### Nuprl Lemma : neg-approx-of-nonneg-real

`∀x:ℝ. ((r0 ≤ x) `` (∀n:ℕ+. (((x n) ≤ 0) `` (|x n| ≤ 2))))`

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` int-to-real: `r(n)` real: `ℝ` absval: `|i|` nat_plus: `ℕ+` le: `A ≤ B` all: `∀x:A. B[x]` implies: `P `` Q` apply: `f a` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` real: `ℝ` nat_plus: `ℕ+` uimplies: `b supposing a` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` rational-approx: `(x within 1/n)` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` subtype_rel: `A ⊆r B` le: `A ≤ B` uiff: `uiff(P;Q)` rge: `x ≥ y` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` rleq: `x ≤ y` rnonneg: `rnonneg(x)` rdiv: `(x/y)` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b`
Lemmas referenced :  rational-approx-property rabs-difference-bound-rleq rational-approx_wf rdiv_wf int-to-real_wf rless-int nat_plus_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf rless_wf le_wf nat_plus_wf rleq_wf real_wf radd_wf int-rdiv_wf intformeq_wf itermMultiply_wf int_formula_prop_eq_lemma int_term_value_mul_lemma equal-wf-base int_subtype_base nequal_wf rleq_functionality req_weakening radd_functionality int-rdiv-req rleq_functionality_wrt_implies rleq_weakening_equal req-int-fractions mul_nat_plus less_than_wf decidable__equal_int rmul_preserves_rleq2 rleq-int decidable__le intformle_wf int_formula_prop_le_lemma less_than'_wf rsub_wf rmul_wf rinv_wf2 req_transitivity radd-rdiv rdiv_functionality radd-int real_term_polynomial itermSubtract_wf real_term_value_const_lemma real_term_value_sub_lemma real_term_value_mul_lemma real_term_value_var_lemma req-iff-rsub-is-0 rmul_functionality rmul-rinv absval_unfold lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot itermMinus_wf itermAdd_wf int_term_value_minus_lemma int_term_value_add_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality because_Cache isectElimination setElimination rename hypothesis natural_numberEquality independent_isectElimination sqequalRule inrFormation productElimination independent_functionElimination unionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll applyEquality dependent_set_memberEquality multiplyEquality baseApply closedConclusion baseClosed imageMemberEquality addEquality isect_memberFormation independent_pairEquality minusEquality axiomEquality equalityTransitivity equalitySymmetry equalityElimination lessCases sqequalAxiom imageElimination promote_hyp instantiate cumulativity

Latex:
\mforall{}x:\mBbbR{}.  ((r0  \mleq{}  x)  {}\mRightarrow{}  (\mforall{}n:\mBbbN{}\msupplus{}.  (((x  n)  \mleq{}  0)  {}\mRightarrow{}  (|x  n|  \mleq{}  2))))

Date html generated: 2017_10_03-AM-08_45_26
Last ObjectModification: 2017_07_28-AM-07_32_04

Theory : reals

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