### Nuprl Lemma : integer-nth-root2

`∀n:{n:ℕ+| (n mod 2) = 1 ∈ ℤ} . ∀x:{...0}.  (∃r:{{...0}| (r - 1^n < x ∧ (x ≤ r^n))})`

Proof

Definitions occuring in Statement :  exp: `i^n` modulus: `a mod n` int_lower: `{...i}` nat_plus: `ℕ+` less_than: `a < b` le: `A ≤ B` all: `∀x:A. B[x]` sq_exists: `∃x:{A| B[x]}` and: `P ∧ Q` set: `{x:A| B[x]} ` subtract: `n - m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` nat: `ℕ` int_lower: `{...i}` uall: `∀[x:A]. B[x]` guard: `{T}` sq_stable: `SqStable(P)` implies: `P `` Q` squash: `↓T` nat_plus: `ℕ+` 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` prop: `ℙ` so_lambda: `λ2x.t[x]` less_than: `a < b` less_than': `less_than'(a;b)` true: `True` subtype_rel: `A ⊆r B` so_apply: `x[s]` sq_exists: `∃x:{A| B[x]}` ge: `i ≥ j ` cand: `A c∧ B` sq_type: `SQType(T)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` eq_int: `(i =z j)` ifthenelse: `if b then t else f fi ` bfalse: `ff` le: `A ≤ B`
Lemmas referenced :  subtract_wf int_formula_prop_less_lemma intformless_wf exp_wf2 decidable__lt set_subtype_base iff_weakening_equal nat_plus_subtype_nat nat_wf exp-minus true_wf squash_wf int_term_value_add_lemma int_term_value_subtract_lemma int_formula_prop_eq_lemma itermAdd_wf itermSubtract_wf intformeq_wf decidable__equal_int int_subtype_base subtype_base_sq nat_properties int-subtype-int_mod int_mod_wf subtype_rel_set less_than_wf modulus_wf_int_mod equal-wf-T-base nat_plus_wf set_wf int_lower_wf le_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_minus_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermMinus_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_plus_properties sq_stable__equal int_lower_properties integer-nth-root-ext
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut lemma_by_obid sqequalHypSubstitution dependent_functionElimination thin setElimination rename hypothesisEquality dependent_set_memberEquality minusEquality isectElimination natural_numberEquality hypothesis because_Cache equalityTransitivity equalitySymmetry independent_functionElimination introduction sqequalRule imageMemberEquality baseClosed imageElimination unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll comment applyEquality dependent_set_memberFormation productElimination instantiate addEquality universeEquality productEquality

Latex:
\mforall{}n:\{n:\mBbbN{}\msupplus{}|  (n  mod  2)  =  1\}  .  \mforall{}x:\{...0\}.    (\mexists{}r:\{\{...0\}|  (r  -  1\^{}n  <  x  \mwedge{}  (x  \mleq{}  r\^{}n))\})

Date html generated: 2016_05_15-PM-05_13_46
Last ObjectModification: 2016_01_16-AM-11_38_53

Theory : general

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