Nuprl Lemma : lsum-split
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ 𝔹]. ∀[f:{x:T| (x ∈ L)} ⟶ ℤ].
(Σ(f[x] | x ∈ L) = (Σ(f[x] | x ∈ filter(λx.P[x];L)) + Σ(f[x] | x ∈ filter(λx.(¬bP[x]);L))) ∈ ℤ)
Proof
Definitions occuring in Statement :
lsum: Σ(f[x] | x ∈ L),
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
bnot: ¬bb,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
lsum: Σ(f[x] | x ∈ L),
guard: {T},
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_type: SQType(T),
true: True,
squash: ↓T,
uimplies: b supposing a,
so_apply: x[s],
ifthenelse: if b then t else f fi ,
bnot: ¬bb
Lemmas referenced :
l_sum-split,
istype-int,
l_member_wf,
bool_wf,
list_wf,
istype-universe,
l_sum_wf,
int_subtype_base,
subtype_base_sq,
true_wf,
squash_wf,
map_wf,
list-subtype,
filter_wf2,
bnot_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
functionIsType,
because_Cache,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
setIsType,
universeIsType,
instantiate,
universeEquality,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
lambdaEquality_alt,
applyEquality,
independent_isectElimination,
intEquality,
cumulativity,
functionExtensionality,
setEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbZ{}].
(\mSigma{}(f[x] | x \mmember{} L) = (\mSigma{}(f[x] | x \mmember{} filter(\mlambda{}x.P[x];L)) + \mSigma{}(f[x] | x \mmember{} filter(\mlambda{}x.(\mneg{}\msubb{}P[x]);L))))
Date html generated:
2020_05_19-PM-09_48_31
Last ObjectModification:
2019_12_31-PM-01_20_57
Theory : list_1
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