### Nuprl Lemma : exp-ratio-property

`∀a:ℕ. ∀b:{a + 1...}. ∀k:ℕ.  (exp-ratio(a;b;0;k;1) ∈ {n:ℕ| k * a^n < b^n} )`

Proof

Definitions occuring in Statement :  exp-ratio: `exp-ratio(a;b;n;p;q)` exp: `i^n` int_upper: `{i...}` nat: `ℕ` less_than: `a < b` all: `∀x:A. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` multiply: `n * m` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` nat_plus: `ℕ+` implies: `P `` Q` prop: `ℙ` guard: `{T}` int_upper: `{i...}` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` and: `P ∧ Q` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` subtract: `n - m` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` squash: `↓T` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` cand: `A c∧ B` sq_type: `SQType(T)` less_than: `a < b` rev_uimplies: `rev_uimplies(P;Q)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut sqequalHypSubstitution hypothesis lemma_by_obid isectElimination thin addEquality setElimination rename hypothesisEquality natural_numberEquality dependent_set_memberEquality dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll multiplyEquality because_Cache applyEquality introduction imageElimination equalityTransitivity equalitySymmetry imageMemberEquality baseClosed universeEquality productElimination independent_functionElimination setEquality pointwiseFunctionality promote_hyp baseApply closedConclusion equalityEquality minusEquality instantiate cumulativity

Latex:
\mforall{}a:\mBbbN{}.  \mforall{}b:\{a  +  1...\}.  \mforall{}k:\mBbbN{}.    (exp-ratio(a;b;0;k;1)  \mmember{}  \{n:\mBbbN{}|  k  *  a\^{}n  <  b\^{}n\}  )

Date html generated: 2016_05_15-PM-04_07_57
Last ObjectModification: 2016_01_16-AM-11_07_12

Theory : general

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