Nuprl Lemma : bag-extensionality
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as,bs:bag(T)]. uiff(as = bs ∈ bag(T);∀x:T. ((#x in as) = (#x in bs) ∈ ℤ))
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag: bag(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
nat: ℕ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
bag: bag(T),
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
istype: istype(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-size: #(bs),
eqof: eqof(d),
bag-filter: [x∈b|p[x]],
deq: EqDecider(T)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
bag-count_wf,
bag_wf,
deq_wf,
nat_wf,
subtype_rel_self,
iff_weakening_equal,
list_wf,
permutation_wf,
permutation_weakening,
quotient-member-eq,
permutation-equiv,
list-subtype-bag,
istype-int,
set_subtype_base,
le_wf,
int_subtype_base,
permutation-iff-count,
filter_functionality,
eta_conv,
bool_wf,
length_wf,
filter_wf5,
subtype_rel_dep_function,
l_member_wf,
bag-count-sqequal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaFormation_alt,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
inhabitedIsType,
universeEquality,
intEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
setElimination,
rename,
because_Cache,
instantiate,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
equalityIsType1,
promote_hyp,
pointwiseFunctionality,
pertypeElimination,
productIsType,
equalityIsType4,
functionIsType,
closedConclusion,
independent_pairEquality,
isect_memberEquality_alt,
hyp_replacement,
setEquality,
setIsType
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[as,bs:bag(T)]. uiff(as = bs;\mforall{}x:T. ((\#x in as) = (\#x in bs)))
Date html generated:
2019_10_16-AM-11_30_35
Last ObjectModification:
2018_10_11-PM-07_08_11
Theory : bags_2
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