Nuprl Lemma : lsum-append
∀[T:Type]. ∀[L1,L2:T List]. ∀[f:{x:T| (x ∈ L1 @ L2)} ⟶ ℤ].
(Σ(f[x] | x ∈ L1 @ L2) = (Σ(f[x] | x ∈ L1) + Σ(f[x] | x ∈ L2)) ∈ ℤ)
Proof
Definitions occuring in Statement :
lsum: Σ(f[x] | x ∈ L),
l_member: (x ∈ l),
append: as @ bs,
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[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,
lsum: Σ(f[x] | x ∈ L),
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T}
Lemmas referenced :
map_append_sq,
istype-void,
l_member_wf,
append_wf,
istype-int,
list_wf,
istype-universe,
list-subtype,
subtype_rel_list_set,
member_append,
l_sum-append,
map_wf,
l_sum_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
functionIsType,
setIsType,
universeIsType,
hypothesisEquality,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality,
applyEquality,
because_Cache,
lambdaEquality_alt,
independent_isectElimination,
setElimination,
rename,
lambdaFormation_alt,
dependent_functionElimination,
productElimination,
independent_functionElimination,
inlFormation_alt,
inrFormation_alt,
setEquality,
intEquality,
addEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L1,L2:T List]. \mforall{}[f:\{x:T| (x \mmember{} L1 @ L2)\} {}\mrightarrow{} \mBbbZ{}].
(\mSigma{}(f[x] | x \mmember{} L1 @ L2) = (\mSigma{}(f[x] | x \mmember{} L1) + \mSigma{}(f[x] | x \mmember{} L2)))
Date html generated:
2020_05_19-PM-09_47_13
Last ObjectModification:
2019_11_27-AM-10_05_52
Theory : list_1
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