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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