### Nuprl Lemma : rless-iff4

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

Proof

Definitions occuring in Statement :  rless: `x < y` real: `ℝ` int_upper: `{i...}` nat_plus: `ℕ+` less_than: `a < b` all: `∀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 :  all: `∀x:A. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` prop: `ℙ` uall: `∀[x:A]. B[x]` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` real: `ℝ` int_upper: `{i...}` le: `A ≤ B` guard: `{T}` uimplies: `b supposing a` so_apply: `x[s]` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` exists: `∃x:A. B[x]` rless: `x < y` sq_exists: `∃x:{A| B[x]}` sq_stable: `SqStable(P)` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` uiff: `uiff(P;Q)`
Lemmas referenced :  le_wf false_wf int_term_value_subtract_lemma int_term_value_constant_lemma int_term_value_add_lemma int_formula_prop_less_lemma itermSubtract_wf itermConstant_wf itermAdd_wf intformless_wf subtract-is-int-iff decidable__lt int_formula_prop_wf int_term_value_var_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le sq_stable__less_than nat_plus_properties int_upper_properties real_wf less_than_transitivity1 less_than_wf int_upper_wf all_wf nat_plus_wf exists_wf rless_wf rless-iff-large-diff
Rules used in proof :  cut lemma_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality productElimination independent_pairFormation independent_functionElimination isectElimination sqequalRule lambdaEquality setElimination rename addEquality applyEquality dependent_set_memberEquality natural_numberEquality independent_isectElimination because_Cache introduction imageMemberEquality baseClosed dependent_pairFormation imageElimination unionElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll pointwiseFunctionality equalityTransitivity equalitySymmetry promote_hyp baseApply closedConclusion dependent_set_memberFormation

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

Date html generated: 2016_05_18-AM-07_03_54
Last ObjectModification: 2016_01_17-AM-01_50_05

Theory : reals

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