Nuprl Lemma : bdd-diff-regular

[x,y:ℕ+ ⟶ ℤ]. ∀[k,l:ℕ+].
  (∀n:ℕ+(|(x n) n| ≤ ((2 k) (2 l)))) supposing (bdd-diff(x;y) and k-regular-seq(x) and l-regular-seq(y))


Definitions occuring in Statement :  bdd-diff: bdd-diff(f;g) regular-int-seq: k-regular-seq(f) absval: |i| nat_plus: + uimplies: supposing a uall: [x:A]. B[x] le: A ≤ B all: x:A. B[x] apply: a function: x:A ⟶ B[x] multiply: m subtract: m add: m natural_number: $n int:
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a all: x:A. B[x] bdd-diff: bdd-diff(f;g) exists: x:A. B[x] le: A ≤ B and: P ∧ Q not: ¬A implies:  Q false: False nat_plus: + subtype_rel: A ⊆B prop: nat: true: True top: Top ge: i ≥  decidable: Dec(P) or: P ∨ Q satisfiable_int_formula: satisfiable_int_formula(fmla) squash: T guard: {T} iff: ⇐⇒ Q rev_implies:  Q rev_uimplies: rev_uimplies(P;Q) subtract: m regular-int-seq: k-regular-seq(f) so_lambda: λ2x.t[x] so_apply: x[s] gt: i > j uiff: uiff(P;Q) less_than': less_than'(a;b)
Lemmas referenced :  le-add-cancel minus-one-mul-top mul-associates mul-commutes mul-swap mul-distributes minus-zero condition-implies-le less-iff-le not-lt-2 false_wf and_wf decidable__lt pos_mul_arg_bounds all_functionality_wrt_uimplies int_formula_prop_less_lemma intformless_wf int_formula_prop_and_lemma intformand_wf multiply_functionality_wrt_le int_term_value_constant_lemma int_term_value_mul_lemma int_term_value_add_lemma itermConstant_wf itermMultiply_wf itermAdd_wf set_subtype_base int_subtype_base nat_plus_subtype_nat absval_pos absval_mul left_mul_subtract_distrib add-zero zero-add zero-mul add-swap add-mul-special add-commutes minus-one-mul add-associates minus-minus minus-add int-triangle-inequality add_functionality_wrt_le le_weakening le_functionality iff_weakening_equal absval_sym add_functionality_wrt_eq true_wf squash_wf le_wf int_formula_prop_wf int_term_value_var_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma itermVar_wf intformle_wf intformnot_wf satisfiable-full-omega-tt less_than_wf decidable__le nat_properties nat_plus_properties nat_wf regular-int-seq_wf bdd-diff_wf subtract_wf absval_wf less_than'_wf nat_plus_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation sqequalHypSubstitution productElimination thin because_Cache lemma_by_obid hypothesis sqequalRule lambdaEquality dependent_functionElimination hypothesisEquality independent_pairEquality isectElimination addEquality multiplyEquality natural_numberEquality setElimination rename applyEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality functionEquality intEquality voidElimination minusEquality voidEquality dependent_set_memberEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality computeAll imageElimination imageMemberEquality baseClosed universeEquality independent_functionElimination sqequalIntensionalEquality independent_pairFormation inlFormation

\mforall{}[x,y:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[k,l:\mBbbN{}\msupplus{}].
    (\mforall{}n:\mBbbN{}\msupplus{}.  (|(x  n)  -  y  n|  \mleq{}  ((2  *  k)  +  (2  *  l))))  supposing 
          (bdd-diff(x;y)  and 
          k-regular-seq(x)  and 

Date html generated: 2016_05_18-AM-06_47_36
Last ObjectModification: 2016_01_17-AM-01_47_24

Theory : reals

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