### 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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