### Nuprl Lemma : assert-regular-upto

`∀[k,n:ℕ]. ∀[f:ℕ+ ⟶ ℤ].  (↑regular-upto(k;n;f) `⇐⇒` ∀i,j:ℕ+n + 1.  (|(i * (f j)) - j * (f i)| ≤ ((2 * k) * (i + j))))`

Proof

Definitions occuring in Statement :  regular-upto: `regular-upto(k;n;f)` absval: `|i|` int_seg: `{i..j-}` nat_plus: `ℕ+` nat: `ℕ` assert: `↑b` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` apply: `f a` function: `x:A ⟶ B[x]` multiply: `n * m` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` all: `∀x:A. B[x]` nat: `ℕ` prop: `ℙ` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` nat_plus: `ℕ+` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` false: `False` uiff: `uiff(P;Q)` uimplies: `b supposing a` lelt: `i ≤ j < k` subtype_rel: `A ⊆r B` less_than': `less_than'(a;b)` true: `True` so_apply: `x[s]` regular-upto: `regular-upto(k;n;f)` subtract: `n - m` top: `Top` guard: `{T}` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` sq_type: `SQType(T)` rev_uimplies: `rev_uimplies(P;Q)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality addEquality setElimination rename hypothesisEquality hypothesis functionExtensionality applyEquality because_Cache sqequalRule lambdaEquality multiplyEquality dependent_set_memberEquality productElimination dependent_functionElimination unionElimination voidElimination independent_functionElimination independent_isectElimination independent_pairEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality intEquality isect_memberEquality addLevel voidEquality minusEquality allFunctionality levelHypothesis promote_hyp approximateComputation dependent_pairFormation int_eqEquality hyp_replacement baseApply closedConclusion baseClosed instantiate cumulativity applyLambdaEquality allLevelFunctionality

Latex:
\mforall{}[k,n:\mBbbN{}].  \mforall{}[f:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}].
(\muparrow{}regular-upto(k;n;f)  \mLeftarrow{}{}\mRightarrow{}  \mforall{}i,j:\mBbbN{}\msupplus{}n  +  1.    (|(i  *  (f  j))  -  j  *  (f  i)|  \mleq{}  ((2  *  k)  *  (i  +  j))))

Date html generated: 2017_10_03-AM-08_42_39
Last ObjectModification: 2017_09_06-PM-04_04_24

Theory : reals

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