Nuprl Lemma : filter-interface-predecessors-first-at
∀[Info:Type]
∀es:EO+(Info)
∀[T:Type]
∀X:EClass(T). ∀P:E(X) ⟶ 𝔹. ∀i:Id.
∀[R:Id ⟶ E(X) ⟶ ℙ]
∀n:ℕ+. ∀L:Id List.
((∀x∈L.∃y:E(X). (R[x;y] ∧ (↑P[y])))
⇒ (∃e:E(X). ((↑P[e]) ∧ e is first@ i s.t. q.||filter(λe.P[e];≤(X)(q))|| = n ∈ ℤ))) supposing
((n ≤ ||L||) and
no_repeats(Id;L))
supposing ∀x1,x2:Id. ∀y:E(X). (R[x1;y] ⇒ R[x2;y] ⇒ (x1 = x2 ∈ Id))
supposing ∀e:E(X). loc(e) = i ∈ Id supposing ↑P[e]
Proof
Definitions occuring in Statement :
es-interface-predecessors: ≤(X)(e),
es-E-interface: E(X),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-first-at: e is first@ i s.t. e.P[e],
es-loc: loc(e),
Id: Id,
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
length: ||as||,
filter: filter(P;l),
list: T List,
nat_plus: ℕ+,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
top: Top,
implies: P ⇒ Q,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
nat_plus: ℕ+,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
es-E-interface: E(X),
cand: A c∧ B,
sq_type: SQType(T),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
strongwellfounded: SWellFounded(R[x; y]),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
bfalse: ff,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
es-first-at: e is first@ i s.t. e.P[e],
alle-lt: ∀e<e'.P[e],
es-locl: (e <loc e')
Latex:
\mforall{}[Info:Type]
\mforall{}es:EO+(Info)
\mforall{}[T:Type]
\mforall{}X:EClass(T). \mforall{}P:E(X) {}\mrightarrow{} \mBbbB{}. \mforall{}i:Id.
\mforall{}[R:Id {}\mrightarrow{} E(X) {}\mrightarrow{} \mBbbP{}]
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}L:Id List.
((\mforall{}x\mmember{}L.\mexists{}y:E(X). (R[x;y] \mwedge{} (\muparrow{}P[y])))
{}\mRightarrow{} (\mexists{}e:E(X)
((\muparrow{}P[e]) \mwedge{} e is first@ i s.t. q.||filter(\mlambda{}e.P[e];\mleq{}(X)(q))|| = n))) supposing
((n \mleq{} ||L||) and
no\_repeats(Id;L))
supposing \mforall{}x1,x2:Id. \mforall{}y:E(X). (R[x1;y] {}\mRightarrow{} R[x2;y] {}\mRightarrow{} (x1 = x2))
supposing \mforall{}e:E(X). loc(e) = i supposing \muparrow{}P[e]
Date html generated:
2016_05_17-AM-07_07_28
Last ObjectModification:
2016_01_17-PM-03_09_44
Theory : event-ordering
Home
Index