Nuprl Lemma : decidable__all-list
∀[T:Type]
((∀x,y:T. Dec(x = y ∈ T))
⇒ (∀L:T List. ∀[P:{x:T| (x ∈ L)} ⟶ ℙ]. ((∀x:{x:T| (x ∈ L)} . Dec(P[x])) ⇒ Dec(∀x:{x:T| (x ∈ L)} . P[x]))))
Proof
Definitions occuring in Statement :
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,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
decidable__l_all,
decidable_functionality,
iff_wf,
l_all_wf,
l_all_iff,
set_wf,
sq_stable__l_member,
equal_wf,
list_wf,
decidable_wf,
l_member_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
independent_pairFormation,
dependent_functionElimination,
setElimination,
rename,
independent_functionElimination,
because_Cache,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
addLevel,
productElimination,
impliesFunctionality
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}L:T List
\mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x:\{x:T| (x \mmember{} L)\} . Dec(P[x])) {}\mRightarrow{} Dec(\mforall{}x:\{x:T| (x \mmember{} L)\} . P[x])\000C)))
Date html generated:
2016_05_14-PM-03_31_48
Last ObjectModification:
2016_01_14-PM-11_20_26
Theory : decidable!equality
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