Nuprl Lemma : bag-member-list-member
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀L:T List. ∀b:bag(T). ∀x:T. ((L = b ∈ bag(T)) ⇒ x ↓∈ b ⇒ (x ∈ L))))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
l_member: (x ∈ l),
list: T List,
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,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
empty-bag: {},
uiff: uiff(P;Q),
false: False,
cons-bag: x.b,
sq_or: a ↓∨ b,
or: P ∨ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P)
Lemmas referenced :
bag-member_wf,
squash_wf,
true_wf,
bag_wf,
iff_weakening_equal,
list_induction,
list-subtype-bag,
l_member_wf,
list_wf,
bag-member-empty-iff,
nil_wf,
bag-member-cons,
cons_wf,
equal_wf,
all_wf,
decidable_wf,
cons_member,
sq_stable__l_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
functionEquality,
cumulativity,
voidElimination,
voidEquality,
rename,
dependent_functionElimination,
unionElimination,
inlFormation,
hyp_replacement,
Error :applyLambdaEquality,
inrFormation
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}L:T List. \mforall{}b:bag(T). \mforall{}x:T. ((L = b) {}\mRightarrow{} x \mdownarrow{}\mmember{} b {}\mRightarrow{} (x \mmember{} L))))
Date html generated:
2016_10_25-AM-10_28_10
Last ObjectModification:
2016_07_12-AM-06_44_25
Theory : bags
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