Nuprl Lemma : decidable__bag-member
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀x:T. ∀bs:bag(T). Dec(x ↓∈ bs)))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ
Lemmas referenced :
deq-exists,
bag-member-count,
bag_wf,
all_wf,
decidable_wf,
equal_wf,
bag-member_wf,
le_wf,
bag-count_wf,
nat_wf,
decidable__le,
decidable_functionality,
iff_weakening_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
productElimination,
independent_functionElimination,
hypothesis,
rename,
hypothesisEquality,
sqequalRule,
lambdaEquality,
universeEquality,
natural_numberEquality,
applyEquality,
setElimination,
dependent_functionElimination
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}x:T. \mforall{}bs:bag(T). Dec(x \mdownarrow{}\mmember{} bs)))
Date html generated:
2016_05_15-PM-07_58_26
Last ObjectModification:
2015_12_27-PM-04_16_28
Theory : bags_2
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