Nuprl Lemma : decidable__l_all
∀[A:Type]. ∀L:A List. ∀[F:{a:A| (a ∈ L)} ⟶ ℙ]. ((∀k:{a:A| (a ∈ L)} . Dec(F[k])) ⇒ Dec((∀k∈L.F[k])))
Proof
Definitions occuring in Statement :
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],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
l-all-decider: l-all-decider(),
ifthenelse: if b then t else f fi ,
genrec-ap: genrec-ap,
it: ⋅,
any: any x,
decidable__false,
decidable__implies,
decidable__not,
int_seg_decide: int_seg_decide(d;i;j),
squash: ↓T,
or: P ∨ Q,
guard: {T},
prop: ℙ,
has-value: (a)↓,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
strict4: strict4(F),
uimplies: b supposing a,
so_apply: x[s],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
uall: ∀[x:A]. B[x],
decidable__exists_int_seg,
decidable__all_int_seg,
decidable__l_all-proof,
member: t ∈ T
Lemmas referenced :
lifting-strict-decide,
is-exception_wf,
base_wf,
has-value_wf_base,
equal_wf,
top_wf,
lifting-strict-callbyvalue,
decidable__l_all-proof,
decidable__false,
decidable__implies,
decidable__not,
decidable__exists_int_seg,
decidable__all_int_seg
Rules used in proof :
inlFormation,
exceptionSqequal,
imageElimination,
imageMemberEquality,
because_Cache,
inrFormation,
decideExceptionCases,
closedConclusion,
baseApply,
independent_functionElimination,
dependent_functionElimination,
sqleReflexivity,
unionElimination,
unionEquality,
equalitySymmetry,
equalityTransitivity,
hypothesisEquality,
callbyvalueDecide,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
baseClosed,
isectElimination,
sqequalHypSubstitution,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
Latex:
\mforall{}[A:Type]. \mforall{}L:A List. \mforall{}[F:\{a:A| (a \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}k:\{a:A| (a \mmember{} L)\} . Dec(F[k])) {}\mRightarrow{} Dec((\mforall{}k\mmember{}L.F[k])\000C))
Date html generated:
2018_05_21-PM-00_35_36
Last ObjectModification:
2018_05_18-PM-04_17_57
Theory : list_1
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