Nuprl Lemma : increasing_split

∀m:ℕ
  ∀[P:ℕm ⟶ ℙ]
    ((∀i:ℕm. Dec(P i))
    ⇒ (∃n,k:ℕ
         ∃f:ℕn ⟶ ℕm
          ∃g:ℕk ⟶ ℕm
           (increasing(f;n)
           ∧ increasing(g;k)
           ∧ (∀i:ℕn. (P (f i)))
           ∧ (∀j:ℕk. (¬(P (g j))))
           ∧ (∀i:ℕm. ((∃j:ℕn. (i = (f j) ∈ ℤ)) ∨ (∃j:ℕk. (i = (g j) ∈ ℤ)))))))


Proof



Definitions occuring in Statement :  increasing: increasing(f;k),  int_seg: {i..j-},  nat: ℕ,  decidable: Dec(P),  uall: ∀[x:A]. B[x],  prop: ℙ,  all: ∀x:A. B[x],  exists: ∃x:A. B[x],  not: ¬A,  implies: P ⇒ Q,  or: P ∨ Q,  and: P ∧ Q,  apply: f a,  function: x:A ⟶ B[x],  natural_number: $n,  int: ℤ,  equal: s = t ∈ T
Definitions unfolded in proof :  all: ∀x:A. B[x],  implies: P ⇒ Q,  uall: ∀[x:A]. B[x],  member: t ∈ T,  prop: ℙ,  exists: ∃x:A. B[x],  nat: ℕ,  and: P ∧ Q,  subtype_rel: A ⊆r B,  int_seg: {i..j-},  not: ¬A,  false: False,  or: P ∨ Q,  so_lambda: λ2x.t[x],  so_apply: x[s],  uimplies: b supposing a,  top: Top,  satisfiable_int_formula: satisfiable_int_formula(fmla),  lelt: i ≤ j < k,  guard: {T},  cand: A c∧ B,  less_than': less_than'(a;b),  le: A ≤ B,  decidable: Dec(P),  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  uiff: uiff(P;Q),  subtract: n - m,  true: True,  ge: i ≥ j ,  fappend: f[n:=x],  increasing: increasing(f;k),  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  ifthenelse: if b then t else f fi ,  bfalse: ff,  sq_type: SQType(T),  bnot: ¬bb,  assert: ↑b,  nequal: a ≠ b ∈ T ,  less_than: a < b,  squash: ↓T
Lemmas referenced :  int_seg_wf,  subtract_wf,  decidable_wf,  istype-nat,  increasing_wf,  subtype_rel_self,  istype-void,  set_subtype_base,  lelt_wf,  int_subtype_base,  istype-int,  istype-less_than,  primrec-wf2,  uall_wf,  subtype_rel_universe1,  all_wf,  exists_wf,  nat_wf,  not_wf,  or_wf,  equal-wf-base,  equal_wf,  int_formula_prop_wf,  int_formula_prop_le_lemma,  int_term_value_constant_lemma,  int_term_value_var_lemma,  int_formula_prop_less_lemma,  int_formula_prop_and_lemma,  intformle_wf,  itermConstant_wf,  itermVar_wf,  intformless_wf,  intformand_wf,  satisfiable-full-omega-tt,  int_seg_properties,  id_increasing,  le_wf,  false_wf,  subtype_rel_function,  int_seg_subtype,  istype-false,  decidable__le,  not-le-2,  condition-implies-le,  add-associates,  minus-add,  minus-one-mul,  add-swap,  minus-one-mul-top,  add-mul-special,  zero-mul,  add-zero,  add-commutes,  le-add-cancel2,  decidable__lt,  full-omega-unsat,  intformnot_wf,  itermSubtract_wf,  int_formula_prop_not_lemma,  int_term_value_subtract_lemma,  less_than_wf,  istype-le,  nat_properties,  itermAdd_wf,  int_term_value_add_lemma,  fappend_wf,  eq_int_wf,  eqtt_to_assert,  assert_of_eq_int,  add-subtract-cancel,  intformeq_wf,  int_formula_prop_eq_lemma,  eqff_to_assert,  bool_cases_sqequal,  subtype_base_sq,  bool_subtype_base,  assert-bnot,  neg_assert_of_eq_int,  bool_wf,  decidable__equal_int,  assert_elim,  bnot_wf,  squash_wf,  true_wf,  istype-universe,  eq_int_eq_true,  iff_weakening_equal,  bfalse_wf,  btrue_neq_bfalse,  assert_wf,  uiff_transitivity,  iff_transitivity,  iff_weakening_uiff,  assert_of_bnot,  btrue_wf
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  Error :lambdaFormation_alt,  cut,  thin,  rename,  setElimination,  sqequalRule,  Error :isectIsType,  Error :functionIsType,  Error :universeIsType,  introduction,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  natural_numberEquality,  hypothesisEquality,  hypothesis,  universeEquality,  because_Cache,  applyEquality,  Error :productIsType,  functionExtensionality,  Error :lambdaEquality_alt,  Error :inhabitedIsType,  equalityTransitivity,  equalitySymmetry,  instantiate,  Error :unionIsType,  Error :equalityIsType4,  intEquality,  closedConclusion,  independent_isectElimination,  Error :setIsType,  functionEquality,  cumulativity,  productEquality,  independent_functionElimination,  computeAll,  voidEquality,  voidElimination,  isect_memberEquality,  dependent_functionElimination,  int_eqEquality,  dependent_pairFormation,  productElimination,  lambdaEquality,  lambdaFormation,  independent_pairFormation,  dependent_set_memberEquality,  isect_memberFormation,  Error :isect_memberFormation_alt,  unionElimination,  addEquality,  minusEquality,  Error :isect_memberEquality_alt,  multiplyEquality,  Error :dependent_set_memberEquality_alt,  approximateComputation,  Error :dependent_pairFormation_alt,  equalityElimination,  promote_hyp,  Error :equalityIsType1,  baseApply,  baseClosed,  applyLambdaEquality,  imageElimination,  Error :inlFormation_alt,  imageMemberEquality,  Error :inrFormation_alt,  Error :equalityIsType2
Latex:
\mforall{}m:\mBbbN{}
    \mforall{}[P:\mBbbN{}m  {}\mrightarrow{}  \mBbbP{}]
        ((\mforall{}i:\mBbbN{}m.  Dec(P  i))
        {}\mRightarrow{}  (\mexists{}n,k:\mBbbN{}
                  \mexists{}f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}m
                    \mexists{}g:\mBbbN{}k  {}\mrightarrow{}  \mBbbN{}m
                      (increasing(f;n)
                      \mwedge{}  increasing(g;k)
                      \mwedge{}  (\mforall{}i:\mBbbN{}n.  (P  (f  i)))
                      \mwedge{}  (\mforall{}j:\mBbbN{}k.  (\mneg{}(P  (g  j))))
                      \mwedge{}  (\mforall{}i:\mBbbN{}m.  ((\mexists{}j:\mBbbN{}n.  (i  =  (f  j)))  \mvee{}  (\mexists{}j:\mBbbN{}k.  (i  =  (g  j))))))))


Date html generated: 2019_06_20-PM-02_29_21
Last ObjectModification: 2018_10_29-PM-06_04_20

Theory : num_thy_1


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