Proof Assistant Projects
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Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
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for T being Noetherian sup-Semilattice for I being Ideal of T
holds ex_sup_of I, T & sup I in I | proof
let T be Noetherian sup-Semilattice;
let I be Ideal of T;
consider a being Element of T such that
A1: a in I and
A2: for b being Element of T st b in I holds not a < b by Def2;
A3: I is_<=_than a
proof
let d be Element of T;
assume d in I;
then a"\/"d in I by A1,WAYBEL_0:40;
then
A4: not a... | theorem | abcmiz_0:Th1 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for A1,A2 being AdjectiveStr st the adjectives of A1 = the adjectives
of A2 holds A1 is void implies A2 is void | proof
let A1,A2 be AdjectiveStr such that
A1: the adjectives of A1 = the adjectives of A2;
assume the adjectives of A1 is empty;
hence the adjectives of A2 is empty by A1;
end; | theorem | abcmiz_0:T2 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for A1,A2 being AdjectiveStr st the AdjectiveStr of A1 = the
AdjectiveStr of A2 for a1 being adjective of A1, a2 being adjective of A2 st a1
= a2 holds non-a1 = non-a2 | theorem | abcmiz_0:T3 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | ||
for a1,a2 being set st a1 <> a2 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 A is non void involutive without_fixpoints | proof
let a1,a2 be set such that
A1: a1 <> a2;
let A be AdjectiveStr such that
A2: the adjectives of A = {a1,a2} and
A3: (the non-op of A).a1 = a2 and
A4: (the non-op of A).a2 = a1;
thus the adjectives of A is non empty by A2;
hereby
let a be adjective of A;
a = a1 or a = a2 by A2,TARSKI:def 2;
henc... | theorem | abcmiz_0:Th4 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for A1,A2 being AdjectiveStr st the AdjectiveStr of A1 = the
AdjectiveStr of A2 holds A1 is involutive implies A2 is involutive | proof
let A1,A2 be AdjectiveStr such that
A1: the AdjectiveStr of A1 = the AdjectiveStr of A2;
assume
A2: for a being adjective of A1 holds non-non-a = a;
let a be adjective of A2;
reconsider b = a as adjective of A1 by A1;
thus non-non-a = non-non-b by A1
.= a by A2;
end; | theorem | abcmiz_0:Th5 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for A1,A2 being AdjectiveStr st the AdjectiveStr of A1 = the
AdjectiveStr of A2 holds A1 is without_fixpoints implies A2 is
without_fixpoints | proof
let A1,A2 be AdjectiveStr such that
A1: the AdjectiveStr of A1 = the AdjectiveStr of A2;
assume
A2: not ex a being adjective of A1 st non-a = a;
given a being adjective of A2 such that
A3: non-a = a;
reconsider b = a as adjective of A1 by A1;
non-b = b by A1,A3;
hence contradiction by A2;
end; | theorem | abcmiz_0:Th6 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T1,T2 being TA-structure st the TA-structure of T1 = the
TA-structure of T2 for t1 being type of T1, t2 being type of T2 st t1 = t2
holds adjs t1 = adjs t2 | theorem | abcmiz_0:T7 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | ||
for T1,T2 being TA-structure st the TA-structure of T1 = the
TA-structure of T2 holds T1 is consistent implies T2 is consistent | proof
let T1,T2 be TA-structure such that
A1: the TA-structure of T1 = the TA-structure of T2 and
A2: for t being type of T1 for a being adjective of T1 st a in adjs t
holds not non-a in adjs t;
let t2 be type of T2, a2 be adjective of T2;
reconsider a1 = a2 as adjective of T1 by A1;
reconsider t1 = t2 as typ... | theorem | abcmiz_0:Th8 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T1,T2 being non empty TA-structure st the TA-structure of T1
= the TA-structure of T2 holds T1 is adj-structured implies T2 is
adj-structured | proof
let T1,T2 be non empty TA-structure such that
A1: the TA-structure of T1 = the TA-structure of T2;
assume the adj-map of T1 is join-preserving Function of T1, (BoolePoset the
adjectives of T1) opp;
then reconsider f = the adj-map of T1 as join-preserving Function of T1, (
BoolePoset the adjectives of T1... | theorem | abcmiz_0:Th9 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being reflexive transitive antisymmetric with_suprema
TA-structure st T is adj-structured for t1,t2 being type of T st t1 <= t2 holds
adjs t2 c= adjs t1 | proof
let T be reflexive transitive antisymmetric with_suprema TA-structure such
that
A1: for t1,t2 being type of T holds adjs(t1"\/"t2) = (adjs t1) /\ (adjs t2);
let t1,t2 be type of T;
assume t1 <= t2;
then t1"\/"t2 = t2 by YELLOW_0:24;
then adjs t2 = (adjs t1)/\(adjs t2) by A1;
hence thesis by XBOOLE_1... | theorem | abcmiz_0:Th10 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T1,T2 being TA-structure st the TA-structure of T1 = the
TA-structure of T2 for a1 being adjective of T1, a2 being adjective of T2 st a1
= a2 holds types a1 = types a2 | proof
let T1,T2 be TA-structure such that
A1: the TA-structure of T1 = the TA-structure of T2;
let a1 be adjective of T1, a2 be adjective of T2 such that
A2: a1 = a2;
now
thus types a1 is Subset of T2 by A1;
let x be set;
hereby
assume x in types a1;
then consider t1 being type of T1 such ... | theorem | abcmiz_0:Th11 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
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
let T be non empty TA-structure;
let a be adjective of T;
set X = {t where t is type of T: a in adjs t};
X c= the carrier of T
proof
let x be set;
assume x in X;
then ex t being type of T st x = t & a in adjs t;
hence thesis;
end;
then reconsider X as Subset of T;
for x being set h... | theorem | abcmiz_0:T12 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being TA-structure for t being type of T, a being
adjective of T holds a in adjs t iff t in types a | proof
let T be TA-structure;
let t be type of T, a be adjective of T;
thus a in adjs t implies t in types a by Def12;
assume t in types a;
then ex t9 being type of T st t = t9 & a in adjs t9 by Def12;
hence thesis;
end; | theorem | abcmiz_0:Th13 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty TA-structure for t being type of T, A
being Subset of the adjectives of T holds A c= adjs t iff t in types A | proof
let T be non empty TA-structure;
let t be type of T, a be Subset of the adjectives of T;
hereby
assume a c= adjs t;
then for b being adjective of T st b in a holds t in types b by Th13;
hence t in types a by Def13;
end;
assume
A1: t in types a;
let x be set;
assume
A2: x in a;
then rec... | theorem | abcmiz_0:Th14 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
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
let T be non void TA-structure;
let t be type of T;
set X = {a where a is adjective of T: t in types a};
thus adjs t c= X
proof
let x be set;
assume
A1: x in adjs t;
then reconsider a = x as adjective of T;
t in types a by A1,Th13;
hence thesis;
end;
let x be set;
assume x in X... | theorem | abcmiz_0:T15 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty TA-structure holds types ({} the
adjectives of T) = the carrier of T | proof
let T be non empty TA-structure;
thus types ({} the adjectives of T) c= the carrier of T;
let x be set;
assume x in the carrier of T;
then reconsider t = x as type of T;
for a being adjective of T st a in {} the adjectives of T holds t in types a;
hence thesis by Def13;
end; | theorem | abcmiz_0:Th16 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T1,T2 being TA-structure st the TA-structure of T1 = the
TA-structure of T2 holds T1 is adjs-typed implies T2 is adjs-typed | proof
let T1,T2 be TA-structure such that
A1: the TA-structure of T1 = the TA-structure of T2 and
A2: for a being adjective of T1 holds types a \/ types non-a is non empty;
let b be adjective of T2;
reconsider a = b as adjective of T1 by A1;
A3: types a \/ types non-a is non empty by A2;
types a = types b by A1... | theorem | abcmiz_0:Th17 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being consistent TA-structure for a being adjective of T holds
types a misses types non-a | proof
let T be consistent TA-structure;
let a be adjective of T;
assume types a meets types non-a;
then consider x being set such that
A1: x in types a and
A2: x in types non-a by XBOOLE_0:3;
A3: ex t2 being type of T st x = t2 & non-a in adjs t2 by A2,Def12;
ex t1 being type of T st x = t1 & a in adjs t1 by ... | theorem | abcmiz_0:T18 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being adj-structured reflexive transitive antisymmetric
with_suprema 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
let T be adj-structured reflexive transitive antisymmetric with_suprema
TA-structure;
let a be adjective of T;
let t be type of T;
given t9 being type of T such that
A1: t9 in types a and
A2: t9 <= t;
t9 in downarrow t by A2,WAYBEL_0:17;
hence thesis by A1,WAYBEL_0:27,44,XBOOLE_0:def 4;
end; | theorem | abcmiz_0:Th19 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema 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
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema TA-structure;
let t be type of T;
let a be adjective of T;
assume a is_applicable_to t;
then types a /\ downarrow t is Ideal of T by Th19;
then sup (types a /\ downarrow t) in types a /\ downarrow t by Th1;
then a a... | theorem | abcmiz_0:Th20 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema 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
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema TA-structure;
let t be type of T;
let a be adjective of T;
assume a is_applicable_to t;
then types a /\ downarrow t is Ideal of T by Th19;
then sup (types a /\ downarrow t) in types a /\ downarrow t by Th1;
then a a... | theorem | abcmiz_0:Th21 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema 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
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema TA-structure;
let t be type of T;
let a be adjective of T;
assume a is_applicable_to t;
then types a /\ downarrow t is Ideal of T by Th19;
then sup (types a /\ downarrow t) in types a /\ downarrow t by Th1;
hence th... | theorem | abcmiz_0:Th22 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema TA-structure for t being type of T for a being
adjective of T for t9 being type of T st t9 <= t & a in adjs t9 holds a
is_applicable_to t & t9 <= a ast t | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema TA-structure;
let t be type of T;
let a be adjective of T;
let t9 be type of T;
assume that
A1: t9 <= t and
A2: a in adjs t9;
A3: t9 in downarrow t by A1,WAYBEL_0:17;
thus a is_applicable_to t
proof
take t9;
... | theorem | abcmiz_0:Th23 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema 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
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema TA-structure;
let t be type of T;
let a be adjective of T;
assume
A1: a in adjs t;
hence a is_applicable_to t by Th23;
then
A2: a ast t <= t by Th20;
t <= a ast t by A1,Th23;
hence thesis by A2,YELLOW_0:def 3;
end... | theorem | abcmiz_0:Th24 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema 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
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema TA-structure;
let t be type of T;
let a,b be adjective of T such that
A1: a is_applicable_to t and
A2: b is_applicable_to a ast t;
consider t9 being type of T such that
A3: t9 in types b and
A4: t9 <= a ast t by A2,Def15;... | theorem | abcmiz_0:T25 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being adj-structured reflexive transitive antisymmetric
with_suprema 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
let T be adj-structured reflexive transitive antisymmetric with_suprema
TA-structure;
let A be Subset of the adjectives of T;
let t be type of T;
given t9 being type of T such that
A1: A c= adjs t9 and
A2: t9 <= t;
A3: t9 in downarrow t by A2,WAYBEL_0:17;
t9 in types A by A1,Th14;
hence thesis by A3... | theorem | abcmiz_0:Th26 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
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
let T be non empty reflexive transitive antisymmetric TA-structure;
let t be type of T;
set A = {} the adjectives of T;
types A = the carrier of T by Th16;
then types A /\ downarrow t = downarrow t by XBOOLE_1:28;
hence thesis by WAYBEL_0:34;
end; | theorem | abcmiz_0:Th27 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive TA-structure for t
being type of T holds apply(<*> the adjectives of T, t) = <*t*> | proof
let T be non empty non void reflexive transitive TA-structure;
let t be type of T;
A1: apply(<*> the adjectives of T, t).1 = t by Def19;
len apply(<*> the adjectives of T, t) = 0+1 by Def19,CARD_1:27;
hence thesis by A1,FINSEQ_1:40;
end; | theorem | abcmiz_0:T28 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive TA-structure
for t being type of T, a be adjective of T holds apply(<*a*>, t) = <*t, a ast t
*> | proof
let T be non empty non void reflexive transitive TA-structure;
let t be type of T, a be adjective of T;
A1: <*a*>.1 = a by FINSEQ_1:40;
A2: apply(<*a*>, t).1 = t by Def19;
A3: len <*a*> = 1 by FINSEQ_1:40;
then
A4: len apply(<*a*>, t) = 1+1 by Def19;
1 in dom <*a*> by A3,FINSEQ_3:25;
then apply(<*a*>, t... | theorem | abcmiz_0:Th29 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive TA-structure for t
being type of T holds (<*> the adjectives of T) ast t = t | by Def19 | theorem | abcmiz_0:T30 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive TA-structure
for t being type of T for a being adjective of T holds <*a*> ast t = a ast t | proof
let T be non empty non void reflexive transitive TA-structure;
let t be type of T;
let a be adjective of T;
A1: len <*a*> = 1 by FINSEQ_1:40;
apply(<*a*>, t) = <*t, a ast t*> by Th29;
hence thesis by A1,FINSEQ_1:44;
end; | theorem | abcmiz_0:Th31 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for p,q being FinSequence for i being Nat st i >= 1 & i <
len p holds (p$^q).i = p.i | proof
let p,q be FinSequence;
let i be Nat;
assume that
A1: i >= 1 and
A2: i < len p;
per cases;
suppose
q = {};
hence thesis by REWRITE1:1;
end;
suppose
q <> {};
then consider j being Element of NAT, r being FinSequence such that
A3: len p = j+1 and
A4: r = p|Seg j and
A5: p$^q = r^q by A... | theorem | abcmiz_0:T32 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for p being non empty FinSequence, q being FinSequence for i
being Nat st i < len q holds (p$^q).(len p+i) = q.(i+1) | proof
let p be non empty FinSequence, q be FinSequence;
let i be Nat;
A1: i+1 >= 1 by NAT_1:11;
assume
A2: i < len q;
then consider j being Element of NAT, r being FinSequence such that
A3: len p = j+1 and
A4: r = p|Seg j and
A5: p$^q = r^q by CARD_1:27,REWRITE1:def 1;
i+1 <= len q by A2,NAT_1:13;
then
A6: ... | theorem | abcmiz_0:Th33 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive 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
let T be non empty non void reflexive transitive TA-structure;
let t be type of T;
let v1,v2 be FinSequence of the adjectives of T;
consider tt being FinSequence of the carrier of T, q being Element of T such
that
A1: apply(v1,t) = tt^<*q*> by HILBERT2:4;
set s = apply(v1, t) $^ apply(v2, v1 ast t), p... | theorem | abcmiz_0:Th34 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive TA-structure
for t being type of T for v1,v2 being FinSequence of the adjectives of T for i
being Nat st i in dom v1 holds apply(v1^v2, t).i = apply(v1, t).i | proof
let T be non empty non void reflexive transitive TA-structure;
let t be type of T;
let v1,v2 be FinSequence of the adjectives of T;
set v = v1^v2;
consider tt being FinSequence of the carrier of T, q being Element of T such
that
A1: apply(v1,t) = tt^<*q*> by HILBERT2:4;
let i be Nat;
A2: len apply(v... | theorem | abcmiz_0:Th35 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive 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
let T be non empty non void reflexive transitive TA-structure;
let t be type of T;
let v1,v2 be FinSequence of the adjectives of T;
set v = v1^v2;
A1: len apply(v2, v1 ast t) = len v2+1 by Def19;
A2: apply(v,t) = apply(v1,t) $^ apply(v2, v1 ast t) by Th34;
len apply(v1,t) = len v1+1 by Def19;
then app... | theorem | abcmiz_0:Th36 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive 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
let T be non empty non void reflexive transitive TA-structure;
let t be type of T;
let v1,v2 be FinSequence of the adjectives of T;
set v = v1^v2;
consider tt being FinSequence of the carrier of T, q being Element of T such
that
A1: apply(v1,t) = tt^<*q*> by HILBERT2:4;
A2: len apply(v1,t) = len v1+1 ... | theorem | abcmiz_0:Th37 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive TA-structure for t
being type of T holds <*> the adjectives of T is_applicable_to t | proof
let T be non empty non void reflexive transitive TA-structure;
let t be type of T;
let i be Nat;
thus thesis;
end; | theorem | abcmiz_0:T38 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive TA-structure for t
being type of T, a being adjective of T holds a is_applicable_to t iff <*a*>
is_applicable_to t | proof
let T be non empty non void reflexive transitive TA-structure;
let t be type of T;
let a be adjective of T;
set v = <*a*>;
A1: v.1 = a by FINSEQ_1:40;
hereby
assume
A2: a is_applicable_to t;
thus <*a*> is_applicable_to t
proof
let i be Nat, b be adjective of T, s be type of T;
as... | theorem | abcmiz_0:T39 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive 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
let T be non empty non void reflexive transitive TA-structure;
let t be type of T;
let v1,v2 be FinSequence of the adjectives of T;
set v = v1^v2;
A1: apply(v,t) = apply(v1,t)$^apply(v2, v1 ast t) by Th34;
A2: len apply(v2, v1 ast t) = len v2+1 by Def19;
assume
A3: for i being Nat, a being adjective of ... | theorem | abcmiz_0:Th40 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TA-structure for t being type of T for v
being FinSequence of the adjectives of T st v is_applicable_to t for i1,i2
being Nat st 1 <= i1 & i1 <= i2 & i2 <= len v+1 for t1,t2 being type
of T st t1 = apply(v,t).i1 & t2 = ... | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TA-structure;
let t be type of T;
let v be FinSequence of the adjectives of T such that
A1: for i being Nat, a being adjective of T, s being type of
T st i in dom v & a = v.i & s = apply(v,t).i holds a is_applica... | theorem | abcmiz_0:Th41 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TA-structure for t being type of T for v
being FinSequence of the adjectives of T st v is_applicable_to t for s being
type of T st s in rng apply(v, t) holds v ast t <= s & s <= t | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TA-structure;
let t be type of T;
let v be FinSequence of the adjectives of T such that
A1: v is_applicable_to t;
A2: len apply(v,t) = len v+1 by Def19;
let s be type of T;
assume s in rng apply(v,t);
then co... | theorem | abcmiz_0:Th42 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TA-structure for t being type of T for v
being FinSequence of the adjectives of T st v is_applicable_to t holds v ast t
<= t | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TA-structure;
let t be type of T;
let v be FinSequence of the adjectives of T such that
A1: v is_applicable_to t;
A2: len v+1 >= 1 by NAT_1:11;
len apply(v,t) = len v+1 by Def19;
then len v+1 in dom apply(v, t)... | theorem | abcmiz_0:Th43 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TA-structure for t being type of T for v
being FinSequence of the adjectives of T st v is_applicable_to t holds rng v c=
adjs (v ast t) | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TA-structure;
let t be type of T;
let v be FinSequence of the adjectives of T such that
A1: v is_applicable_to t;
let a be set;
assume
A2: a in rng v;
then consider x being set such that
A3: x in dom v and
A4... | theorem | abcmiz_0:Th44 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TA-structure for t being type of T for v
being FinSequence of the adjectives of T st v is_applicable_to t for A being
Subset of the adjectives of T st A = rng v holds A is_applicable_to t | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TA-structure;
let t be type of T;
let v be FinSequence of the adjectives of T;
assume
A1: v is_applicable_to t;
then
A2: rng v c= adjs (v ast t) by Th44;
v ast t <= t by A1,Th43;
hence thesis by A2,Def16;
e... | theorem | abcmiz_0:Th45 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void 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 for s being type of T st s in rng apply(v2,t) holds v1 ast t <= s | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TA-structure;
let t be type of T;
let v,v9 be FinSequence of the adjectives of T such that
A1: v is_applicable_to t and
A2: rng v9 c= rng v;
defpred P[Nat] means $1 <= len apply(v9,t) implies for s being type of ... | theorem | abcmiz_0:Th46 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema 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
v2^v1 is_applicable_to t | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TA-structure;
let t be type of T;
let v1,v2 be FinSequence of the adjectives of T;
A1: rng (v1^v2) = rng v1 \/ rng v2 by FINSEQ_1:31;
assume
A2: v1^v2 is_applicable_to t;
then
A3: rng (v1^v2) c= adjs ((v1^v2) a... | theorem | abcmiz_0:Th47 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema 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^v2 ast t = v2^v1 ast t | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TA-structure;
let t be type of T;
let v1,v2 be FinSequence of the adjectives of T;
assume
A1: v1^v2 is_applicable_to t;
A2: len (v1^v2) = len v1+len v2 by FINSEQ_1:22;
A3: rng (v1^v2) = rng v1 \/ rng v2 by FINSEQ... | theorem | abcmiz_0:T48 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema 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
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema TA-structure;
let t be type of T;
let a be Subset of the adjectives of T;
assume a is_applicable_to t;
then types a /\ downarrow t is Ideal of T by Th26;
then sup (types a /\ downarrow t) in types a /\ downarrow t by ... | theorem | abcmiz_0:Th49 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema 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 c= adjs(A ast t) | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema TA-structure;
let t be type of T;
let a be Subset of the adjectives of T;
assume a is_applicable_to t;
then types a /\ downarrow t is Ideal of T by Th26;
then sup (types a /\ downarrow t) in types a /\ downarrow t by ... | theorem | abcmiz_0:Th50 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema 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 in types A | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema TA-structure;
let t be type of T;
let a be Subset of the adjectives of T;
assume a is_applicable_to t;
then types a /\ downarrow t is Ideal of T by Th26;
then sup (types a /\ downarrow t) in types a /\ downarrow t by ... | theorem | abcmiz_0:T51 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema TA-structure for t being type of T for A being
Subset of the adjectives of T for t9 being type of T st t9 <= t & A c= adjs t9
holds A is_applicable_to t & t9 <= A ast t | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema TA-structure;
let t be type of T;
let a be Subset of the adjectives of T;
let t9 be type of T;
assume that
A1: t9 <= t and
A2: a c= adjs t9;
A3: t9 in downarrow t by A1,WAYBEL_0:17;
thus a is_applicable_to t
proof
... | theorem | abcmiz_0:Th52 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema 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
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema TA-structure;
let t be type of T;
let a be Subset of the adjectives of T;
assume
A1: a c= adjs t;
hence a is_applicable_to t by Th52;
then
A2: a ast t <= t by Th49;
t <= a ast t by A1,Th52;
hence thesis by A2,YELL... | theorem | abcmiz_0:T53 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being TA-structure, 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
let T be TA-structure;
let t be type of T;
let A,B be Subset of the adjectives of T;
given t9 being type of T such that
A1: A c= adjs t9 and
A2: t9 <= t;
assume
A3: B c= A;
take t9;
thus thesis by A1,A2,A3,XBOOLE_1:1;
end; | theorem | abcmiz_0:Th54 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TA-structure for t being type of T, 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
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TA-structure;
let t be type of T, a be adjective of T;
let A,B be Subset of the adjectives of T such that
A1: B = A \/ {a} and
A2: B is_applicable_to t;
A3: A is_applicable_to t by A1,A2,Th54,XBOOLE_1:7;
A4: {a} c=... | theorem | abcmiz_0:Th55 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TA-structure for t being type of T for v
being FinSequence of the adjectives of T st v is_applicable_to t for A being
Subset of the adjectives of T st A = rng v holds v ast t = A ast t | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TA-structure;
defpred P[Element of NAT] means for t being type of T, v being FinSequence
of the adjectives of T st $1 = len v & v is_applicable_to t for A being Subset
of the adjectives of T st A = rng v holds v as... | theorem | abcmiz_0:Th56 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive
TAS-structure for t being type of T, v being FinSequence of the adjectives of T
st v is_properly_applicable_to t holds v is_applicable_to t | proof
let T be non empty non void reflexive transitive TAS-structure;
let t be type of T;
let v be FinSequence of the adjectives of T;
assume
A1: for i being Nat, a being adjective of T, 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;
let i be Nat, a be adje... | theorem | abcmiz_0:Th57 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive TAS-structure for
t being type of T holds <*> the adjectives of T is_properly_applicable_to t | proof
let T be non empty non void reflexive transitive TAS-structure;
let t be type of T;
let i be Nat;
thus thesis;
end; | theorem | abcmiz_0:T58 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive TAS-structure for
t being type of T, a being adjective of T holds a is_properly_applicable_to t
iff <*a*> is_properly_applicable_to t | proof
let T be non empty non void reflexive transitive TAS-structure;
let t be type of T;
let a be adjective of T;
set v = <*a*>;
A1: v.1 = a by FINSEQ_1:40;
hereby
assume
A2: a is_properly_applicable_to t;
thus <*a*> is_properly_applicable_to t
proof
let i be Nat, b be adjective of T, s be ... | theorem | abcmiz_0:T59 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive
TAS-structure for t being type of T, 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
let T be non empty non void reflexive transitive TAS-structure;
let t be type of T;
let v1,v2 be FinSequence of the adjectives of T;
set v = v1^v2;
A1: apply(v,t) = apply(v1,t)$^apply(v2, v1 ast t) by Th34;
A2: len apply(v2, v1 ast t) = len v2+1 by Def19;
assume
A3: for i being Nat, a being adjective of... | theorem | abcmiz_0:Th60 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive
TAS-structure for t being type of T, 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
let T be non empty non void reflexive transitive TAS-structure;
let t be type of T;
let v1,v2 be FinSequence of the adjectives of T;
set v = v1^v2;
assume
A1: for i being Nat, a being adjective of T, s being type of
T st i in dom v1 & a = v1.i & s = apply(v1,t).i holds a
is_properly_applicable_to s;... | theorem | abcmiz_0:Th61 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive
TAS-structure for t being type of T, A being Subset of the adjectives of T st A
is_properly_applicable_to t holds A is finite | proof
let T be non empty non void reflexive transitive TAS-structure;
let t be type of T, A be Subset of the adjectives of T;
assume ex s being FinSequence of the adjectives of T st rng s = A & s
is_properly_applicable_to t;
hence thesis;
end; | theorem | abcmiz_0:Th62 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive
TAS-structure for t being type of T holds {} the adjectives of T
is_properly_applicable_to t | proof
let T be non empty non void reflexive transitive TAS-structure;
let t be type of T;
take s = <*> the adjectives of T;
thus rng s = {} the adjectives of T;
let i be Nat;
thus thesis;
end; | theorem | abcmiz_0:Th63 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being non empty non void reflexive transitive
TAS-structure for t being type of T, A being Subset of the adjectives of T st A
is_properly_applicable_to t 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 &... | proof
let T be non empty non void reflexive transitive TAS-structure;
let t be type of T, A be Subset of the adjectives of T;
defpred P[set] means ex B being Subset of the adjectives of T st $1 = B & $1
c= A & B is_properly_applicable_to t & A ast t = B ast t;
assume
A1: A is_properly_applicable_to t;
then ... | theorem | abcmiz_0:Th64 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TAS-structure for t being type of T, A
being Subset of the adjectives of T st A is_properly_applicable_to t ex s being
one-to-one FinSequence of the adjectives of T st rng s = A & s
is_properly_applicable_to t | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TAS-structure;
let t be type of T, A be Subset of the adjectives of T;
given s9 being FinSequence of the adjectives of T such that
A1: rng s9 = A and
A2: s9 is_properly_applicable_to t;
defpred P[Nat] means ex s ... | theorem | abcmiz_0:Th65 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being adj-structured antisymmetric non void reflexive
transitive with_suprema Noetherian TAS-structure holds T@--> c= the
InternalRel of T | proof
let T be adj-structured with_suprema antisymmetric non empty non void
reflexive transitive Noetherian TAS-structure;
let t1,t2 be Element of T;
reconsider q1 = t1, q2 = t2 as type of T;
assume [t1,t2] in T@-->;
then consider a being adjective of T such that
not a in adjs q2 and
A1: a is_properly_app... | theorem | abcmiz_0:Th66 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being adj-structured antisymmetric non void reflexive
transitive with_suprema Noetherian TAS-structure for t1,t2 being type of T st
T@--> reduces t1,t2 holds t1 <= t2 | proof
let T be adj-structured with_suprema antisymmetric non empty non void
reflexive transitive Noetherian TAS-structure;
let t1,t2 be type of T;
set R = T@-->;
defpred P[Element of T, Element of T] means $1 <= $2;
A1: for x,y,z be Element of T holds P[x, y] & P[y, z] implies P[x, z] by
YELLOW_0:def 2;
A2: n... | theorem | abcmiz_0:Th67 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric non void with_suprema TAS-structure holds T@--> is irreflexive | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric non
void with_suprema TAS-structure;
set R = T@-->;
let x be set;
assume x in field R;
assume
A1: [x,x] in R;
then reconsider x as type of T by ZFMISC_1:87;
consider a being adjective of T such that
A2: not a in adjs x and
A3: a... | theorem | abcmiz_0:Th68 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being adj-structured antisymmetric non void reflexive
transitive with_suprema Noetherian TAS-structure holds T@--> is
strongly-normalizing | proof
let T be adj-structured with_suprema antisymmetric non empty non void
reflexive transitive Noetherian TAS-structure;
set R = T@-->, Q = the InternalRel of T;
A1: field R c= field Q by Th66,RELAT_1:16;
A2: R c= Q by Th66;
R is co-well_founded
proof
let Y be set;
assume that
A3: Y c= field R and
A... | theorem | abcmiz_0:Th69 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TAS-structure for t being type of T, 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 for s being one-to... | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TAS-structure;
let t be type of T, A be finite Subset of the adjectives of T such that
A1: 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;
let ... | theorem | abcmiz_0:Th70 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TAS-structure for t being type of T, 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 for s being one-to... | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TAS-structure;
let t be type of T, A be finite Subset of the adjectives of T such that
A1: 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;
let ... | theorem | abcmiz_0:Th71 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TAS-structure for t being type of T, A
being finite Subset of the adjectives of T st A is_properly_applicable_to t
holds T@--> reduces A ast t, t | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TAS-structure;
set R = T@-->;
let t be type of T, A be finite Subset of the adjectives of T;
assume A is_properly_applicable_to t;
then consider A9 being Subset of the adjectives of T such that
A9 c= A and
A1... | theorem | abcmiz_0:Th72 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
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
let X be non empty set;
let R be Relation of X;
let p be RedSequence of R such that
A1: p.1 in X;
let x be set;
assume x in rng p;
then consider i being set such that
A2: i in dom p and
A3: x = p.i by FUNCT_1:def 3;
reconsider i as Element of NAT by A2;
A4: i >= 1 by A2,FINSEQ_3:25;
per cases by A... | theorem | abcmiz_0:Th73 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for X being non empty set for R being Relation of X for x be
Element of X, y being set st R reduces x,y holds y in X | proof
let X be non empty set;
let R be Relation of X;
let x be Element of X, y be set;
given p being RedSequence of R such that
A1: p.1 = x and
A2: p.len p = y;
len p >= 0+1 by NAT_1:13;
then len p in dom p by FINSEQ_3:25;
then
A3: y in rng p by A2,FUNCT_1:3;
p is FinSequence of X by A1,Th73;
then rng... | theorem | abcmiz_0:Th74 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for X being non empty set for R being Relation of X st R is
with_UN_property weakly-normalizing for x be Element of X holds nf(x, R) in X | proof
let X be non empty set;
let R be Relation of X such that
A1: R is with_UN_property weakly-normalizing;
let x be Element of X;
nf(x,R) is_a_normal_form_of x, R by A1,REWRITE1:54;
then R reduces x, nf(x,R) by REWRITE1:def 6;
hence thesis by Th74;
end; | theorem | abcmiz_0:Th75 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TAS-structure for t1, t2 being type of T
st T@--> reduces t1, t2 ex A being finite Subset of the adjectives of T st A
is_properly_applicable_to t2 & t1 = A ast t2 | proof
let T be Noetherian adj-structured reflexive transitive antisymmetric
with_suprema non void TAS-structure;
let t1,t2 be type of T;
set R = T@-->;
given p being RedSequence of R such that
A1: p.1 = t1 and
A2: t2 = p.len p;
defpred P[set,set] means ex j being Element of NAT, a being adjective of T,
t ... | theorem | abcmiz_0:Th76 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being adj-structured antisymmetric commutative non void
reflexive transitive with_suprema Noetherian TAS-structure holds T@--> is
with_Church-Rosser_property with_UN_property | proof
let T be adj-structured with_suprema antisymmetric commutative non empty
non void reflexive transitive Noetherian TAS-structure;
set R = T@-->;
R is locally-confluent
proof
let a,b,c be set;
assume that
A1: [a,b] in R and
A2: [a,c] in R;
reconsider t = a, t1 = b, t2 = c as type of T by A1,A2... | theorem | abcmiz_0:Th77 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being adj-structured with_suprema antisymmetric
commutative non empty non void reflexive transitive Noetherian TAS-structure
for t being type of T holds T@--> reduces t, radix t | proof
let T be adj-structured with_suprema antisymmetric commutative non empty
non void reflexive transitive Noetherian TAS-structure;
let t be type of T;
set R = T@-->;
R is with_Church-Rosser_property with_UN_property strongly-normalizing
Relation by Th69,Th77;
then nf(t, R) is_a_normal_form_of t, R by ... | theorem | abcmiz_0:Th78 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being adj-structured with_suprema antisymmetric commutative non
empty non void reflexive transitive Noetherian TAS-structure for t being type
of T holds t <= radix t | by Th67,Th78 | theorem | abcmiz_0:T79 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being adj-structured with_suprema antisymmetric
commutative non empty non void reflexive transitive Noetherian TAS-structure
for t being type of T for X being set st X = {t9 where t9 is type of T: ex A
being finite Subset of the adjectives of T st A is_properly_applicable_to t9 &
A ast t9 = t} holds ex_sup_of... | proof
let T be adj-structured with_suprema antisymmetric commutative non empty
non void reflexive transitive Noetherian TAS-structure;
let t be type of T;
set R = T@-->;
set AA = {t9 where t9 is type of T: ex A being finite Subset of the
adjectives of T st A is_properly_applicable_to t9 & A ast t9 = t};
A1:... | theorem | abcmiz_0:Th80 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being adj-structured with_suprema antisymmetric
commutative non empty non void reflexive transitive Noetherian TAS-structure
for t1,t2 being type of T, a being adjective of T st a
is_properly_applicable_to t1 & a ast t1 <= radix t2 holds t1 <= radix t2 | proof
let T be adj-structured with_suprema antisymmetric commutative non empty
non void reflexive transitive Noetherian TAS-structure;
let t1,t2 be type of T, a be adjective of T;
set R = T@-->;
set AA = {t9 where t9 is type of T: ex A being finite Subset of the
adjectives of T st A is_properly_applicable_t... | theorem | abcmiz_0:Th81 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being adj-structured with_suprema antisymmetric commutative non
empty non void reflexive transitive Noetherian TAS-structure for t1,t2 being
type of T st t1 <= t2 holds radix t1 <= radix t2 | proof
let T be adj-structured with_suprema antisymmetric commutative non empty
non void reflexive transitive Noetherian TAS-structure;
set R = T@-->;
let t1, t2 be type of T such that
A1: t1 <= t2;
t2 <= radix t2 by Th67,Th78;
then
A2: t1 <= radix t2 by A1,YELLOW_0:def 2;
set X = the carrier of T;
defpr... | theorem | abcmiz_0:T82 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for T being adj-structured with_suprema antisymmetric commutative non
empty non void reflexive transitive Noetherian TAS-structure for t being type
of T, a being adjective of T st a is_properly_applicable_to t holds radix (a
ast t) = radix t | proof
let T be adj-structured with_suprema antisymmetric commutative non empty
non void reflexive transitive Noetherian TAS-structure;
let t be type of T, a be adjective of T;
A1: a in adjs t or not a in adjs t;
assume a is_properly_applicable_to t;
then a ast t = t or [a ast t, t] in T@--> by A1,Def31,Th24;
... | theorem | abcmiz_0:T83 | mml | mml/abcmiz_0.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for f being Function holds f.x c= Union f | proof
let f be Function;
x in dom f or not x in dom f;
then f.x in rng f or f.x = {} by FUNCT_1:3,def 2;
hence thesis by XBOOLE_1:2,ZFMISC_1:74;
end; | theorem | abcmiz_1:Th1 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for f being Function st Union f = {} holds f.x = {} | by Th1,XBOOLE_1:3 | theorem | abcmiz_1:T2 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for f being Function for x,y being set st f = [x,y] holds x = y | proof
let f be Function, x,y be set;
assume
A1: f = [x,y];
then
A2: {x} in f by TARSKI:def 2;
A3: {x,y} in f by A1,TARSKI:def 2;
consider a,b being set such that
A4: {x} = [a,b] by A2,RELAT_1:def 1;
A5: {a} = {a,b} by A4,ZFMISC_1:5;
A6: x = {a} by A4,ZFMISC_1:4;
consider c,d being set such that
A7: {x,y} = [c... | theorem | abcmiz_1:Th3 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
(id X).:Y c= Y | proof
let x be set;
assume x in (id X).:Y;
then ex y being set st [y,x] in id X & y in Y by RELAT_1:def 13;
hence thesis by RELAT_1:def 10;
end; | theorem | abcmiz_1:Th4 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for S being non void Signature
for X being non-empty ManySortedSet of the carrier of S
for t being Term of S, X
holds t is non pair | proof
let S be non void Signature;
let X be non-empty ManySortedSet of the carrier of S;
let t be Term of S, X;
given x,y being set such that
A1: t = [x,y];
(ex s being SortSymbol of S, v being Element of X.s st t.{} = [v,s])
or t.{} in [:the carrier' of S,{the carrier of S}:]
by MSATERM:2;
then (ex s b... | theorem | abcmiz_1:Th5 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for x,y,z being set st x in {z}* & y in {z}* & card x = card y holds x = y | proof
let x,y,z be set such that
A1: x in {z}* and
A2: y in {z}* and
A3: card x = card y;
reconsider x, y as FinSequence of {z} by A1,A2,FINSEQ_1:def 11;
A4: dom x = Seg len x by FINSEQ_1:def 3
.= dom y by A3,FINSEQ_1:def 3;
now
let i be Nat;
assume
A5: i in dom x;
then
A6: x .i in rng x by FUNCT_... | theorem | abcmiz_1:Th6 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for S being non void Signature
for X being non empty-yielding ManySortedSet of the carrier of S
for t being Element of Free(S,X) holds
(ex s being SortSymbol of S, v being set st
t = root-tree [v,s] & v in X.s) or
ex o being OperSymbol of S,
p being FinSequence of Free(S,X) st
t = [o,the carrier of S]-tre... | proof
let S be non void Signature;
let X be non empty-yielding ManySortedSet of the carrier of S;
let t be Element of Free(S,X);
set V = X\/((the carrier of S)-->{0});
reconsider t9 = t as Term of S,V by MSAFREE3:8;
defpred P[set] means $1 is Element of Free(S,X) implies
(ex s being SortSymbol of S, v bei... | theorem | abcmiz_1:Th7 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
varcl {} = {} | proof
A1: for x,y st [x,y] in {} holds x c= {};
for B being set st {} c= B & for x,y st [x,y] in B holds x c= B holds {}
c= B;
hence thesis by A1,Def1;
end; | theorem | abcmiz_1:Th8 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for A,B being set st A c= B holds varcl A c= varcl B | proof
let A, B be set such that
A1: A c= B;
B c= varcl B by Def1;
then
A2: A c= varcl B by A1,XBOOLE_1:1;
for x,y st [x,y] in varcl B holds x c= varcl B by Def1;
hence thesis by A2,Def1;
end; | theorem | abcmiz_1:Th9 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for A being set holds
varcl union A = union {varcl a where a is Element of A: not contradiction} | proof
let A be set;
set X = {varcl a where a is Element of A: not contradiction};
A1: union A c= union X
proof
let x;
assume x in union A;
then consider Y such that
A2: x in Y and
A3: Y in A by TARSKI:def 4;
reconsider Y as Element of A by A3;
A4: Y c= varcl Y by Def1;
varcl Y in X;
hence ... | theorem | abcmiz_1:Th10 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
varcl (X \/ Y) = (varcl X) \/ (varcl Y) | proof
set A = {varcl a where a is Element of {X,Y}: not contradiction};
X \/ Y = union {X,Y} by ZFMISC_1:75;
then
A1: varcl (X \/ Y) = union A by Th10;
A = {varcl X, varcl Y}
proof
thus
now
let x;
assume x in A;
then consider a being Element of {X,Y} such that
A2: x = varcl a;
... | theorem | abcmiz_1:Th11 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for A being non empty set st for a being Element of A holds varcl a = a
holds varcl meet A = meet A | proof
let B be non empty set;
set A = meet B;
assume
A1: for a being Element of B holds varcl a = a;
now
thus A c= A;
let x,y;
assume
A2: [x,y] in A;
now
let Y;
assume
A3: Y in B;
then
A4: [x,y] in Y by A2,SETFAM_1:def 1;
Y = varcl Y by A1,A3;
hence x c= Y by A4... | theorem | abcmiz_1:Th12 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
varcl ((varcl X) /\ (varcl Y)) = (varcl X) /\ (varcl Y) | proof
set A = (varcl X) /\ (varcl Y);
now
thus A c= A;
let x,y;
assume
A1: [x,y] in A;
then
A2: [x,y] in varcl X by XBOOLE_0:def 4;
A3: [x,y] in varcl Y by A1,XBOOLE_0:def 4;
A4: x c= varcl X by A2,Def1;
x c= varcl Y by A3,Def1;
hence x c= A by A4,XBOOLE_1:19;
end;
hence varcl ((varcl X)... | theorem | abcmiz_1:Th13 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for V being ManySortedSet of NAT st
V.0 = {[{}, i] where i is Element of NAT: not contradiction} &
for n being Nat holds
V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT:
A is finite}
for i,j being Element of NAT st i <= j holds V.i c= V.j | proof
let V be ManySortedSet of NAT such that
A1: V.0 = {[{}, i] where i is Element of NAT: not contradiction} and
A2: for n being Nat holds
V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT:
A is finite};
defpred Q[Nat] means V.0 c= V.$1;
A3: now
let j;
assume Q[j];
A4: V.(j+1) = {[... | theorem | abcmiz_1:Th14 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for V being ManySortedSet of NAT st
V.0 = {[{}, i] where i is Element of NAT: not contradiction} &
for n being Nat holds
V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT:
A is finite}
for A being finite Subset of Vars
ex i being Element of NAT st A c= V.i | proof
let V be ManySortedSet of NAT such that
A1: V.0 = {[{}, i] where i is Element of NAT: not contradiction} and
A2: for n being Nat holds
V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT:
A is finite};
let A be finite Subset of Vars;
A3: Vars = Union V by A1,A2,Def2;
defpred P[set,set]... | theorem | abcmiz_1:Th15 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
{[{}, i] where i is Element of NAT: not contradiction} c= Vars | proof consider V being ManySortedSet of NAT such that
A1: Vars = Union V and
A2: V.0 = {[{}, i] where i is Element of NAT: not contradiction} and
for n being Nat holds
V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT:
A is finite} by Def2;
dom V = NAT by PARTFUN1:def 2;
then V.0 in rng V ... | theorem | abcmiz_1:Th16 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff | |
for A being finite Subset of Vars, i being Nat holds [varcl A, i] in Vars | proof
let A be finite Subset of Vars, i be Nat;
consider V being ManySortedSet of NAT such that
A1: Vars = Union V and
A2: V.0 = {[{}, k] where k is Element of NAT: not contradiction} and
A3: for n being Nat holds
V.(n+1) = {[varcl b, j] where b is Subset of V.n, j is Element of NAT:
b is finite} by Def2;
con... | theorem | abcmiz_1:Th17 | mml | mml/abcmiz_1.miz | [] | [] | https://github.com/MizarSystem/MML | 047822c4d814630b28eec8ca6b455e9eb912d5ff |
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Mizar
A structured dataset of theorems and schemes from the Mizar Mathematical Library (MML), one of the largest libraries of formalized mathematics.
Source
- Repository: https://github.com/MizarSystem/MML
- Commit:
047822c4d814630b28eec8ca6b455e9eb912d5ff - Files: 1153
- License: gpl-3.0
Schema
| Column | Type | Description |
|---|---|---|
| statement | string | Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof |
| proof | string | Verbatim proof/body, empty if the declaration has none |
| type | string | Declaration keyword |
| symbolic_name | string | Declaration identifier |
| library | string | Sub-library |
| filename | string | Repository-relative source path |
| imports | list[string] | File-level Require/Import modules |
| deps | list[string] | Intra-corpus identifiers referenced |
| docstring | string | Preceding documentation comment, empty if absent |
| source_url | string | Upstream repository |
| commit | string | Upstream commit extracted |
Statistics
- Entries: 51,673
- With proof: 50,494 (97.7%)
- With docstring: 0 (0.0%)
- Libraries: 1
By type
| Type | Count |
|---|---|
| theorem | 51,673 |
Example
for A1,A2 being AdjectiveStr st the adjectives of A1 = the adjectives
of A2 holds A1 is void implies A2 is void
proof
let A1,A2 be AdjectiveStr such that
A1: the adjectives of A1 = the adjectives of A2;
assume the adjectives of A1 is empty;
hence the adjectives of A2 is empty by A1;
end;
- type: theorem | symbolic_name:
abcmiz_0:T2| mml/abcmiz_0.miz
Use
Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint
and together form the complete declaration, for proof modeling, autoformalization, retrieval, and
dependency analysis via deps.
Citation
@misc{mizar_dataset,
title = {Mizar},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/MizarSystem/MML, commit 047822c4d814},
url = {https://huggingface.co/datasets/phanerozoic/Mizar}
}
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