Nuprl Lemma : l_sum-mapfilter-upto
∀[n:ℕ]. ∀[P:ℕn ⟶ 𝔹]. ∀[f:{x:ℕn| ↑(P x)} ⟶ ℤ].
(l_sum(mapfilter(f;P;upto(n))) = Σ(if P i then f i else 0 fi | i < n) ∈ ℤ)
Proof
Definitions occuring in Statement :
l_sum: l_sum(L),
upto: upto(n),
mapfilter: mapfilter(f;P;L),
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
and: P ∧ Q,
prop: ℙ,
squash: ↓T,
so_lambda: λ2x.t[x],
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),
uimplies: b supposing a,
bfalse: ff,
so_apply: x[s],
true: True
Lemmas referenced :
int_seg_wf,
istype-assert,
istype-int,
bool_wf,
istype-nat,
l_sum-mapfilter,
upto_wf,
l_member_wf,
assert_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
length_upto,
sum_wf,
select-upto,
eqtt_to_assert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
functionIsType,
setIsType,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
applyEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
functionExtensionality,
dependent_set_memberEquality_alt,
productElimination,
setEquality,
closedConclusion,
productEquality,
hyp_replacement,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
instantiate,
universeEquality,
intEquality,
lambdaFormation_alt,
unionElimination,
equalityElimination,
independent_isectElimination,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[P:\mBbbN{}n {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{x:\mBbbN{}n| \muparrow{}(P x)\} {}\mrightarrow{} \mBbbZ{}].
(l\_sum(mapfilter(f;P;upto(n))) = \mSigma{}(if P i then f i else 0 fi | i < n))
Date html generated:
2020_05_19-PM-09_46_06
Last ObjectModification:
2020_01_01-AM-10_05_43
Theory : list_1
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