Nuprl Lemma : l-all-decider_wf
∀[A:Type]
∀L:A List. ∀[F:{a:A| (a ∈ L)} ⟶ ℙ]. ∀dcd:∀k:{a:A| (a ∈ L)} . Dec(F[k]). (l-all-decider() L dcd ∈ Dec((∀k∈L.F[k])))
Proof
Definitions occuring in Statement :
l-all-decider: l-all-decider(),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable__l_all,
subtype_rel: A ⊆r B,
implies: P ⇒ Q
Lemmas referenced :
all_wf,
l_member_wf,
decidable_wf,
list_wf,
decidable__l_all,
l_all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
isectElimination,
thin,
setEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
isect_memberEquality,
because_Cache,
instantiate,
isectEquality
Latex:
\mforall{}[A:Type]
\mforall{}L:A List
\mforall{}[F:\{a:A| (a \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]
\mforall{}dcd:\mforall{}k:\{a:A| (a \mmember{} L)\} . Dec(F[k]). (l-all-decider() L dcd \mmember{} Dec((\mforall{}k\mmember{}L.F[k])))
Date html generated:
2018_05_21-PM-00_35_46
Last ObjectModification:
2018_05_19-AM-06_43_05
Theory : list_1
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