Nuprl Lemma : filter-interface-predecessors-lower-bound
∀[Info:Type]. ∀[es:EO+(Info)]. ∀[T:Type]. ∀[X:EClass(T)]. ∀[P:E(X) ⟶ 𝔹]. ∀[n:ℕ]. ∀[f:ℕn ⟶ {e:E(X)| ↑P[e]} ].
∀[k:ℕn]. n ≤ ||filter(λe.P[e];≤(X)(f k))|| supposing ∀i:ℕn. f i ≤loc f k supposing Inj(ℕn;{e:E(X)| ↑P[e]} ;f)
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-le: e ≤loc e' ,
length: ||as||,
filter: filter(P;l),
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
es-E-interface: E(X),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
pi1: fst(t),
inject: Inj(A;B;f)
Latex:
\mforall{}[Info:Type]. \mforall{}[es:EO+(Info)]. \mforall{}[T:Type]. \mforall{}[X:EClass(T)]. \mforall{}[P:E(X) {}\mrightarrow{} \mBbbB{}]. \mforall{}[n:\mBbbN{}].
\mforall{}[f:\mBbbN{}n {}\mrightarrow{} \{e:E(X)| \muparrow{}P[e]\} ].
\mforall{}[k:\mBbbN{}n]. n \mleq{} ||filter(\mlambda{}e.P[e];\mleq{}(X)(f k))|| supposing \mforall{}i:\mBbbN{}n. f i \mleq{}loc f k
supposing Inj(\mBbbN{}n;\{e:E(X)| \muparrow{}P[e]\} ;f)
Date html generated:
2016_05_17-AM-07_05_55
Last ObjectModification:
2016_01_17-PM-03_07_27
Theory : event-ordering
Home
Index