Nuprl Lemma : bag-restrict-disjoint
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[b:bag(T)]. (b|x) ~ {} supposing ¬x ↓∈ b
Proof
Definitions occuring in Statement :
bag-restrict: (b|x),
bag-member: x ↓∈ bs,
empty-bag: {},
bag: bag(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
not: ¬A,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
empty-bag: {},
bag-restrict: (b|x),
bag-filter: [x∈b|p[x]],
deq: EqDecider(T),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
false: False,
uiff: uiff(P;Q),
eqof: eqof(d)
Lemmas referenced :
equal-empty-bag,
bag-restrict_wf,
bag_to_squash_list,
not_wf,
bag-member_wf,
bag-member-list,
decidable-equal-deq,
l_member_wf,
equal-wf-T-base,
bag_wf,
deq_wf,
filter_is_nil,
nil_wf,
list-subtype-bag,
l_all_iff,
assert_wf,
and_wf,
equal_wf,
safe-assert-deq,
eqof_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
cumulativity,
hypothesisEquality,
hypothesis,
independent_functionElimination,
imageElimination,
productElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
sqequalRule,
rename,
lambdaFormation,
baseClosed,
sqequalAxiom,
isect_memberEquality,
equalityTransitivity,
universeEquality,
lambdaEquality,
applyEquality,
setElimination,
independent_isectElimination,
voidEquality,
voidElimination,
setEquality,
dependent_set_memberEquality,
independent_pairFormation,
addLevel,
impliesFunctionality,
levelHypothesis,
impliesLevelFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[b:bag(T)]. (b|x) \msim{} \{\} supposing \mneg{}x \mdownarrow{}\mmember{} b
Date html generated:
2016_10_25-AM-11_31_38
Last ObjectModification:
2016_07_12-AM-07_36_40
Theory : bags_2
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