Nuprl Lemma : sum-l_sum
∀[n:ℕ]. ∀[a:ℕn ⟶ ℤ]. (Σ(a[i] | i < n) = l_sum(map(λi.a[i];upto(n))) ∈ ℤ)
Proof
Definitions occuring in Statement :
l_sum: l_sum(L),
upto: upto(n),
sum: Σ(f[x] | x < k),
map: map(f;as),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
prop: ℙ,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x.t[x],
label: ...$L... t,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
l_sum-sum,
int_seg_wf,
upto_wf,
lelt_wf,
l_member_wf,
subtype_base_sq,
int_subtype_base,
nat_wf,
sum_wf,
squash_wf,
true_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
length_upto,
le_wf,
equal_wf,
select_upto,
int_seg_properties,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
dependent_set_memberEquality,
setEquality,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalRule,
functionEquality,
isect_memberEquality,
axiomEquality,
imageElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
universeEquality,
productElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. (\mSigma{}(a[i] | i < n) = l\_sum(map(\mlambda{}i.a[i];upto(n))))
Date html generated:
2017_04_17-AM-08_38_49
Last ObjectModification:
2017_02_27-PM-04_57_16
Theory : list_1
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