Nuprl Lemma : bag-filter-map2
∀[T,A:Type]. ∀[f:A ⟶ T]. ∀[p:T ⟶ 𝔹]. ∀[as:bag(A)].
([x∈bag-map(f;as)|p[x]] = bag-map(f;[x∈as|p[f x]]) ∈ bag({x:T| ↑p[x]} ))
Proof
Definitions occuring in Statement :
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
squash: ↓T,
exists: ∃x:A. B[x]
Lemmas referenced :
bag-filter-map,
bag-map_wf,
assert_wf,
bag_to_squash_list,
istype-assert,
bag-filter_wf,
bag_wf,
bool_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
setEquality,
hypothesisEquality,
applyEquality,
imageElimination,
productElimination,
promote_hyp,
rename,
functionExtensionality_alt,
setElimination,
dependent_set_memberEquality_alt,
setIsType,
because_Cache,
lambdaEquality_alt,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[f:A {}\mrightarrow{} T]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[as:bag(A)].
([x\mmember{}bag-map(f;as)|p[x]] = bag-map(f;[x\mmember{}as|p[f x]]))
Date html generated:
2020_05_20-AM-08_01_31
Last ObjectModification:
2019_12_31-PM-07_06_26
Theory : bags
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