73: j in dom z & x = z.j by A4,FUNCT_1:def 3;
reconsider j as Nat by A73;
f.(s,m:=(\0(T,I),A)\;for-do(i:=(\1(T,I),A),W,K,J)).I.i <= len z &
f.(s,m:=(\0(T,I),A)\;for-do(i:=(\1(T,I),A),W,K,J)).I.i >= len z
by A24,Th61,A66,A67;
then f.(s,m:=(\0(T,I),A)\;for-do(i:=(\1(T,I),A),W,K,J)).I.i = len z &
j < len z by A73,AFINSQ_1:86,XXREAL_0:1;
hence thesis by A71,A73,A66,A67;
end;
for x being UpperBound of X holds (M.(n,I)) value_at(C,s) <= x
proof
let x be UpperBound of X;
n in dom z & M.(n,I) value_at(C,s) = z.n
by A66,A67,A71,A3,AFINSQ_1:86;
then M.(n,I) value_at(C,s) in X by A4,FUNCT_1:def 3;
hence (M.(n,I)) value_at(C,s) <= x by XXREAL_2:def 1;
end;
hence (M.(n,I)) value_at(C,s) = max X by A72,XXREAL_2:def 3;
end;
theorem Th94:
for A being elementary IfWhileAlgebra of the generators of G
for f being ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G,b) st
f in C-Execution(A,b,\falseC) & G is C-supported
for t0,t1 being Element of T,I for J being Algorithm of A
for P being set st
P is_invariant_wrt i:=(t0,A),f &
P is_invariant_wrt b gt(t1,@i,A),f &
P is_invariant_wrt i:=(@i+\1(T,I),A),f &
P is_invariant_wrt J,f & J is_terminating_wrt f,P &
for s holds f.(s,J).I.i = s.I.i & f.(s,b gt(t1,@i,A)).I.i = s.I.i &
t1 value_at(C,f.(s, b gt(t1,@i,A))) = t1 value_at(C,s) &
t1 value_at(C,f.(s, J\;i:=(@i+\1(T,I),A))) =
t1 value_at(C,s)
holds
for-do(i:=(t0,A),b gt(t1,@i,A),i:=(@i+\1(T,I),A), J) is_terminating_wrt f, P
proof
let A be elementary IfWhileAlgebra of the generators of G;
let f be ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G,b);
assume
A1: f in C-Execution(A,b,\falseC) & G is C-supported;
let t0,t1 be Element of T,I;
let J be Algorithm of A;
let P be set;
assume A2: P is_invariant_wrt i:=(t0,A),f;
assume A3: P is_invariant_wrt b gt(t1,@i,A),f;
assume A4: P is_invariant_wrt i:=(@i+\1(T,I),A),f;
assume A5: P is_invariant_wrt J,f;
assume A6: J is_terminating_wrt f,P;
set W = b gt(t1,@i,A);
set L = i:=(@i+\1(T,I),A);
set K = i:=(t0,A);
set ST = C-States(the generators of G);
set TV = (\falseC)-States(the generators of G, b);
assume A7: for s holds f.(s,J).I.i = s.I.i & f.(s,W).I.i = s.I.i &
t1 value_at(C,f.(s, W)) = t1 value_at(C,s) &
t1 value_at(C,f.(s, J\;L)) = t1 value_at(C,s);
A8: K is_terminating_wrt f, P by AOFA_000:107;
L is_terminating_wrt f,P by AOFA_000:107;
then
A9: W is_terminating_wrt f & J\;L is_terminating_wrt f,P
by A5,A6,AOFA_000:104,111;
A10: for s st s in P & f.(f.(s,J\;L),W) in TV holds f.(s,J\;L) in P
by A5,A4,AOFA_000:109,def 39;
for s st f.(s,W) in P holds f iteration_terminates_for J\;L\;W, f.(s,W)
proof let s; assume
f.(s,W) in P;
defpred P[Element of ST] means $1.I.i < t1 value_at(C,$1);
deffunc F(Element of ST) = In((t1 value_at(C,$1))-$1.I.i, NAT);
@i value_at(C,s) = s.I.i &
f.(s,W).I.i = s.I.i & t1 value_at(C,f.(s,W)) = t1 value_at(C,s)
by A7,Th61;
then
A11: f.(s,W) in TV iff P[f.(s,W)] by A1,Th66;
A12: for s being Element of ST st P[s]
holds (P[f.(s,J\;L\;W)] iff f.(s,J\;L\;W) in TV) &
F(f.(s,J\;L\;W)) < F(s)
proof
let s; assume
A13: P[s];
A14: f.(s,J\;L\;W) = f.(f.(s, J\;L), W) by AOFA_000:def 29;
then @i value_at(C,f.(s,J\;L)) = f.(s,J\;L).I.i &
f.(s,J\;L\;W).I.i = f.(s,J\;L).I.i &
t1 value_at(C,f.(s,J\;L\;W)) = t1 value_at(C,f.(s,J\;L))
by A7,Th61;
hence P[f.(s,J\;L\;W)] iff f.(s,J\;L\;W) in TV by A14,A1,Th66;
A15: s.I.i+1 <= t1 value_at(C,s) by A13,INT_1:7;
A16: (@i value_at(C,f.(s,J)))+(\1(T,I)value_at(C,f.(s,J)))
= (@i value_at(C,f.(s,J)) qua Integer)+(\1(T,I)value_at(C,f.(s,J)))
by AOFA_A00:55;
F(f.(s,J\;L\;W))
= In((t1 value_at(C,f.(f.(s,J\;L),W)))-f.(s,J\;L\;W).I.i, NAT)
by AOFA_000:def 29
.= In((t1 value_at(C,f.(s,J\;L)))-f.(s,J\;L\;W).I.i, NAT) by A7
.= In((t1 value_at(C,s))-f.(s,J\;L\;W).I.i, NAT) by A7
.= In((t1 value_at(C,s))-f.(f.(s,J\;L),W).I.i, NAT) by AOFA_000:def 29
.= In((t1 value_at(C,s))-f.(s,J\;L).I.i, NAT) by A7
.= In((t1 value_at(C,s))-f.(f.(s,J),L).I.i, NAT) by AOFA_000:def 29
.= In((t1 value_at(C,s) qua Integer)
-((@i+\1(T,I))value_at(C,f.(s,J))), NAT) by A1,Th65
.= In((t1 value_at(C,s))-((@i value_at(C,f.(s,J)) qua Integer)+
(\1(T,I)value_at(C,f.(s,J)))), NAT) by A16,Th39
.= In((t1 value_at(C,s))-(f.(s,J).I.i+
(\1(T,I)value_at(C,f.(s,J)))), NAT) by Th61
.= In((t1 value_at(C,s))-(s.I.i+(\1(T,I)value_at(C,f.(s,J)))), NAT)
by A7
.= In((t1 value_at(C,s))-(s.I.i+1), NAT) by Th37
.= (t1 value_at(C,s))-s.I.i-1 by A15,INT_1:5,SUBSET_1:def 8
.= F(s)-1 by A13,INT_1:5,SUBSET_1:def 8;
hence F(f.(s,J\;L\;W)) < F(s) by XREAL_1:44;
end;
thus f iteration_terminates_for J\;L\;W, f.(s,W)
from AOFA_000:sch 3(A11,A12);
end;
then while(W,J\;L) is_terminating_wrt f, P by A3,A9,A10,AOFA_000:118;
hence for-do(K,W,L, J) is_terminating_wrt f, P by A2,A8,AOFA_000:111;
end;
theorem
for A being elementary IfWhileAlgebra of the generators of G
for f being ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G,b) holds
f in C-Execution(A,b,\falseC) & G is C-supported & i <> m implies
m:=(\0(T,I),A)\;
for-do(i:=(\1(T,I),A), b gt(length(@M,I),@i,A), i:=(@i+\1(T,I),A),
if-then(b gt(@M.(@i), @M.(@m), A), m:=(@i,A)))
is_terminating_wrt f, {s: s.(the_array_sort_of S).M <> {}}
proof
let A be elementary IfWhileAlgebra of the generators of G;
let f be ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G,b);
assume A1: f in C-Execution(A,b,\falseC);
assume A2: G is C-supported;
assume A3: i <> m;
set J = m:=(\0(T,I),A);
set K = i:=(\1(T,I),A);
set W = b gt(length(@M,I),@i,A);
set L = i:=(@i+\1(T,I),A);
set N = b gt(@M.(@i), @M.(@m), A);
set O = m:=(@i,A);
set a = the_array_sort_of S;
set P = {s: s.(a).M <> {}};
A4: (the Sorts of C).the bool-sort of S = BOOLEAN &
(the Sorts of C).a = INT^omega by Th74,AOFA_A00:def 32;
then
A5: the bool-sort of S <> a & I <> a & the bool-sort of S <> I
by Th73,AOFA_A00:53;
A6: P is_invariant_wrt J, f
proof
let s; assume s in P;
then consider s1 such that
A7: s = s1 & s1.a.M <> {};
f.(s,J).a.M = s.a.M by A1,A2,A5,Th65;
hence thesis by A7;
end;
A8: P is_invariant_wrt K,f
proof
let s; assume s in P;
then consider s1 such that
A9: s = s1 & s1.a.M <> {};
f.(s,K).a.M = s.a.M by A1,A2,A5,Th65;
hence thesis by A9;
end;
A10: P is_invariant_wrt W,f
proof
let s; assume s in P;
then consider s1 such that
A11: s = s1 & s1.a.M <> {};
f.(s,W).a.M = s.a.M by A1,A2,A4,Th65;
hence thesis by A11;
end;
A12: P is_invariant_wrt L,f
proof
let s; assume s in P;
then consider s1 such that
A13: s = s1 & s1.a.M <> {};
f.(s,L).a.M = s.a.M by A1,A2,A5,Th65;
hence thesis by A13;
end;
A14: P is_invariant_wrt N,f
proof
let s; assume s in P;
then consider s1 such that
A15: s = s1 & s1.a.M <> {};
f.(s,N).a.M = s.a.M by A1,A2,A4,Th65;
hence thesis by A15;
end;
A16: P is_invariant_wrt O,f
proof
let s; assume s in P;
then consider s1 such that
A17: s = s1 & s1.a.M <> {};
f.(s,O).a.M = s.a.M by A1,A2,A5,Th65;
hence thesis by A17;
end;
set ST = C-States(the generators of G);
set TV = (\falseC)-States(the generators of G, b);
A18: f complies_with_if_wrt TV by AOFA_000:def 32;
A19: P is_invariant_wrt if-then(N,O), f
proof
let s; assume s in P;
then
A20: f.(s,N) in P by A14;
per cases;
suppose f.(s, N) in TV;
then f.(s,if-then(N,O)) = f.(f.(s,N),O) by A18;
hence thesis by A20,A16;
end;
suppose f.(s, N) nin TV;
then f.(s,if-then(N,O)) = f.(f.(s,N),EmptyIns A) by A18;
hence thesis by A20,AOFA_000:def 28;
end;
end;
A21: J is_terminating_wrt f,P by AOFA_000:107;
for s holds f.(s,if-then(N,O)).I.i = s.I.i & f.(s,W).I.i = s.I.i &
length(@M,I) value_at(C,f.(s,W)) = length(@M,I) value_at(C,s) &
length(@M,I) value_at(C,f.(s,if-then(N,O)\;L)) =
length(@M,I) value_at(C,s)
proof
let s;
hereby
per cases;
suppose f.(s,N) in TV;
then f.(s,if-then(N,O)) = f.(f.(s,N),O) by A18;
hence f.(s,if-then(N,O)).I.i = f.(s,N).I.i by A1,A2,A3,Th65
.= s.I.i by A1,A2,A5,Th65;
end;
suppose f.(s,N) nin TV;
then f.(s,if-then(N,O)) = f.(f.(s,N),EmptyIns A)
by A18;
hence f.(s,if-then(N,O)).I.i = f.(s,N).I.i by AOFA_000:def 28
.= s.I.i by A1,A2,A5,Th65;
end;
end;
A22: now
per cases;
suppose f.(s,N) in TV;
then f.(s,if-then(N,O)) = f.(f.(s,N),O) by A18;
hence f.(s,if-then(N,O)).a.M = f.(s,N).a.M by A1,A2,A5,Th65
.= s.a.M by A1,A2,A4,Th65;
end;
suppose f.(s,N) nin TV;
then f.(s,if-then(N,O)) = f.(f.(s,N),EmptyIns A)
by A18;
hence f.(s,if-then(N,O)).a.M = f.(s,N).a.M by AOFA_000:def 28
.= s.a.M by A1,A2,A4,Th65;
end;
end;
thus f.(s,W).I.i = s.I.i by A1,A2,A5,Th65;
A23: @M value_at(C,f.(s,if-then(N,O)\;L))
= f.(s,if-then(N,O)\;L).a.M &
@M value_at(C,f.(s,W)) = f.(s,W).a.M & @M value_at(C,s) = s.a.M by Th61;
thus length(@M,I) value_at(C,f.(s,W))
= length(@M value_at(C,f.(s,W)),I) by Th81
.= len(f.(s,W).a.M) by A23,Th74
.= len(s.a.M) by A1,A2,A4,Th65
.= length(@M value_at(C,s),I) by A23,Th74
.= length(@M,I) value_at(C,s) by Th81;
thus length(@M,I) value_at(C,f.(s,if-then(N,O)\;L))
= length(@M value_at(C,f.(s,if-then(N,O)\;L)),I) by Th81
.= len(f.(s,if-then(N,O)\;L).a.M) by A23,Th74
.= len(f.(f.(s,if-then(N,O)),L).a.M) by AOFA_000:def 29
.= len(f.(s,if-then(N,O)).a.M) by A1,A2,A5,Th65
.= length(@M value_at(C,s),I) by A23,A22,Th74
.= length(@M,I) value_at(C,s) by Th81;
end;
then for-do(K,W,L,if-then(N,O)) is_terminating_wrt f, P
by A1,A2,A8,A10,A12,A19,Th94,AOFA_000:107;
hence thesis by A6,A21,AOFA_000:111;
end;
begin :: Sorting by exchanging
reserve i1,i2 for pure Element of (the generators of G).I;
definition
let S,X,T,G;
attr G is integer-array means
for I holds
the set of all @M.t where t is Element of T,I
c= (the generators of G).I &
for M for t being Element of T,I
for g being Element of G,I st g = @M.t
ex x st x nin (vf t).I & supp-var g = x &
(supp-term g).(the_array_sort_of S).M = (@M,t)<-@x &
for s being SortSymbol of S
for y st y in (vf g).s & (s = the_array_sort_of S implies y <> M)
holds (supp-term g).s.y = y;
end;
theorem Th96:
G is integer-array implies
for t being Element of T,I holds @M.t in (the generators of G).I
proof
assume
A1: G is integer-array;
let t be Element of T,I;
A2: @M.t in the set of all @N.q where q is Element of T,I;
the set of all @N.q where q is Element of T,I
c= (the generators of G).I by A1;
hence @M.t in (the generators of G).I by A2;
end;
definition
func (#INT,<=#) -> strict real non empty Poset equals RealPoset INT;
coherence;
end;
definition
let S,X,T,G;
let A be elementary IfWhileAlgebra of the generators of G;
let a be SortSymbol of S;
let t1,t2 be Element of T,a such that
A1: t1 in (the generators of G).a;
func t1:=(t2,A) -> absolutely-terminating Algorithm of A equals: Def19:
(the assignments of A).[t1,t2];
coherence
proof
reconsider x = t1 as Element of (the generators of G).a by A1;
(the assignments of A).[t1,t2] = x:=(t2,A);
hence thesis;
end;
end;
theorem Th97:
for X being countable non-empty ManySortedSet of the carrier of S
for T being vf-free all_vars_including inheriting_operations free_in_itself
(X,S)-terms integer-array non-empty VarMSAlgebra over S
for G being basic GeneratorSystem over S,X,T
for M being pure Element of (the generators of G).the_array_sort_of S
for i,x being pure Element of (the generators of G).I holds
@M.@i <> x
proof
let X be countable non-empty ManySortedSet of the carrier of S;
let T be vf-free all_vars_including inheriting_operations free_in_itself
(X,S)-terms integer-array non-empty VarMSAlgebra over S;
let G be basic GeneratorSystem over S,X,T;
let M be pure Element of (the generators of G).the_array_sort_of S;
let i,x be pure Element of (the generators of G).I;
set C = the (11,1,1)-array (4,1) integer bool-correct non-empty image of T;
set ST = C-States the generators of G;
assume
A1: @M.@i = x;
set q = the ManySortedFunction of FreeGen T, the Sorts of C;
set g = q+*(I,x,0)+*(the_array_sort_of S,M,<%1%>)+*(I,i,0);
set a = the_array_sort_of S;
consider h being ManySortedFunction of T,C such that
A2: h is_homomorphism T,C & h||FreeGen T = g by MSAFREE4:def 12;
reconsider s = h||the generators of G as Element of ST
by A2,AOFA_A00:def 19;
A3: the_array_sort_of S <> I by Th73;
A4: @M value_at(C,s) = s.(the_array_sort_of S).M &
@i value_at(C,s) = s.I.i &
@M.@i value_at(C,s) = (@M value_at(C,s)).(@i value_at(C,s)) by Th79,Th61;
A5: i in (FreeGen T).I & x in (FreeGen T).I & M in (FreeGen T).a by Def4;
A6: dom ((q+*(I,x,0)+*(a,M,<%1%>)).I) = (FreeGen T).I &
0 in INT & INT = (the Sorts of C).I
by INT_1:def 2,AOFA_A00:55,FUNCT_2:def 1;
((h.I)|((the generators of G).I)).i = h.I.i by FUNCT_1:49;
then
A7: s.I.i = h.I.i by MSAFREE:def 1
.= ((h.I)|((FreeGen T).I)).i by Def4,FUNCT_1:49
.= g.I.i by A2,MSAFREE:def 1
.= (((q+*(I,x,0)+*(a,M,<%1%>)).I)+*(i,0)).i by A5,A6,AOFA_A00:def 2
.= 0 by Def4,A6,FUNCT_7:31;
reconsider 01 = 1 as Element of INT by INT_1:def 2;
A8: <%01%> in INT^omega & (the Sorts of C).a = INT^omega &
dom ((q+*(I,x,0)).a) = (FreeGen T).a & dom (q.I) = (FreeGen T).I &
dom ((q+*(I,x,0)+*(a,M,<%1%>)).a) = (FreeGen T).a
by Th74,AFINSQ_1:def 7,FUNCT_2:def 1;
A9: s.a.M = ((h.a)|((the generators of G).a)).M by MSAFREE:def 1
.= h.a.M by FUNCT_1:49
.= ((h.a)|((FreeGen T).a)).M by Def4,FUNCT_1:49
.= g.a.M by A2,MSAFREE:def 1
.= (q+*(I,x,0)+*(a,M,<%1%>)).a.M by A5,A6,A3,AOFA_A00:def 2
.= (q+*(I,x,0)).a+*(M,<%1%>).M by A8,A5,AOFA_A00:def 2
.= <%1%> by Def4,A8,FUNCT_7:31;
0 < len(s.a.M) by A9,AFINSQ_1:34;
then 0 in dom(s.a.M) by AFINSQ_1:86;
then @x value_at(C,s) = s.a.M.(s.I.i) by A1,A4,Th74,A7;
then
A10: s.I.x = <%1%>.(s.I.i) by A9,Th61
.= 1 by A7;
s.I.x = ((h.I)|((the generators of G).I)).x by MSAFREE:def 1
.= h.I.x by FUNCT_1:49
.= ((h.I)|((FreeGen T).I)).x by Def4,FUNCT_1:49
.= g.I.x by A2,MSAFREE:def 1
.= ((q+*(I,x,0)+*(a,M,<%1%>)).I+*(i,0)).x by A5,A6,AOFA_A00:def 2
.= (q+*(I,x,0)+*(a,M,<%1%>)).I.x by A7,A10,FUNCT_7:32
.= (q+*(I,x,0)).I.x by A3,A5,A8,AOFA_A00:def 2
.= ((q.I)+*(x,0)).x by A5,A6,AOFA_A00:def 2
.= 0 by Def4,A8,FUNCT_7:31;
hence contradiction by A10;
end;
registration
let S be non empty non void ManySortedSign;
let A be disjoint_valued MSAlgebra over S;
cluster the Sorts of A -> disjoint_valued;
coherence by MSAFREE1:def 2;
end;
definition
let S,X;
let T be all_vars_including inheriting_operations free_in_itself
(X,S)-terms MSAlgebra over S;
attr T is array-degenerated means
ex I st
ex M being Element of (FreeGen T).the_array_sort_of S st
ex t being Element of T,I st
@M.t <> Sym(In((the connectives of S).11, the carrier' of S),X)-tree<*M,t*>;
end;
registration
let S,X;
cluster Free(S,X) -> non array-degenerated;
coherence
proof set T = Free(S,X);
let I;
let M be Element of (FreeGen T).the_array_sort_of S;
let t be Element of T,I;
set o = In((the connectives of S).11, the carrier' of S);
A1: Free(S,X) = FreeMSA X by MSAFREE3:31;
consider J1,K1,L1 being Element of S such that
A2: L1 = 1 & K1 = 1 & J1 <> L1 & J1 <> K1 &
(the connectives of S).11 is_of_type <*J1,K1*>, L1 &
(the connectives of S).(11+1) is_of_type <*J1,K1,L1*>, J1 &
(the connectives of S).(11+2) is_of_type <*J1*>, K1 &
(the connectives of S).(11+3) is_of_type <*K1,L1*>, J1 by AOFA_A00:def 51;
A3: I = 1 by AOFA_A00:def 40;
11+3 <= len the connectives of S by AOFA_A00:def 51;
then 11 <= len the connectives of S by XXREAL_0:2;
then 11 in dom the connectives of S by FINSEQ_3:25;
then (the connectives of S).11 = o by FUNCT_1:102,SUBSET_1:def 8;
then the_arity_of o = <*the_array_sort_of S, I*>
by A2,A3;
then
Args(o, T) = product <*(the Sorts of T).the_array_sort_of S,
(the Sorts of T).I*> by Th23;
then
<*@M,t*> in Args(o,T) by FINSEQ_3:124;
then
@M.t = [In((the connectives of S).11, the carrier' of S),the carrier of S]
-tree<*M,t*> by A1,INSTALG1:3;
hence thesis by MSAFREE:def 9;
end;
end;
registration
let S,X;
cluster non array-degenerated for all_vars_including inheriting_operations
free_in_itself (X,S)-terms MSAlgebra over S;
existence
proof
take Free(S,X);
thus thesis;
end;
end;
theorem Th98:
T is non array-degenerated implies
vf (@M.@i) = (I-singleton i) (\/) ((the_array_sort_of S)-singleton M)
proof set t = @M.@i;
assume
A1: T is non array-degenerated;
reconsider N = M as Element of (FreeGen T).the_array_sort_of S by Def4;
@N = @M; then
A2: @M.@i = Sym(In((the connectives of S).11, the carrier' of S),X)-tree<*M,i*>
by A1;
A3: <*M,i*>.(0+1) = M & <*M,i*>.(1+1) = i & len <*M,i*> = 2 by FINSEQ_1:44;
then
A4: t|<*0*> = M & t|<*1*> = i by A2,TREES_4:def 4;
M in (FreeGen T).the_array_sort_of S by Def4;
then M in FreeGen(the_array_sort_of S, X) by MSAFREE:def 16;
then consider m being set such that
A5: m in X.the_array_sort_of S & M = root-tree[m,the_array_sort_of S]
by MSAFREE:def 15;
i in (FreeGen T).I by Def4;
then i in FreeGen(I, X) by MSAFREE:def 16;
then consider j being set such that
A6: j in X.I & i = root-tree[j,I] by MSAFREE:def 15;
(t|<*0*>).{} = [m,the_array_sort_of S] & (t|<*1*>).{} = [j,I]
by A4,A5,A6,TREES_4:3;
then
A7: ((t|<*0*>).{})`2 = the_array_sort_of S & ((t|<*1*>).{})`2 = I;
{} in dom root-tree[m,the_array_sort_of S] & {} in dom root-tree[j,I]
by TREES_1:22;
then <*0*>^{} in dom t & <*1*>^{} in dom t by A2,A5,A6,A3,TREES_4:11;
then
A8: <*0*> in dom t & <*1*> in dom t by FINSEQ_1:34;
A9: the_array_sort_of S <> I by Th73;
A10: {M} = (vf t).the_array_sort_of S
proof
set A = {t|p where p is Element of dom t: ((t|p).{})`2
= the_array_sort_of S};
A11: M in A & A = (vf t).the_array_sort_of S
by A4,A7,A8,AOFA_A00:def 12;
hence {M} c= (vf t).the_array_sort_of S by ZFMISC_1:31;
let x be object; assume x in (vf t).the_array_sort_of S;
then consider p being Element of dom t such that
A12: x = t|p & ((t|p).{})`2 = the_array_sort_of S by A11;
per cases by A2,TREES_4:11;
suppose p = {};
then t|p = t by TREES_9:1;
then (t|p).{} = Sym(In((the connectives of S).11, the carrier' of S),X)
by A2,TREES_4:def 4
.= [In((the connectives of S).11, the carrier' of S), the carrier of S]
by MSAFREE:def 9;
then ((t|p).{})`2 = the carrier of S &
the_array_sort_of S in the carrier of S;
hence thesis by A12;
end;
suppose ex k being Nat, T being DecoratedTree,
q being Node of T st k < len <*M,i*> & T = <*M,i*>.(k+1) & p = <*k*>^q;
then consider k being Nat, Q being DecoratedTree,
q being Element of dom Q such that
A13: k < len <*M,i*> & Q = <*M,i*>.(k+1) & p = <*k*>^q;
A14: k = 0 or k = 1 by A3,A13,NAT_1:23;
then q in dom root-tree [m,the_array_sort_of S] or
q in dom root-tree [j,I] by A13,A5,A6,A3;
then q in {{}} by TREES_4:3,TREES_1:29;
then q = {} by TARSKI:def 1;
then p = <*0*> by A12,A14,A9,A7,A13,FINSEQ_1:34;
hence x in {M} by A12,A4,TARSKI:def 1;
end;
end;
A15: {i} = (vf t).I
proof
set A = {t|p where p is Element of dom t: ((t|p).{})`2 = I};
A16: i in A & A = (vf t).I by A4,A7,A8,AOFA_A00:def 12;
hence {i} c= (vf t).I by ZFMISC_1:31;
let x be object; assume x in (vf t).I;
then consider p being Element of dom t such that
A17: x = t|p & ((t|p).{})`2 = I by A16;
per cases by A2,TREES_4:11;
suppose p = {};
then t|p = t by TREES_9:1;
then (t|p).{} = Sym(In((the connectives of S).11, the carrier' of S),X)
by A2,TREES_4:def 4
.= [In((the connectives of S).11, the carrier' of S), the carrier of S]
by MSAFREE:def 9;
then ((t|p).{})`2 = the carrier of S & I in the carrier of S;
hence thesis by A17;
end;
suppose ex k being Nat, T being DecoratedTree,
q being Node of T st k < len <*M,i*> & T = <*M,i*>.(k+1) & p = <*k*>^q;
then consider k being Nat, Q being DecoratedTree,
q being Element of dom Q such that
A18: k < len <*M,i*> & Q = <*M,i*>.(k+1) & p = <*k*>^q;
A19: k = 0 or k = 1 by A3,A18,NAT_1:23;
then q in dom root-tree [m,the_array_sort_of S] or
q in dom root-tree [j,I] by A18,A5,A6,A3;
then q in {{}} by TREES_4:3,TREES_1:29;
then q = {} by TARSKI:def 1;
then p = <*1*> by A17,A19,A9,A7,A18,FINSEQ_1:34;
hence x in {i} by A17,A4,TARSKI:def 1;
end;
end;
A20: for s being SortSymbol of S st s <> the_array_sort_of S & s <> I
holds {} = (vf t).s
proof
let s be SortSymbol of S;
assume A21: s <> the_array_sort_of S;
assume A22: s <> I;
set A = {t|p where p is Element of dom t: ((t|p).{})`2 = s};
A23: A = (vf t).s by AOFA_A00:def 12;
thus {} c= (vf t).s;
let x be object; assume x in (vf t).s;
then consider p being Element of dom t such that
A24: x = t|p & ((t|p).{})`2 = s by A23;
per cases by A2,TREES_4:11;
suppose p = {};
then t|p = t by TREES_9:1;
then (t|p).{} = Sym(In((the connectives of S).11, the carrier' of S),X)
by A2,TREES_4:def 4
.= [In((the connectives of S).11, the carrier' of S), the carrier of S]
by MSAFREE:def 9;
then ((t|p).{})`2 = the carrier of S & s in the carrier of S;
hence thesis by A24;
end;
suppose ex k being Nat, T being DecoratedTree,
q being Node of T st k < len <*M,i*> & T = <*M,i*>.(k+1) & p = <*k*>^q;
then consider k being Nat, Q being DecoratedTree,
q being Element of dom Q such that
A25: k < len <*M,i*> & Q = <*M,i*>.(k+1) & p = <*k*>^q;
A26: k = 0 or k = 1 by A3,A25,NAT_1:23;
then q in dom root-tree [m,the_array_sort_of S] or
q in dom root-tree [j,I] by A25,A5,A6,A3;
then q in {{}} by TREES_4:3,TREES_1:29;
then q = {} by TARSKI:def 1;
hence thesis by A24,A26,A7,A21,A22,A25,FINSEQ_1:34;
end;
end;
let a be SortSymbol of S;
per cases;
suppose
A27: a = the_array_sort_of S;
thus (vf t).a
= {} \/ ((the_array_sort_of S)-singleton M).a by A10,A27,AOFA_A00:6
.= ((I-singleton i).a)\/((the_array_sort_of S)-singleton M).a
by A27,A9,AOFA_A00:6
.= ((I-singleton i) (\/) ((the_array_sort_of S)-singleton M)).a
by PBOOLE:def 4;
end;
suppose
A28: a = I;
hence (vf t).a = (I-singleton i).a \/ {} by A15,AOFA_A00:6
.= ((I-singleton i).a)\/((the_array_sort_of S)-singleton M).a
by A28,Th73,AOFA_A00:6
.= ((I-singleton i) (\/) ((the_array_sort_of S)-singleton M)).a
by PBOOLE:def 4;
end;
suppose
A29: a <> the_array_sort_of S & a <> I;
hence (vf t).a = {} by A20
.= (I-singleton i).a \/ {} by A29,AOFA_A00:6
.= ((I-singleton i).a)\/((the_array_sort_of S)-singleton M).a
by A29,AOFA_A00:6
.= ((I-singleton i) (\/) ((the_array_sort_of S)-singleton M)).a
by PBOOLE:def 4;
end;
end;
theorem Th99:
for A being elementary IfWhileAlgebra of the generators of G
for f being ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G, b) st
G is integer-array C-supported & f in C-Execution(A,b,\falseC) &
X is countable & T is non array-degenerated
for t being Element of T,I holds
f.(s,@M.@i:=(t,A)) = f.(s,M:=((@M,@i)<-t,A))
proof
let A be elementary IfWhileAlgebra of the generators of G;
let f be ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G, b);
assume A1: G is integer-array;
assume A2: G is C-supported;
assume A3: f in C-Execution(A,b,\falseC);
assume A4: X is countable;
assume A5: T is non array-degenerated;
let t be Element of T,I;
reconsider H = FreeGen T as ManySortedSubset of the generators of G
by Def3,PBOOLE:def 18;
set v = t value_at(C,s);
reconsider p = @M.@i as Element of G,I by A1,Th96,AOFA_A00:def 22;
reconsider g = s as ManySortedFunction of the generators of G,
the Sorts of C by AOFA_A00:48;
reconsider g1 = f.(s,@M.@i:=(t,A)), g2 = f.(s,M:=((@M,@i)<-t,A))
as ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
reconsider Mi = @M.@i as Element of (the generators of G).I by A1,Th96;
M in (the generators of G).the_array_sort_of S &
(the generators of G).the_array_sort_of S
c= (the Sorts of T).the_array_sort_of S by PBOOLE:def 2,def 18;
then reconsider m = M as Element of G,the_array_sort_of S
by AOFA_A00:def 22;
Mi:=(t,A) = @M.@i:=(t,A) by Def19;
then
A6: g1 = succ(s,p,v) & g2 = succ(s,m,(@M,@i)<-t value_at(C,s))
by A3,AOFA_A00:def 28;
A7: vf p = ((the_array_sort_of S)-singleton M)(\/)(I-singleton i) by A5,Th98;
M in (FreeGen T).the_array_sort_of S by Def4;
then
A8: vf @M = (the_array_sort_of S)-singleton M by AOFA_A00:41;
i in (FreeGen T).I by Def4;
then
A9:vf @i = I-singleton i by AOFA_A00:41;
consider x such that
A10: x nin (vf @i).I & supp-var p = x &
(supp-term p).(the_array_sort_of S).M = (@M,@i)<-@x &
for s being SortSymbol of S
for y st y in (vf p).s & (s = the_array_sort_of S implies y <> M)
holds (supp-term p).s.y = y by A1;
g1 = g2
proof
let a be SortSymbol of S;
A11: now
let a be SortSymbol of S;
A12: (vf @M).a c= (vf p).a by A7,A8,PBOOLE:14,def 2;
let b be Element of (the generators of G).a;
assume
A13: b in (FreeGen T).a;
per cases by A13;
suppose
A14: b in (FreeGen T).a & b nin (vf p).a;
then
A15: b nin (vf @M).a by A12;
A16: now
assume a = the_array_sort_of S;
then (vf @M).a = {M} by A8,AOFA_A00:6;
hence b <> m by A15,TARSKI:def 1;
end;
b is pure by A14;
then a = I implies b <> p by A4,Th97;
then g1.a.b = s.a.b & g2.a.b = s.a.b
by A2,A6,A14,A15,A16,AOFA_A00:def 27;
hence g1.a.b = g2.a.b;
end;
suppose
A17: b in (FreeGen T).a & b in (vf p).a & b nin (vf @M).a;
A18: now
assume a = the_array_sort_of S;
then (vf @M).a = {M} by A8,AOFA_A00:6;
hence b <> m by A17,TARSKI:def 1;
end;
consider u being ManySortedFunction of FreeGen T, the Sorts of C
such that
A19: u = (g||H)+*(I,supp-var p,v);
A20: a = I & b = i or a = the_array_sort_of S & b = M by A7,A17,Th4;
A21: (supp-term p).a.b = @i by A20,A17,A18,A10;
A22: i in (vf @i).a by A20,A8,A17,Th3,A9;
b is pure by A17;
then a = I implies b <> p by A4,Th97;
then g1.a.b = @i value_at(C,u)
by A2,A6,A17,A19,A20,A21,A8,Th3,AOFA_A00:def 27
.= u.a.i by A20,A8,A17,Th3,Th62
.= (((g||H).a)+*(supp-var p,v)).b by A19,A20,A8,A17,Th3
,AOFA_A00:def 2
.= (g||H).a.b by A20,A22,A10,A8,A17,Th3,FUNCT_7:32
.= ((g.a)|(H.a)).b by MSAFREE:def 1
.= g.a.b by A17,FUNCT_1:49;
hence g1.a.b = g2.a.b by A2,A6,A17,A18,AOFA_A00:def 27;
end;
suppose
A23: b in (FreeGen T).a & b in (vf p).a & b in (vf @M).a;
then
A24: b = M & a = the_array_sort_of S by A8,Th3;
A25: I <> the_array_sort_of S by Th73;
set u = (g||H)+*(I,supp-var p,v);
reconsider uIx = u.I.x as Element of C,I by Def4,FUNCT_2:5;
A26: dom((g||H).I) = (FreeGen T).I by FUNCT_2:def 1;
(vf @i).I = {i} by A9,AOFA_A00:6;
then
A27: i in (vf @i).I by TARSKI:def 1;
A28: uIx = ((g||H).I+*(x,v)).x by A10,AOFA_A00:def 2
.= v by Def4,A26,FUNCT_7:31;
A29: @M value_at(C,u) = u.(the_array_sort_of S).M by Th62
.= (g||H).a.b by A24,A25,AOFA_A00:def 2
.= ((g.a)|(H.a)).b by MSAFREE:def 1
.= g.a.b by A23,FUNCT_1:49
.= @M value_at(C,s) by A24,Th61;
A30: @i value_at(C,u) = u.I.i by Th62
.= ((g||H).I+*(x,v)).i by A10,AOFA_A00:def 2
.= (g||H).I.i by A27,A10,FUNCT_7:32
.= ((g.I)|(H.I)).i by MSAFREE:def 1
.= g.I.i by Def4,FUNCT_1:49
.= @i value_at(C,s) by Th61;
b is pure by A23;
then a = I implies b <> p by A4,Th97;
then
A31: g1.a.b = (@M,@i)<-@x value_at(C,u)
by A2,A6,A23,A24,A10,AOFA_A00:def 27
.= (@M value_at(C,u),@i value_at(C,u))<-(@x value_at(C,u)) by Th84
.= (@M value_at(C,u),@i value_at(C,u))<-v by A28,Th62;
g2.a.b = (@M,@i)<-t value_at(C,s) by A24,A2,A3,Th65
.= (@M value_at(C,s),@i value_at(C,s))<-(t value_at(C,s)) by Th80;
hence g1.a.b = g2.a.b by A31,A29,A30;
end;
end;
thus g1.a = g2.a
proof
let b be Element of (the generators of G).a;
per cases;
suppose b in (FreeGen T).a;
hence thesis by A11;
end;
suppose
b nin (FreeGen T).a;
consider h1 being ManySortedFunction of T,C such that
A32: h1 is_homomorphism T,C & g1 = h1||the generators of G
by AOFA_A00:def 19;
consider h2 being ManySortedFunction of T,C such that
A33: h2 is_homomorphism T,C & g2 = h2||the generators of G
by AOFA_A00:def 19;
h1||FreeGen T = h2||FreeGen T
proof
let a be SortSymbol of S;
thus (h1||FreeGen T).a = (h2||FreeGen T).a
proof
let b be Element of (FreeGen T).a;
A34: H.a c= (the generators of G).a & b in (FreeGen T).a
by PBOOLE:def 2,def 18;
thus (h1||FreeGen T).a.b
= ((h1.a)|((FreeGen T).a)).b by MSAFREE:def 1
.= h1.a.b by FUNCT_1:49
.= ((h1.a)|((the generators of G).a)).b by A34,FUNCT_1:49
.= g1.a.b by A32,MSAFREE:def 1
.= g2.a.b by A34,A11
.= ((h2.a)|((the generators of G).a)).b by A33,MSAFREE:def 1
.= h2.a.b by A34,FUNCT_1:49
.= ((h2.a)|((FreeGen T).a)).b by FUNCT_1:49
.= (h2||FreeGen T).a.b by MSAFREE:def 1;
end;
end;
hence thesis by A32,A33,EXTENS_1:19;
end;
end;
end;
hence f.(s,@M.@i:=(t,A)) = f.(s,M:=((@M,@i)<-t,A));
end;
registration
let S,X,T,G,C,s,b;
cluster s.(the bool-sort of S).b -> boolean;
coherence
proof
reconsider s as ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
(the Sorts of C).(the bool-sort of S) = BOOLEAN by AOFA_A00:def 32;
then s.(the bool-sort of S).b in BOOLEAN;
hence thesis;
end;
end;
theorem
for A being elementary IfWhileAlgebra of the generators of G
for f being ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G, b) st
G is integer-array C-supported & f in C-Execution(A,b,\falseC) &
T is non array-degenerated & X is countable
for J being Algorithm of A st
for s holds f.(s,J).(the_array_sort_of S).M = s.(the_array_sort_of S).M &
for D being array of (#INT,<=#) st D = s.(the_array_sort_of S).M holds
(D <> {} implies f.(s,J).I.i1 in dom D & f.(s,J).I.i2 in dom D) &
(inversions D <> {} implies [f.(s,J).I.i1,f.(s,J).I.i2] in inversions D) &
(f.(s,J).(the bool-sort of S).b = TRUE iff inversions D <> {})
for D being 0-based finite array of (#INT,<=#)
st D = s.(the_array_sort_of S).M & y <> i1 & y <> i2
holds
f.(s,while(J, y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A))).
(the_array_sort_of S).M is ascending permutation of D &
(J is absolutely-terminating implies
while(J, y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A))
is_terminating_wrt f, {s1: s1.(the_array_sort_of S).M <> {}})
proof
let A be elementary IfWhileAlgebra of the generators of G;
let f be ExecutionFunction of A, C-States(the generators of G),
(\falseC)-States(the generators of G, b);
assume A1: G is integer-array;
assume A2: G is C-supported;
assume A3: f in C-Execution(A,b,\falseC);
assume A4: T is non array-degenerated;
assume A5: X is countable;
let J be Algorithm of A;
assume A6: for s
holds f.(s,J).(the_array_sort_of S).M = s.(the_array_sort_of S).M &
for D being array of (#INT,<=#) st D = s.(the_array_sort_of S).M
holds (D <> {} implies f.(s,J).I.i1 in dom D & f.(s,J).I.i2 in dom D) &
(inversions D <> {} implies [f.(s,J).I.i1,f.(s,J).I.i2] in inversions D) &
(f.(s,J).(the bool-sort of S).b = TRUE iff inversions D <> {});
let D be 0-based finite array of (#INT,<=#);
assume A7: D = s.(the_array_sort_of S).M;
assume A8: y <> i1 & y <> i2;
deffunc F(Nat,Element of C-States(the generators of G))
= f.($2,J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A));
set ST = C-States(the generators of G);
A9: the_array_sort_of S <> I by Th73;
consider g being Function of NAT, ST such that
A10: g.0 = s & for i being Nat holds g.(i+1) = F(i,g.i qua Element of ST)
from NAT_1:sch 12;
A11: the carrier of (#INT,<=#) = INT by LFUZZY_0:def 3;
deffunc G(object) = g.In($1,NAT).(the_array_sort_of S).M;
A12: for x being object st x in NAT holds G(x) in INT^omega by AFINSQ_1:def 7;
consider h being Function of NAT, INT^omega such that
A13: for i being object st i in NAT holds h.i = G(i) from FUNCT_2:sch 2(A12);
A14: dom h = NAT & dom g = NAT by FUNCT_2:def 1;
then
A15: h is non empty Sequence by ORDINAL1:def 7;
then
A16: base-h = 0 by EXCHSORT:24;
then
A17: h.(base-h) = g.In(0,NAT).(the_array_sort_of S).M by A13
.= D by A7,A10;
A18: for a being Ordinal st a in dom g holds h.a is array of (#INT,<=#)
proof
let a be Ordinal;
assume a in dom g;
then a in NAT by FUNCT_2:def 1;
then h.a in INT^omega by FUNCT_2:5;
hence h.a is array of (#INT,<=#) by A11;
end;
set TV = (\falseC)-States(the generators of G, b);
hereby per cases;
suppose
A19: D = {};
then inversions D = {};
then f.(s,J).(the bool-sort of S).b <> TRUE by A6,A7;
then
A20: f.(s,J).(the bool-sort of S).b = FALSE by XBOOLEAN:def 3
.= \falseC by Th10;
f.(s,J) is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
then f.(s,J) nin TV & f complies_with_while_wrt TV
by A20,AOFA_000:def 32,AOFA_A00:def 20;
then
A21: f.(s, while(J, y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A))) =
f.(s,J);
f.(s,J).(the_array_sort_of S).M = D by A6,A7;
hence f.(s,while(J, y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A))).
(the_array_sort_of S).M is ascending permutation of D
by A19,A21,EXCHSORT:38;
end;
suppose
A22: D <> {};
defpred T[Nat] means h.$1 <> {};
A23: T[0] by A22,A17,A15,EXCHSORT:24;
A24: for i being Nat holds T[i] implies T[i+1]
proof
let i be Nat;
assume A25: T[i];
A26: I <> the_array_sort_of S by Th73;
A27: h.i = g.(In(i,NAT)).(the_array_sort_of S).M by A13
.= g.i.(the_array_sort_of S).M;
reconsider R = h.i as array of (#INT,<=#) by A18,A14,ORDINAL1:def 12;
A28: h.(i+1) = g.(In(i+1,NAT)).(the_array_sort_of S).M by A13
.= f.(g.i, J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A))
.(the_array_sort_of S).M by A10
.= f.(f.(g.i, J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)),@M.@i2:=(@y,A))
.(the_array_sort_of S).M by AOFA_000:def 29
.= f.(f.(f.(g.i, J\;y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),@M.@i2:=(@y,A))
.(the_array_sort_of S).M by AOFA_000:def 29
.= f.(f.(f.(f.(g.i, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),
@M.@i2:=(@y,A)).(the_array_sort_of S).M by AOFA_000:def 29;
@i1 value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A)))
= f.(f.(g.i, J),y:=(@M.@i1,A)).I.i1 by Th61
.= f.(g.i, J).I.i1 by A2,A3,A8,Th65;
then
A29: @i1 value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A))) in dom R by A6,A25,A27;
A30: @M value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A)))
= f.(f.(g.i, J),y:=(@M.@i1,A)).(the_array_sort_of S).M by Th61;
f.(g.i, J).(the_array_sort_of S).M = g.i.(the_array_sort_of S).M by A6;
then
A31: (f.(f.(g.i, J),y:=(@M.@i1,A))).(the_array_sort_of S).M
= g.i.(the_array_sort_of S).M by A2,A3,A26,Th65;
f.(f.(f.(g.i, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))
.(the_array_sort_of S).M
= f.(f.(f.(g.i, J),y:=(@M.@i1,A)),M:=((@M,@i1)<-(@M.@i2),A))
.(the_array_sort_of S).M by A1,A2,A3,A4,A5,Th99
.= (@M,@i1)<-(@M.@i2) value_at(C, f.(f.(g.i, J),y:=(@M.@i1,A)))
by A2,A3,Th65
.= (@M value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A))),
@i1 value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A))))
<-(@M.@i2 value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A)))) by Th80
.= (f.(f.(g.i, J),y:=(@M.@i1,A)).(the_array_sort_of S).M)+*
(@i1 value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A))),
(@M.@i2 value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A)))))
by A29,A31,A27,A30,Th74;
then
A32: dom (f.(f.(f.(g.i, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))
.(the_array_sort_of S).M)
= dom (g.i.(the_array_sort_of S).M) by A31,FUNCT_7:30;
A33: @M value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)))
= f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)).
(the_array_sort_of S).M by Th61;
@i2 value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)))
= f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)).I.i2 by Th61
.= f.(f.(f.(g.i,J),y:=(@M.@i1,A)),M:=((@M,@i1)<-(@M.@i2),A)).I.i2
by A1,A2,A3,A4,A5,Th99
.= f.(f.(g.i,J),y:=(@M.@i1,A)).I.i2 by A2,A3,Th73,Th65
.= (f.(g.i,J)).I.i2 by A2,A3,A8,Th65;
then
A34: @i2 value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)))
in dom R by A6,A25,A27;
A35: f.(f.(f.(f.(g.i, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),
@M.@i2:=(@y,A)).(the_array_sort_of S).M
= f.(f.(f.(f.(g.i, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),
M:=((@M,@i2)<-@y,A)).(the_array_sort_of S).M by A1,A2,A3,A4,A5,Th99
.= (@M,@i2)<-@y value_at(C,f.(f.(f.(g.i, J),y:=(@M.@i1,A)),
@M.@i1:=(@M.@i2,A))) by A2,A3,Th65
.= (@M value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))),
@i2 value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))))
<-(@y value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))))
by Th80
.= (@M value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))))
+*(@i2 value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))),
@y value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))))
by A32,A33,A34,A27,Th74;
dom((@M value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),
@M.@i1:=(@M.@i2,A))))
+*(@i2 value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))),
@y value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)))))
= dom (f.(f.(f.(g.i, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))
.(the_array_sort_of S).M) by A33,FUNCT_7:30;
hence h.(i+1) <> {} by A35,A28,A27,A25,A32;
end;
A36: for i being Nat holds T[i] from NAT_1:sch 2(A23,A24);
A37: for a being Nat
for R being (array of (#INT,<=#)) st R = h.a
for s st g.a = s
ex x,y being set st x = f.(s,J).I.i1 & y = f.(s,J).I.i2 &
x in dom R & y in dom R & h.(a+1) = Swap(R,x,y)
proof
let a be Nat;
let R be array of (#INT,<=#) such that
A38: R = h.a;
let s1 such that
A39: g.a = s1;
reconsider i = a as Element of NAT by ORDINAL1:def 12;
reconsider s = g.i as Element of ST;
set y1 = f.(s,J).I.i1, y2 = f.(s,J).I.i2;
take y1,y2; thus y1 = f.(s1,J).I.i1 & y2 = f.(s1,J).I.i2 by A39;
In(i,NAT) = i;
then
A40: h.i = g.i.(the_array_sort_of S).M by A13;
R <> {} by A36,A38;
hence
A41: y1 in dom R & y2 in dom R by A40,A6,A38;
A42: succ Segm a = Segm(i+1) & In(i+1,NAT) = i+1 by NAT_1:38;
then
A43: h.succ a = g.(i+1).(the_array_sort_of S).M by A13;
A44: g.(i+1) = f.(s, J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A))
by A10
.= f.(f.(s, J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)),@M.@i2:=(@y,A))
by AOFA_000:def 29
.= f.(f.(f.(s, J\;y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),@M.@i2:=(@y,A))
by AOFA_000:def 29
.= f.(f.(f.(f.(s, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),@M.@i2:=(@y,A))
by AOFA_000:def 29
.= f.(f.(f.(f.(s, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),
M:=((@M,@i2)<-@y,A)) by A1,A2,A3,A4,A5,Th99
.= f.(f.(f.(f.(s, J),y:=(@M.@i1,A)),M:=((@M,@i1)<-(@M.@i2),A)),
M:=((@M,@i2)<-@y,A)) by A1,A2,A3,A4,A5,Th99;
set s1 = f.(s,J);
set s2 = f.(s1,y:=(@M.@i1,A));
set s3 = f.(s2,M:=((@M,@i1)<-(@M.@i2),A));
set s4 = f.(s3,M:=((@M,@i2)<-@y,A));
A45: @i1 value_at(C,f.(s,J)) = f.(s,J).I.i1 by Th61;
A46: s1.(the_array_sort_of S).M = s.(the_array_sort_of S).M by A6;
A47: @M value_at(C,s1) = s1.(the_array_sort_of S).M by Th61
.= s.(the_array_sort_of S).M by A6;
A48: s2.I.y = @M.@i1 value_at(C,s1) by A2,A3,Th65
.= (@M value_at(C,s1)).(@i1 value_at(C,s1)) by Th79
.= R.y1 by A38,A41,A45,A40,A47,Th74;
A49: s2.I.i1 = y1 by A2,A3,A8,Th65;
A50: s2.I.i2 = y2 by A2,A3,A8,Th65;
A51: s2.(the_array_sort_of S).M = s1.(the_array_sort_of S).M
by A2,A3,A9,Th65;
A52: s3.I.y = s2.I.y by A2,A3,Th73,Th65;
A53: s3.I.i2 = s2.I.i2 by A2,A3,Th73,Th65;
A54: @M value_at(C,s2) = s2.(the_array_sort_of S).M by Th61;
A55: @i1 value_at(C,s2) = s2.I.i1 by Th61;
A56: @i2 value_at(C,s2) = s2.I.i2 by Th61;
A57: s3.(the_array_sort_of S).M = (@M,@i1)<-(@M.@i2) value_at(C,s2)
by A2,A3,Th65
.= (@M value_at(C,s2),@i1 value_at(C,s2))<-(@M.@i2 value_at(C,s2))
by Th80
.= R+*(@i1 value_at(C,s2),@M.@i2 value_at(C,s2))
by A38,A54,A55,A46,A51,A49,A41,A40,Th74;
A58: @M value_at(C,s3) = s3.(the_array_sort_of S).M by Th61;
A59: @i2 value_at(C,s3) = s3.I.i2 by Th61;
A60: @y value_at(C,s3) = s3.I.y by Th61;
A61: dom R = dom(R+*(y1,@M.@i2 value_at(C,s2))) by FUNCT_7:30;
A62: s4.(the_array_sort_of S).M = (@M,@i2)<-@y value_at(C,s3) by A2,A3,Th65
.= (@M value_at(C,s3),@i2 value_at(C,s3))<-(@y value_at(C,s3))
by Th80
.= R+*(y1,@M.@i2 value_at(C,s2))+*(y2, @y value_at(C,s3))
by A58,A59,A61,A55,A41,A57,A49,A53,A50,Th74;
@M.@i2 value_at(C,s2) = (@M value_at(C,s2)).(@i2 value_at(C,s2))
by Th79 .= R.y2 by A38,A41,A54,A56,A50,A51,A46,A40,Th74;
hence h.(a+1) = Swap(R,y1,y2)
by A42,A43,A44,A41,A60,A62,A52,A48,FUNCT_7:def 12;
end;
defpred Q[Nat] means h.$1 is permutation of D;
A63: Q[0] by A17,A16,EXCHSORT:38;
A64: now
let i be Nat; assume
A65: Q[i];
thus Q[i+1]
proof
reconsider R = h.i as array of (#INT,<=#) by A18,A14,ORDINAL1:def 12;
reconsider s = g.i as Element of ST;
consider x,y being set such that
x = f.(s,J).I.i1 & y = f.(s,J).I.i2 and
A66: x in dom R & y in dom R & h.(i+1) = Swap(R,x,y) by A37;
thus h.(i+1) is permutation of D by A65,A66,EXCHSORT:44;
end;
end;
A67: for i being Nat holds Q[i] from NAT_1:sch 2(A63,A64);
defpred P[Nat] means
g.$1.(the_array_sort_of S).M is ascending permutation of D;
A68: ex i being Nat st P[i]
proof
assume
A69: not thesis;
for a being Ordinal st a in dom h & succ a in dom h
ex R being (array of (#INT,<=#)), x,y being set
st [x,y] in inversions R & h.a = R & h.succ a = Swap(R,x,y)
proof
let a be Ordinal;
assume A70: a in dom h;
assume succ a in dom h;
reconsider i = a as Element of NAT by A70,FUNCT_2:def 1;
reconsider R = h.i as array of (#INT,<=#) by A11;
reconsider s = g.i as Element of ST;
set y1 = f.(s,J).I.i1, y2 = f.(s,J).I.i2;
take R, y1,y2;
In(i,NAT) = i;
then
A71: Q[i] & not P[i] & h.i = g.i.(the_array_sort_of S).M by A67,A69,A13;
then inversions R <> {} by EXCHSORT:48;
hence [y1,y2] in inversions R by A71,A6;
thus h.a = R;
A72: succ Segm i = Segm(i+1) by NAT_1:38;
consider x,y being set such that
A73: x = f.(s,J).I.i1 & y = f.(s,J).I.i2 &
x in dom R & y in dom R & h.(i+1) = Swap(R,x,y) by A37;
thus h.succ a = Swap(R,y1,y2) by A73,A72;
end;
then h is 0-based arr_computation of D by A15,A14,A17,A18
,EXCHSORT:def 14;
then h is finite by EXCHSORT:76;
hence contradiction by A14;
end;
consider B being Nat such that
A74: P[B] & for i being Nat st P[i] holds B <= i from NAT_1:sch 5(A68);
reconsider h as Sequence of INT^omega by A14,ORDINAL1:def 7;
reconsider c = h|succ B as array of INT^omega;
deffunc H(Nat) = f.(g.($1-1),J);
consider r being FinSequence such that
A75: len r = B+1 &
for i being Nat st i in dom r holds r.i = H(i) from FINSEQ_1:sch 2;
rng r c= ST
proof
let x be object; assume x in rng r;
then consider y being object such that
A76: y in dom r & x = r.y by FUNCT_1:def 3;
reconsider y as Nat by A76;
consider i being Nat such that
A77: y = 1+i by A76,FINSEQ_3:25,NAT_1:10;
x = H(y) by A75,A76 .= f.(g.i,J) by A77;
hence thesis;
end;
then reconsider r as non empty FinSequence of ST by A75,FINSEQ_1:def 4;
A78: 1 <= B+1 by NAT_1:11;
A79: r.1 = f.(g.(1-1),J) by A75,A78,FINSEQ_3:25 .= f.(s,J) by A10;
A80: r.len r = f.(g.(B+1-1),J) by A75,A78,FINSEQ_3:25 .= f.(g.B,J);
reconsider R = g.B.(the_array_sort_of S).M as ascending permutation of D
by A74;
A81: f.(g.B,J).(the_array_sort_of S).M = R by A6;
A82: f.(g.B,J) is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
inversions R = {} by EXCHSORT:48;
then f.(g.B,J).(the bool-sort of S).b <> TRUE by A6;
then f.(g.B,J).(the bool-sort of S).b = FALSE by XBOOLEAN:def 3
.= \falseC by Th10;
then
A83: r.len r nin TV by A80,A82,AOFA_A00:def 20;
for i being Nat st 1 <= i & i < len r
holds r.i in TV &
r.(i+1) = f.(r.i, y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A)\;J)
proof
let i be Nat;
assume A84: 1 <= i;
assume A85: i < len r;
consider j being Nat such that
A86: i = 1+j by A84,NAT_1:10;
A87: r.i = H(i) by A75,A84,A85,FINSEQ_3:25 .= f.(g.j,J) by A86;
In(j,NAT) = j;
then h.j = g.j.(the_array_sort_of S).M by A13;
then reconsider R = g.j.(the_array_sort_of S).M as permutation of D
by A67;
A88: f.(g.j,J) is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
i <= B by A75,A85,NAT_1:13;
then R is not ascending by A74,A86,NAT_1:13;
then inversions R <> {} by EXCHSORT:48;
then f.(g.j,J).(the bool-sort of S).b <> FALSE & FALSE = \falseC
by A6,Th10;
hence r.i in TV by A87,A88,AOFA_A00:def 20;
1 <= i+1 & i+1 <= len r by A85,NAT_1:11,13;
hence r.(i+1) = H(i+1) by A75,FINSEQ_3:25
.= f.(f.(g.j,J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A)),J)
by A10,A86
.= f.(f.(f.(g.j,J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)),@M.@i2:=(@y,A)),
J) by AOFA_000:def 29
.= f.(f.(f.(f.(g.j,J\;y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),
@M.@i2:=(@y,A)),J) by AOFA_000:def 29
.= f.(f.(f.(f.(f.(g.j,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),
@M.@i2:=(@y,A)),J) by AOFA_000:def 29
.= f.(f.(f.(f.(g.j,J),y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)),
@M.@i2:=(@y,A)),J) by AOFA_000:def 29
.= f.(f.(f.(g.j,J),y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A)),
J) by AOFA_000:def 29
.= f.(r.i,y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A)\;J)
by A87,AOFA_000:def 29;
end;
hence f.(s,while(J, y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A))).
(the_array_sort_of S).M is ascending permutation of D
by A80,A81,A79,A83,AOFA_000:86;
end;
end;
assume
A89:J is absolutely-terminating;
let s; assume s in {s1: s1.(the_array_sort_of S).M <> {}};
then consider s1 such that
A90: s = s1 & s1.(the_array_sort_of S).M <> {};
A91: the carrier of (#INT,<=#) = INT by LFUZZY_0:def 3;
reconsider D = s.(the_array_sort_of S).M as 0-based finite non empty
array of (#INT,<=#) by A91,A90;
consider g being Function of NAT, ST such that
A92: g.0 = s & for i being Nat holds g.(i+1) = F(i,g.i qua Element of ST)
from NAT_1:sch 12;
deffunc G(object) = g.In($1,NAT).(the_array_sort_of S).M;
A93: for x being object st x in NAT holds G(x) in INT^omega by AFINSQ_1:def 7;
consider h being Function of NAT, INT^omega such that
A94: for i being object st i in NAT holds h.i = G(i) from FUNCT_2:sch 2(A93);
A95: dom h = NAT & dom g = NAT by FUNCT_2:def 1;
then
A96: h is non empty Sequence by ORDINAL1:def 7;
then
A97: base-h = 0 by EXCHSORT:24;
then
A98: h.(base-h) = g.In(0,NAT).(the_array_sort_of S).M by A94
.= D by A92;
A99: for a being Ordinal st a in dom g holds h.a is array of (#INT,<=#)
proof
let a be Ordinal;
assume a in dom g;
then a in NAT by FUNCT_2:def 1;
then h.a in INT^omega by FUNCT_2:5;
hence h.a is array of (#INT,<=#) by A91;
end;
defpred T[Nat] means h.$1 <> {};
A100: T[0] by A98,A96,EXCHSORT:24;
A101: for i being Nat holds T[i] implies T[i+1]
proof
let i be Nat;
assume A102: T[i];
A103: I <> the_array_sort_of S by Th73;
A104: h.i = g.(In(i,NAT)).(the_array_sort_of S).M by A94
.= g.i.(the_array_sort_of S).M;
reconsider R = h.i as array of (#INT,<=#) by A99,A95,ORDINAL1:def 12;
A105: h.(i+1) = g.(In(i+1,NAT)).(the_array_sort_of S).M by A94
.= f.(g.i, J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A))
.(the_array_sort_of S).M by A92
.= f.(f.(g.i, J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)),@M.@i2:=(@y,A))
.(the_array_sort_of S).M by AOFA_000:def 29
.= f.(f.(f.(g.i, J\;y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),@M.@i2:=(@y,A))
.(the_array_sort_of S).M by AOFA_000:def 29
.= f.(f.(f.(f.(g.i, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),
@M.@i2:=(@y,A)).(the_array_sort_of S).M by AOFA_000:def 29;
@i1 value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A)))
= f.(f.(g.i, J),y:=(@M.@i1,A)).I.i1 by Th61
.= f.(g.i, J).I.i1 by A2,A3,A8,Th65;
then
A106: @i1 value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A))) in dom R by A6,A102,A104;
A107: @M value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A)))
= f.(f.(g.i, J),y:=(@M.@i1,A)).(the_array_sort_of S).M by Th61;
f.(g.i, J).(the_array_sort_of S).M = g.i.(the_array_sort_of S).M by A6;
then
A108: (f.(f.(g.i, J),y:=(@M.@i1,A))).(the_array_sort_of S).M
= g.i.(the_array_sort_of S).M by A2,A3,A103,Th65;
f.(f.(f.(g.i, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))
.(the_array_sort_of S).M
= f.(f.(f.(g.i, J),y:=(@M.@i1,A)),M:=((@M,@i1)<-(@M.@i2),A))
.(the_array_sort_of S).M by A1,A2,A3,A4,A5,Th99
.= (@M,@i1)<-(@M.@i2) value_at(C, f.(f.(g.i, J),y:=(@M.@i1,A)))
by A2,A3,Th65
.= (@M value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A))),
@i1 value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A))))
<-(@M.@i2 value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A)))) by Th80
.= (f.(f.(g.i, J),y:=(@M.@i1,A)).(the_array_sort_of S).M)+*
(@i1 value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A))),
(@M.@i2 value_at(C,f.(f.(g.i, J),y:=(@M.@i1,A)))))
by A106,A108,A104,A107,Th74;
then
A109: dom (f.(f.(f.(g.i, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))
.(the_array_sort_of S).M)
= dom (g.i.(the_array_sort_of S).M) by A108,FUNCT_7:30;
A110: @M value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)))
= f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)).
(the_array_sort_of S).M by Th61;
@i2 value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)))
= f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)).I.i2 by Th61
.= f.(f.(f.(g.i,J),y:=(@M.@i1,A)),M:=((@M,@i1)<-(@M.@i2),A)).I.i2
by A1,A2,A3,A4,A5,Th99
.= f.(f.(g.i,J),y:=(@M.@i1,A)).I.i2 by A2,A3,Th73,Th65
.= (f.(g.i,J)).I.i2 by A2,A3,A8,Th65;
then
A111: @i2 value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)))
in dom R by A6,A102,A104;
A112: f.(f.(f.(f.(g.i, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),
@M.@i2:=(@y,A)).(the_array_sort_of S).M
= f.(f.(f.(f.(g.i, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),
M:=((@M,@i2)<-@y,A)).(the_array_sort_of S).M by A1,A2,A3,A4,A5,Th99
.= (@M,@i2)<-@y value_at(C,f.(f.(f.(g.i, J),y:=(@M.@i1,A)),
@M.@i1:=(@M.@i2,A))) by A2,A3,Th65
.= (@M value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))),
@i2 value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))))
<-(@y value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))))
by Th80
.= (@M value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))))
+*(@i2 value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))),
@y value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))))
by A109,A110,A111,A104,Th74;
dom((@M value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),
@M.@i1:=(@M.@i2,A))))
+*(@i2 value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))),
@y value_at(C,f.(f.(f.(g.i,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)))))
= dom (f.(f.(f.(g.i, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A))
.(the_array_sort_of S).M) by A110,FUNCT_7:30;
hence h.(i+1) <> {} by A112,A105,A109,A104,A102;
end;
A113: for i being Nat holds T[i] from NAT_1:sch 2(A100,A101);
A114: for a being Nat
for R being (array of (#INT,<=#)) st R = h.a
for s st g.a = s
ex x,y being set st x = f.(s,J).I.i1 & y = f.(s,J).I.i2 &
x in dom R & y in dom R & h.(a+1) = Swap(R,x,y)
proof
let a be Nat;
let R be array of (#INT,<=#) such that
A115: R = h.a;
let s1 such that
A116: g.a = s1;
reconsider i = a as Element of NAT by ORDINAL1:def 12;
reconsider s = g.i as Element of ST;
set y1 = f.(s,J).I.i1, y2 = f.(s,J).I.i2;
take y1,y2; thus y1 = f.(s1,J).I.i1 & y2 = f.(s1,J).I.i2 by A116;
In(i,NAT) = i;
then
A117: h.i = g.i.(the_array_sort_of S).M by A94;
R <> {} by A115,A113;
hence
A118: y1 in dom R & y2 in dom R by A117,A6,A115;
A119: succ Segm i = Segm(i+1) & In(i+1,NAT) = i+1 by NAT_1:38;
then
A120: h.succ a = g.(i+1).(the_array_sort_of S).M by A94;
A121: g.(i+1) = f.(s, J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A))
by A92
.= f.(f.(s, J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)),@M.@i2:=(@y,A))
by AOFA_000:def 29
.= f.(f.(f.(s, J\;y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),@M.@i2:=(@y,A))
by AOFA_000:def 29
.= f.(f.(f.(f.(s, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),@M.@i2:=(@y,A))
by AOFA_000:def 29
.= f.(f.(f.(f.(s, J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),
M:=((@M,@i2)<-@y,A)) by A1,A2,A3,A4,A5,Th99
.= f.(f.(f.(f.(s, J),y:=(@M.@i1,A)),M:=((@M,@i1)<-(@M.@i2),A)),
M:=((@M,@i2)<-@y,A)) by A1,A2,A3,A4,A5,Th99;
set s1 = f.(s,J);
set s2 = f.(s1,y:=(@M.@i1,A));
set s3 = f.(s2,M:=((@M,@i1)<-(@M.@i2),A));
set s4 = f.(s3,M:=((@M,@i2)<-@y,A));
A122: @i1 value_at(C,f.(s,J)) = f.(s,J).I.i1 by Th61;
A123: s1.(the_array_sort_of S).M = s.(the_array_sort_of S).M by A6;
A124: @M value_at(C,s1) = s1.(the_array_sort_of S).M by Th61
.= s.(the_array_sort_of S).M by A6;
A125: s2.I.y = @M.@i1 value_at(C,s1) by A2,A3,Th65
.= (@M value_at(C,s1)).(@i1 value_at(C,s1)) by Th79
.= R.y1 by A115,A118,A122,A117,A124,Th74;
A126: s2.I.i1 = y1 by A2,A3,A8,Th65;
A127: s2.I.i2 = y2 by A2,A3,A8,Th65;
A128: s2.(the_array_sort_of S).M = s1.(the_array_sort_of S).M
by A2,A3,A9,Th65;
A129: s3.I.y = s2.I.y by A2,A3,Th73,Th65;
A130:s3.I.i2 = s2.I.i2 by A2,A3,Th73,Th65;
A131: @M value_at(C,s2) = s2.(the_array_sort_of S).M by Th61;
A132: @i1 value_at(C,s2) = s2.I.i1 by Th61;
A133: @i2 value_at(C,s2) = s2.I.i2 by Th61;
A134: s3.(the_array_sort_of S).M = (@M,@i1)<-(@M.@i2) value_at(C,s2)
by A2,A3,Th65
.= (@M value_at(C,s2),@i1 value_at(C,s2))<-(@M.@i2 value_at(C,s2))
by Th80
.= R+*(@i1 value_at(C,s2),@M.@i2 value_at(C,s2))
by A115,A131,A132,A123,A128,A126,A118,A117,Th74;
A135: @M value_at(C,s3) = s3.(the_array_sort_of S).M by Th61;
A136: @i2 value_at(C,s3) = s3.I.i2 by Th61;
A137: @y value_at(C,s3) = s3.I.y by Th61;
A138: dom R = dom(R+*(y1,@M.@i2 value_at(C,s2))) by FUNCT_7:30;
A139: s4.(the_array_sort_of S).M = (@M,@i2)<-@y value_at(C,s3) by A2,A3,Th65
.= (@M value_at(C,s3),@i2 value_at(C,s3))<-(@y value_at(C,s3))
by Th80
.= R+*(y1,@M.@i2 value_at(C,s2))+*(y2, @y value_at(C,s3))
by A135,A136,A138,A132,A118,A134,A126,A130,A127,Th74;
@M.@i2 value_at(C,s2) = (@M value_at(C,s2)).(@i2 value_at(C,s2))
by Th79 .= R.y2 by A115,A118,A131,A133,A127,A128,A123,A117,Th74;
hence h.(a+1) = Swap(R,y1,y2)
by A119,A120,A121,A118,A137,A139,A129,A125,FUNCT_7:def 12;
end;
defpred Q[Nat] means h.$1 is permutation of D;
A140: Q[0] by A98,A97,EXCHSORT:38;
A141: now
let i be Nat; assume
A142: Q[i];
thus Q[i+1]
proof
reconsider R = h.i as array of (#INT,<=#) by A99,A95,ORDINAL1:def 12;
reconsider s = g.i as Element of ST;
consider x,y being set such that
x = f.(s,J).I.i1 & y = f.(s,J).I.i2 and
A143: x in dom R & y in dom R & h.(i+1) = Swap(R,x,y) by A114;
thus h.(i+1) is permutation of D by A142,A143,EXCHSORT:44;
end;
end;
A144: for i being Nat holds Q[i] from NAT_1:sch 2(A140,A141);
defpred P[Nat] means
g.$1.(the_array_sort_of S).M is ascending permutation of D;
A145: ex i being Nat st P[i]
proof
assume
A146: not thesis;
for a being Ordinal st a in dom h & succ a in dom h
ex R being (array of (#INT,<=#)), x,y being set
st [x,y] in inversions R & h.a = R & h.succ a = Swap(R,x,y)
proof
let a be Ordinal;
assume A147: a in dom h;
assume succ a in dom h;
reconsider i = a as Element of NAT by A147,FUNCT_2:def 1;
reconsider R = h.i as array of (#INT,<=#) by A91;
reconsider s = g.i as Element of ST;
set y1 = f.(s,J).I.i1, y2 = f.(s,J).I.i2;
take R, y1,y2;
In(i,NAT) = i;
then
A148: Q[i] & not P[i] & h.i = g.i.(the_array_sort_of S).M by A144,A146,A94;
then inversions R <> {} by EXCHSORT:48;
hence [y1,y2] in inversions R by A148,A6;
thus h.a = R;
consider x,y being set such that
A149: x = f.(s,J).I.i1 & y = f.(s,J).I.i2 &
x in dom R & y in dom R & h.(i+1) = Swap(R,x,y) by A114;
succ Segm i = Segm(i+1) by NAT_1:38;
hence h.succ a = Swap(R,y1,y2) by A149;
end;
then h is 0-based arr_computation of D by A96,A95,A98,A99
,EXCHSORT:def 14;
then h is finite by EXCHSORT:76;
hence contradiction by A95;
end;
consider B being Nat such that
A150: P[B] & for i being Nat st P[i] holds B <= i from NAT_1:sch 5(A145);
reconsider h as Sequence of INT^omega by A95,ORDINAL1:def 7;
reconsider c = h|succ B as array of INT^omega;
set TV = (\falseC)-States(the generators of G, b);
deffunc H(Nat) = f.(g.($1-1),J);
consider r being FinSequence such that
A151: len r = B+1 &
for i being Nat st i in dom r holds r.i = H(i) from FINSEQ_1:sch 2;
rng r c= ST
proof
let x be object; assume x in rng r;
then consider y being object such that
A152: y in dom r & x = r.y by FUNCT_1:def 3;
reconsider y as Nat by A152;
consider i being Nat such that
A153: y = 1+i by A152,FINSEQ_3:25,NAT_1:10;
x = H(y) by A151,A152 .= f.(g.i,J) by A153;
hence thesis;
end;
then reconsider r as non empty FinSequence of ST by A151,FINSEQ_1:def 4;
A154:1 <= B+1 by NAT_1:11;
A155: r.1 = f.(g.(1-1),J) by A151,A154,FINSEQ_3:25 .= f.(s,J) by A92;
A156: r.len r = f.(g.(B+1-1),J) by A151,A154,FINSEQ_3:25 .= f.(g.B,J);
reconsider R = g.B.(the_array_sort_of S).M as ascending permutation of D
by A150;
A157: f.(g.B,J) is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
inversions R = {} by EXCHSORT:48;
then f.(g.B,J).(the bool-sort of S).b <> TRUE by A6;
then f.(g.B,J).(the bool-sort of S).b = FALSE by XBOOLEAN:def 3
.= \falseC by Th10;
then
A158: r.len r nin TV by A156,A157,AOFA_A00:def 20;
for i being Nat st 1 <= i & i < len r
holds r.i in TV &
r.(i+1) = f.(r.i, y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A)\;J)
proof
let i be Nat;
assume A159: 1 <= i;
assume A160: i < len r;
consider j being Nat such that
A161: i = 1+j by A159,NAT_1:10;
A162: r.i = H(i) by A151,A159,A160,FINSEQ_3:25 .= f.(g.j,J) by A161;
In(j,NAT) = j;
then h.j = g.j.(the_array_sort_of S).M by A94;
then reconsider R = g.j.(the_array_sort_of S).M as permutation of D
by A144;
A163: f.(g.j,J) is ManySortedFunction of the generators of G, the Sorts of C
by AOFA_A00:48;
i <= B by A151,A160,NAT_1:13;
then R is not ascending by A150,A161,NAT_1:13;
then inversions R <> {} by EXCHSORT:48;
then f.(g.j,J).(the bool-sort of S).b <> FALSE & FALSE = \falseC
by A6,Th10;
hence r.i in TV by A162,A163,AOFA_A00:def 20;
1 <= i+1 & i+1 <= len r by A160,NAT_1:11,13;
hence r.(i+1) = H(i+1) by A151,FINSEQ_3:25
.= f.(f.(g.j,J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A)),J)
by A92,A161
.= f.(f.(f.(g.j,J\;y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)),@M.@i2:=(@y,A)),J)
by AOFA_000:def 29
.= f.(f.(f.(f.(g.j,J\;y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),
@M.@i2:=(@y,A)),J) by AOFA_000:def 29
.= f.(f.(f.(f.(f.(g.j,J),y:=(@M.@i1,A)),@M.@i1:=(@M.@i2,A)),
@M.@i2:=(@y,A)),J) by AOFA_000:def 29
.= f.(f.(f.(f.(g.j,J),y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)),
@M.@i2:=(@y,A)),J) by AOFA_000:def 29
.= f.(f.(f.(g.j,J),y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A)),J)
by AOFA_000:def 29
.= f.(r.i,y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A)\;J)
by A162,AOFA_000:def 29;
end;
hence [s,while(J, y:=(@M.@i1,A)\;@M.@i1:=(@M.@i2,A)\;@M.@i2:=(@y,A))]
in TerminatingPrograms(A,ST,TV,f) by A89,A155,A158,AOFA_000:def 33,101;
end;