Nuprl Lemma : sub-bag-filter
∀[T:Type]. ∀p:T ⟶ 𝔹. ∀b,c:bag(T). (sub-bag(T;b;[x∈c|p[x]]) ⇐⇒ sub-bag(T;b;c) ∧ (∀x:T. (x ↓∈ b ⇒ (↑p[x]))))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
sub-bag: sub-bag(T;as;bs),
bag-filter: [x∈b|p[x]],
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
cand: A c∧ B,
member: t ∈ T,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
sub-bag: sub-bag(T;as;bs),
exists: ∃x:A. B[x],
top: Top,
respects-equality: respects-equality(S;T),
true: True,
squash: ↓T,
guard: {T}
Lemmas referenced :
assert_witness,
bag-member_wf,
sub-bag_wf,
bag-filter_wf,
istype-assert,
bag_wf,
bool_wf,
istype-universe,
bag-split,
sub-bag-append-trivial1,
bnot_wf,
sub-bag-member,
bag-member-filter,
equal_wf,
subtype_rel_bag,
assert_wf,
bag-filter-append,
istype-void,
bag-append_wf,
respects-equality-bag,
subtype-respects-equality,
bag-filter-trivial2,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
independent_functionElimination,
universeIsType,
because_Cache,
sqequalRule,
lambdaEquality_alt,
productElimination,
productIsType,
functionIsType,
inhabitedIsType,
instantiate,
universeEquality,
dependent_functionElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
independent_isectElimination,
equalityTransitivity,
setEquality,
setElimination,
rename,
setIsType,
isect_memberEquality_alt,
voidElimination,
dependent_pairFormation_alt,
equalityIstype,
productEquality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]
\mforall{}p:T {}\mrightarrow{} \mBbbB{}. \mforall{}b,c:bag(T).
(sub-bag(T;b;[x\mmember{}c|p[x]]) \mLeftarrow{}{}\mRightarrow{} sub-bag(T;b;c) \mwedge{} (\mforall{}x:T. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (\muparrow{}p[x]))))
Date html generated:
2019_10_15-AM-11_02_00
Last ObjectModification:
2018_11_30-AM-10_08_19
Theory : bags
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