### Nuprl Lemma : integer-nth-root

`∀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 :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` prop: `ℙ` so_apply: `x[s]` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` uimplies: `b supposing a` less_than: `a < b` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` nat_plus: `ℕ+` nequal: `a ≠ b ∈ T ` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` sq_exists: `∃x:{A| B[x]}` cand: `A c∧ B` rev_implies: `P `` Q` int_nzero: `ℤ-o` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` sq_stable: `SqStable(P)` sq_type: `SQType(T)`
Lemmas referenced :  set_subtype_base equal-wf-base mul_preserves_le multiply-is-int-iff add-is-int-iff decidable__equal_int int_subtype_base subtype_base_sq int_term_value_mul_lemma int_term_value_add_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma itermMultiply_wf itermAdd_wf intformle_wf intformnot_wf decidable__le sq_stable__less_than exp-fastexp fastexp_wf le-add-cancel zero-add add-commutes add-swap add-associates add_functionality_wrt_le le_antisymmetry_iff less-iff-le not-lt-2 decidable__lt rem_bounds_1 nequal_wf subtype_rel_sets div_rem_sum exp-of-mul and_wf exp-positive exp-zero nat_plus_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_and_lemma intformless_wf itermConstant_wf itermVar_wf intformeq_wf intformand_wf satisfiable-full-omega-tt nat_plus_properties nat_properties nat_plus_subtype_nat nat_wf sq_exists_wf div_nat_induction int-value-type equal_wf set-value-type iff_weakening_equal exp-one true_wf squash_wf exp_wf2 le_wf false_wf exp_preserves_lt less_than_wf set_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin intEquality sqequalRule lambdaEquality natural_numberEquality hypothesisEquality hypothesis dependent_set_memberEquality independent_pairFormation because_Cache independent_isectElimination introduction imageMemberEquality baseClosed applyEquality imageElimination equalityTransitivity equalitySymmetry universeEquality productElimination independent_functionElimination cutEval equalityEquality setElimination rename dependent_functionElimination productEquality addEquality divideEquality setEquality dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality computeAll dependent_set_memberFormation multiplyEquality unionElimination instantiate cumulativity pointwiseFunctionality promote_hyp baseApply closedConclusion

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

Date html generated: 2016_05_15-PM-05_13_25
Last ObjectModification: 2016_01_16-AM-11_41_43

Theory : general

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