Article ABCMIZ_0, MML version 4.99.1005
:: ABCMIZ_0:condreg 1
registration
cluster non empty trivial reflexive -> complete (RelStr);
end;
:: ABCMIZ_0:modenot 1
definition
let a1 be RelStr;
mode type of a1 is Element of the carrier of a1;
end;
:: ABCMIZ_0:attrnot 1 => ABCMIZ_0:attr 1
definition
let a1 be RelStr;
attr a1 is Noetherian means
the InternalRel of a1 is co-well_founded;
end;
:: ABCMIZ_0:dfs 1
definiens
let a1 be RelStr;
To prove
a1 is Noetherian
it is sufficient to prove
thus the InternalRel of a1 is co-well_founded;
:: ABCMIZ_0:def 1
theorem
for b1 being RelStr holds
b1 is Noetherian
iff
the InternalRel of b1 is co-well_founded;
:: ABCMIZ_0:condreg 2
registration
cluster non empty trivial -> Noetherian (RelStr);
end;
:: ABCMIZ_0:attrnot 2 => ABCMIZ_0:attr 1
definition
let a1 be RelStr;
attr a1 is Noetherian means
for b1 being non empty Element of bool the carrier of a1 holds
ex b2 being Element of the carrier of a1 st
b2 in b1 &
(for b3 being Element of the carrier of a1
st b3 in b1
holds not b2 < b3);
end;
:: ABCMIZ_0:dfs 2
definiens
let a1 be non empty RelStr;
To prove
a1 is Noetherian
it is sufficient to prove
thus for b1 being non empty Element of bool the carrier of a1 holds
ex b2 being Element of the carrier of a1 st
b2 in b1 &
(for b3 being Element of the carrier of a1
st b3 in b1
holds not b2 < b3);
:: ABCMIZ_0:def 2
theorem
for b1 being non empty RelStr holds
b1 is Noetherian
iff
for b2 being non empty Element of bool the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
b3 in b2 &
(for b4 being Element of the carrier of b1
st b4 in b2
holds not b3 < b4);
:: ABCMIZ_0:attrnot 3 => ABCMIZ_0:attr 2
definition
let a1 be reflexive transitive antisymmetric RelStr;
attr a1 is Mizar-widening-like means
a1 is reflexive transitive antisymmetric with_suprema RelStr & a1 is Noetherian;
end;
:: ABCMIZ_0:dfs 3
definiens
let a1 be reflexive transitive antisymmetric RelStr;
To prove
a1 is Mizar-widening-like
it is sufficient to prove
thus a1 is reflexive transitive antisymmetric with_suprema RelStr & a1 is Noetherian;
:: ABCMIZ_0:def 3
theorem
for b1 being reflexive transitive antisymmetric RelStr holds
b1 is Mizar-widening-like
iff
b1 is reflexive transitive antisymmetric with_suprema RelStr & b1 is Noetherian;
:: ABCMIZ_0:condreg 3
registration
cluster reflexive transitive antisymmetric Mizar-widening-like -> with_suprema upper-bounded Noetherian (RelStr);
end;
:: ABCMIZ_0:condreg 4
registration
cluster reflexive transitive antisymmetric with_suprema Noetherian -> Mizar-widening-like (RelStr);
end;
:: ABCMIZ_0:exreg 1
registration
cluster non empty reflexive transitive antisymmetric with_suprema with_infima complete lower-bounded upper-bounded bounded up-complete /\-complete Mizar-widening-like RelStr;
end;
:: ABCMIZ_0:funcreg 1
registration
let a1 be Noetherian RelStr;
cluster the InternalRel of a1 -> co-well_founded;
end;
:: ABCMIZ_0:th 1
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian RelStr
for b2 being non empty directed lower Element of bool the carrier of b1 holds
ex_sup_of b2,b1 & "\/"(b2,b1) in b2;
:: ABCMIZ_0:structnot 1 => ABCMIZ_0:struct 1
definition
struct() AdjectiveStr(#
adjectives -> set,
non-op -> Function-like quasi_total Relation of the adjectives of it,the adjectives of it
#);
end;
:: ABCMIZ_0:attrnot 4 => ABCMIZ_0:attr 3
definition
let a1 be AdjectiveStr;
attr a1 is strict;
end;
:: ABCMIZ_0:exreg 2
registration
cluster strict AdjectiveStr;
end;
:: ABCMIZ_0:aggrnot 1 => ABCMIZ_0:aggr 1
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of a1,a1;
aggr AdjectiveStr(#a1,a2#) -> strict AdjectiveStr;
end;
:: ABCMIZ_0:selnot 1 => ABCMIZ_0:sel 1
definition
let a1 be AdjectiveStr;
sel the adjectives of a1 -> set;
end;
:: ABCMIZ_0:selnot 2 => ABCMIZ_0:sel 2
definition
let a1 be AdjectiveStr;
sel the non-op of a1 -> Function-like quasi_total Relation of the adjectives of a1,the adjectives of a1;
end;
:: ABCMIZ_0:attrnot 5 => ABCMIZ_0:attr 4
definition
let a1 be AdjectiveStr;
attr a1 is void means
the adjectives of a1 is empty;
end;
:: ABCMIZ_0:dfs 4
definiens
let a1 be AdjectiveStr;
To prove
a1 is void
it is sufficient to prove
thus the adjectives of a1 is empty;
:: ABCMIZ_0:def 4
theorem
for b1 being AdjectiveStr holds
b1 is void
iff
the adjectives of b1 is empty;
:: ABCMIZ_0:modenot 2
definition
let a1 be AdjectiveStr;
mode adjective of a1 is Element of the adjectives of a1;
end;
:: ABCMIZ_0:th 2
theorem
for b1, b2 being AdjectiveStr
st the adjectives of b1 = the adjectives of b2 & b1 is void
holds b2 is void;
:: ABCMIZ_0:funcnot 1 => ABCMIZ_0:func 1
definition
let a1 be AdjectiveStr;
let a2 be Element of the adjectives of a1;
func non- A2 -> Element of the adjectives of a1 equals
(the non-op of a1) . a2;
end;
:: ABCMIZ_0:def 5
theorem
for b1 being AdjectiveStr
for b2 being Element of the adjectives of b1 holds
non- b2 = (the non-op of b1) . b2;
:: ABCMIZ_0:th 3
theorem
for b1, b2 being AdjectiveStr
st AdjectiveStr(#the adjectives of b1,the non-op of b1#) = AdjectiveStr(#the adjectives of b2,the non-op of b2#)
for b3 being Element of the adjectives of b1
for b4 being Element of the adjectives of b2
st b3 = b4
holds non- b3 = non- b4;
:: ABCMIZ_0:attrnot 6 => ABCMIZ_0:attr 5
definition
let a1 be AdjectiveStr;
attr a1 is involutive means
for b1 being Element of the adjectives of a1 holds
non- non- b1 = b1;
end;
:: ABCMIZ_0:dfs 6
definiens
let a1 be AdjectiveStr;
To prove
a1 is involutive
it is sufficient to prove
thus for b1 being Element of the adjectives of a1 holds
non- non- b1 = b1;
:: ABCMIZ_0:def 6
theorem
for b1 being AdjectiveStr holds
b1 is involutive
iff
for b2 being Element of the adjectives of b1 holds
non- non- b2 = b2;
:: ABCMIZ_0:attrnot 7 => ABCMIZ_0:attr 6
definition
let a1 be AdjectiveStr;
attr a1 is without_fixpoints means
for b1 being Element of the adjectives of a1 holds
non- b1 <> b1;
end;
:: ABCMIZ_0:dfs 7
definiens
let a1 be AdjectiveStr;
To prove
a1 is without_fixpoints
it is sufficient to prove
thus for b1 being Element of the adjectives of a1 holds
non- b1 <> b1;
:: ABCMIZ_0:def 7
theorem
for b1 being AdjectiveStr holds
b1 is without_fixpoints
iff
for b2 being Element of the adjectives of b1 holds
non- b2 <> b2;
:: ABCMIZ_0:th 4
theorem
for b1, b2 being set
st b1 <> b2
for b3 being AdjectiveStr
st the adjectives of b3 = {b1,b2} & (the non-op of b3) . b1 = b2 & (the non-op of b3) . b2 = b1
holds b3 is not void & b3 is involutive & b3 is without_fixpoints;
:: ABCMIZ_0:th 5
theorem
for b1, b2 being AdjectiveStr
st AdjectiveStr(#the adjectives of b1,the non-op of b1#) = AdjectiveStr(#the adjectives of b2,the non-op of b2#) &
b1 is involutive
holds b2 is involutive;
:: ABCMIZ_0:th 6
theorem
for b1, b2 being AdjectiveStr
st AdjectiveStr(#the adjectives of b1,the non-op of b1#) = AdjectiveStr(#the adjectives of b2,the non-op of b2#) &
b1 is without_fixpoints
holds b2 is without_fixpoints;
:: ABCMIZ_0:exreg 3
registration
cluster strict non void involutive without_fixpoints AdjectiveStr;
end;
:: ABCMIZ_0:funcreg 2
registration
let a1 be non void AdjectiveStr;
cluster the adjectives of a1 -> non empty;
end;
:: ABCMIZ_0:structnot 2 => ABCMIZ_0:struct 2
definition
struct(RelStr, AdjectiveStr) TA-structure(#
carrier -> set,
adjectives -> set,
InternalRel -> Relation of the carrier of it,the carrier of it,
non-op -> Function-like quasi_total Relation of the adjectives of it,the adjectives of it,
adj-map -> Function-like quasi_total Relation of the carrier of it,Fin the adjectives of it
#);
end;
:: ABCMIZ_0:attrnot 8 => ABCMIZ_0:attr 7
definition
let a1 be TA-structure;
attr a1 is strict;
end;
:: ABCMIZ_0:exreg 4
registration
cluster strict TA-structure;
end;
:: ABCMIZ_0:aggrnot 2 => ABCMIZ_0:aggr 2
definition
let a1, a2 be set;
let a3 be Relation of a1,a1;
let a4 be Function-like quasi_total Relation of a2,a2;
let a5 be Function-like quasi_total Relation of a1,Fin a2;
aggr TA-structure(#a1,a2,a3,a4,a5#) -> strict TA-structure;
end;
:: ABCMIZ_0:selnot 3 => ABCMIZ_0:sel 3
definition
let a1 be TA-structure;
sel the adj-map of a1 -> Function-like quasi_total Relation of the carrier of a1,Fin the adjectives of a1;
end;
:: ABCMIZ_0:funcreg 3
registration
let a1 be non empty set;
let a2 be set;
let a3 be Relation of a1,a1;
let a4 be Function-like quasi_total Relation of a2,a2;
let a5 be Function-like quasi_total Relation of a1,Fin a2;
cluster TA-structure(#a1,a2,a3,a4,a5#) -> non empty strict;
end;
:: ABCMIZ_0:funcreg 4
registration
let a1 be set;
let a2 be non empty set;
let a3 be Relation of a1,a1;
let a4 be Function-like quasi_total Relation of a2,a2;
let a5 be Function-like quasi_total Relation of a1,Fin a2;
cluster TA-structure(#a1,a2,a3,a4,a5#) -> non void strict;
end;
:: ABCMIZ_0:exreg 5
registration
cluster non empty trivial reflexive non void involutive without_fixpoints strict TA-structure;
end;
:: ABCMIZ_0:funcnot 2 => ABCMIZ_0:func 2
definition
let a1 be TA-structure;
let a2 be Element of the carrier of a1;
func adjs A2 -> Element of bool the adjectives of a1 equals
(the adj-map of a1) . a2;
end;
:: ABCMIZ_0:def 8
theorem
for b1 being TA-structure
for b2 being Element of the carrier of b1 holds
adjs b2 = (the adj-map of b1) . b2;
:: ABCMIZ_0:th 7
theorem
for b1, b2 being TA-structure
st TA-structure(#the carrier of b1,the adjectives of b1,the InternalRel of b1,the non-op of b1,the adj-map of b1#) = TA-structure(#the carrier of b2,the adjectives of b2,the InternalRel of b2,the non-op of b2,the adj-map of b2#)
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
st b3 = b4
holds adjs b3 = adjs b4;
:: ABCMIZ_0:attrnot 9 => ABCMIZ_0:attr 8
definition
let a1 be TA-structure;
attr a1 is consistent means
for b1 being Element of the carrier of a1
for b2 being Element of the adjectives of a1
st b2 in adjs b1
holds not non- b2 in adjs b1;
end;
:: ABCMIZ_0:dfs 9
definiens
let a1 be TA-structure;
To prove
a1 is consistent
it is sufficient to prove
thus for b1 being Element of the carrier of a1
for b2 being Element of the adjectives of a1
st b2 in adjs b1
holds not non- b2 in adjs b1;
:: ABCMIZ_0:def 9
theorem
for b1 being TA-structure holds
b1 is consistent
iff
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1
st b3 in adjs b2
holds not non- b3 in adjs b2;
:: ABCMIZ_0:th 8
theorem
for b1, b2 being TA-structure
st TA-structure(#the carrier of b1,the adjectives of b1,the InternalRel of b1,the non-op of b1,the adj-map of b1#) = TA-structure(#the carrier of b2,the adjectives of b2,the InternalRel of b2,the non-op of b2,the adj-map of b2#) &
b1 is consistent
holds b2 is consistent;
:: ABCMIZ_0:attrnot 10 => ABCMIZ_0:attr 9
definition
let a1 be non empty TA-structure;
attr a1 is adj-structured means
the adj-map of a1 is Function-like quasi_total join-preserving Relation of the carrier of a1,the carrier of (BoolePoset the adjectives of a1) ~;
end;
:: ABCMIZ_0:dfs 10
definiens
let a1 be non empty TA-structure;
To prove
a1 is adj-structured
it is sufficient to prove
thus the adj-map of a1 is Function-like quasi_total join-preserving Relation of the carrier of a1,the carrier of (BoolePoset the adjectives of a1) ~;
:: ABCMIZ_0:def 10
theorem
for b1 being non empty TA-structure holds
b1 is adj-structured
iff
the adj-map of b1 is Function-like quasi_total join-preserving Relation of the carrier of b1,the carrier of (BoolePoset the adjectives of b1) ~;
:: ABCMIZ_0:th 9
theorem
for b1, b2 being non empty TA-structure
st TA-structure(#the carrier of b1,the adjectives of b1,the InternalRel of b1,the non-op of b1,the adj-map of b1#) = TA-structure(#the carrier of b2,the adjectives of b2,the InternalRel of b2,the non-op of b2,the adj-map of b2#) &
b1 is adj-structured
holds b2 is adj-structured;
:: ABCMIZ_0:attrnot 11 => ABCMIZ_0:attr 9
definition
let a1 be non empty TA-structure;
attr a1 is adj-structured means
for b1, b2 being Element of the carrier of a1 holds
adjs (b1 "\/" b2) = (adjs b1) /\ adjs b2;
end;
:: ABCMIZ_0:dfs 11
definiens
let a1 be reflexive transitive antisymmetric with_suprema TA-structure;
To prove
a1 is adj-structured
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
adjs (b1 "\/" b2) = (adjs b1) /\ adjs b2;
:: ABCMIZ_0:def 11
theorem
for b1 being reflexive transitive antisymmetric with_suprema TA-structure holds
b1 is adj-structured
iff
for b2, b3 being Element of the carrier of b1 holds
adjs (b2 "\/" b3) = (adjs b2) /\ adjs b3;
:: ABCMIZ_0:th 10
theorem
for b1 being reflexive transitive antisymmetric with_suprema TA-structure
st b1 is adj-structured
for b2, b3 being Element of the carrier of b1
st b2 <= b3
holds adjs b3 c= adjs b2;
:: ABCMIZ_0:funcnot 3 => ABCMIZ_0:func 3
definition
let a1 be TA-structure;
let a2 be Element of the adjectives of a1;
func types A2 -> Element of bool the carrier of a1 means
for b1 being set holds
b1 in it
iff
ex b2 being Element of the carrier of a1 st
b1 = b2 & a2 in adjs b2;
end;
:: ABCMIZ_0:def 12
theorem
for b1 being TA-structure
for b2 being Element of the adjectives of b1
for b3 being Element of bool the carrier of b1 holds
b3 = types b2
iff
for b4 being set holds
b4 in b3
iff
ex b5 being Element of the carrier of b1 st
b4 = b5 & b2 in adjs b5;
:: ABCMIZ_0:funcnot 4 => ABCMIZ_0:func 4
definition
let a1 be non empty TA-structure;
let a2 be Element of bool the adjectives of a1;
func types A2 -> Element of bool the carrier of a1 means
for b1 being Element of the carrier of a1 holds
b1 in it
iff
for b2 being Element of the adjectives of a1
st b2 in a2
holds b1 in types b2;
end;
:: ABCMIZ_0:def 13
theorem
for b1 being non empty TA-structure
for b2 being Element of bool the adjectives of b1
for b3 being Element of bool the carrier of b1 holds
b3 = types b2
iff
for b4 being Element of the carrier of b1 holds
b4 in b3
iff
for b5 being Element of the adjectives of b1
st b5 in b2
holds b4 in types b5;
:: ABCMIZ_0:th 11
theorem
for b1, b2 being TA-structure
st TA-structure(#the carrier of b1,the adjectives of b1,the InternalRel of b1,the non-op of b1,the adj-map of b1#) = TA-structure(#the carrier of b2,the adjectives of b2,the InternalRel of b2,the non-op of b2,the adj-map of b2#)
for b3 being Element of the adjectives of b1
for b4 being Element of the adjectives of b2
st b3 = b4
holds types b3 = types b4;
:: ABCMIZ_0:th 12
theorem
for b1 being non empty TA-structure
for b2 being Element of the adjectives of b1 holds
types b2 = {b3 where b3 is Element of the carrier of b1: b2 in adjs b3};
:: ABCMIZ_0:th 13
theorem
for b1 being TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1 holds
b3 in adjs b2
iff
b2 in types b3;
:: ABCMIZ_0:th 14
theorem
for b1 being non empty TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of bool the adjectives of b1 holds
b3 c= adjs b2
iff
b2 in types b3;
:: ABCMIZ_0:th 15
theorem
for b1 being non void TA-structure
for b2 being Element of the carrier of b1 holds
adjs b2 = {b3 where b3 is Element of the adjectives of b1: b2 in types b3};
:: ABCMIZ_0:th 16
theorem
for b1 being non empty TA-structure holds
types {} the adjectives of b1 = the carrier of b1;
:: ABCMIZ_0:attrnot 12 => ABCMIZ_0:attr 10
definition
let a1 be TA-structure;
attr a1 is adjs-typed means
for b1 being Element of the adjectives of a1 holds
(types b1) \/ types non- b1 is not empty;
end;
:: ABCMIZ_0:dfs 14
definiens
let a1 be TA-structure;
To prove
a1 is adjs-typed
it is sufficient to prove
thus for b1 being Element of the adjectives of a1 holds
(types b1) \/ types non- b1 is not empty;
:: ABCMIZ_0:def 14
theorem
for b1 being TA-structure holds
b1 is adjs-typed
iff
for b2 being Element of the adjectives of b1 holds
(types b2) \/ types non- b2 is not empty;
:: ABCMIZ_0:th 17
theorem
for b1, b2 being TA-structure
st TA-structure(#the carrier of b1,the adjectives of b1,the InternalRel of b1,the non-op of b1,the adj-map of b1#) = TA-structure(#the carrier of b2,the adjectives of b2,the InternalRel of b2,the non-op of b2,the adj-map of b2#) &
b1 is adjs-typed
holds b2 is adjs-typed;
:: ABCMIZ_0:exreg 6
registration
cluster non empty trivial finite reflexive transitive antisymmetric with_suprema with_infima complete lower-bounded upper-bounded bounded connected up-complete /\-complete Noetherian Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure;
end;
:: ABCMIZ_0:th 18
theorem
for b1 being consistent TA-structure
for b2 being Element of the adjectives of b1 holds
types b2 misses types non- b2;
:: ABCMIZ_0:funcreg 5
registration
let a1 be reflexive transitive antisymmetric with_suprema adj-structured TA-structure;
let a2 be Element of the adjectives of a1;
cluster types a2 -> directed lower;
end;
:: ABCMIZ_0:funcreg 6
registration
let a1 be reflexive transitive antisymmetric with_suprema adj-structured TA-structure;
let a2 be Element of bool the adjectives of a1;
cluster types a2 -> directed lower;
end;
:: ABCMIZ_0:prednot 1 => ABCMIZ_0:pred 1
definition
let a1 be TA-structure;
let a2 be Element of the carrier of a1;
let a3 be Element of the adjectives of a1;
pred A3 is_applicable_to A2 means
ex b1 being Element of the carrier of a1 st
b1 in types a3 & b1 <= a2;
end;
:: ABCMIZ_0:dfs 15
definiens
let a1 be TA-structure;
let a2 be Element of the carrier of a1;
let a3 be Element of the adjectives of a1;
To prove
a3 is_applicable_to a2
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
b1 in types a3 & b1 <= a2;
:: ABCMIZ_0:def 15
theorem
for b1 being TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1 holds
b3 is_applicable_to b2
iff
ex b4 being Element of the carrier of b1 st
b4 in types b3 & b4 <= b2;
:: ABCMIZ_0:prednot 2 => ABCMIZ_0:pred 2
definition
let a1 be TA-structure;
let a2 be Element of the carrier of a1;
let a3 be Element of bool the adjectives of a1;
pred A3 is_applicable_to A2 means
ex b1 being Element of the carrier of a1 st
a3 c= adjs b1 & b1 <= a2;
end;
:: ABCMIZ_0:dfs 16
definiens
let a1 be TA-structure;
let a2 be Element of the carrier of a1;
let a3 be Element of bool the adjectives of a1;
To prove
a3 is_applicable_to a2
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
a3 c= adjs b1 & b1 <= a2;
:: ABCMIZ_0:def 16
theorem
for b1 being TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of bool the adjectives of b1 holds
b3 is_applicable_to b2
iff
ex b4 being Element of the carrier of b1 st
b3 c= adjs b4 & b4 <= b2;
:: ABCMIZ_0:th 20
theorem
for b1 being reflexive transitive antisymmetric with_suprema adj-structured TA-structure
for b2 being Element of the adjectives of b1
for b3 being Element of the carrier of b1
st b2 is_applicable_to b3
holds (types b2) /\ downarrow b3 is non empty directed lower Element of bool the carrier of b1;
:: ABCMIZ_0:funcnot 5 => ABCMIZ_0:func 5
definition
let a1 be non empty reflexive transitive TA-structure;
let a2 be Element of the carrier of a1;
let a3 be Element of the adjectives of a1;
func A3 ast A2 -> Element of the carrier of a1 equals
"\/"((types a3) /\ downarrow a2,a1);
end;
:: ABCMIZ_0:def 17
theorem
for b1 being non empty reflexive transitive TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1 holds
b3 ast b2 = "\/"((types b3) /\ downarrow b2,b1);
:: ABCMIZ_0:th 21
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1
st b3 is_applicable_to b2
holds b3 ast b2 <= b2;
:: ABCMIZ_0:th 22
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1
st b3 is_applicable_to b2
holds b3 in adjs (b3 ast b2);
:: ABCMIZ_0:th 23
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1
st b3 is_applicable_to b2
holds b3 ast b2 in types b3;
:: ABCMIZ_0:th 24
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1
for b4 being Element of the carrier of b1
st b4 <= b2 & b3 in adjs b4
holds b3 is_applicable_to b2 & b4 <= b3 ast b2;
:: ABCMIZ_0:th 25
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1
st b3 in adjs b2
holds b3 is_applicable_to b2 & b3 ast b2 = b2;
:: ABCMIZ_0:th 26
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3, b4 being Element of the adjectives of b1
st b3 is_applicable_to b2 & b4 is_applicable_to b3 ast b2
holds b4 is_applicable_to b2 &
b3 is_applicable_to b4 ast b2 &
b3 ast (b4 ast b2) = b4 ast (b3 ast b2);
:: ABCMIZ_0:th 27
theorem
for b1 being reflexive transitive antisymmetric with_suprema adj-structured TA-structure
for b2 being Element of bool the adjectives of b1
for b3 being Element of the carrier of b1
st b2 is_applicable_to b3
holds (types b2) /\ downarrow b3 is non empty directed lower Element of bool the carrier of b1;
:: ABCMIZ_0:funcnot 6 => ABCMIZ_0:func 6
definition
let a1 be non empty reflexive transitive TA-structure;
let a2 be Element of the carrier of a1;
let a3 be Element of bool the adjectives of a1;
func A3 ast A2 -> Element of the carrier of a1 equals
"\/"((types a3) /\ downarrow a2,a1);
end;
:: ABCMIZ_0:def 18
theorem
for b1 being non empty reflexive transitive TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of bool the adjectives of b1 holds
b3 ast b2 = "\/"((types b3) /\ downarrow b2,b1);
:: ABCMIZ_0:th 28
theorem
for b1 being non empty reflexive transitive antisymmetric TA-structure
for b2 being Element of the carrier of b1 holds
({} the adjectives of b1) ast b2 = b2;
:: ABCMIZ_0:funcnot 7 => ABCMIZ_0:func 7
definition
let a1 be non empty reflexive transitive non void TA-structure;
let a2 be Element of the carrier of a1;
let a3 be FinSequence of the adjectives of a1;
func apply(A3,A2) -> FinSequence of the carrier of a1 means
len it = (len a3) + 1 &
it . 1 = a2 &
(for b1 being Element of NAT
for b2 being Element of the adjectives of a1
for b3 being Element of the carrier of a1
st b1 in dom a3 & b2 = a3 . b1 & b3 = it . b1
holds it . (b1 + 1) = b2 ast b3);
end;
:: ABCMIZ_0:def 19
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1
for b3 being FinSequence of the adjectives of b1
for b4 being FinSequence of the carrier of b1 holds
b4 = apply(b3,b2)
iff
len b4 = (len b3) + 1 &
b4 . 1 = b2 &
(for b5 being Element of NAT
for b6 being Element of the adjectives of b1
for b7 being Element of the carrier of b1
st b5 in dom b3 & b6 = b3 . b5 & b7 = b4 . b5
holds b4 . (b5 + 1) = b6 ast b7);
:: ABCMIZ_0:funcreg 7
registration
let a1 be non empty reflexive transitive non void TA-structure;
let a2 be Element of the carrier of a1;
let a3 be FinSequence of the adjectives of a1;
cluster apply(a3,a2) -> non empty;
end;
:: ABCMIZ_0:th 29
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1 holds
apply(<*> the adjectives of b1,b2) = <*b2*>;
:: ABCMIZ_0:th 30
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1 holds
apply(<*b3*>,b2) = <*b2,b3 ast b2*>;
:: ABCMIZ_0:funcnot 8 => ABCMIZ_0:func 8
definition
let a1 be non empty reflexive transitive non void TA-structure;
let a2 be Element of the carrier of a1;
let a3 be FinSequence of the adjectives of a1;
func A3 ast A2 -> Element of the carrier of a1 equals
(apply(a3,a2)) . ((len a3) + 1);
end;
:: ABCMIZ_0:def 20
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1
for b3 being FinSequence of the adjectives of b1 holds
b3 ast b2 = (apply(b3,b2)) . ((len b3) + 1);
:: ABCMIZ_0:th 31
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1 holds
(<*> the adjectives of b1) ast b2 = b2;
:: ABCMIZ_0:th 32
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1 holds
<*b3*> ast b2 = b3 ast b2;
:: ABCMIZ_0:th 33
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3 being natural set
st 1 <= b3 & b3 < len b1
holds (b1 $^ b2) . b3 = b1 . b3;
:: ABCMIZ_0:th 34
theorem
for b1 being non empty Relation-like Function-like FinSequence-like set
for b2 being Relation-like Function-like FinSequence-like set
for b3 being natural set
st b3 < len b2
holds (b1 $^ b2) . ((len b1) + b3) = b2 . (b3 + 1);
:: ABCMIZ_0:th 35
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1
for b3, b4 being FinSequence of the adjectives of b1 holds
apply(b3 ^ b4,b2) = (apply(b3,b2)) $^ apply(b4,b3 ast b2);
:: ABCMIZ_0:th 36
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1
for b3, b4 being FinSequence of the adjectives of b1
for b5 being natural set
st b5 in dom b3
holds (apply(b3 ^ b4,b2)) . b5 = (apply(b3,b2)) . b5;
:: ABCMIZ_0:th 37
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1
for b3, b4 being FinSequence of the adjectives of b1 holds
(apply(b3 ^ b4,b2)) . ((len b3) + 1) = b3 ast b2;
:: ABCMIZ_0:th 38
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1
for b3, b4 being FinSequence of the adjectives of b1 holds
b4 ast (b3 ast b2) = (b3 ^ b4) ast b2;
:: ABCMIZ_0:prednot 3 => ABCMIZ_0:pred 3
definition
let a1 be non empty reflexive transitive non void TA-structure;
let a2 be Element of the carrier of a1;
let a3 be FinSequence of the adjectives of a1;
pred A3 is_applicable_to A2 means
for b1 being natural set
for b2 being Element of the adjectives of a1
for b3 being Element of the carrier of a1
st b1 in dom a3 & b2 = a3 . b1 & b3 = (apply(a3,a2)) . b1
holds b2 is_applicable_to b3;
end;
:: ABCMIZ_0:dfs 21
definiens
let a1 be non empty reflexive transitive non void TA-structure;
let a2 be Element of the carrier of a1;
let a3 be FinSequence of the adjectives of a1;
To prove
a3 is_applicable_to a2
it is sufficient to prove
thus for b1 being natural set
for b2 being Element of the adjectives of a1
for b3 being Element of the carrier of a1
st b1 in dom a3 & b2 = a3 . b1 & b3 = (apply(a3,a2)) . b1
holds b2 is_applicable_to b3;
:: ABCMIZ_0:def 21
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1
for b3 being FinSequence of the adjectives of b1 holds
b3 is_applicable_to b2
iff
for b4 being natural set
for b5 being Element of the adjectives of b1
for b6 being Element of the carrier of b1
st b4 in dom b3 & b5 = b3 . b4 & b6 = (apply(b3,b2)) . b4
holds b5 is_applicable_to b6;
:: ABCMIZ_0:th 39
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1 holds
<*> the adjectives of b1 is_applicable_to b2;
:: ABCMIZ_0:th 40
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1 holds
b3 is_applicable_to b2
iff
<*b3*> is_applicable_to b2;
:: ABCMIZ_0:th 41
theorem
for b1 being non empty reflexive transitive non void TA-structure
for b2 being Element of the carrier of b1
for b3, b4 being FinSequence of the adjectives of b1
st b3 ^ b4 is_applicable_to b2
holds b3 is_applicable_to b2 & b4 is_applicable_to b3 ast b2;
:: ABCMIZ_0:th 42
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being FinSequence of the adjectives of b1
st b3 is_applicable_to b2
for b4, b5 being natural set
st 1 <= b4 & b4 <= b5 & b5 <= (len b3) + 1
for b6, b7 being Element of the carrier of b1
st b6 = (apply(b3,b2)) . b4 & b7 = (apply(b3,b2)) . b5
holds b7 <= b6;
:: ABCMIZ_0:th 43
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being FinSequence of the adjectives of b1
st b3 is_applicable_to b2
for b4 being Element of the carrier of b1
st b4 in rng apply(b3,b2)
holds b3 ast b2 <= b4 & b4 <= b2;
:: ABCMIZ_0:th 44
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being FinSequence of the adjectives of b1
st b3 is_applicable_to b2
holds b3 ast b2 <= b2;
:: ABCMIZ_0:th 45
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being FinSequence of the adjectives of b1
st b3 is_applicable_to b2
holds rng b3 c= adjs (b3 ast b2);
:: ABCMIZ_0:th 46
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being FinSequence of the adjectives of b1
st b3 is_applicable_to b2
for b4 being Element of bool the adjectives of b1
st b4 = rng b3
holds b4 is_applicable_to b2;
:: ABCMIZ_0:th 47
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3, b4 being FinSequence of the adjectives of b1
st b3 is_applicable_to b2 & rng b4 c= rng b3
for b5 being Element of the carrier of b1
st b5 in rng apply(b4,b2)
holds b3 ast b2 <= b5;
:: ABCMIZ_0:th 48
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3, b4 being FinSequence of the adjectives of b1
st b3 ^ b4 is_applicable_to b2
holds b4 ^ b3 is_applicable_to b2;
:: ABCMIZ_0:th 49
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3, b4 being FinSequence of the adjectives of b1
st b3 ^ b4 is_applicable_to b2
holds (b3 ^ b4) ast b2 = (b4 ^ b3) ast b2;
:: ABCMIZ_0:th 50
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of bool the adjectives of b1
st b3 is_applicable_to b2
holds b3 ast b2 <= b2;
:: ABCMIZ_0:th 51
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of bool the adjectives of b1
st b3 is_applicable_to b2
holds b3 c= adjs (b3 ast b2);
:: ABCMIZ_0:th 52
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of bool the adjectives of b1
st b3 is_applicable_to b2
holds b3 ast b2 in types b3;
:: ABCMIZ_0:th 53
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of bool the adjectives of b1
for b4 being Element of the carrier of b1
st b4 <= b2 & b3 c= adjs b4
holds b3 is_applicable_to b2 & b4 <= b3 ast b2;
:: ABCMIZ_0:th 54
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of bool the adjectives of b1
st b3 c= adjs b2
holds b3 is_applicable_to b2 & b3 ast b2 = b2;
:: ABCMIZ_0:th 55
theorem
for b1 being TA-structure
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the adjectives of b1
st b3 is_applicable_to b2 & b4 c= b3
holds b4 is_applicable_to b2;
:: ABCMIZ_0:th 56
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1
for b4, b5 being Element of bool the adjectives of b1
st b5 = b4 \/ {b3} & b5 is_applicable_to b2
holds b3 ast (b4 ast b2) = b5 ast b2;
:: ABCMIZ_0:th 57
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure
for b2 being Element of the carrier of b1
for b3 being FinSequence of the adjectives of b1
st b3 is_applicable_to b2
for b4 being Element of bool the adjectives of b1
st b4 = rng b3
holds b3 ast b2 = b4 ast b2;
:: ABCMIZ_0:funcnot 9 => ABCMIZ_0:func 9
definition
let a1 be non empty non void TA-structure;
func sub A1 -> Function-like quasi_total Relation of the adjectives of a1,the carrier of a1 means
for b1 being Element of the adjectives of a1 holds
it . b1 = "\/"((types b1) \/ types non- b1,a1);
end;
:: ABCMIZ_0:def 22
theorem
for b1 being non empty non void TA-structure
for b2 being Function-like quasi_total Relation of the adjectives of b1,the carrier of b1 holds
b2 = sub b1
iff
for b3 being Element of the adjectives of b1 holds
b2 . b3 = "\/"((types b3) \/ types non- b3,b1);
:: ABCMIZ_0:structnot 3 => ABCMIZ_0:struct 3
definition
struct(TA-structure) TAS-structure(#
carrier -> set,
adjectives -> set,
InternalRel -> Relation of the carrier of it,the carrier of it,
non-op -> Function-like quasi_total Relation of the adjectives of it,the adjectives of it,
adj-map -> Function-like quasi_total Relation of the carrier of it,Fin the adjectives of it,
sub-map -> Function-like quasi_total Relation of the adjectives of it,the carrier of it
#);
end;
:: ABCMIZ_0:attrnot 13 => ABCMIZ_0:attr 11
definition
let a1 be TAS-structure;
attr a1 is strict;
end;
:: ABCMIZ_0:exreg 7
registration
cluster strict TAS-structure;
end;
:: ABCMIZ_0:aggrnot 3 => ABCMIZ_0:aggr 3
definition
let a1, a2 be set;
let a3 be Relation of a1,a1;
let a4 be Function-like quasi_total Relation of a2,a2;
let a5 be Function-like quasi_total Relation of a1,Fin a2;
let a6 be Function-like quasi_total Relation of a2,a1;
aggr TAS-structure(#a1,a2,a3,a4,a5,a6#) -> strict TAS-structure;
end;
:: ABCMIZ_0:selnot 4 => ABCMIZ_0:sel 4
definition
let a1 be TAS-structure;
sel the sub-map of a1 -> Function-like quasi_total Relation of the adjectives of a1,the carrier of a1;
end;
:: ABCMIZ_0:exreg 8
registration
cluster non empty trivial reflexive non void strict TAS-structure;
end;
:: ABCMIZ_0:funcnot 10 => ABCMIZ_0:func 10
definition
let a1 be non empty non void TAS-structure;
let a2 be Element of the adjectives of a1;
func sub A2 -> Element of the carrier of a1 equals
(the sub-map of a1) . a2;
end;
:: ABCMIZ_0:def 23
theorem
for b1 being non empty non void TAS-structure
for b2 being Element of the adjectives of b1 holds
sub b2 = (the sub-map of b1) . b2;
:: ABCMIZ_0:attrnot 14 => ABCMIZ_0:attr 12
definition
let a1 be non empty non void TAS-structure;
attr a1 is non-absorbing means
(the sub-map of a1) * the non-op of a1 = the sub-map of a1;
end;
:: ABCMIZ_0:dfs 24
definiens
let a1 be non empty non void TAS-structure;
To prove
a1 is non-absorbing
it is sufficient to prove
thus (the sub-map of a1) * the non-op of a1 = the sub-map of a1;
:: ABCMIZ_0:def 24
theorem
for b1 being non empty non void TAS-structure holds
b1 is non-absorbing
iff
(the sub-map of b1) * the non-op of b1 = the sub-map of b1;
:: ABCMIZ_0:attrnot 15 => ABCMIZ_0:attr 13
definition
let a1 be non empty non void TAS-structure;
attr a1 is subjected means
for b1 being Element of the adjectives of a1 holds
(types b1) \/ types non- b1 is_<=_than sub b1 &
(types b1 <> {} & types non- b1 <> {} implies sub b1 = "\/"((types b1) \/ types non- b1,a1));
end;
:: ABCMIZ_0:dfs 25
definiens
let a1 be non empty non void TAS-structure;
To prove
a1 is subjected
it is sufficient to prove
thus for b1 being Element of the adjectives of a1 holds
(types b1) \/ types non- b1 is_<=_than sub b1 &
(types b1 <> {} & types non- b1 <> {} implies sub b1 = "\/"((types b1) \/ types non- b1,a1));
:: ABCMIZ_0:def 25
theorem
for b1 being non empty non void TAS-structure holds
b1 is subjected
iff
for b2 being Element of the adjectives of b1 holds
(types b2) \/ types non- b2 is_<=_than sub b2 &
(types b2 <> {} & types non- b2 <> {} implies sub b2 = "\/"((types b2) \/ types non- b2,b1));
:: ABCMIZ_0:attrnot 16 => ABCMIZ_0:attr 12
definition
let a1 be non empty non void TAS-structure;
attr a1 is non-absorbing means
for b1 being Element of the adjectives of a1 holds
sub non- b1 = sub b1;
end;
:: ABCMIZ_0:dfs 26
definiens
let a1 be non empty non void TAS-structure;
To prove
a1 is non-absorbing
it is sufficient to prove
thus for b1 being Element of the adjectives of a1 holds
sub non- b1 = sub b1;
:: ABCMIZ_0:def 26
theorem
for b1 being non empty non void TAS-structure holds
b1 is non-absorbing
iff
for b2 being Element of the adjectives of b1 holds
sub non- b2 = sub b2;
:: ABCMIZ_0:prednot 4 => ABCMIZ_0:pred 4
definition
let a1 be non empty non void TAS-structure;
let a2 be Element of the carrier of a1;
let a3 be Element of the adjectives of a1;
pred A3 is_properly_applicable_to A2 means
a2 <= sub a3 & a3 is_applicable_to a2;
end;
:: ABCMIZ_0:dfs 27
definiens
let a1 be non empty non void TAS-structure;
let a2 be Element of the carrier of a1;
let a3 be Element of the adjectives of a1;
To prove
a3 is_properly_applicable_to a2
it is sufficient to prove
thus a2 <= sub a3 & a3 is_applicable_to a2;
:: ABCMIZ_0:def 27
theorem
for b1 being non empty non void TAS-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1 holds
b3 is_properly_applicable_to b2
iff
b2 <= sub b3 & b3 is_applicable_to b2;
:: ABCMIZ_0:prednot 5 => ABCMIZ_0:pred 5
definition
let a1 be non empty reflexive transitive non void TAS-structure;
let a2 be Element of the carrier of a1;
let a3 be FinSequence of the adjectives of a1;
pred A3 is_properly_applicable_to A2 means
for b1 being natural set
for b2 being Element of the adjectives of a1
for b3 being Element of the carrier of a1
st b1 in dom a3 & b2 = a3 . b1 & b3 = (apply(a3,a2)) . b1
holds b2 is_properly_applicable_to b3;
end;
:: ABCMIZ_0:dfs 28
definiens
let a1 be non empty reflexive transitive non void TAS-structure;
let a2 be Element of the carrier of a1;
let a3 be FinSequence of the adjectives of a1;
To prove
a3 is_properly_applicable_to a2
it is sufficient to prove
thus for b1 being natural set
for b2 being Element of the adjectives of a1
for b3 being Element of the carrier of a1
st b1 in dom a3 & b2 = a3 . b1 & b3 = (apply(a3,a2)) . b1
holds b2 is_properly_applicable_to b3;
:: ABCMIZ_0:def 28
theorem
for b1 being non empty reflexive transitive non void TAS-structure
for b2 being Element of the carrier of b1
for b3 being FinSequence of the adjectives of b1 holds
b3 is_properly_applicable_to b2
iff
for b4 being natural set
for b5 being Element of the adjectives of b1
for b6 being Element of the carrier of b1
st b4 in dom b3 & b5 = b3 . b4 & b6 = (apply(b3,b2)) . b4
holds b5 is_properly_applicable_to b6;
:: ABCMIZ_0:th 58
theorem
for b1 being non empty reflexive transitive non void TAS-structure
for b2 being Element of the carrier of b1
for b3 being FinSequence of the adjectives of b1
st b3 is_properly_applicable_to b2
holds b3 is_applicable_to b2;
:: ABCMIZ_0:th 59
theorem
for b1 being non empty reflexive transitive non void TAS-structure
for b2 being Element of the carrier of b1 holds
<*> the adjectives of b1 is_properly_applicable_to b2;
:: ABCMIZ_0:th 60
theorem
for b1 being non empty reflexive transitive non void TAS-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1 holds
b3 is_properly_applicable_to b2
iff
<*b3*> is_properly_applicable_to b2;
:: ABCMIZ_0:th 61
theorem
for b1 being non empty reflexive transitive non void TAS-structure
for b2 being Element of the carrier of b1
for b3, b4 being FinSequence of the adjectives of b1
st b3 ^ b4 is_properly_applicable_to b2
holds b3 is_properly_applicable_to b2 & b4 is_properly_applicable_to b3 ast b2;
:: ABCMIZ_0:th 62
theorem
for b1 being non empty reflexive transitive non void TAS-structure
for b2 being Element of the carrier of b1
for b3, b4 being FinSequence of the adjectives of b1
st b3 is_properly_applicable_to b2 & b4 is_properly_applicable_to b3 ast b2
holds b3 ^ b4 is_properly_applicable_to b2;
:: ABCMIZ_0:prednot 6 => ABCMIZ_0:pred 6
definition
let a1 be non empty reflexive transitive non void TAS-structure;
let a2 be Element of the carrier of a1;
let a3 be Element of bool the adjectives of a1;
pred A3 is_properly_applicable_to A2 means
ex b1 being FinSequence of the adjectives of a1 st
rng b1 = a3 & b1 is_properly_applicable_to a2;
end;
:: ABCMIZ_0:dfs 29
definiens
let a1 be non empty reflexive transitive non void TAS-structure;
let a2 be Element of the carrier of a1;
let a3 be Element of bool the adjectives of a1;
To prove
a3 is_properly_applicable_to a2
it is sufficient to prove
thus ex b1 being FinSequence of the adjectives of a1 st
rng b1 = a3 & b1 is_properly_applicable_to a2;
:: ABCMIZ_0:def 29
theorem
for b1 being non empty reflexive transitive non void TAS-structure
for b2 being Element of the carrier of b1
for b3 being Element of bool the adjectives of b1 holds
b3 is_properly_applicable_to b2
iff
ex b4 being FinSequence of the adjectives of b1 st
rng b4 = b3 & b4 is_properly_applicable_to b2;
:: ABCMIZ_0:th 63
theorem
for b1 being non empty reflexive transitive non void TAS-structure
for b2 being Element of the carrier of b1
for b3 being Element of bool the adjectives of b1
st b3 is_properly_applicable_to b2
holds b3 is finite;
:: ABCMIZ_0:th 64
theorem
for b1 being non empty reflexive transitive non void TAS-structure
for b2 being Element of the carrier of b1 holds
{} the adjectives of b1 is_properly_applicable_to b2;
:: ABCMIZ_0:sch 1
scheme ABCMIZ_0:sch 1
ex b1 being finite set st
P1[b1] &
(for b2 being set
st b2 c= b1 & P1[b2]
holds b2 = b1)
provided
ex b1 being finite set st
P1[b1];
:: ABCMIZ_0:th 65
theorem
for b1 being non empty reflexive transitive non void TAS-structure
for b2 being Element of the carrier of b1
for b3 being Element of bool the adjectives of b1
st b3 is_properly_applicable_to b2
holds ex b4 being Element of bool the adjectives of b1 st
b4 c= b3 &
b4 is_properly_applicable_to b2 &
b3 ast b2 = b4 ast b2 &
(for b5 being Element of bool the adjectives of b1
st b5 c= b4 & b5 is_properly_applicable_to b2 & b3 ast b2 = b5 ast b2
holds b5 = b4);
:: ABCMIZ_0:attrnot 17 => ABCMIZ_0:attr 14
definition
let a1 be non empty reflexive transitive non void TAS-structure;
attr a1 is commutative means
for b1, b2 being Element of the carrier of a1
for b3 being Element of the adjectives of a1
st b3 is_properly_applicable_to b1 & b3 ast b1 <= b2
holds ex b4 being finite Element of bool the adjectives of a1 st
b4 is_properly_applicable_to b1 "\/" b2 & b4 ast (b1 "\/" b2) = b2;
end;
:: ABCMIZ_0:dfs 30
definiens
let a1 be non empty reflexive transitive non void TAS-structure;
To prove
a1 is commutative
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
for b3 being Element of the adjectives of a1
st b3 is_properly_applicable_to b1 & b3 ast b1 <= b2
holds ex b4 being finite Element of bool the adjectives of a1 st
b4 is_properly_applicable_to b1 "\/" b2 & b4 ast (b1 "\/" b2) = b2;
:: ABCMIZ_0:def 30
theorem
for b1 being non empty reflexive transitive non void TAS-structure holds
b1 is commutative
iff
for b2, b3 being Element of the carrier of b1
for b4 being Element of the adjectives of b1
st b4 is_properly_applicable_to b2 & b4 ast b2 <= b3
holds ex b5 being finite Element of bool the adjectives of b1 st
b5 is_properly_applicable_to b2 "\/" b3 & b5 ast (b2 "\/" b3) = b3;
:: ABCMIZ_0:exreg 9
registration
cluster non empty trivial finite reflexive transitive antisymmetric with_suprema with_infima complete lower-bounded upper-bounded bounded connected up-complete /\-complete Noetherian Mizar-widening-like non void involutive without_fixpoints consistent adj-structured adjs-typed strict non-absorbing subjected commutative TAS-structure;
end;
:: ABCMIZ_0:th 66
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure
for b2 being Element of the carrier of b1
for b3 being Element of bool the adjectives of b1
st b3 is_properly_applicable_to b2
holds ex b4 being one-to-one FinSequence of the adjectives of b1 st
rng b4 = b3 & b4 is_properly_applicable_to b2;
:: ABCMIZ_0:funcnot 11 => ABCMIZ_0:func 11
definition
let a1 be non empty reflexive transitive non void TAS-structure;
func A1 @--> -> Relation of the carrier of a1,the carrier of a1 means
for b1, b2 being Element of the carrier of a1 holds
[b1,b2] in it
iff
ex b3 being Element of the adjectives of a1 st
not b3 in adjs b2 & b3 is_properly_applicable_to b2 & b3 ast b2 = b1;
end;
:: ABCMIZ_0:def 31
theorem
for b1 being non empty reflexive transitive non void TAS-structure
for b2 being Relation of the carrier of b1,the carrier of b1 holds
b2 = b1 @-->
iff
for b3, b4 being Element of the carrier of b1 holds
[b3,b4] in b2
iff
ex b5 being Element of the adjectives of b1 st
not b5 in adjs b4 & b5 is_properly_applicable_to b4 & b5 ast b4 = b3;
:: ABCMIZ_0:th 67
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure holds
b1 @--> c= the InternalRel of b1;
:: ABCMIZ_0:sch 2
scheme ABCMIZ_0:sch 2
{F1 -> non empty set,
F2 -> Relation of F1(),F1()}:
for b1, b2 being Element of F1()
st F2() reduces b1,b2
holds P1[b1, b2]
provided
for b1, b2 being Element of F1()
st [b1,b2] in F2()
holds P1[b1, b2]
and
for b1 being Element of F1() holds
P1[b1, b1]
and
for b1, b2, b3 being Element of F1()
st P1[b1, b2] & P1[b2, b3]
holds P1[b1, b3];
:: ABCMIZ_0:th 68
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure
for b2, b3 being Element of the carrier of b1
st b1 @--> reduces b2,b3
holds b2 <= b3;
:: ABCMIZ_0:th 69
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure holds
b1 @--> is irreflexive;
:: ABCMIZ_0:th 70
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure holds
b1 @--> is strongly-normalizing;
:: ABCMIZ_0:th 71
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure
for b2 being Element of the carrier of b1
for b3 being finite Element of bool the adjectives of b1
st for b4 being Element of bool the adjectives of b1
st b4 c= b3 & b4 is_properly_applicable_to b2 & b3 ast b2 = b4 ast b2
holds b4 = b3
for b4 being one-to-one FinSequence of the adjectives of b1
st rng b4 = b3 & b4 is_properly_applicable_to b2
for b5 being natural set
st 1 <= b5 & b5 <= len b4
holds [(apply(b4,b2)) . (b5 + 1),(apply(b4,b2)) . b5] in b1 @-->;
:: ABCMIZ_0:th 72
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure
for b2 being Element of the carrier of b1
for b3 being finite Element of bool the adjectives of b1
st for b4 being Element of bool the adjectives of b1
st b4 c= b3 & b4 is_properly_applicable_to b2 & b3 ast b2 = b4 ast b2
holds b4 = b3
for b4 being one-to-one FinSequence of the adjectives of b1
st rng b4 = b3 & b4 is_properly_applicable_to b2
holds Rev apply(b4,b2) is RedSequence of b1 @-->;
:: ABCMIZ_0:th 73
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure
for b2 being Element of the carrier of b1
for b3 being finite Element of bool the adjectives of b1
st b3 is_properly_applicable_to b2
holds b1 @--> reduces b3 ast b2,b2;
:: ABCMIZ_0:th 74
theorem
for b1 being non empty set
for b2 being Relation of b1,b1
for b3 being RedSequence of b2
st b3 . 1 in b1
holds b3 is FinSequence of b1;
:: ABCMIZ_0:th 75
theorem
for b1 being non empty set
for b2 being Relation of b1,b1
for b3 being Element of b1
for b4 being set
st b2 reduces b3,b4
holds b4 in b1;
:: ABCMIZ_0:th 76
theorem
for b1 being non empty set
for b2 being Relation of b1,b1
st b2 is with_UN_property & b2 is weakly-normalizing
for b3 being Element of b1 holds
nf(b3,b2) in b1;
:: ABCMIZ_0:th 77
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure
for b2, b3 being Element of the carrier of b1
st b1 @--> reduces b2,b3
holds ex b4 being finite Element of bool the adjectives of b1 st
b4 is_properly_applicable_to b3 & b2 = b4 ast b3;
:: ABCMIZ_0:th 78
theorem
for b1 being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure holds
b1 @--> is with_Church-Rosser_property & b1 @--> is with_UN_property;
:: ABCMIZ_0:funcnot 12 => ABCMIZ_0:func 12
definition
let a1 be non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure;
let a2 be Element of the carrier of a1;
func radix A2 -> Element of the carrier of a1 equals
nf(a2,a1 @-->);
end;
:: ABCMIZ_0:def 32
theorem
for b1 being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for b2 being Element of the carrier of b1 holds
radix b2 = nf(b2,b1 @-->);
:: ABCMIZ_0:th 79
theorem
for b1 being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for b2 being Element of the carrier of b1 holds
b1 @--> reduces b2,radix b2;
:: ABCMIZ_0:th 80
theorem
for b1 being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for b2 being Element of the carrier of b1 holds
b2 <= radix b2;
:: ABCMIZ_0:th 81
theorem
for b1 being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for b2 being Element of the carrier of b1
for b3 being set
st b3 = {b4 where b4 is Element of the carrier of b1: ex b5 being finite Element of bool the adjectives of b1 st
b5 is_properly_applicable_to b4 & b5 ast b4 = b2}
holds ex_sup_of b3,b1 & radix b2 = "\/"(b3,b1);
:: ABCMIZ_0:th 82
theorem
for b1 being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for b2, b3 being Element of the carrier of b1
for b4 being Element of the adjectives of b1
st b4 is_properly_applicable_to b2 & b4 ast b2 <= radix b3
holds b2 <= radix b3;
:: ABCMIZ_0:th 83
theorem
for b1 being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for b2, b3 being Element of the carrier of b1
st b2 <= b3
holds radix b2 <= radix b3;
:: ABCMIZ_0:th 84
theorem
for b1 being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure
for b2 being Element of the carrier of b1
for b3 being Element of the adjectives of b1
st b3 is_properly_applicable_to b2
holds radix (b3 ast b2) = radix b2;