Nuprl Lemma : bag-mapfilter-fast-eq2
∀[A,B:Type]. ∀[bs:bag(A)]. ∀[P:A ⟶ 𝔹]. ∀[f:{x:A| ↑P[x]} ⟶ B].
(bag-mapfilter(f;P;bs) = bag-mapfilter-fast(f;P;bs) ∈ bag(B))
Proof
Definitions occuring in Statement :
bag-mapfilter-fast: bag-mapfilter-fast(f;P;bs),
bag-mapfilter: bag-mapfilter(f;P;bs),
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
bag_wf,
bag-mapfilter_wf,
assert_wf,
bag-mapfilter-fast-eq,
iff_weakening_equal,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
cumulativity,
dependent_functionElimination,
functionExtensionality,
because_Cache,
sqequalRule,
setEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
functionEquality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[bs:bag(A)]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{x:A| \muparrow{}P[x]\} {}\mrightarrow{} B].
(bag-mapfilter(f;P;bs) = bag-mapfilter-fast(f;P;bs))
Date html generated:
2017_10_01-AM-08_58_39
Last ObjectModification:
2017_07_26-PM-04_40_32
Theory : bags
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