### Nuprl Lemma : rless-iff

`∀x,y:ℝ.  (x < y `⇐⇒` ∃n:ℕ+. ∀m:ℕ+. ((n ≤ m) `` (m ≤ (n * ((y m) - x m)))))`

Proof

Definitions occuring in Statement :  rless: `x < y` real: `ℝ` nat_plus: `ℕ+` le: `A ≤ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` apply: `f a` multiply: `n * m` subtract: `n - m`
Definitions unfolded in proof :  rless: `x < y` all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` prop: `ℙ` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` real: `ℝ` so_apply: `x[s]` int_upper: `{i...}` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` exists: `∃x:A. B[x]` sq_exists: `∃x:{A| B[x]}` sq_stable: `SqStable(P)` squash: `↓T` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` less_than: `a < b` true: `True` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` subtype_rel: `A ⊆r B` nat: `ℕ` ge: `i ≥ j ` rev_uimplies: `rev_uimplies(P;Q)` subtract: `n - m`
Lemmas referenced :  regular-less-iff rless_wf exists_wf nat_plus_wf all_wf le_wf subtract_wf real_wf false_wf nat_plus_properties sq_stable__less_than decidable__le satisfiable-full-omega-tt intformnot_wf intformle_wf itermVar_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf mul_nat_plus less_than_wf regular-consistency absval_unfold lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf add-is-int-iff intformand_wf intformless_wf itermConstant_wf itermSubtract_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_add_lemma 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 nat_plus_subtype_nat le_functionality multiply_functionality_wrt_le le_weakening multiply-is-int-iff itermMultiply_wf int_term_value_mul_lemma subtract-is-int-iff mul_preserves_le int_upper_wf decidable__lt not-lt-2 condition-implies-le minus-add minus-one-mul zero-add minus-one-mul-top add-commutes add_functionality_wrt_le add-associates le-add-cancel less_than_transitivity1 int_upper_properties subtype_rel_sets mul_cancel_in_lt
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep lambdaFormation hypothesis isectElimination thin hypothesisEquality independent_pairFormation productElimination independent_functionElimination lambdaEquality functionEquality setElimination rename because_Cache multiplyEquality applyEquality dependent_functionElimination dependent_set_memberEquality natural_numberEquality addEquality imageMemberEquality baseClosed imageElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll minusEquality equalityElimination equalityTransitivity equalitySymmetry lessCases isect_memberFormation sqequalAxiom pointwiseFunctionality promote_hyp baseApply closedConclusion instantiate cumulativity setEquality applyLambdaEquality

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

Date html generated: 2017_10_03-AM-08_24_48
Last ObjectModification: 2017_07_28-AM-07_23_29

Theory : reals

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