Nuprl Lemma : rsum-telescopes
∀[n:ℤ]. ∀[m:{n...}]. ∀[x,y:{n..m + 1-} ⟶ ℝ].
Σ{x[k] - y[k] | n≤k≤m} = (x[m] - y[n]) supposing ∀i:{n..m-}. (y[i + 1] = x[i])
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rsub: x - y,
req: x = y,
real: ℝ,
int_upper: {i...},
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
less_than: a < b,
squash: ↓T,
nat: ℕ,
ge: i ≥ j ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
subtract: n - m
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[m:\{n...\}]. \mforall{}[x,y:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}].
\mSigma{}\{x[k] - y[k] | n\mleq{}k\mleq{}m\} = (x[m] - y[n]) supposing \mforall{}i:\{n..m\msupminus{}\}. (y[i + 1] = x[i])
Date html generated:
2020_05_20-AM-11_10_52
Last ObjectModification:
2020_02_07-PM-01_44_35
Theory : reals
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