### Nuprl Lemma : rleq-iff4

`∀[x,y:ℝ].  (x ≤ y `⇐⇒` ∀n:ℕ+. ((x n) ≤ ((y n) + 4)))`

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` real: `ℝ` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀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]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` all: `∀x:A. B[x]` prop: `ℙ` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` real: `ℝ` so_apply: `x[s]` le: `A ≤ B` not: `¬A` false: `False` rleq: `x ≤ y` rnonneg: `rnonneg(x)` subtype_rel: `A ⊆r B` decidable: `Dec(P)` or: `P ∨ Q` rless: `x < y` sq_exists: `∃x:{A| B[x]}` nat_plus: `ℕ+` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` uiff: `uiff(P;Q)` rsub: `x - y` rminus: `-(x)` radd: `a + b` accelerate: `accelerate(k;f)` has-value: `(a)↓` nequal: `a ≠ b ∈ T ` sq_type: `SQType(T)` guard: `{T}` int_nzero: `ℤ-o` nat: `ℕ` sq_stable: `SqStable(P)` lt_int: `i <z j` ifthenelse: `if b then t else f fi ` btrue: `tt` bfalse: `ff`
Lemmas referenced :  assert_of_bnot iff_weakening_uiff iff_transitivity eqff_to_assert assert_of_lt_int eqtt_to_assert bool_subtype_base bool_wf bool_cases not_wf bnot_wf assert_wf int_term_value_minus_lemma itermMinus_wf add-is-int-iff sq_stable__less_than lt_int_wf absval_ifthenelse subtract_wf sq_stable__le set_wf nat_wf absval_wf rem_bounds_absval nequal_wf div_rem_sum2 mul_cancel_in_le l_sum_nil_lemma l_sum_cons_lemma map_nil_lemma map_cons_lemma iff_weakening_equal equal_wf int_subtype_base subtype_base_sq int_formula_prop_eq_lemma intformeq_wf decidable__equal_int nil_wf cons_wf reg-seq-list-add-as-l_sum true_wf squash_wf int-value-type value-type-has-value false_wf int_term_value_subtract_lemma itermSubtract_wf subtract-is-int-iff rleq-implies int_term_value_mul_lemma itermMultiply_wf mul_nat_plus rless-iff-large-diff int_formula_prop_wf int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformle_wf itermConstant_wf itermVar_wf itermAdd_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt less_than_wf decidable__lt nat_plus_properties decidable__le real_wf rsub_wf less_than'_wf le_wf all_wf rleq_wf nat_plus_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation lambdaFormation lemma_by_obid hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality applyEquality setElimination rename addEquality natural_numberEquality productElimination independent_pairEquality dependent_functionElimination because_Cache axiomEquality equalityTransitivity equalitySymmetry minusEquality isect_memberEquality voidElimination unionElimination dependent_set_memberFormation dependent_set_memberEquality independent_isectElimination dependent_pairFormation int_eqEquality intEquality voidEquality computeAll independent_functionElimination imageMemberEquality baseClosed multiplyEquality pointwiseFunctionality promote_hyp baseApply closedConclusion callbyvalueReduce sqleReflexivity imageElimination divideEquality functionEquality addLevel instantiate cumulativity universeEquality remainderEquality setEquality impliesFunctionality

Latex:
\mforall{}[x,y:\mBbbR{}].    (x  \mleq{}  y  \mLeftarrow{}{}\mRightarrow{}  \mforall{}n:\mBbbN{}\msupplus{}.  ((x  n)  \mleq{}  ((y  n)  +  4)))

Date html generated: 2016_05_18-AM-07_04_48
Last ObjectModification: 2016_01_17-AM-01_53_20

Theory : reals

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