:: General theory and tools for proving algorithms in nominative data systems
:: by Adrian Jaszczak
::
:: Received October 25, 2020
:: Copyright (c) 2020-2021 Association of Mizar Users


registration
let D be non empty set ;
cluster non empty Relation-like NAT -defined D -valued Function-like V34() FinSequence-like FinSubsequence-like for set ;
existence
ex b1 being FinSequence st
( not b1 is empty & b1 is D -valued )
proof end;
end;

registration
let D be non empty set ;
let n be Nat;
cluster Relation-like NAT -defined D -valued Function-like V34() n -element FinSequence-like FinSubsequence-like for set ;
existence
ex b1 being FinSequence st
( b1 is D -valued & b1 is n -element )
proof end;
end;

definition
let D be non empty set ;
let f1, f2, f3, f4, f5, f6, f7 be BinominativeFunction of D;
func PP_composition (f1,f2,f3,f4,f5,f6,f7) -> BinominativeFunction of D equals :: NOMIN_8:def 1
 PP_composition ((PP_composition (f1,f2,f3,f4,f5,f6)),f7);
coherence
PP_composition ((PP_composition (f1,f2,f3,f4,f5,f6)),f7) is BinominativeFunction of D ;
end;

:: deftheorem defines PP_composition NOMIN_8:def 1 :
for D being non empty set
for f1, f2, f3, f4, f5, f6, f7 being BinominativeFunction of D holds PP_composition (f1,f2,f3,f4,f5,f6,f7) = PP_composition ((PP_composition (f1,f2,f3,f4,f5,f6)),f7);

:: WP: Unconditional composition rule for 7 programs
theorem Th1: :: NOMIN_8:1
for D being non empty set
for f1, f2, f3, f4, f5, f6, f7 being BinominativeFunction of D
for p1, p2, p3, p4, p5, p6, p7, p8 being PartialPredicate of D st <*p1,f1,p2*> is SFHT of D & <*p2,f2,p3*> is SFHT of D & <*p3,f3,p4*> is SFHT of D & <*p4,f4,p5*> is SFHT of D & <*p5,f5,p6*> is SFHT of D & <*p6,f6,p7*> is SFHT of D & <*p7,f7,p8*> is SFHT of D & <*(PP_inversion p2),f2,p3*> is SFHT of D & <*(PP_inversion p3),f3,p4*> is SFHT of D & <*(PP_inversion p4),f4,p5*> is SFHT of D & <*(PP_inversion p5),f5,p6*> is SFHT of D & <*(PP_inversion p6),f6,p7*> is SFHT of D & <*(PP_inversion p7),f7,p8*> is SFHT of D holds
<*p1,(PP_composition (f1,f2,f3,f4,f5,f6,f7)),p8*> is SFHT of D
proof end;

definition
let D be non empty set ;
let f1, f2, f3, f4, f5, f6, f7, f8 be BinominativeFunction of D;
func PP_composition (f1,f2,f3,f4,f5,f6,f7,f8) -> BinominativeFunction of D equals :: NOMIN_8:def 2
 PP_composition ((PP_composition (f1,f2,f3,f4,f5,f6,f7)),f8);
coherence
PP_composition ((PP_composition (f1,f2,f3,f4,f5,f6,f7)),f8) is BinominativeFunction of D ;
end;

:: deftheorem defines PP_composition NOMIN_8:def 2 :
for D being non empty set
for f1, f2, f3, f4, f5, f6, f7, f8 being BinominativeFunction of D holds PP_composition (f1,f2,f3,f4,f5,f6,f7,f8) = PP_composition ((PP_composition (f1,f2,f3,f4,f5,f6,f7)),f8);

:: WP: Unconditional composition rule for 8 programs
theorem Th2: :: NOMIN_8:2
for D being non empty set
for f1, f2, f3, f4, f5, f6, f7, f8 being BinominativeFunction of D
for p1, p2, p3, p4, p5, p6, p7, p8, p9 being PartialPredicate of D st <*p1,f1,p2*> is SFHT of D & <*p2,f2,p3*> is SFHT of D & <*p3,f3,p4*> is SFHT of D & <*p4,f4,p5*> is SFHT of D & <*p5,f5,p6*> is SFHT of D & <*p6,f6,p7*> is SFHT of D & <*p7,f7,p8*> is SFHT of D & <*p8,f8,p9*> is SFHT of D & <*(PP_inversion p2),f2,p3*> is SFHT of D & <*(PP_inversion p3),f3,p4*> is SFHT of D & <*(PP_inversion p4),f4,p5*> is SFHT of D & <*(PP_inversion p5),f5,p6*> is SFHT of D & <*(PP_inversion p6),f6,p7*> is SFHT of D & <*(PP_inversion p7),f7,p8*> is SFHT of D & <*(PP_inversion p8),f8,p9*> is SFHT of D holds
<*p1,(PP_composition (f1,f2,f3,f4,f5,f6,f7,f8)),p9*> is SFHT of D
proof end;

definition
let D be non empty set ;
let f1, f2, f3, f4, f5, f6, f7, f8, f9 be BinominativeFunction of D;
func PP_composition (f1,f2,f3,f4,f5,f6,f7,f8,f9) -> BinominativeFunction of D equals :: NOMIN_8:def 3
 PP_composition ((PP_composition (f1,f2,f3,f4,f5,f6,f7,f8)),f9);
coherence
PP_composition ((PP_composition (f1,f2,f3,f4,f5,f6,f7,f8)),f9) is BinominativeFunction of D ;
end;

:: deftheorem defines PP_composition NOMIN_8:def 3 :
for D being non empty set
for f1, f2, f3, f4, f5, f6, f7, f8, f9 being BinominativeFunction of D holds PP_composition (f1,f2,f3,f4,f5,f6,f7,f8,f9) = PP_composition ((PP_composition (f1,f2,f3,f4,f5,f6,f7,f8)),f9);

:: WP: Unconditional composition rule for 9 programs
theorem Th3: :: NOMIN_8:3
for D being non empty set
for f1, f2, f3, f4, f5, f6, f7, f8, f9 being BinominativeFunction of D
for p1, p2, p3, p4, p5, p6, p7, p8, p9, p10 being PartialPredicate of D st <*p1,f1,p2*> is SFHT of D & <*p2,f2,p3*> is SFHT of D & <*p3,f3,p4*> is SFHT of D & <*p4,f4,p5*> is SFHT of D & <*p5,f5,p6*> is SFHT of D & <*p6,f6,p7*> is SFHT of D & <*p7,f7,p8*> is SFHT of D & <*p8,f8,p9*> is SFHT of D & <*p9,f9,p10*> is SFHT of D & <*(PP_inversion p2),f2,p3*> is SFHT of D & <*(PP_inversion p3),f3,p4*> is SFHT of D & <*(PP_inversion p4),f4,p5*> is SFHT of D & <*(PP_inversion p5),f5,p6*> is SFHT of D & <*(PP_inversion p6),f6,p7*> is SFHT of D & <*(PP_inversion p7),f7,p8*> is SFHT of D & <*(PP_inversion p8),f8,p9*> is SFHT of D & <*(PP_inversion p9),f9,p10*> is SFHT of D holds
<*p1,(PP_composition (f1,f2,f3,f4,f5,f6,f7,f8,f9)),p10*> is SFHT of D
proof end;

definition
let D be non empty set ;
let f1, f2, f3, f4, f5, f6, f7, f8, f9, f10 be BinominativeFunction of D;
func PP_composition (f1,f2,f3,f4,f5,f6,f7,f8,f9,f10) -> BinominativeFunction of D equals :: NOMIN_8:def 4
 PP_composition ((PP_composition (f1,f2,f3,f4,f5,f6,f7,f8,f9)),f10);
coherence
PP_composition ((PP_composition (f1,f2,f3,f4,f5,f6,f7,f8,f9)),f10) is BinominativeFunction of D ;
end;

:: deftheorem defines PP_composition NOMIN_8:def 4 :
for D being non empty set
for f1, f2, f3, f4, f5, f6, f7, f8, f9, f10 being BinominativeFunction of D holds PP_composition (f1,f2,f3,f4,f5,f6,f7,f8,f9,f10) = PP_composition ((PP_composition (f1,f2,f3,f4,f5,f6,f7,f8,f9)),f10);

:: WP: Unconditional composition rule for 10 programs
theorem :: NOMIN_8:4
for D being non empty set
for f1, f2, f3, f4, f5, f6, f7, f8, f9, f10 being BinominativeFunction of D
for p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11 being PartialPredicate of D st <*p1,f1,p2*> is SFHT of D & <*p2,f2,p3*> is SFHT of D & <*p3,f3,p4*> is SFHT of D & <*p4,f4,p5*> is SFHT of D & <*p5,f5,p6*> is SFHT of D & <*p6,f6,p7*> is SFHT of D & <*p7,f7,p8*> is SFHT of D & <*p8,f8,p9*> is SFHT of D & <*p9,f9,p10*> is SFHT of D & <*p10,f10,p11*> is SFHT of D & <*(PP_inversion p2),f2,p3*> is SFHT of D & <*(PP_inversion p3),f3,p4*> is SFHT of D & <*(PP_inversion p4),f4,p5*> is SFHT of D & <*(PP_inversion p5),f5,p6*> is SFHT of D & <*(PP_inversion p6),f6,p7*> is SFHT of D & <*(PP_inversion p7),f7,p8*> is SFHT of D & <*(PP_inversion p8),f8,p9*> is SFHT of D & <*(PP_inversion p9),f9,p10*> is SFHT of D & <*(PP_inversion p10),f10,p11*> is SFHT of D holds
<*p1,(PP_composition (f1,f2,f3,f4,f5,f6,f7,f8,f9,f10)),p11*> is SFHT of D
proof end;

theorem Th5: :: NOMIN_8:5
for D being non empty set
for f1, f2 being BinominativeFunction of D
for p1, p2 being PartialPredicate of D
for q1 being total PartialPredicate of D st <*p1,f1,q1*> is SFHT of D & <*q1,f2,p2*> is SFHT of D holds
<*p1,(PP_composition (f1,f2)),p2*> is SFHT of D
proof end;

theorem Th6: :: NOMIN_8:6
for D being non empty set
for f1, f2, f3 being BinominativeFunction of D
for p1, p2 being PartialPredicate of D
for q1, q2 being total PartialPredicate of D st <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D & <*q2,f3,p2*> is SFHT of D holds
<*p1,(PP_composition (f1,f2,f3)),p2*> is SFHT of D
proof end;

theorem Th7: :: NOMIN_8:7
for D being non empty set
for f1, f2, f3, f4 being BinominativeFunction of D
for p1, p2 being PartialPredicate of D
for q1, q2, q3 being total PartialPredicate of D st <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D & <*q2,f3,q3*> is SFHT of D & <*q3,f4,p2*> is SFHT of D holds
<*p1,(PP_composition (f1,f2,f3,f4)),p2*> is SFHT of D
proof end;

theorem Th8: :: NOMIN_8:8
for D being non empty set
for f1, f2, f3, f4, f5 being BinominativeFunction of D
for p1, p2 being PartialPredicate of D
for q1, q2, q3, q4 being total PartialPredicate of D st <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D & <*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D & <*q4,f5,p2*> is SFHT of D holds
<*p1,(PP_composition (f1,f2,f3,f4,f5)),p2*> is SFHT of D
proof end;

theorem Th9: :: NOMIN_8:9
for D being non empty set
for f1, f2, f3, f4, f5, f6 being BinominativeFunction of D
for p1, p2 being PartialPredicate of D
for q1, q2, q3, q4, q5 being total PartialPredicate of D st <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D & <*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D & <*q4,f5,q5*> is SFHT of D & <*q5,f6,p2*> is SFHT of D holds
<*p1,(PP_composition (f1,f2,f3,f4,f5,f6)),p2*> is SFHT of D
proof end;

theorem Th10: :: NOMIN_8:10
for D being non empty set
for f1, f2, f3, f4, f5, f6, f7 being BinominativeFunction of D
for p1, p2 being PartialPredicate of D
for q1, q2, q3, q4, q5, q6 being total PartialPredicate of D st <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D & <*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D & <*q4,f5,q5*> is SFHT of D & <*q5,f6,q6*> is SFHT of D & <*q6,f7,p2*> is SFHT of D holds
<*p1,(PP_composition (f1,f2,f3,f4,f5,f6,f7)),p2*> is SFHT of D
proof end;

theorem Th11: :: NOMIN_8:11
for D being non empty set
for f1, f2, f3, f4, f5, f6, f7, f8 being BinominativeFunction of D
for p1, p2 being PartialPredicate of D
for q1, q2, q3, q4, q5, q6, q7 being total PartialPredicate of D st <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D & <*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D & <*q4,f5,q5*> is SFHT of D & <*q5,f6,q6*> is SFHT of D & <*q6,f7,q7*> is SFHT of D & <*q7,f8,p2*> is SFHT of D holds
<*p1,(PP_composition (f1,f2,f3,f4,f5,f6,f7,f8)),p2*> is SFHT of D
proof end;

theorem Th12: :: NOMIN_8:12
for D being non empty set
for f1, f2, f3, f4, f5, f6, f7, f8, f9 being BinominativeFunction of D
for p1, p2 being PartialPredicate of D
for q1, q2, q3, q4, q5, q6, q7, q8 being total PartialPredicate of D st <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D & <*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D & <*q4,f5,q5*> is SFHT of D & <*q5,f6,q6*> is SFHT of D & <*q6,f7,q7*> is SFHT of D & <*q7,f8,q8*> is SFHT of D & <*q8,f9,p2*> is SFHT of D holds
<*p1,(PP_composition (f1,f2,f3,f4,f5,f6,f7,f8,f9)),p2*> is SFHT of D
proof end;

theorem :: NOMIN_8:13
for D being non empty set
for f1, f2, f3, f4, f5, f6, f7, f8, f9, f10 being BinominativeFunction of D
for p1, p2 being PartialPredicate of D
for q1, q2, q3, q4, q5, q6, q7, q8, q9 being total PartialPredicate of D st <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D & <*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D & <*q4,f5,q5*> is SFHT of D & <*q5,f6,q6*> is SFHT of D & <*q6,f7,q7*> is SFHT of D & <*q7,f8,q8*> is SFHT of D & <*q8,f9,q9*> is SFHT of D & <*q9,f10,p2*> is SFHT of D holds
<*p1,(PP_composition (f1,f2,f3,f4,f5,f6,f7,f8,f9,f10)),p2*> is SFHT of D
proof end;

definition
let D be non empty set ;
let fD be PFuncs (D,D) -valued FinSequence;
assume A1: 0 < len fD ;
reconsider X = fD . 1 as Element of PFuncs (D,D) by PARTFUN1:45;
defpred S1[ Nat, object , object ] means ex g being PartFunc of D,D st
( g = $2 & $3 = PP_composition (g,(fD . ($1 + 1))) );
func PP_compositionSeq fD -> FinSequence of PFuncs (D,D) means :Def5: :: NOMIN_8:def 5
( len it = len fD & it . 1 = fD . 1 & ( for n being Nat st 1 <= n & n < len fD holds
it . (n + 1) = PP_composition ((it . n),(fD . (n + 1))) ) );
existence
ex b1 being FinSequence of PFuncs (D,D) st
( len b1 = len fD & b1 . 1 = fD . 1 & ( for n being Nat st 1 <= n & n < len fD holds
b1 . (n + 1) = PP_composition ((b1 . n),(fD . (n + 1))) ) )
proof end;
uniqueness
for b1, b2 being FinSequence of PFuncs (D,D) st len b1 = len fD & b1 . 1 = fD . 1 & ( for n being Nat st 1 <= n & n < len fD holds
b1 . (n + 1) = PP_composition ((b1 . n),(fD . (n + 1))) ) & len b2 = len fD & b2 . 1 = fD . 1 & ( for n being Nat st 1 <= n & n < len fD holds
b2 . (n + 1) = PP_composition ((b2 . n),(fD . (n + 1))) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines PP_compositionSeq NOMIN_8:def 5 :
for D being non empty set
for fD being PFuncs (b1,b1) -valued FinSequence st 0 < len fD holds
for b3 being FinSequence of PFuncs (D,D) holds
( b3 = PP_compositionSeq fD iff ( len b3 = len fD & b3 . 1 = fD . 1 & ( for n being Nat st 1 <= n & n < len fD holds
b3 . (n + 1) = PP_composition ((b3 . n),(fD . (n + 1))) ) ) );

definition
let D be non empty set ;
let fD be PFuncs (D,D) -valued FinSequence;
func PP_composition fD -> BinominativeFunction of D equals :: NOMIN_8:def 6
(PP_compositionSeq fD) . (len (PP_compositionSeq fD));
coherence
(PP_compositionSeq fD) . (len (PP_compositionSeq fD)) is BinominativeFunction of D ;
end;

:: deftheorem defines PP_composition NOMIN_8:def 6 :
for D being non empty set
for fD being PFuncs (b1,b1) -valued FinSequence holds PP_composition fD = (PP_compositionSeq fD) . (len (PP_compositionSeq fD));

definition
let D be non empty set ;
let fD be PFuncs (D,D) -valued FinSequence;
let fB be PFuncs (D,BOOLEAN) -valued FinSequence;
pred fD,fB are_composable means :: NOMIN_8:def 7
( 1 <= len fD & len fB = (len fD) + 1 & ( for n being Nat st 1 <= n & n <= len fD holds
<*(fB . n),(fD . n),(fB . (n + 1))*> is SFHT of D ) & ( for n being Nat st 2 <= n & n <= len fD holds
<*(PP_inversion (fB . n)),(fD . n),(fB . (n + 1))*> is SFHT of D ) );
end;

:: deftheorem defines are_composable NOMIN_8:def 7 :
for D being non empty set
for fD being PFuncs (b1,b1) -valued FinSequence
for fB being PFuncs (b1,BOOLEAN) -valued FinSequence holds
( fD,fB are_composable iff ( 1 <= len fD & len fB = (len fD) + 1 & ( for n being Nat st 1 <= n & n <= len fD holds
<*(fB . n),(fD . n),(fB . (n + 1))*> is SFHT of D ) & ( for n being Nat st 2 <= n & n <= len fD holds
<*(PP_inversion (fB . n)),(fD . n),(fB . (n + 1))*> is SFHT of D ) ) );

:: WP: Composition rule for sequences of programs
theorem :: NOMIN_8:14
for D being non empty set
for fD being PFuncs (b1,b1) -valued FinSequence
for fB being PFuncs (b1,BOOLEAN) -valued FinSequence st fD,fB are_composable holds
<*(fB . 1),(PP_composition fD),(fB . (len fB))*> is SFHT of D
proof end;

definition
let V, A be set ;
let val be FinSequence;
set size = len val;
set D = PFuncs ((ND (V,A)),(ND (V,A)));
defpred S1[ Nat, object ] means $2 = denaming (V,A,(val . $1));
func denamingSeq (V,A,val) -> FinSequence of PFuncs ((ND (V,A)),(ND (V,A))) means :: NOMIN_8:def 8
( len it = len val & ( for n being Nat st 1 <= n & n <= len it holds
it . n = denaming (V,A,(val . n)) ) );
existence
ex b1 being FinSequence of PFuncs ((ND (V,A)),(ND (V,A))) st
( len b1 = len val & ( for n being Nat st 1 <= n & n <= len b1 holds
b1 . n = denaming (V,A,(val . n)) ) )
proof end;
uniqueness
for b1, b2 being FinSequence of PFuncs ((ND (V,A)),(ND (V,A))) st len b1 = len val & ( for n being Nat st 1 <= n & n <= len b1 holds
b1 . n = denaming (V,A,(val . n)) ) & len b2 = len val & ( for n being Nat st 1 <= n & n <= len b2 holds
b2 . n = denaming (V,A,(val . n)) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines denamingSeq NOMIN_8:def 8 :
for V, A being set
for val being FinSequence
for b4 being FinSequence of PFuncs ((ND (V,A)),(ND (V,A))) holds
( b4 = denamingSeq (V,A,val) iff ( len b4 = len val & ( for n being Nat st 1 <= n & n <= len b4 holds
b4 . n = denaming (V,A,(val . n)) ) ) );

definition
let V, A be set ;
let loc be V -valued Function;
let val be FinSequence;
assume A1: len val > 0 ;
let p be SCPartialNominativePredicate of V,A;
set D = PFuncs ((ND (V,A)),BOOLEAN);
set size = len val;
set X = SC_Psuperpos (p,(denaming (V,A,(val . (len val)))),(loc /. (len val)));
defpred S1[ Nat, object , object ] means ex f being SCPartialNominativePredicate of V,A st
( f = $2 & $3 = SC_Psuperpos (f,(denaming (V,A,(val . ((len val) - $1)))),(loc /. ((len val) - $1))) );
func SC_Psuperpos_Seq (loc,val,p) -> FinSequence of PFuncs ((ND (V,A)),BOOLEAN) means :Def9: :: NOMIN_8:def 9
( len it = len val & it . 1 = SC_Psuperpos (p,(denaming (V,A,(val . (len val)))),(loc /. (len val))) & ( for n being Nat st 1 <= n & n < len it holds
it . (n + 1) = SC_Psuperpos ((it . n),(denaming (V,A,(val . ((len val) - n)))),(loc /. ((len val) - n))) ) );
existence
ex b1 being FinSequence of PFuncs ((ND (V,A)),BOOLEAN) st
( len b1 = len val & b1 . 1 = SC_Psuperpos (p,(denaming (V,A,(val . (len val)))),(loc /. (len val))) & ( for n being Nat st 1 <= n & n < len b1 holds
b1 . (n + 1) = SC_Psuperpos ((b1 . n),(denaming (V,A,(val . ((len val) - n)))),(loc /. ((len val) - n))) ) )
proof end;
uniqueness
for b1, b2 being FinSequence of PFuncs ((ND (V,A)),BOOLEAN) st len b1 = len val & b1 . 1 = SC_Psuperpos (p,(denaming (V,A,(val . (len val)))),(loc /. (len val))) & ( for n being Nat st 1 <= n & n < len b1 holds
b1 . (n + 1) = SC_Psuperpos ((b1 . n),(denaming (V,A,(val . ((len val) - n)))),(loc /. ((len val) - n))) ) & len b2 = len val & b2 . 1 = SC_Psuperpos (p,(denaming (V,A,(val . (len val)))),(loc /. (len val))) & ( for n being Nat st 1 <= n & n < len b2 holds
b2 . (n + 1) = SC_Psuperpos ((b2 . n),(denaming (V,A,(val . ((len val) - n)))),(loc /. ((len val) - n))) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def9 defines SC_Psuperpos_Seq NOMIN_8:def 9 :
for V, A being set
for loc being b1 -valued Function
for val being FinSequence st len val > 0 holds
for p being SCPartialNominativePredicate of V,A
for b6 being FinSequence of PFuncs ((ND (V,A)),BOOLEAN) holds
( b6 = SC_Psuperpos_Seq (loc,val,p) iff ( len b6 = len val & b6 . 1 = SC_Psuperpos (p,(denaming (V,A,(val . (len val)))),(loc /. (len val))) & ( for n being Nat st 1 <= n & n < len b6 holds
b6 . (n + 1) = SC_Psuperpos ((b6 . n),(denaming (V,A,(val . ((len val) - n)))),(loc /. ((len val) - n))) ) ) );

theorem Th15: :: NOMIN_8:15
for n, m being Nat
for V, A being set
for loc being b3 -valued Function
for d1 being NonatomicND of V,A
for size being non zero Nat
for val being b7 -element FinSequence st loc,val,size are_correct_wrt d1 & 1 <= n & n <= len (LocalOverlapSeq (A,loc,val,d1,size)) & 1 <= m & m <= len (LocalOverlapSeq (A,loc,val,d1,size)) holds
(LocalOverlapSeq (A,loc,val,d1,size)) . n in dom (denaming (V,A,(val . m)))
proof end;

definition
let V, A be set ;
let inp be FinSequence;
let d be object ;
let val be FinSequence;
pred inp is_valid_input V,A,val,d means :: NOMIN_8:def 10
 ex d1 being NonatomicND of V,A st
( d = d1 & val is_valid_wrt d1 & ( for n being Nat st 1 <= n & n <= len inp holds
d1 . (val . n) = inp . n ) );
end;

:: deftheorem defines is_valid_input NOMIN_8:def 10 :
for V, A being set
for inp being FinSequence
for d being object
for val being FinSequence holds
( inp is_valid_input V,A,val,d iff ex d1 being NonatomicND of V,A st
( d = d1 & val is_valid_wrt d1 & ( for n being Nat st 1 <= n & n <= len inp holds
d1 . (val . n) = inp . n ) ) );

definition
let V, A be set ;
let inp, val be FinSequence;
defpred S1[ object ] means inp is_valid_input V,A,val,$1;
func valid_input (V,A,val,inp) -> SCPartialNominativePredicate of V,A means :Def11: :: NOMIN_8:def 11
( dom it = ND (V,A) & ( for d being object st d in dom it holds
( ( inp is_valid_input V,A,val,d implies it . d = TRUE ) & ( not inp is_valid_input V,A,val,d implies it . d = FALSE ) ) ) );
existence
ex b1 being SCPartialNominativePredicate of V,A st
( dom b1 = ND (V,A) & ( for d being object st d in dom b1 holds
( ( inp is_valid_input V,A,val,d implies b1 . d = TRUE ) & ( not inp is_valid_input V,A,val,d implies b1 . d = FALSE ) ) ) )
proof end;
uniqueness
for b1, b2 being SCPartialNominativePredicate of V,A st dom b1 = ND (V,A) & ( for d being object st d in dom b1 holds
( ( inp is_valid_input V,A,val,d implies b1 . d = TRUE ) & ( not inp is_valid_input V,A,val,d implies b1 . d = FALSE ) ) ) & dom b2 = ND (V,A) & ( for d being object st d in dom b2 holds
( ( inp is_valid_input V,A,val,d implies b2 . d = TRUE ) & ( not inp is_valid_input V,A,val,d implies b2 . d = FALSE ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines valid_input NOMIN_8:def 11 :
for V, A being set
for inp, val being FinSequence
for b5 being SCPartialNominativePredicate of V,A holds
( b5 = valid_input (V,A,val,inp) iff ( dom b5 = ND (V,A) & ( for d being object st d in dom b5 holds
( ( inp is_valid_input V,A,val,d implies b5 . d = TRUE ) & ( not inp is_valid_input V,A,val,d implies b5 . d = FALSE ) ) ) ) );

registration
let V, A be set ;
let inp, val be FinSequence;
cluster valid_input (V,A,val,inp) -> total ;
coherence
valid_input (V,A,val,inp) is total by Def11;
end;

definition
let V, A be set ;
let d be object ;
let Z, res be FinSequence;
pred res is_valid_output V,A,Z,d means :: NOMIN_8:def 12
 ex d1 being NonatomicND of V,A st
( d = d1 & Z is_valid_wrt d1 & ( for n being Nat st 1 <= n & n <= len Z holds
d1 . (Z . n) = res . n ) );
end;

:: deftheorem defines is_valid_output NOMIN_8:def 12 :
for V, A being set
for d being object
for Z, res being FinSequence holds
( res is_valid_output V,A,Z,d iff ex d1 being NonatomicND of V,A st
( d = d1 & Z is_valid_wrt d1 & ( for n being Nat st 1 <= n & n <= len Z holds
d1 . (Z . n) = res . n ) ) );

definition
let V, A be set ;
let Z, res be object ;
set D = { d where d is TypeSCNominativeData of V,A : d in dom (denaming (V,A,Z)) } ;
defpred S1[ object ] means <*res*> is_valid_output V,A,<*Z*>,$1;
func valid_output (V,A,Z,res) -> SCPartialNominativePredicate of V,A means :: NOMIN_8:def 13
( dom it = { d where d is TypeSCNominativeData of V,A : d in dom (denaming (V,A,Z)) } & ( for d being object st d in dom it holds
( ( <*res*> is_valid_output V,A,<*Z*>,d implies it . d = TRUE ) & ( not <*res*> is_valid_output V,A,<*Z*>,d implies it . d = FALSE ) ) ) );
existence
ex b1 being SCPartialNominativePredicate of V,A st
( dom b1 = { d where d is TypeSCNominativeData of V,A : d in dom (denaming (V,A,Z)) } & ( for d being object st d in dom b1 holds
( ( <*res*> is_valid_output V,A,<*Z*>,d implies b1 . d = TRUE ) & ( not <*res*> is_valid_output V,A,<*Z*>,d implies b1 . d = FALSE ) ) ) )
proof end;
uniqueness
for b1, b2 being SCPartialNominativePredicate of V,A st dom b1 = { d where d is TypeSCNominativeData of V,A : d in dom (denaming (V,A,Z)) } & ( for d being object st d in dom b1 holds
( ( <*res*> is_valid_output V,A,<*Z*>,d implies b1 . d = TRUE ) & ( not <*res*> is_valid_output V,A,<*Z*>,d implies b1 . d = FALSE ) ) ) & dom b2 = { d where d is TypeSCNominativeData of V,A : d in dom (denaming (V,A,Z)) } & ( for d being object st d in dom b2 holds
( ( <*res*> is_valid_output V,A,<*Z*>,d implies b2 . d = TRUE ) & ( not <*res*> is_valid_output V,A,<*Z*>,d implies b2 . d = FALSE ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines valid_output NOMIN_8:def 13 :
for V, A being set
for Z, res being object
for b5 being SCPartialNominativePredicate of V,A holds
( b5 = valid_output (V,A,Z,res) iff ( dom b5 = { d where d is TypeSCNominativeData of V,A : d in dom (denaming (V,A,Z)) } & ( for d being object st d in dom b5 holds
( ( <*res*> is_valid_output V,A,<*Z*>,d implies b5 . d = TRUE ) & ( not <*res*> is_valid_output V,A,<*Z*>,d implies b5 . d = FALSE ) ) ) ) );

theorem Th16: :: NOMIN_8:16
for n being Nat
for V, A being set
for loc being b2 -valued Function
for d1 being NonatomicND of V,A
for p being SCPartialNominativePredicate of V,A
for d being object
for size being non zero Nat
for val being b8 -element FinSequence st loc,val,size are_correct_wrt d1 & d = (LocalOverlapSeq (A,loc,val,d1,size)) . (size - 1) & 2 <= n + 1 & n + 1 < size & local_overlapping (V,A,d,((denaming (V,A,(val . (len val)))) . d),(loc /. (len val))) in dom p holds
local_overlapping (V,A,((LocalOverlapSeq (A,loc,val,d1,size)) . ((size - n) - 1)),((denaming (V,A,(val . ((len val) - n)))) . ((LocalOverlapSeq (A,loc,val,d1,size)) . ((size - n) - 1))),(loc /. ((len val) - n))) in dom ((SC_Psuperpos_Seq (loc,val,p)) . n)
proof end;

theorem Th17: :: NOMIN_8:17
for V, A being set
for loc being b1 -valued Function
for d1 being NonatomicND of V,A
for p being SCPartialNominativePredicate of V,A
for d being object
for size being non zero Nat
for val being b7 -element FinSequence st loc,val,size are_correct_wrt d1 & d = (LocalOverlapSeq (A,loc,val,d1,size)) . (size - 1) & local_overlapping (V,A,d,((denaming (V,A,(val . (len val)))) . d),(loc /. (len val))) in dom p holds
for m, n being Nat st 1 <= m & m < size & 1 <= n & n < size holds
((SC_Psuperpos_Seq (loc,val,p)) . m) . ((LocalOverlapSeq (A,loc,val,d1,size)) . (size - m)) = ((SC_Psuperpos_Seq (loc,val,p)) . n) . ((LocalOverlapSeq (A,loc,val,d1,size)) . (size - n))
proof end;

theorem :: NOMIN_8:18
for V, A being set
for loc being b1 -valued Function
for d1 being NonatomicND of V,A
for p being SCPartialNominativePredicate of V,A
for size being non zero Nat
for val being b6 -element FinSequence
for dx, dy being object
for NN being Nat st NN = size - 2 & loc,val,size are_correct_wrt d1 & dx = (LocalOverlapSeq (A,loc,val,d1,size)) . (size - 1) & local_overlapping (V,A,dx,((denaming (V,A,(val . (len val)))) . dx),(loc /. (len val))) in dom p & dy = local_overlapping (V,A,((LocalOverlapSeq (A,loc,val,d1,size)) . NN),((denaming (V,A,(val . (NN + 1)))) . ((LocalOverlapSeq (A,loc,val,d1,size)) . NN)),(loc /. (NN + 1))) & local_overlapping (V,A,dy,((denaming (V,A,(val . (len val)))) . dy),(loc /. (len val))) in dom p holds
((SC_Psuperpos_Seq (loc,val,p)) . 1) . ((LocalOverlapSeq (A,loc,val,d1,size)) . (size - 1)) = p . ((LocalOverlapSeq (A,loc,val,d1,size)) . size)
proof end;

theorem :: NOMIN_8:19
for V, A being set
for loc being b1 -valued Function
for d1 being NonatomicND of V,A
for size being non zero Nat
for val being b5 -element FinSequence
for p being SCPartialNominativePredicate of V,A st 3 <= size & loc,val,size are_correct_wrt d1 & local_overlapping (V,A,((LocalOverlapSeq (A,loc,val,d1,size)) . (size - 1)),((denaming (V,A,(val . (len val)))) . ((LocalOverlapSeq (A,loc,val,d1,size)) . (size - 1))),(loc /. (len val))) in dom p & local_overlapping (V,A,d1,((denaming (V,A,(val . 1))) . d1),(loc /. 1)) in dom ((SC_Psuperpos_Seq (loc,val,p)) . (size - 1)) holds
((SC_Psuperpos_Seq (loc,val,p)) . (len (SC_Psuperpos_Seq (loc,val,p)))) . d1 = (SC_Psuperpos (((SC_Psuperpos_Seq (loc,val,p)) . (size - 2)),(denaming (V,A,(val . 2))),(loc /. 2))) . ((LocalOverlapSeq (A,loc,val,d1,size)) . 1)
proof end;

definition
let V, A be set ;
let loc be V -valued Function;
let d1 be NonatomicND of V,A;
let pos be FinSequence;
let prg be FPrg (ND (V,A)) -valued FinSequence;
assume A1: len prg > 0 ;
defpred S1[ Nat, object , object ] means $3 = local_overlapping (V,A,$2,((prg . ($1 + 1)) . $2),(loc /. (pos . ($1 + 1))));
set X = local_overlapping (V,A,d1,((prg . 1) . d1),(loc /. (pos . 1)));
func PrgLocalOverlapSeq (A,loc,d1,prg,pos) -> FinSequence of ND (V,A) means :Def14: :: NOMIN_8:def 14
( len it = len prg & it . 1 = local_overlapping (V,A,d1,((prg . 1) . d1),(loc /. (pos . 1))) & ( for n being Nat st 1 <= n & n < len it holds
it . (n + 1) = local_overlapping (V,A,(it . n),((prg . (n + 1)) . (it . n)),(loc /. (pos . (n + 1)))) ) );
existence
ex b1 being FinSequence of ND (V,A) st
( len b1 = len prg & b1 . 1 = local_overlapping (V,A,d1,((prg . 1) . d1),(loc /. (pos . 1))) & ( for n being Nat st 1 <= n & n < len b1 holds
b1 . (n + 1) = local_overlapping (V,A,(b1 . n),((prg . (n + 1)) . (b1 . n)),(loc /. (pos . (n + 1)))) ) )
proof end;
uniqueness
for b1, b2 being FinSequence of ND (V,A) st len b1 = len prg & b1 . 1 = local_overlapping (V,A,d1,((prg . 1) . d1),(loc /. (pos . 1))) & ( for n being Nat st 1 <= n & n < len b1 holds
b1 . (n + 1) = local_overlapping (V,A,(b1 . n),((prg . (n + 1)) . (b1 . n)),(loc /. (pos . (n + 1)))) ) & len b2 = len prg & b2 . 1 = local_overlapping (V,A,d1,((prg . 1) . d1),(loc /. (pos . 1))) & ( for n being Nat st 1 <= n & n < len b2 holds
b2 . (n + 1) = local_overlapping (V,A,(b2 . n),((prg . (n + 1)) . (b2 . n)),(loc /. (pos . (n + 1)))) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def14 defines PrgLocalOverlapSeq NOMIN_8:def 14 :
for V, A being set
for loc being b1 -valued Function
for d1 being NonatomicND of V,A
for pos being FinSequence
for prg being FPrg (ND (b1,b2)) -valued FinSequence st len prg > 0 holds
for b7 being FinSequence of ND (V,A) holds
( b7 = PrgLocalOverlapSeq (A,loc,d1,prg,pos) iff ( len b7 = len prg & b7 . 1 = local_overlapping (V,A,d1,((prg . 1) . d1),(loc /. (pos . 1))) & ( for n being Nat st 1 <= n & n < len b7 holds
b7 . (n + 1) = local_overlapping (V,A,(b7 . n),((prg . (n + 1)) . (b7 . n)),(loc /. (pos . (n + 1)))) ) ) );

registration
let V, A be set ;
let loc be V -valued Function;
let d1 be NonatomicND of V,A;
let prg be non empty FPrg (ND (V,A)) -valued FinSequence;
let pos be FinSequence;
cluster PrgLocalOverlapSeq (A,loc,d1,prg,pos) -> V,A -NonatomicND-yielding ;
coherence
PrgLocalOverlapSeq (A,loc,d1,prg,pos) is V,A -NonatomicND-yielding
proof end;
end;

registration
let V, A be set ;
let loc be V -valued Function;
let d1 be NonatomicND of V,A;
let prg be non empty FPrg (ND (V,A)) -valued FinSequence;
let pos be FinSequence;
let n be Nat;
cluster (PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . n -> Relation-like Function-like ;
coherence
( (PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . n is Function-like & (PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . n is Relation-like )
proof end;
end;

definition
let V, A be set ;
let loc be V -valued Function;
let d1 be NonatomicND of V,A;
let prg be non empty FPrg (ND (V,A)) -valued FinSequence;
let pos be FinSequence;
pred prg_doms_of loc,d1,prg,pos means :: NOMIN_8:def 15
 for n being Nat st 1 <= n & n < len prg holds
(PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . n in dom (prg . (n + 1));
end;

:: deftheorem defines prg_doms_of NOMIN_8:def 15 :
for V, A being set
for loc being b1 -valued Function
for d1 being NonatomicND of V,A
for prg being non empty FPrg (ND (b1,b2)) -valued FinSequence
for pos being FinSequence holds
( prg_doms_of loc,d1,prg,pos iff for n being Nat st 1 <= n & n < len prg holds
(PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . n in dom (prg . (n + 1)) );

theorem Th20: :: NOMIN_8:20
for n, m being Nat
for V, A being set
for loc being b3 -valued Function
for d1 being NonatomicND of V,A
for pos being FinSequence
for prg being non empty FPrg (ND (b3,b4)) -valued FinSequence st 1 <= n & n <= len prg & (PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . m in dom (prg . n) holds
(prg . n) . ((PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . m) is TypeSCNominativeData of V,A
proof end;

theorem Th21: :: NOMIN_8:21
for V, A being set
for loc being b1 -valued Function
for d1 being NonatomicND of V,A
for pos being FinSequence
for prg being non empty FPrg (ND (b1,b2)) -valued FinSequence st not V is empty & A is_without_nonatomicND_wrt V holds
for n being Nat st 1 <= n & n < len prg & (PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . n in dom (prg . (n + 1)) holds
dom ((PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . (n + 1)) = {(loc /. (pos . (n + 1)))} \/ (dom ((PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . n))
proof end;

theorem Th22: :: NOMIN_8:22
for V, A being set
for loc being b1 -valued Function
for d1 being NonatomicND of V,A
for pos being FinSequence
for prg being non empty FPrg (ND (b1,b2)) -valued FinSequence st not V is empty & A is_without_nonatomicND_wrt V holds
for n being Nat st 1 <= n & n < len prg & (PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . n in dom (prg . (n + 1)) holds
dom ((PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . n) c= dom ((PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . (n + 1))
proof end;

theorem :: NOMIN_8:23
for V, A being set
for loc being b1 -valued Function
for d1 being NonatomicND of V,A
for pos being FinSequence
for prg being non empty FPrg (ND (b1,b2)) -valued FinSequence st not V is empty & A is_without_nonatomicND_wrt V & dom (PrgLocalOverlapSeq (A,loc,d1,prg,pos)) c= dom prg & d1 in dom (prg . 1) & prg_doms_of loc,d1,prg,pos holds
for n being Nat st 1 <= n & n <= len prg holds
dom d1 c= dom ((PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . n)
proof end;

theorem Th24: :: NOMIN_8:24
for V, A being set
for loc being b1 -valued Function
for d1 being NonatomicND of V,A
for pos being FinSequence
for prg being non empty FPrg (ND (b1,b2)) -valued FinSequence st not V is empty & A is_without_nonatomicND_wrt V & prg_doms_of loc,d1,prg,pos holds
for m, n being Nat st 1 <= n & n <= m & m <= len prg holds
dom ((PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . n) c= dom ((PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . m)
proof end;

theorem :: NOMIN_8:25
for V, A being set
for loc being b1 -valued Function
for d1 being NonatomicND of V,A
for pos being FinSequence
for prg being non empty FPrg (ND (b1,b2)) -valued FinSequence st not V is empty & A is_without_nonatomicND_wrt V & dom (PrgLocalOverlapSeq (A,loc,d1,prg,pos)) c= dom prg & d1 in dom (prg . 1) & prg_doms_of loc,d1,prg,pos holds
for m, n being Nat st 1 <= n & n <= m & m <= len prg holds
loc /. (pos . n) in dom ((PrgLocalOverlapSeq (A,loc,d1,prg,pos)) . m)
proof end;