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]` subtype_rel: `A ⊆r B` int_upper: `{i...}` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` uimplies: `b supposing a` nat: `ℕ` nat_plus: `ℕ+` 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: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than: `a < b` 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_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

Home Index