### Nuprl Lemma : regular-less-iff

`∀[x,y:ℝ].  (∃n:{ℕ+| (x n) + 4 < y n} `⇐⇒` ∀b:{4...}. ∃n:ℕ+. ∀m:{n...}. (x m) + b < y m)`

Proof

Definitions occuring in Statement :  real: `ℝ` int_upper: `{i...}` nat_plus: `ℕ+` less_than: `a < b` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` sq_exists: `∃x:{A| B[x]}` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` apply: `f a` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` real: `ℝ` so_apply: `x[s]` rev_implies: `P `` Q` nat_plus: `ℕ+` int_upper: `{i...}` le: `A ≤ B` guard: `{T}` uimplies: `b supposing a` sq_exists: `∃x:{A| B[x]}` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` subtype_rel: `A ⊆r B` uiff: `uiff(P;Q)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` less_than: `a < b` less_than': `less_than'(a;b)` true: `True` squash: `↓T` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` nat: `ℕ` ge: `i ≥ j `
Lemmas referenced :  int_upper_wf sq_exists_wf nat_plus_wf less_than_wf all_wf exists_wf less_than_transitivity1 real_wf sq_stable__less_than int_upper_properties nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_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_add_lemma int_term_value_var_lemma int_formula_prop_wf mul_preserves_le nat_plus_subtype_nat decidable__lt intformless_wf itermMultiply_wf int_formula_prop_less_lemma int_term_value_mul_lemma multiply-is-int-iff int_subtype_base regular-consistency subtype_rel_sets le_wf absval_unfold subtract_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf add-is-int-iff itermSubtract_wf int_term_value_subtract_lemma false_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot itermMinus_wf int_term_value_minus_lemma absval_wf nat_wf int_upper_subtype_nat le_functionality multiply_functionality_wrt_le le_weakening add_functionality_wrt_le regular-less absval_ifthenelse subtract-is-int-iff assert_wf bnot_wf not_wf minus-is-int-iff bool_cases iff_transitivity iff_weakening_uiff assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation independent_pairFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality hypothesis sqequalRule lambdaEquality addEquality applyEquality setElimination rename hypothesisEquality because_Cache dependent_set_memberEquality productElimination independent_isectElimination promote_hyp dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll multiplyEquality baseApply closedConclusion baseClosed setEquality applyLambdaEquality minusEquality equalityElimination equalityTransitivity equalitySymmetry lessCases sqequalAxiom imageMemberEquality imageElimination independent_functionElimination pointwiseFunctionality instantiate cumulativity impliesFunctionality dependent_set_memberFormation

Latex:
\mforall{}[x,y:\mBbbR{}].    (\mexists{}n:\{\mBbbN{}\msupplus{}|  (x  n)  +  4  <  y  n\}  \mLeftarrow{}{}\mRightarrow{}  \mforall{}b:\{4...\}.  \mexists{}n:\mBbbN{}\msupplus{}.  \mforall{}m:\{n...\}.  (x  m)  +  b  <  y  m)

Date html generated: 2017_10_02-PM-07_13_52
Last ObjectModification: 2017_07_28-AM-07_20_05

Theory : reals

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