### Nuprl Lemma : rmaximum-select

`∀n,m:ℤ.  ∀x:{n..m + 1-} ⟶ ℝ. ∀e:ℝ.  ((r0 < e) `` (∃i:{n..m + 1-}. ((rmaximum(n;m;i.x[i]) - e) < x[i]))) supposing n ≤ m`

Proof

Definitions occuring in Statement :  rmaximum: `rmaximum(n;m;k.x[k])` rless: `x < y` rsub: `x - y` int-to-real: `r(n)` real: `ℝ` int_seg: `{i..j-}` uimplies: `b supposing a` so_apply: `x[s]` le: `A ≤ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  cand: `A c∧ B` rge: `x ≥ y` req_int_terms: `t1 ≡ t2` true: `True` less_than': `less_than'(a;b)` subtract: `n - m` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` bfalse: `ff` ifthenelse: `if b then t else f fi ` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` uiff: `uiff(P;Q)` so_apply: `x[s]` nat_plus: `ℕ+` squash: `↓T` sq_stable: `SqStable(P)` real: `ℝ` subtype_rel: `A ⊆r B` lelt: `i ≤ j < k` sq_exists: `∃x:A [B[x]]` rless: `x < y` int_seg: `{i..j-}` so_lambda: `λ2x.t[x]` sq_type: `SQType(T)` ge: `i ≥ j ` guard: `{T}` top: `Top` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` or: `P ∨ Q` decidable: `Dec(P)` nat: `ℕ` rmaximum: `rmaximum(n;m;k.x[k])` prop: `ℙ` uall: `∀[x:A]. B[x]` false: `False` implies: `P `` Q` not: `¬A` and: `P ∧ Q` le: `A ≤ B` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]`
Rules used in proof :  multiplyEquality minusEquality closedConclusion baseApply equalityElimination functionExtensionality imageElimination baseClosed imageMemberEquality applyEquality addEquality functionEquality setElimination applyLambdaEquality equalitySymmetry equalityTransitivity cumulativity instantiate independent_pairFormation voidEquality isect_memberEquality int_eqEquality dependent_pairFormation independent_functionElimination approximateComputation independent_isectElimination unionElimination natural_numberEquality because_Cache dependent_set_memberEquality intEquality rename axiomEquality hypothesis isectElimination extract_by_obid voidElimination hypothesisEquality dependent_functionElimination lambdaEquality independent_pairEquality thin productElimination sqequalHypSubstitution sqequalRule introduction cut isect_memberFormation lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}n,m:\mBbbZ{}.
\mforall{}x:\{n..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}.  \mforall{}e:\mBbbR{}.    ((r0  <  e)  {}\mRightarrow{}  (\mexists{}i:\{n..m  +  1\msupminus{}\}.  ((rmaximum(n;m;i.x[i])  -  e)  <  x[i])))
supposing  n  \mleq{}  m

Date html generated: 2018_05_22-PM-01_57_29
Last ObjectModification: 2018_05_21-AM-00_16_42

Theory : reals

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