Nuprl Lemma : sum-consecutive-cubes
∀[n:ℕ]. (Σ((i * i) * i | i < n) = ((((n - 1) * (n - 1)) * n * n) ÷ 4) ∈ ℤ)
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
nat: ℕ,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
multiply: n * m,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s],
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
sum-of-consecutive-cubes,
subtype_base_sq,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
sum_wf,
int_seg_wf,
divide-exact,
equal-wf-base,
nequal_wf,
iff_weakening_equal,
nat_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
sqequalRule,
multiplyEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
dependent_set_memberEquality,
addLevel,
lambdaFormation,
voidElimination,
baseClosed,
imageMemberEquality,
productElimination
Latex:
\mforall{}[n:\mBbbN{}]. (\mSigma{}((i * i) * i | i < n) = ((((n - 1) * (n - 1)) * n * n) \mdiv{} 4))
Date html generated:
2017_04_14-AM-09_21_43
Last ObjectModification:
2017_02_27-PM-03_57_17
Theory : int_2
Home
Index