Nuprl Lemma : decidable__bag-member2
∀[T:Type]. ∀x:T. ((∀y:T. Dec(x = y ∈ 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],
all: ∀x:A. B[x],
implies: P ⇒ Q,
decidable: Dec(P),
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
guard: {T},
not: ¬A,
uimplies: b supposing a,
assert: ↑b,
ifthenelse: if b then t else f fi ,
isl: isl(x),
btrue: tt,
true: True,
bfalse: ff,
false: False,
squash: ↓T,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
bag-filter: [x∈b|p[x]],
bag-null: bag-null(bs),
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
decidable__assert,
bag-null_wf,
assert_wf,
isl_wf,
equal_wf,
not_wf,
bag-filter_wf,
bag_wf,
all_wf,
decidable_wf,
bag-filter-empty,
or_wf,
bag-member_wf,
bag_to_squash_list,
sq_stable__bag-member,
list-member-bag-member,
assert_of_null,
equal-wf-T-base,
list_wf,
filter_wf5,
l_member_wf,
member_exists,
member_filter,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
cut,
sqequalHypSubstitution,
sqequalRule,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
isectElimination,
because_Cache,
hypothesis,
applyEquality,
functionExtensionality,
lambdaEquality,
unionElimination,
universeEquality,
inrFormation,
independent_isectElimination,
independent_functionElimination,
natural_numberEquality,
voidElimination,
equalityTransitivity,
equalitySymmetry,
inlFormation,
imageElimination,
productElimination,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
setElimination,
baseClosed,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}x:T. ((\mforall{}y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}bs:bag(T). Dec(x \mdownarrow{}\mmember{} bs)))
Date html generated:
2018_05_21-PM-09_46_23
Last ObjectModification:
2017_07_26-PM-06_29_59
Theory : bags_2
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