Nuprl Lemma : bag-combine-mapfilter
∀[A,B,C:Type]. ∀[b:bag(A)]. ∀[P:A ⟶ 𝔹]. ∀[f:{x:A| ↑P[x]} ⟶ B]. ∀[g:B ⟶ bag(C)].
(⋃x∈bag-mapfilter(f;P;b).g[x] = ⋃x∈b.if P[x] then g[f[x]] else {} fi ∈ bag(C))
Proof
Definitions occuring in Statement :
bag-combine: ⋃x∈bs.f[x],
bag-mapfilter: bag-mapfilter(f;P;bs),
empty-bag: {},
bag: bag(T),
assert: ↑b,
ifthenelse: if b then t else f fi ,
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,
bag-mapfilter: bag-mapfilter(f;P;bs),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
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,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
bag_wf,
bag-combine-map,
assert_wf,
bag-filter_wf,
bag-combine_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
empty-bag_wf,
iff_weakening_equal,
bag-combine-filter
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,
because_Cache,
setEquality,
cumulativity,
functionExtensionality,
sqequalRule,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_set_memberEquality,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
setElimination,
rename,
functionEquality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[b:bag(A)]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{x:A| \muparrow{}P[x]\} {}\mrightarrow{} B]. \mforall{}[g:B {}\mrightarrow{} bag(C)].
(\mcup{}x\mmember{}bag-mapfilter(f;P;b).g[x] = \mcup{}x\mmember{}b.if P[x] then g[f[x]] else \{\} fi )
Date html generated:
2017_10_01-AM-08_47_43
Last ObjectModification:
2017_07_26-PM-04_32_07
Theory : bags
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