Nuprl Lemma : integer-nth-root

n:ℕ+. ∀x:ℕ.  (∃r:{ℕ((r^n ≤ x) ∧ x < 1^n)})


Proof




Definitions occuring in Statement :  exp: i^n nat_plus: + nat: less_than: a < b le: A ≤ B all: x:A. B[x] sq_exists: x:{A| B[x]} and: P ∧ Q add: 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: so_apply: x[s] nat: le: A ≤ B and: P ∧ Q less_than': less_than'(a;b) false: False not: ¬A implies:  Q uimplies: supposing a less_than: a < b squash: T true: True subtype_rel: A ⊆B guard: {T} iff: ⇐⇒ Q nat_plus: + nequal: a ≠ b ∈  ge: i ≥  satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] top: Top sq_exists: x:{A| B[x]} cand: c∧ B rev_implies:  Q int_nzero: -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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