Nuprl Lemma : fl-filter-subset
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Q:fset(T + T) ⟶ 𝔹]. ∀[s:fset(fset(T + T))]. fl-filter(s;x.Q[x]) ⊆ s
Proof
Definitions occuring in Statement :
fl-filter: fl-filter(s;x.Q[x]),
deq-fset: deq-fset(eq),
f-subset: xs ⊆ ys,
fset: fset(T),
union-deq: union-deq(A;B;a;b),
deq: EqDecider(T),
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
fl-filter: fl-filter(s;x.Q[x]),
cal-filter: cal-filter(s;x.P[x]),
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
deq_wf,
bool_wf,
union-deq_wf,
deq-fset_wf,
fset_wf,
fset-filter-subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
unionEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[Q:fset(T + T) {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(fset(T + T))].
fl-filter(s;x.Q[x]) \msubseteq{} s
Date html generated:
2016_05_18-AM-11_40_46
Last ObjectModification:
2016_01_19-AM-09_58_41
Theory : lattices
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