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: + nat: 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: so_apply: x[s] subtype_rel: A ⊆r B int_upper: {i...} le: A ≤ B and: P ∧ Q less_than': less_than'(a;b) false: False not: implies:  Q uimplies: b supposing a nat: nat_plus: + nequal: a ≠ b ∈ T  squash: ↓T guard: {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 true: True iff: ⇐⇒ Q rev_implies:  Q less_than: a < b int_nzero: -o decidable: Dec(P) or: P ∨ Q uiff: uiff(P;Q) sq_stable: SqStable(P) sq_type: SQType(T)
Lemmas referenced :  set_wf less_than_wf exp_wf2 nat_plus_subtype_nat exp-ge-1 false_wf le_wf equal_wf set-value-type int-value-type div_nat_induction sq_exists_wf nat_wf nat_properties nat_plus_properties satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_wf equal-wf-base int_subtype_base equal-wf-T-base set_subtype_base nat_plus_wf squash_wf true_wf exp-zero iff_weakening_equal exp-positive exp-of-mul div_rem_sum subtype_rel_sets nequal_wf rem_bounds_1 decidable__lt not-lt-2 less-iff-le le_antisymmetry_iff add_functionality_wrt_le add-associates add-swap add-commutes zero-add le-add-cancel fastexp_wf exp-fastexp sq_stable__less_than decidable__le intformnot_wf intformle_wf itermAdd_wf itermMultiply_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_add_lemma int_term_value_mul_lemma subtype_base_sq decidable__equal_int add-is-int-iff multiply-is-int-iff mul_preserves_le
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin intEquality sqequalRule lambdaEquality natural_numberEquality hypothesisEquality hypothesis dependent_set_memberEquality applyEquality independent_pairFormation cutEval equalityTransitivity equalitySymmetry independent_isectElimination setElimination rename dependent_functionElimination productEquality because_Cache addEquality independent_functionElimination divideEquality applyLambdaEquality imageMemberEquality baseClosed imageElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality computeAll productElimination dependent_set_memberFormation universeEquality multiplyEquality setEquality 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: 2018_05_21-PM-07_50_08
Last ObjectModification: 2017_07_26-PM-05_27_54

Theory : general


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