### Nuprl Lemma : rmaximum-split

`∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[k:ℤ].`
`  (rmaximum(n;m;i.x[i]) = rmax(rmaximum(n;k;i.x[i]);rmaximum(k + 1;m;i.x[i]))) supposing (k < m and (n ≤ k))`

Proof

Definitions occuring in Statement :  rmaximum: `rmaximum(n;m;k.x[k])` rmax: `rmax(x;y)` req: `x = y` real: `ℝ` int_seg: `{i..j-}` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` 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` rmaximum: `rmaximum(n;m;k.x[k])` nat: `ℕ` 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` and: `P ∧ Q` prop: `ℙ` guard: `{T}` ge: `i ≥ j ` sq_type: `SQType(T)` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` cand: `A c∧ B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` true: `True` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff`

Latex:
\mforall{}[n,m:\mBbbZ{}].  \mforall{}[x:\{n..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}].  \mforall{}[k:\mBbbZ{}].
(rmaximum(n;m;i.x[i])  =  rmax(rmaximum(n;k;i.x[i]);rmaximum(k  +  1;m;i.x[i])))  supposing
(k  <  m  and
(n  \mleq{}  k))

Date html generated: 2020_05_20-AM-11_14_11
Last ObjectModification: 2019_12_14-PM-00_56_15

Theory : reals

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