:: ABCMIZ_0 semantic presentation

registration

cluster non empty trivial reflexive -> non empty complete RelStr ;
coherence

proof end;

end;

definition

let T be RelStr ;
mode type of T is Element of T;

end;

definition

let T be RelStr ;
attr T is Noetherian means :Def1: :: ABCMIZ_0:def 1
the InternalRel of T is co-well_founded;

end;

:: deftheorem Def1 defines Noetherian ABCMIZ_0:def 1 :

registration

cluster non empty trivial -> non empty Noetherian RelStr ;
coherence

proof end;

end;

definition

let T be non empty RelStr ;
redefine attr T is Noetherian means :Def2: :: ABCMIZ_0:def 2
for A being non empty Subset of T ex a being Element of T st
( a in A & ( for b being Element of T st b in A holds
not a < b ) );
compatibility

proof end;

end;

:: deftheorem Def2 defines Noetherian ABCMIZ_0:def 2 :

definition

let T be Poset;
attr T is Mizar-widening-like means :: ABCMIZ_0:def 3
( T is sup-Semilattice & T is Noetherian );

end;

:: deftheorem defines Mizar-widening-like ABCMIZ_0:def 3 :

registration

cluster Mizar-widening-like -> with_suprema upper-bounded Noetherian RelStr ;
coherence

proof end;

end;

registration

cluster Noetherian -> upper-bounded Noetherian Mizar-widening-like RelStr ;
coherence

proof end;

end;

registration

cluster complete Noetherian Mizar-widening-like RelStr ;
existence

proof end;

end;

registration

let T be Noetherian RelStr ;
cluster the InternalRel of T -> co-well_founded ;
coherence by Def1;

end;

theorem Th1: :: ABCMIZ_0:1

for T being Noetherian sup-Semilattice
for I being Ideal of T holds
( ex_sup_of I,T & sup I in I )

proof end;

definition

attr c₁ is strict;
struct AdjectiveStr -> ;
aggr AdjectiveStr(# adjectives, non-op #) -> AdjectiveStr ;
sel adjectives c₁ -> set ;
sel non-op c₁ -> UnOp of the adjectives of c₁;

end;

definition

let A be AdjectiveStr ;
attr A is void means :Def4: :: ABCMIZ_0:def 4
the adjectives of A is empty;
mode adjective of A is Element of the adjectives of A;

end;

:: deftheorem Def4 defines void ABCMIZ_0:def 4 :

theorem :: ABCMIZ_0:2

for A1, A2 being AdjectiveStr st the adjectives of A1 = the adjectives of A2 & A1 is void holds
A2 is void

proof end;

definition

let A be AdjectiveStr ;
let a be Element of the adjectives of A;
func non- a -> adjective of A equals :: ABCMIZ_0:def 5
the non-op of A . a;
coherence

proof end;

end;

:: deftheorem defines non- ABCMIZ_0:def 5 :

theorem :: ABCMIZ_0:3

for A1, A2 being AdjectiveStr st AdjectiveStr(# the adjectives of A1,the non-op of A1 #) = AdjectiveStr(# the adjectives of A2,the non-op of A2 #) holds
for a1 being adjective of A1
for a2 being adjective of A2 st a1 = a2 holds
non- a1 = non- a2 ;

definition

let A be AdjectiveStr ;
attr A is involutive means :Def6: :: ABCMIZ_0:def 6
for a being adjective of A holds non- (non- a) = a;
attr A is without_fixpoints means :: ABCMIZ_0:def 7
for a being adjective of A holds not non- a = a;

end;

:: deftheorem Def6 defines involutive ABCMIZ_0:def 6 :

:: deftheorem defines without_fixpoints ABCMIZ_0:def 7 :

theorem Th4: :: ABCMIZ_0:4

for a1, a2 being set st a1 <> a2 holds
for A being AdjectiveStr st the adjectives of A = {a1,a2} & the non-op of A . a1 = a2 & the non-op of A . a2 = a1 holds
( not A is void & A is involutive & A is without_fixpoints )

proof end;

theorem Th5: :: ABCMIZ_0:5

for A1, A2 being AdjectiveStr st AdjectiveStr(# the adjectives of A1,the non-op of A1 #) = AdjectiveStr(# the adjectives of A2,the non-op of A2 #) & A1 is involutive holds
A2 is involutive

proof end;

theorem Th6: :: ABCMIZ_0:6

for A1, A2 being AdjectiveStr st AdjectiveStr(# the adjectives of A1,the non-op of A1 #) = AdjectiveStr(# the adjectives of A2,the non-op of A2 #) & A1 is without_fixpoints holds
A2 is without_fixpoints

proof end;

registration

cluster strict non void involutive without_fixpoints AdjectiveStr ;
existence

proof end;

end;

registration

let A be non void AdjectiveStr ;
cluster the adjectives of A -> non empty ;
coherence by Def4;

end;

definition

attr c₁ is strict;
struct TA-structure -> RelStr , AdjectiveStr ;
aggr TA-structure(# carrier, adjectives, InternalRel, non-op, adj-map #) -> TA-structure ;
sel adj-map c₁ -> Function of the carrier of c₁, Fin the adjectives of c₁;

end;

registration

let X be non empty set ;
let A be set ;
let r be Relation of X;
let n be UnOp of A;
let a be Function of X, Fin A;
cluster TA-structure(# X,A,r,n,a #) -> non empty ;
coherence by STRUCT_0:def 1;

end;

registration

let X be set ;
let A be non empty set ;
let r be Relation of X;
let n be UnOp of A;
let a be Function of X, Fin A;
cluster TA-structure(# X,A,r,n,a #) -> non void ;
coherence by Def4;

end;

registration

cluster non empty trivial reflexive complete Noetherian Mizar-widening-like non void involutive without_fixpoints strict TA-structure ;
existence

proof end;

end;

definition

let T be TA-structure ;
let t be Element of T;
func adjs t -> Subset of the adjectives of T equals :: ABCMIZ_0:def 8
the adj-map of T . t;
coherence

proof end;

end;

:: deftheorem defines adjs ABCMIZ_0:def 8 :

theorem :: ABCMIZ_0:7

for T1, T2 being TA-structure st TA-structure(# the carrier of T1,the adjectives of T1,the InternalRel of T1,the non-op of T1,the adj-map of T1 #) = TA-structure(# the carrier of T2,the adjectives of T2,the InternalRel of T2,the non-op of T2,the adj-map of T2 #) holds
for t1 being type of T1
for t2 being type of T2 st t1 = t2 holds
adjs t1 = adjs t2 ;

definition

let T be TA-structure ;
attr T is consistent means :Def9: :: ABCMIZ_0:def 9
for t being type of T
for a being adjective of T st a in adjs t holds
not non- a in adjs t;

end;

:: deftheorem Def9 defines consistent ABCMIZ_0:def 9 :

theorem Th8: :: ABCMIZ_0:8

proof end;

definition

let T be non empty TA-structure ;
attr T is adj-structured means :: ABCMIZ_0:def 10
the adj-map of T is join-preserving Function of T,((BoolePoset the adjectives of T) opp );

end;

:: deftheorem defines adj-structured ABCMIZ_0:def 10 :

theorem Th9: :: ABCMIZ_0:9

for T1, T2 being non empty TA-structure st TA-structure(# the carrier of T1,the adjectives of T1,the InternalRel of T1,the non-op of T1,the adj-map of T1 #) = TA-structure(# the carrier of T2,the adjectives of T2,the InternalRel of T2,the non-op of T2,the adj-map of T2 #) & T1 is adj-structured holds
T2 is adj-structured

proof end;

definition

let T be reflexive transitive antisymmetric with_suprema TA-structure ;
redefine attr T is adj-structured means :Def11: :: ABCMIZ_0:def 11
for t1, t2 being type of T holds adjs (t1 "\/" t2) = (adjs t1) /\ (adjs t2);
compatibility

proof end;

end;

:: deftheorem Def11 defines adj-structured ABCMIZ_0:def 11 :

theorem Th10: :: ABCMIZ_0:10

for T being reflexive transitive antisymmetric with_suprema TA-structure st T is adj-structured holds
for t1, t2 being type of T st t1 <= t2 holds
adjs t2 c= adjs t1

proof end;

definition

let T be TA-structure ;
let a be Element of the adjectives of T;
func types a -> Subset of T means :Def12: :: ABCMIZ_0:def 12
for x being set holds
( x in it iff ex t being type of T st
( x = t & a in adjs t ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def12 defines types ABCMIZ_0:def 12 :

definition

let T be non empty TA-structure ;
let A be Subset of the adjectives of T;
func types A -> Subset of T means :Def13: :: ABCMIZ_0:def 13
for t being type of T holds
( t in it iff for a being adjective of T st a in A holds
t in types a );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def13 defines types ABCMIZ_0:def 13 :

theorem Th11: :: ABCMIZ_0:11

for T1, T2 being TA-structure st TA-structure(# the carrier of T1,the adjectives of T1,the InternalRel of T1,the non-op of T1,the adj-map of T1 #) = TA-structure(# the carrier of T2,the adjectives of T2,the InternalRel of T2,the non-op of T2,the adj-map of T2 #) holds
for a1 being adjective of T1
for a2 being adjective of T2 st a1 = a2 holds
types a1 = types a2

proof end;

theorem :: ABCMIZ_0:12

for T being non empty TA-structure
for a being adjective of T holds types a = { t where t is type of T : a in adjs t }

proof end;

theorem Th13: :: ABCMIZ_0:13

for T being TA-structure
for t being type of T
for a being adjective of T holds
( a in adjs t iff t in types a )

proof end;

theorem Th14: :: ABCMIZ_0:14

for T being non empty TA-structure
for t being type of T
for A being Subset of the adjectives of T holds
( A c= adjs t iff t in types A )

proof end;

theorem :: ABCMIZ_0:15

for T being non void TA-structure
for t being type of T holds adjs t = { a where a is adjective of T : t in types a }

proof end;

theorem Th16: :: ABCMIZ_0:16

for T being non empty TA-structure holds types ({} the adjectives of T) = the carrier of T

proof end;

definition

let T be TA-structure ;
attr T is adjs-typed means :: ABCMIZ_0:def 14
for a being adjective of T holds not (types a) \/ (types (non- a)) is empty;

end;

:: deftheorem defines adjs-typed ABCMIZ_0:def 14 :

theorem Th17: :: ABCMIZ_0:17

proof end;

registration

cluster non empty trivial reflexive transitive antisymmetric complete upper-bounded Noetherian Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure ;
existence

proof end;

end;

theorem :: ABCMIZ_0:18

for T being consistent TA-structure
for a being adjective of T holds types a misses types (non- a)

proof end;

registration

let T be reflexive transitive antisymmetric with_suprema adj-structured TA-structure ;
let a be adjective of T;
cluster types a -> directed lower ;
coherence

proof end;

end;

registration

let T be reflexive transitive antisymmetric with_suprema adj-structured TA-structure ;
let A be Subset of the adjectives of T;
cluster types A -> directed lower ;
coherence

proof end;

end;

definition

let T be TA-structure ;
let t be Element of T;
let a be adjective of T;
pred a is_applicable_to t means :Def15: :: ABCMIZ_0:def 15
ex t' being type of T st
( t' in types a & t' <= t );

end;

:: deftheorem Def15 defines is_applicable_to ABCMIZ_0:def 15 :

definition

let T be TA-structure ;
let t be type of T;
let A be Subset of the adjectives of T;
pred A is_applicable_to t means :Def16: :: ABCMIZ_0:def 16
ex t' being type of T st
( A c= adjs t' & t' <= t );

end;

:: deftheorem Def16 defines is_applicable_to ABCMIZ_0:def 16 :

theorem :: ABCMIZ_0:19

canceled;

theorem Th20: :: ABCMIZ_0:20

for T being reflexive transitive antisymmetric with_suprema adj-structured TA-structure
for a being adjective of T
for t being type of T st a is_applicable_to t holds
(types a) /\ (downarrow t) is Ideal of T

proof end;

definition

let T be non empty reflexive transitive TA-structure ;
let t be Element of T;
let a be adjective of T;
func a ast t -> type of T equals :: ABCMIZ_0:def 17
sup ((types a) /\ (downarrow t));
coherence ;

end;

:: deftheorem defines ast ABCMIZ_0:def 17 :

theorem Th21: :: ABCMIZ_0:21

for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for t being type of T
for a being adjective of T st a is_applicable_to t holds
a ast t <= t

proof end;

theorem Th22: :: ABCMIZ_0:22

for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for t being type of T
for a being adjective of T st a is_applicable_to t holds
a in adjs (a ast t)

proof end;

theorem Th23: :: ABCMIZ_0:23

for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for t being type of T
for a being adjective of T st a is_applicable_to t holds
a ast t in types a

proof end;

theorem Th24: :: ABCMIZ_0:24

for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for t being type of T
for a being adjective of T
for t' being type of T st t' <= t & a in adjs t' holds
( a is_applicable_to t & t' <= a ast t )

proof end;

theorem Th25: :: ABCMIZ_0:25

for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for t being type of T
for a being adjective of T st a in adjs t holds
( a is_applicable_to t & a ast t = t )

proof end;

theorem :: ABCMIZ_0:26

for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for t being type of T
for a, b being adjective of T st a is_applicable_to t & b is_applicable_to a ast t holds
( b is_applicable_to t & a is_applicable_to b ast t & a ast (b ast t) = b ast (a ast t) )

proof end;

theorem Th27: :: ABCMIZ_0:27

for T being reflexive transitive antisymmetric with_suprema adj-structured TA-structure
for A being Subset of the adjectives of T
for t being type of T st A is_applicable_to t holds
(types A) /\ (downarrow t) is Ideal of T

proof end;

definition

let T be non empty reflexive transitive TA-structure ;
let t be type of T;
let A be Subset of the adjectives of T;
func A ast t -> type of T equals :: ABCMIZ_0:def 18
sup ((types A) /\ (downarrow t));
coherence ;

end;

:: deftheorem defines ast ABCMIZ_0:def 18 :

theorem Th28: :: ABCMIZ_0:28

for T being non empty reflexive transitive antisymmetric TA-structure
for t being type of T holds ({} the adjectives of T) ast t = t

proof end;

definition

let T be non empty reflexive transitive non void TA-structure ;
let t be type of T;
let p be FinSequence of the adjectives of T;
func apply p,t -> FinSequence of the carrier of T means :Def19: :: ABCMIZ_0:def 19
( len it = (len p) + 1 & it . 1 = t & ( for i being Nat
for a being adjective of T
for t being type of T st i in dom p & a = p . i & t = it . i holds
it . (i + 1) = a ast t ) );
existence

proof end;

correctness

proof end;

end;

:: deftheorem Def19 defines apply ABCMIZ_0:def 19 :

registration

let T be non empty reflexive transitive non void TA-structure ;
let t be type of T;
let p be FinSequence of the adjectives of T;
cluster apply p,t -> non empty ;
coherence

proof end;

end;

theorem :: ABCMIZ_0:29

for T being non empty reflexive transitive non void TA-structure
for t being type of T holds apply (<*> the adjectives of T),t = <*t*>

proof end;

theorem Th30: :: ABCMIZ_0:30

for T being non empty reflexive transitive non void TA-structure
for t being type of T
for a being adjective of T holds apply <*a*>,t = <*t,(a ast t)*>

proof end;

definition

let T be non empty reflexive transitive non void TA-structure ;
let t be type of T;
let v be FinSequence of the adjectives of T;
func v ast t -> type of T equals :: ABCMIZ_0:def 20
(apply v,t) . ((len v) + 1);
coherence

proof end;

end;

:: deftheorem defines ast ABCMIZ_0:def 20 :

theorem Th31: :: ABCMIZ_0:31

for T being non empty reflexive transitive non void TA-structure
for t being type of T holds (<*> the adjectives of T) ast t = t

proof end;

theorem Th32: :: ABCMIZ_0:32

for T being non empty reflexive transitive non void TA-structure
for t being type of T
for a being adjective of T holds <*a*> ast t = a ast t

proof end;

theorem :: ABCMIZ_0:33

for p, q being FinSequence
for i being natural number st i >= 1 & i < len p holds
(p $^ q) . i = p . i

proof end;

theorem Th34: :: ABCMIZ_0:34

for p being non empty FinSequence
for q being FinSequence
for i being natural number st i < len q holds
(p $^ q) . ((len p) + i) = q . (i + 1)

proof end;

theorem Th35: :: ABCMIZ_0:35

for T being non empty reflexive transitive non void TA-structure
for t being type of T
for v1, v2 being FinSequence of the adjectives of T holds apply (v1 ^ v2),t = (apply v1,t) $^ (apply v2,(v1 ast t))

proof end;

theorem Th36: :: ABCMIZ_0:36

for T being non empty reflexive transitive non void TA-structure
for t being type of T
for v1, v2 being FinSequence of the adjectives of T
for i being natural number st i in dom v1 holds
(apply (v1 ^ v2),t) . i = (apply v1,t) . i

proof end;

theorem Th37: :: ABCMIZ_0:37

for T being non empty reflexive transitive non void TA-structure
for t being type of T
for v1, v2 being FinSequence of the adjectives of T holds (apply (v1 ^ v2),t) . ((len v1) + 1) = v1 ast t

proof end;

theorem Th38: :: ABCMIZ_0:38

for T being non empty reflexive transitive non void TA-structure
for t being type of T
for v1, v2 being FinSequence of the adjectives of T holds v2 ast (v1 ast t) = (v1 ^ v2) ast t

proof end;

definition

let T be non empty reflexive transitive non void TA-structure ;
let t be type of T;
let v be FinSequence of the adjectives of T;
pred v is_applicable_to t means :Def21: :: ABCMIZ_0:def 21
for i being natural number
for a being adjective of T
for s being type of T st i in dom v & a = v . i & s = (apply v,t) . i holds
a is_applicable_to s;

end;

:: deftheorem Def21 defines is_applicable_to ABCMIZ_0:def 21 :

theorem :: ABCMIZ_0:39

for T being non empty reflexive transitive non void TA-structure
for t being type of T holds <*> the adjectives of T is_applicable_to t

proof end;

theorem Th40: :: ABCMIZ_0:40

for T being non empty reflexive transitive non void TA-structure
for t being type of T
for a being adjective of T holds
( a is_applicable_to t iff <*a*> is_applicable_to t )

proof end;

theorem Th41: :: ABCMIZ_0:41

for T being non empty reflexive transitive non void TA-structure
for t being type of T
for v1, v2 being FinSequence of the adjectives of T st v1 ^ v2 is_applicable_to t holds
( v1 is_applicable_to t & v2 is_applicable_to v1 ast t )

proof end;

theorem Th42: :: ABCMIZ_0:42

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for t being type of T
for v being FinSequence of the adjectives of T st v is_applicable_to t holds
for i1, i2 being natural number st 1 <= i1 & i1 <= i2 & i2 <= (len v) + 1 holds
for t1, t2 being type of T st t1 = (apply v,t) . i1 & t2 = (apply v,t) . i2 holds
t2 <= t1

proof end;

theorem Th43: :: ABCMIZ_0:43

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for t being type of T
for v being FinSequence of the adjectives of T st v is_applicable_to t holds
for s being type of T st s in rng (apply v,t) holds
( v ast t <= s & s <= t )

proof end;

theorem Th44: :: ABCMIZ_0:44

proof end;

theorem Th45: :: ABCMIZ_0:45

proof end;

theorem Th46: :: ABCMIZ_0:46

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for t being type of T
for v being FinSequence of the adjectives of T st v is_applicable_to t holds
for A being Subset of the adjectives of T st A = rng v holds
A is_applicable_to t

proof end;

theorem Th47: :: ABCMIZ_0:47

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for t being type of T
for v1, v2 being FinSequence of the adjectives of T st v1 is_applicable_to t & rng v2 c= rng v1 holds
for s being type of T st s in rng (apply v2,t) holds
v1 ast t <= s

proof end;

theorem Th48: :: ABCMIZ_0:48

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for t being type of T
for v1, v2 being FinSequence of the adjectives of T st v1 ^ v2 is_applicable_to t holds
v2 ^ v1 is_applicable_to t

proof end;

theorem :: ABCMIZ_0:49

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for t being type of T
for v1, v2 being FinSequence of the adjectives of T st v1 ^ v2 is_applicable_to t holds
(v1 ^ v2) ast t = (v2 ^ v1) ast t

proof end;

theorem Th50: :: ABCMIZ_0:50

for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for t being type of T
for A being Subset of the adjectives of T st A is_applicable_to t holds
A ast t <= t

proof end;

theorem Th51: :: ABCMIZ_0:51

proof end;

theorem :: ABCMIZ_0:52

proof end;

theorem Th53: :: ABCMIZ_0:53

for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for t being type of T
for A being Subset of the adjectives of T
for t' being type of T st t' <= t & A c= adjs t' holds
( A is_applicable_to t & t' <= A ast t )

proof end;

theorem :: ABCMIZ_0:54

for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for t being type of T
for A being Subset of the adjectives of T st A c= adjs t holds
( A is_applicable_to t & A ast t = t )

proof end;

theorem Th55: :: ABCMIZ_0:55

for T being TA-structure
for t being type of T
for A, B being Subset of the adjectives of T st A is_applicable_to t & B c= A holds
B is_applicable_to t

proof end;

theorem Th56: :: ABCMIZ_0:56

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for t being type of T
for a being adjective of T
for A, B being Subset of the adjectives of T st B = A \/ {a} & B is_applicable_to t holds
a ast (A ast t) = B ast t

proof end;

theorem Th57: :: ABCMIZ_0:57

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for t being type of T
for v being FinSequence of the adjectives of T st v is_applicable_to t holds
for A being Subset of the adjectives of T st A = rng v holds
v ast t = A ast t

proof end;

definition

let T be non empty non void TA-structure ;
func sub T -> Function of the adjectives of T,the carrier of T means :Def22: :: ABCMIZ_0:def 22
for a being adjective of T holds it . a = sup ((types a) \/ (types (non- a)));
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def22 defines sub ABCMIZ_0:def 22 :

definition

attr c₁ is strict;
struct TAS-structure -> TA-structure ;
aggr TAS-structure(# carrier, adjectives, InternalRel, non-op, adj-map, sub-map #) -> TAS-structure ;
sel sub-map c₁ -> Function of the adjectives of c₁,the carrier of c₁;

end;

registration

cluster non empty trivial reflexive complete Noetherian Mizar-widening-like non void strict TAS-structure ;
existence

proof end;

end;

definition

let T be non empty non void TAS-structure ;
let a be adjective of T;
func sub a -> type of T equals :: ABCMIZ_0:def 23
the sub-map of T . a;
coherence ;

end;

:: deftheorem defines sub ABCMIZ_0:def 23 :

definition

let T be non empty non void TAS-structure ;
attr T is non-absorbing means :: ABCMIZ_0:def 24
the sub-map of T * the non-op of T = the sub-map of T;
attr T is subjected means :: ABCMIZ_0:def 25
for a being adjective of T holds
( (types a) \/ (types (non- a)) is_<=_than sub a & ( types a <> {} & types (non- a) <> {} implies sub a = sup ((types a) \/ (types (non- a))) ) );

end;

:: deftheorem defines non-absorbing ABCMIZ_0:def 24 :

:: deftheorem defines subjected ABCMIZ_0:def 25 :

definition

let T be non empty non void TAS-structure ;
redefine attr T is non-absorbing means :: ABCMIZ_0:def 26
for a being adjective of T holds sub (non- a) = sub a;
compatibility

proof end;

end;

:: deftheorem defines non-absorbing ABCMIZ_0:def 26 :

definition

let T be non empty non void TAS-structure ;
let t be Element of T;
let a be adjective of T;
pred a is_properly_applicable_to t means :Def27: :: ABCMIZ_0:def 27
( t <= sub a & a is_applicable_to t );

end;

:: deftheorem Def27 defines is_properly_applicable_to ABCMIZ_0:def 27 :

definition

let T be non empty reflexive transitive non void TAS-structure ;
let t be type of T;
let v be FinSequence of the adjectives of T;
pred v is_properly_applicable_to t means :Def28: :: ABCMIZ_0:def 28
for i being natural number
for a being adjective of T
for s being type of T st i in dom v & a = v . i & s = (apply v,t) . i holds
a is_properly_applicable_to s;

end;

:: deftheorem Def28 defines is_properly_applicable_to ABCMIZ_0:def 28 :

theorem Th58: :: ABCMIZ_0:58

for T being non empty reflexive transitive non void TAS-structure
for t being type of T
for v being FinSequence of the adjectives of T st v is_properly_applicable_to t holds
v is_applicable_to t

proof end;

theorem :: ABCMIZ_0:59

for T being non empty reflexive transitive non void TAS-structure
for t being type of T holds <*> the adjectives of T is_properly_applicable_to t

proof end;

theorem :: ABCMIZ_0:60

for T being non empty reflexive transitive non void TAS-structure
for t being type of T
for a being adjective of T holds
( a is_properly_applicable_to t iff <*a*> is_properly_applicable_to t )

proof end;

theorem Th61: :: ABCMIZ_0:61

for T being non empty reflexive transitive non void TAS-structure
for t being type of T
for v1, v2 being FinSequence of the adjectives of T st v1 ^ v2 is_properly_applicable_to t holds
( v1 is_properly_applicable_to t & v2 is_properly_applicable_to v1 ast t )

proof end;

theorem Th62: :: ABCMIZ_0:62

for T being non empty reflexive transitive non void TAS-structure
for t being type of T
for v1, v2 being FinSequence of the adjectives of T st v1 is_properly_applicable_to t & v2 is_properly_applicable_to v1 ast t holds
v1 ^ v2 is_properly_applicable_to t

proof end;

definition

let T be non empty reflexive transitive non void TAS-structure ;
let t be type of T;
let A be Subset of the adjectives of T;
pred A is_properly_applicable_to t means :Def29: :: ABCMIZ_0:def 29
ex s being FinSequence of the adjectives of T st
( rng s = A & s is_properly_applicable_to t );

end;

:: deftheorem Def29 defines is_properly_applicable_to ABCMIZ_0:def 29 :

theorem Th63: :: ABCMIZ_0:63

for T being non empty reflexive transitive non void TAS-structure
for t being type of T
for A being Subset of the adjectives of T st A is_properly_applicable_to t holds
A is finite

proof end;

theorem Th64: :: ABCMIZ_0:64

for T being non empty reflexive transitive non void TAS-structure
for t being type of T holds {} the adjectives of T is_properly_applicable_to t

proof end;

scheme :: ABCMIZ_0:sch 1
MinimalFiniteSet{ P₁[ set ] } :

ex A being finite set st
( P₁[A] & ( for B being set st B c= A & P₁[B] holds
B = A ) )

provided

A1: ex A being finite set st P₁[A]

proof end;

theorem Th65: :: ABCMIZ_0:65

for T being non empty reflexive transitive non void TAS-structure
for t being type of T
for A being Subset of the adjectives of T st A is_properly_applicable_to t holds
ex B being Subset of the adjectives of T st
( B c= A & B is_properly_applicable_to t & A ast t = B ast t & ( for C being Subset of the adjectives of T st C c= B & C is_properly_applicable_to t & A ast t = C ast t holds
C = B ) )

proof end;

definition

let T be non empty reflexive transitive non void TAS-structure ;
attr T is commutative means :Def30: :: ABCMIZ_0:def 30
for t1, t2 being type of T
for a being adjective of T st a is_properly_applicable_to t1 & a ast t1 <= t2 holds
ex A being finite Subset of the adjectives of T st
( A is_properly_applicable_to t1 "\/" t2 & A ast (t1 "\/" t2) = t2 );

end;

:: deftheorem Def30 defines commutative ABCMIZ_0:def 30 :

registration

cluster non empty trivial reflexive transitive antisymmetric complete upper-bounded Noetherian Mizar-widening-like non void involutive without_fixpoints consistent adj-structured adjs-typed strict non-absorbing subjected commutative TAS-structure ;
existence

proof end;

end;

theorem Th66: :: ABCMIZ_0:66

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure
for t being type of T
for A being Subset of the adjectives of T st A is_properly_applicable_to t holds
ex s being one-to-one FinSequence of the adjectives of T st
( rng s = A & s is_properly_applicable_to t )

proof end;

definition

let T be non empty reflexive transitive non void TAS-structure ;
func T @--> -> Relation of T means :Def31: :: ABCMIZ_0:def 31
for t1, t2 being type of T holds
( [t1,t2] in it iff ex a being adjective of T st
( not a in adjs t2 & a is_properly_applicable_to t2 & a ast t2 = t1 ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def31 defines @--> ABCMIZ_0:def 31 :

theorem Th67: :: ABCMIZ_0:67

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure holds T @--> c= the InternalRel of T

proof end;

scheme :: ABCMIZ_0:sch 2
RedInd{ F₁() -> non empty set , P₁[ set , set ], F₂() -> Relation of F₁() } :

for x, y being Element of F₁() st F₂() reduces x,y holds
P₁[x,y]

provided

A1: for x, y being Element of F₁() st [x,y] in F₂() holds
P₁[x,y] and
A2: for x being Element of F₁() holds P₁[x,x] and
A3: for x, y, z being Element of F₁() st P₁[x,y] & P₁[y,z] holds
P₁[x,z]

proof end;

theorem Th68: :: ABCMIZ_0:68

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure
for t1, t2 being type of T st T @--> reduces t1,t2 holds
t1 <= t2

proof end;

theorem Th69: :: ABCMIZ_0:69

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure holds T @--> is irreflexive

proof end;

theorem Th70: :: ABCMIZ_0:70

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure holds T @--> is strongly-normalizing

proof end;

theorem Th71: :: ABCMIZ_0:71

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure
for t being type of T
for A being finite Subset of the adjectives of T st ( for C being Subset of the adjectives of T st C c= A & C is_properly_applicable_to t & A ast t = C ast t holds
C = A ) holds
for s being one-to-one FinSequence of the adjectives of T st rng s = A & s is_properly_applicable_to t holds
for i being natural number st 1 <= i & i <= len s holds
[((apply s,t) . (i + 1)),((apply s,t) . i)] in T @-->

proof end;

theorem Th72: :: ABCMIZ_0:72

proof end;

theorem Th73: :: ABCMIZ_0:73

proof end;

theorem Th74: :: ABCMIZ_0:74

for X being non empty set
for R being Relation of X
for r being RedSequence of R st r . 1 in X holds
r is FinSequence of X

proof end;

theorem Th75: :: ABCMIZ_0:75

for X being non empty set
for R being Relation of X
for x being Element of X
for y being set st R reduces x,y holds
y in X

proof end;

theorem Th76: :: ABCMIZ_0:76

for X being non empty set
for R being Relation of X st R is with_UN_property & R is weakly-normalizing holds
for x being Element of X holds nf x,R in X

proof end;

theorem Th77: :: ABCMIZ_0:77

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure
for t1, t2 being type of T st T @--> reduces t1,t2 holds
ex A being finite Subset of the adjectives of T st
( A is_properly_applicable_to t2 & t1 = A ast t2 )

proof end;

theorem Th78: :: ABCMIZ_0:78

for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure holds
( T @--> is with_Church-Rosser_property & T @--> is with_UN_property )

proof end;

definition

let T be non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure ;
let t be type of T;
func radix t -> type of T equals :: ABCMIZ_0:def 32
nf t,(T @--> );
coherence

proof end;

end;

:: deftheorem defines radix ABCMIZ_0:def 32 :

theorem Th79: :: ABCMIZ_0:79

for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for t being type of T holds T @--> reduces t, radix t

proof end;

theorem Th80: :: ABCMIZ_0:80

for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for t being type of T holds t <= radix t

proof end;

theorem Th81: :: ABCMIZ_0:81

for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for t being type of T
for X being set st X = { t' where t' is type of T : ex A being finite Subset of the adjectives of T st
( A is_properly_applicable_to t' & A ast t' = t ) } holds
( ex_sup_of X,T & radix t = "\/" X,T )

proof end;

theorem Th82: :: ABCMIZ_0:82

for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for t1, t2 being type of T
for a being adjective of T st a is_properly_applicable_to t1 & a ast t1 <= radix t2 holds
t1 <= radix t2

proof end;

theorem :: ABCMIZ_0:83

for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for t1, t2 being type of T st t1 <= t2 holds
radix t1 <= radix t2

proof end;

theorem :: ABCMIZ_0:84

for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for t being type of T
for a being adjective of T st a is_properly_applicable_to t holds
radix (a ast t) = radix t

proof end;