### Nuprl Lemma : rroot-regularity-lemma

`∀[k:{2...}]. ∀[n,m:ℕ+]. ∀[a,b,c,d:ℤ].`
`  (((m ≤ a) ∨ ((a = 0 ∈ ℤ) ∧ (c = 0 ∈ ℤ)))`
`  `` ((n ≤ b) ∨ ((b = 0 ∈ ℤ) ∧ (d = 0 ∈ ℤ)))`
`  `` (a^k ≤ c)`
`  `` c < a + m^k`
`  `` (b^k ≤ d)`
`  `` d < b + n^k`
`  `` (|c - d| ≤ (2^k * (n^k + m^k)))`
`  `` (|a - b| ≤ (2 * (n + m))))`

Proof

Definitions occuring in Statement :  exp: `i^n` absval: `|i|` int_upper: `{i...}` nat_plus: `ℕ+` less_than: `a < b` uall: `∀[x:A]. B[x]` le: `A ≤ B` implies: `P `` Q` or: `P ∨ Q` and: `P ∧ Q` multiply: `n * m` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` prop: `ℙ` subtype_rel: `A ⊆r B` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` nat_plus: `ℕ+` or: `P ∨ Q` all: `∀x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` uimplies: `b supposing a` less_than: `a < b` top: `Top` true: `True` squash: `↓T` guard: `{T}` int_upper: `{i...}` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` sq_type: `SQType(T)` bfalse: `ff` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` int_iseg: `{i...j}` so_apply: `x[s]` lelt: `i ≤ j < k` subtract: `n - m` choose: `choose(n;i)` ycomb: `Y` eq_int: `(i =z j)` bor: `p ∨bq` cand: `A c∧ B` rev_uimplies: `rev_uimplies(P;Q)` ge: `i ≥ j ` label: `...\$L... t`
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis applyEquality because_Cache sqequalRule multiplyEquality dependent_set_memberEquality natural_numberEquality independent_pairFormation addEquality setElimination rename productEquality intEquality baseClosed lambdaEquality dependent_functionElimination productElimination independent_pairEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality voidElimination minusEquality unionElimination equalityElimination independent_isectElimination lessCases sqequalAxiom voidEquality imageMemberEquality imageElimination independent_functionElimination instantiate cumulativity dependent_pairFormation int_eqEquality computeAll promote_hyp baseApply closedConclusion setEquality applyLambdaEquality universeEquality pointwiseFunctionality equalityUniverse levelHypothesis

Latex:
\mforall{}[k:\{2...\}].  \mforall{}[n,m:\mBbbN{}\msupplus{}].  \mforall{}[a,b,c,d:\mBbbZ{}].
(((m  \mleq{}  a)  \mvee{}  ((a  =  0)  \mwedge{}  (c  =  0)))
{}\mRightarrow{}  ((n  \mleq{}  b)  \mvee{}  ((b  =  0)  \mwedge{}  (d  =  0)))
{}\mRightarrow{}  (a\^{}k  \mleq{}  c)
{}\mRightarrow{}  c  <  a  +  m\^{}k
{}\mRightarrow{}  (b\^{}k  \mleq{}  d)
{}\mRightarrow{}  d  <  b  +  n\^{}k
{}\mRightarrow{}  (|c  -  d|  \mleq{}  (2\^{}k  *  (n\^{}k  +  m\^{}k)))
{}\mRightarrow{}  (|a  -  b|  \mleq{}  (2  *  (n  +  m))))

Date html generated: 2017_10_03-AM-10_39_47
Last ObjectModification: 2017_07_28-AM-08_16_22

Theory : reals

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