Nuprl Lemma : bag-remove-trivial
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs:bag(T)]. ∀[x:T]. bs - x = bs ∈ bag(T) supposing ¬x ↓∈ bs
Proof
Definitions occuring in Statement :
bag-remove: bs - x,
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag-remove: bs - x,
prop: ℙ,
deq: EqDecider(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
eqof: eqof(d),
true: True,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
false: False
Lemmas referenced :
not_wf,
bag-member_wf,
bag_wf,
deq_wf,
bnot_wf,
equal_wf,
iff_transitivity,
assert_wf,
eqof_wf,
iff_weakening_uiff,
assert_of_bnot,
safe-assert-deq,
squash_wf,
true_wf,
bag-filter-trivial2,
iff_weakening_equal,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
lambdaEquality,
applyEquality,
setElimination,
rename,
lambdaFormation,
addLevel,
independent_functionElimination,
independent_pairFormation,
impliesFunctionality,
productElimination,
independent_isectElimination,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
voidElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)]. \mforall{}[x:T]. bs - x = bs supposing \mneg{}x \mdownarrow{}\mmember{} bs
Date html generated:
2018_05_21-PM-09_47_44
Last ObjectModification:
2017_07_26-PM-06_30_24
Theory : bags_2
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