### Nuprl Lemma : greatest-p-zero-property

`∀p:ℕ+. ∀a:p-adics(p). ∀n:ℕ.`
`  ((greatest-p-zero(n;a) ≤ n)`
`  ∧ (∀i:ℕ+n + 1. (((i ≤ greatest-p-zero(n;a)) `` ((a i) = 0 ∈ ℤ)) ∧ (greatest-p-zero(n;a) < i `` (¬((a i) = 0 ∈ ℤ))))))`

Proof

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

Latex:
((greatest-p-zero(n;a)  \mleq{}  n)
\mwedge{}  (\mforall{}i:\mBbbN{}\msupplus{}n  +  1
(((i  \mleq{}  greatest-p-zero(n;a))  {}\mRightarrow{}  ((a  i)  =  0))
\mwedge{}  (greatest-p-zero(n;a)  <  i  {}\mRightarrow{}  (\mneg{}((a  i)  =  0))))))

Date html generated: 2018_05_21-PM-03_22_12
Last ObjectModification: 2018_05_19-AM-08_21_03

Theory : rings_1

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