Nuprl Lemma : bag-mapfilter-fast_wf
∀[A,B:Type]. ∀[bs:bag(A)]. ∀[P:A ⟶ 𝔹]. ∀[f:{x:A| ↑P[x]} ⟶ B]. (bag-mapfilter-fast(f;P;bs) ∈ bag(B))
Proof
Definitions occuring in Statement :
bag-mapfilter-fast: bag-mapfilter-fast(f;P;bs),
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bag-mapfilter-fast: bag-mapfilter-fast(f;P;bs),
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
bfalse: ff,
so_apply: x[s1;s2],
top: Top,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
bag-accum_wf,
bag_wf,
empty-bag_wf,
bool_wf,
eqtt_to_assert,
cons-bag_wf,
assert_wf,
equal_wf,
cons-bag-as-append,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bag-append_wf,
squash_wf,
true_wf,
bag-append-comm,
single-bag_wf,
iff_weakening_equal,
bag-append-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
because_Cache,
lambdaEquality,
applyEquality,
functionExtensionality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
setEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
axiomEquality,
functionEquality,
universeEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
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-fast(f;P;bs) \mmember{} bag(B))
Date html generated:
2017_10_01-AM-08_58_25
Last ObjectModification:
2017_07_26-PM-04_40_24
Theory : bags
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