Nuprl Lemma : filter-interface-predecessors-lower-bound2
∀[Info:Type]
∀es:EO+(Info)
∀[T:Type]
∀X:EClass(T). ∀P:E(X) ⟶ 𝔹.
∀[R:Id ⟶ E(X) ⟶ ℙ]
∀L:Id List
((∀x∈L.∃y:E(X). (R[x;y] ∧ (↑P[y])))
⇒ (∃e:{e:E(X)| ↑P[e]} . (||L|| ≤ ||filter(λe.P[e];≤(X)(e))||))) supposing
(0 < ||L|| and
no_repeats(Id;L))
supposing ∀x1,x2:Id. ∀y:E(X). (R[x1;y] ⇒ R[x2;y] ⇒ (x1 = x2 ∈ Id))
supposing ∀e1,e2:E(X). (loc(e1) = loc(e2) ∈ Id) supposing ((↑P[e2]) and (↑P[e1]))
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-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,
assert: ↑b,
bool: 𝔹,
less_than: a < 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,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
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,
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
nat_plus: ℕ+,
es-E-interface: E(X),
and: P ∧ Q,
cand: A c∧ B,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True
Latex:
\mforall{}[Info:Type]
\mforall{}es:EO+(Info)
\mforall{}[T:Type]
\mforall{}X:EClass(T). \mforall{}P:E(X) {}\mrightarrow{} \mBbbB{}.
\mforall{}[R:Id {}\mrightarrow{} E(X) {}\mrightarrow{} \mBbbP{}]
\mforall{}L:Id List
((\mforall{}x\mmember{}L.\mexists{}y:E(X). (R[x;y] \mwedge{} (\muparrow{}P[y])))
{}\mRightarrow{} (\mexists{}e:\{e:E(X)| \muparrow{}P[e]\} . (||L|| \mleq{} ||filter(\mlambda{}e.P[e];\mleq{}(X)(e))||))) supposing
(0 < ||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{}e1,e2:E(X). (loc(e1) = loc(e2)) supposing ((\muparrow{}P[e2]) and (\muparrow{}P[e1]))
Date html generated:
2016_05_17-AM-07_06_53
Last ObjectModification:
2015_12_29-AM-00_09_37
Theory : event-ordering
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