### Nuprl Lemma : bdd-diff-regular

`∀[x,y:ℕ+ ⟶ ℤ]. ∀[k,l:ℕ+].`
`  (∀n:ℕ+. (|(x n) - y n| ≤ ((2 * k) + (2 * l)))) supposing (bdd-diff(x;y) and k-regular-seq(x) and l-regular-seq(y))`

Proof

Definitions occuring in Statement :  bdd-diff: `bdd-diff(f;g)` regular-int-seq: `k-regular-seq(f)` absval: `|i|` nat_plus: `ℕ+` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` 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` uimplies: `b supposing a` all: `∀x:A. B[x]` bdd-diff: `bdd-diff(f;g)` exists: `∃x:A. B[x]` le: `A ≤ B` and: `P ∧ Q` not: `¬A` implies: `P `` Q` false: `False` nat_plus: `ℕ+` subtype_rel: `A ⊆r B` prop: `ℙ` nat: `ℕ` true: `True` top: `Top` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` squash: `↓T` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rev_uimplies: `rev_uimplies(P;Q)` subtract: `n - m` regular-int-seq: `k-regular-seq(f)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` gt: `i > j` uiff: `uiff(P;Q)` less_than': `less_than'(a;b)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation sqequalHypSubstitution productElimination thin because_Cache lemma_by_obid hypothesis sqequalRule lambdaEquality dependent_functionElimination hypothesisEquality independent_pairEquality isectElimination addEquality multiplyEquality natural_numberEquality setElimination rename applyEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality functionEquality intEquality voidElimination minusEquality voidEquality dependent_set_memberEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality computeAll imageElimination imageMemberEquality baseClosed universeEquality independent_functionElimination sqequalIntensionalEquality independent_pairFormation inlFormation

Latex:
\mforall{}[x,y:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[k,l:\mBbbN{}\msupplus{}].
(\mforall{}n:\mBbbN{}\msupplus{}.  (|(x  n)  -  y  n|  \mleq{}  ((2  *  k)  +  (2  *  l))))  supposing
(bdd-diff(x;y)  and
k-regular-seq(x)  and
l-regular-seq(y))

Date html generated: 2016_05_18-AM-06_47_36
Last ObjectModification: 2016_01_17-AM-01_47_24

Theory : reals

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