Nuprl Lemma : bag-member-list
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀x:T. ∀L:T List. (x ↓∈ L ⇐⇒ (x ∈ L))))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
l_member: (x ∈ l),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
bag: bag(T),
quotient: x,y:A//B[x; y],
sq_stable: SqStable(P),
guard: {T}
Lemmas referenced :
bag-member_wf,
list-subtype-bag,
l_member_wf,
list_wf,
all_wf,
decidable_wf,
equal_wf,
member_wf,
permutation_wf,
sq_stable__l_member,
member-permutation,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
applyEquality,
because_Cache,
independent_isectElimination,
lambdaEquality,
hypothesis,
sqequalRule,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
pertypeElimination,
productEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}x:T. \mforall{}L:T List. (x \mdownarrow{}\mmember{} L \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L))))
Date html generated:
2017_10_01-AM-08_53_39
Last ObjectModification:
2017_07_26-PM-04_35_23
Theory : bags
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