### Nuprl Lemma : rmaximum_functionality

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

Proof

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

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

Date html generated: 2018_05_22-PM-01_56_46
Last ObjectModification: 2018_05_21-AM-00_12_03

Theory : reals

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