Nuprl Lemma : rroot-regularity-lemma

∀[k:{2...}]. ∀[n,m:ℕ+]. ∀[a,b,c,d:ℤ].
  (((m ≤ a) ∨ ((a = 0 ∈ ℤ) ∧ (c = 0 ∈ ℤ)))
  ⇒ ((n ≤ b) ∨ ((b = 0 ∈ ℤ) ∧ (d = 0 ∈ ℤ)))
  ⇒ (a^k ≤ c)
  ⇒ c < a + m^k
  ⇒ (b^k ≤ d)
  ⇒ d < b + n^k
  ⇒ (|c - d| ≤ (2^k * (n^k + m^k)))
  ⇒ (|a - b| ≤ (2 * (n + m))))


Proof



Definitions occuring in Statement :  exp: i^n,  absval: |i|,  int_upper: {i...},  nat_plus: ℕ+,  less_than: a < b,  uall: ∀[x:A]. B[x],  le: A ≤ B,  implies: P ⇒ Q,  or: P ∨ Q,  and: P ∧ Q,  multiply: n * m,  subtract: n - m,  add: n + m,  natural_number: $n,  int: ℤ,  equal: s = t ∈ T
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  implies: P ⇒ Q,  prop: ℙ,  subtype_rel: A ⊆r B,  nat: ℕ,  le: A ≤ B,  and: P ∧ Q,  less_than': less_than'(a;b),  false: False,  not: ¬A,  nat_plus: ℕ+,  or: P ∨ Q,  all: ∀x:A. B[x],  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  uiff: uiff(P;Q),  uimplies: b supposing a,  less_than: a < b,  top: Top,  true: True,  squash: ↓T,  guard: {T},  int_upper: {i...},  decidable: Dec(P),  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  sq_type: SQType(T),  bfalse: ff,  bnot: ¬bb,  ifthenelse: if b then t else f fi ,  assert: ↑b,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  so_lambda: λ2x.t[x],  int_seg: {i..j-},  int_iseg: {i...j},  so_apply: x[s],  lelt: i ≤ j < k,  subtract: n - m,  choose: choose(n;i),  ycomb: Y,  eq_int: (i =z j),  bor: p ∨bq,  cand: A c∧ B,  rev_uimplies: rev_uimplies(P;Q),  ge: i ≥ j ,  label: ...$L... t
Lemmas referenced :  le_wf,  absval_wf,  subtract_wf,  exp_wf2,  int_upper_subtype_nat,  false_wf,  less_than_wf,  or_wf,  equal-wf-base,  int_subtype_base,  less_than'_wf,  nat_plus_wf,  int_upper_wf,  absval_unfold,  lt_int_wf,  bool_wf,  eqtt_to_assert,  assert_of_lt_int,  top_wf,  subtype_base_sq,  nat_plus_properties,  int_upper_properties,  decidable__equal_int,  satisfiable-full-omega-tt,  intformand_wf,  intformnot_wf,  intformeq_wf,  itermSubtract_wf,  itermVar_wf,  intformless_wf,  itermConstant_wf,  intformle_wf,  int_formula_prop_and_lemma,  int_formula_prop_not_lemma,  int_formula_prop_eq_lemma,  int_term_value_subtract_lemma,  int_term_value_var_lemma,  int_formula_prop_less_lemma,  int_term_value_constant_lemma,  int_formula_prop_le_lemma,  int_formula_prop_wf,  eqff_to_assert,  equal_wf,  bool_cases_sqequal,  bool_subtype_base,  assert-bnot,  itermMinus_wf,  int_term_value_minus_lemma,  decidable__lt,  exp_preserves_lt,  multiply-is-int-iff,  add-is-int-iff,  not-lt-2,  add_functionality_wrt_le,  add-commutes,  zero-add,  le-add-cancel,  decidable__le,  itermAdd_wf,  itermMultiply_wf,  int_term_value_add_lemma,  int_term_value_mul_lemma,  sum_wf,  choose_wf,  subtype_rel_sets,  lelt_wf,  int_seg_properties,  nat_wf,  int_seg_subtype_nat,  int_seg_wf,  sum_split_first,  condition-implies-le,  minus-add,  minus-one-mul,  minus-one-mul-top,  add-associates,  iff_weakening_equal,  exp0_lemma,  squash_wf,  true_wf,  add_functionality_wrt_eq,  binomial-int,  add-subtract-cancel,  sum_le,  exp_preserves_le,  mul_bounds_1a,  exp_wf4,  le_functionality,  le_weakening,  multiply_functionality_wrt_le,  subtract-is-int-iff,  binomial-inequality1,  less_than_functionality,  mul-distributes,  exp-of-mul,  exp-2-3-fact,  mul_preserves_lt,  exp_wf_nat_plus,  le_weakening2,  exp-positive,  set_subtype_base,  absval_sym,  absval_pos,  minus-zero,  minus-minus,  nat_plus_subtype_nat,  mul_preserves_le,  absval-diff-symmetry
Rules used in proof :  cut,  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation,  introduction,  lambdaFormation,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  hypothesisEquality,  hypothesis,  applyEquality,  because_Cache,  sqequalRule,  multiplyEquality,  dependent_set_memberEquality,  natural_numberEquality,  independent_pairFormation,  addEquality,  setElimination,  rename,  productEquality,  intEquality,  baseClosed,  lambdaEquality,  dependent_functionElimination,  productElimination,  independent_pairEquality,  axiomEquality,  equalityTransitivity,  equalitySymmetry,  isect_memberEquality,  voidElimination,  minusEquality,  unionElimination,  equalityElimination,  independent_isectElimination,  lessCases,  sqequalAxiom,  voidEquality,  imageMemberEquality,  imageElimination,  independent_functionElimination,  instantiate,  cumulativity,  dependent_pairFormation,  int_eqEquality,  computeAll,  promote_hyp,  baseApply,  closedConclusion,  setEquality,  applyLambdaEquality,  universeEquality,  pointwiseFunctionality,  equalityUniverse,  levelHypothesis
Latex:
\mforall{}[k:\{2...\}].  \mforall{}[n,m:\mBbbN{}\msupplus{}].  \mforall{}[a,b,c,d:\mBbbZ{}].
    (((m  \mleq{}  a)  \mvee{}  ((a  =  0)  \mwedge{}  (c  =  0)))
    {}\mRightarrow{}  ((n  \mleq{}  b)  \mvee{}  ((b  =  0)  \mwedge{}  (d  =  0)))
    {}\mRightarrow{}  (a\^{}k  \mleq{}  c)
    {}\mRightarrow{}  c  <  a  +  m\^{}k
    {}\mRightarrow{}  (b\^{}k  \mleq{}  d)
    {}\mRightarrow{}  d  <  b  +  n\^{}k
    {}\mRightarrow{}  (|c  -  d|  \mleq{}  (2\^{}k  *  (n\^{}k  +  m\^{}k)))
    {}\mRightarrow{}  (|a  -  b|  \mleq{}  (2  *  (n  +  m))))


Date html generated: 2017_10_03-AM-10_39_47
Last ObjectModification: 2017_07_28-AM-08_16_22

Theory : reals


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