### Nuprl Lemma : integer-nth-root-ext

`∀n:ℕ+. ∀x:ℕ.  (∃r:ℕ [((r^n ≤ x) ∧ x < (r + 1)^n)])`

Proof

Definitions occuring in Statement :  exp: `i^n` nat_plus: `ℕ+` nat: `ℕ` less_than: `a < b` le: `A ≤ B` all: `∀x:A. B[x]` sq_exists: `∃x:A [B[x]]` and: `P ∧ Q` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  member: `t ∈ T` integer-nth-root div_nat_induction natrec: natrec genrec: genrec so_apply: `x[s1;s2]` decidable__equal_int decidable__int_equal uall: `∀[x:A]. B[x]` so_lambda: `so_lambda(x,y,z,w.t[x; y; z; w])` so_apply: `x[s1;s2;s3;s4]` so_lambda: `λ2x.t[x]` top: `Top` so_apply: `x[s]` uimplies: `b supposing a` strict4: `strict4(F)` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` has-value: `(a)↓` prop: `ℙ` guard: `{T}` or: `P ∨ Q` squash: `↓T` rem_bounds_1 decidable__lt decidable__squash so_lambda: `λ2x y.t[x; y]` decidable_functionality iff_preserves_decidability decidable__and decidable__less_than' genrec-ap: genrec-ap
Lemmas referenced :  integer-nth-root lifting-strict-int_eq top_wf equal_wf has-value_wf_base base_wf is-exception_wf lifting-strict-spread lifting-strict-decide lifting-strict-less div_nat_induction decidable__equal_int decidable__int_equal rem_bounds_1 decidable__lt decidable__squash decidable_functionality iff_preserves_decidability decidable__and decidable__less_than'
Rules used in proof :  introduction sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut instantiate extract_by_obid hypothesis sqequalRule thin sqequalHypSubstitution isectElimination baseClosed isect_memberEquality voidElimination voidEquality independent_isectElimination independent_pairFormation lambdaFormation callbyvalueDecide hypothesisEquality equalityTransitivity equalitySymmetry unionEquality unionElimination sqleReflexivity dependent_functionElimination independent_functionElimination baseApply closedConclusion decideExceptionCases inrFormation because_Cache imageMemberEquality imageElimination exceptionSqequal inlFormation callbyvalueApply applyExceptionCases

Latex:
\mforall{}n:\mBbbN{}\msupplus{}.  \mforall{}x:\mBbbN{}.    (\mexists{}r:\mBbbN{}  [((r\^{}n  \mleq{}  x)  \mwedge{}  x  <  (r  +  1)\^{}n)])

Date html generated: 2018_05_21-PM-07_50_18
Last ObjectModification: 2017_07_26-PM-05_28_05

Theory : general

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