Nuprl Lemma : sub-bag-remove-if
∀[T:Type]. ∀[bs:bag(T)]. ∀eq:EqDecider(T). ∀as:bag(T). ∀x:T. (sub-bag(T;as;bs) ⇒ sub-bag(T;as - x;bs))
Proof
Definitions occuring in Statement :
bag-remove: bs - x,
sub-bag: sub-bag(T;as;bs),
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
sub-bag: sub-bag(T;as;bs),
exists: ∃x:A. B[x],
member: t ∈ T,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
bag-remove: bs - x,
top: Top,
deq: EqDecider(T),
squash: ↓T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
eqof: eqof(d),
true: True,
guard: {T}
Lemmas referenced :
decidable__assert,
bag-deq-member_wf,
sub-bag_wf,
deq_wf,
bag-append_wf,
bag-filter_wf,
eqof_wf,
subtype_rel_bag,
assert_wf,
bag-append-comm,
bag-append-assoc,
equal_wf,
bag_wf,
bnot_wf,
bag-remove_wf,
bag-filter-split,
squash_wf,
true_wf,
bool_wf,
iff_imp_equal_bool,
not_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
safe-assert-deq,
iff_wf,
bag-remove-trivial,
assert-bag-deq-member,
bag-member_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
unionElimination,
because_Cache,
dependent_pairFormation,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
isect_memberEquality,
voidElimination,
voidEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
equalityTransitivity,
imageElimination,
functionEquality,
universeEquality,
independent_pairFormation,
independent_functionElimination,
addLevel,
impliesFunctionality,
levelHypothesis,
andLevelFunctionality,
impliesLevelFunctionality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityUniverse,
promote_hyp
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)].
\mforall{}eq:EqDecider(T). \mforall{}as:bag(T). \mforall{}x:T. (sub-bag(T;as;bs) {}\mRightarrow{} sub-bag(T;as - x;bs))
Date html generated:
2018_05_21-PM-09_47_47
Last ObjectModification:
2017_07_26-PM-06_30_26
Theory : bags_2
Home
Index