### Nuprl Lemma : rroot-abs-non-neg

`∀i:{2...}. ∀x:ℝ. ∀n:ℕ+.  (0 ≤ (rroot-abs(i;x) n))`

Proof

Definitions occuring in Statement :  rroot-abs: `rroot-abs(i;x)` real: `ℝ` int_upper: `{i...}` nat_plus: `ℕ+` le: `A ≤ B` all: `∀x:A. B[x]` apply: `f a` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` rroot-abs: `rroot-abs(i;x)` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` int_upper: `{i...}` nat_plus: `ℕ+` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` le: `A ≤ B` less_than': `less_than'(a;b)` has-value: `(a)↓` so_lambda: `λ2x.t[x]` so_apply: `x[s]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtype_rel: `A ⊆r B` true: `True` ge: `i ≥ j ` squash: `↓T` real: `ℝ`
Lemmas referenced :  exp-fastexp subtract_wf nat_plus_properties int_upper_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_wf le_wf exp_wf4 false_wf nat_wf value-type-has-value set-value-type int-value-type exp_preserves_lt decidable__lt not-lt-2 add_functionality_wrt_le add-commutes zero-add le-add-cancel less_than_wf nat_plus_subtype_nat nat_properties intformless_wf int_formula_prop_less_lemma squash_wf true_wf exp-zero exp_wf2 iff_weakening_equal fastexp_wf int_upper_subtype_nat nat_plus_wf rabs_wf real_wf zero-le-nat iroot_wf equal_wf int_upper_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalRule cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality dependent_set_memberEquality setElimination rename hypothesisEquality hypothesis dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll because_Cache callbyvalueReduce productElimination independent_functionElimination applyEquality equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination imageMemberEquality baseClosed universeEquality

Latex:
\mforall{}i:\{2...\}.  \mforall{}x:\mBbbR{}.  \mforall{}n:\mBbbN{}\msupplus{}.    (0  \mleq{}  (rroot-abs(i;x)  n))

Date html generated: 2017_10_03-AM-10_41_02
Last ObjectModification: 2017_07_28-AM-08_17_03

Theory : reals

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