Nuprl Lemma : interleaving_singleton

∀[T:Type]
  ∀L:T List. ∀i:ℕ||L||.
    ∃L2:T List
     ∃f1:ℕ1 ⟶ ℕ||L||. ∃f2:ℕ||L2|| ⟶ ℕ||L||. (interleaving_occurence(T;[L[i]];L2;L;f1;f2) ∧ ((f1 0) = i ∈ ℤ))


Proof



Definitions occuring in Statement :  interleaving_occurence: interleaving_occurence(T;L1;L2;L;f1;f2),  select: L[n],  length: ||as||,  cons: [a / b],  nil: [],  list: T List,  int_seg: {i..j-},  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  exists: ∃x:A. B[x],  and: P ∧ Q,  apply: f a,  function: x:A ⟶ B[x],  natural_number: $n,  int: ℤ,  universe: Type,  equal: s = t ∈ T
Definitions unfolded in proof :  and: P ∧ Q,  exists: ∃x:A. B[x],  implies: P ⇒ Q,  int_seg: {i..j-},  member: t ∈ T,  all: ∀x:A. B[x],  uall: ∀[x:A]. B[x],  prop: ℙ,  top: Top,  not: ¬A,  false: False,  satisfiable_int_formula: satisfiable_int_formula(fmla),  uimplies: b supposing a,  squash: ↓T,  less_than: a < b,  or: P ∨ Q,  decidable: Dec(P),  lelt: i ≤ j < k,  guard: {T},  ge: i ≥ j ,  nat: ℕ,  less_than': less_than'(a;b),  le: A ≤ B,  increasing: increasing(f;k),  interleaving_occurence: interleaving_occurence(T;L1;L2;L;f1;f2),  cons: [a / b],  select: L[n],  sq_type: SQType(T),  rev_implies: P ⇐ Q,  iff: P ⇐⇒ Q,  subtype_rel: A ⊆r B,  true: True,  subtract: n - m,  sq_stable: SqStable(P),  uiff: uiff(P;Q),  so_apply: x[s],  so_lambda: λ2x.t[x],  length: ||as||,  list_ind: list_ind,  nil: [],  it: ⋅,  cand: A c∧ B
Lemmas referenced :  list_wf,  decidable__int_equal,  length_wf,  int_seg_wf,  equal_wf,  interleaving_split,  int_formula_prop_wf,  int_formula_prop_le_lemma,  int_term_value_var_lemma,  int_term_value_constant_lemma,  int_formula_prop_less_lemma,  int_formula_prop_not_lemma,  int_formula_prop_and_lemma,  intformle_wf,  itermVar_wf,  itermConstant_wf,  intformless_wf,  intformnot_wf,  intformand_wf,  satisfiable-full-omega-tt,  decidable__lt,  int_seg_properties,  lelt_wf,  int_term_value_subtract_lemma,  itermSubtract_wf,  subtract_wf,  nat_properties,  false_wf,  int_formula_prop_eq_lemma,  intformeq_wf,  decidable__equal_int,  nat_wf,  less_than_wf,  length_of_nil_lemma,  length_of_cons_lemma,  nil_wf,  decidable__le,  select_wf,  cons_wf,  list_extensionality,  int_subtype_base,  subtype_base_sq,  iff_weakening_equal,  true_wf,  squash_wf,  length_wf_nat,  non_neg_length,  exists_wf,  le-add-cancel,  zero-add,  not-lt-2,  length-singleton,  interleaving_occurence_wf,  subtype_rel_self,  le-add-cancel2,  add-zero,  add_functionality_wrt_le,  add-commutes,  add-associates,  minus-one-mul-top,  add-swap,  minus-one-mul,  minus-add,  condition-implies-le,  sq_stable__le,  not-le-2,  product_subtype_list,  cons_neq_nil,  list-cases,  int_seg_subtype,  subtype_rel_dep_function,  equal-wf-base,  set_subtype_base,  le_wf,  istype-int,  add-is-int-iff,  full-omega-unsat,  itermAdd_wf,  int_term_value_add_lemma,  istype-le,  istype-universe,  istype-less_than
Rules used in proof :  universeEquality,  productElimination,  because_Cache,  sqequalRule,  independent_functionElimination,  cumulativity,  natural_numberEquality,  hypothesis,  rename,  setElimination,  intEquality,  lambdaEquality,  dependent_functionElimination,  hypothesisEquality,  thin,  isectElimination,  sqequalHypSubstitution,  extract_by_obid,  introduction,  cut,  lambdaFormation,  isect_memberFormation,  sqequalReflexivity,  computationStep,  sqequalTransitivity,  sqequalSubstitution,  computeAll,  independent_pairFormation,  voidEquality,  voidElimination,  isect_memberEquality,  int_eqEquality,  dependent_pairFormation,  independent_isectElimination,  imageElimination,  unionElimination,  applyLambdaEquality,  equalitySymmetry,  equalityTransitivity,  dependent_set_memberEquality,  instantiate,  baseClosed,  imageMemberEquality,  applyEquality,  functionExtensionality,  functionEquality,  productEquality,  minusEquality,  addEquality,  hypothesis_subsumption,  promote_hyp,  hyp_replacement,  lambdaEquality_alt,  pointwiseFunctionality,  baseApply,  closedConclusion,  approximateComputation,  dependent_pairFormation_alt,  Error :memTop,  universeIsType,  dependent_set_memberEquality_alt,  lambdaFormation_alt,  productIsType,  inhabitedIsType,  equalityIstype,  sqequalBase
Latex:
\mforall{}[T:Type]
    \mforall{}L:T  List.  \mforall{}i:\mBbbN{}||L||.
        \mexists{}L2:T  List
          \mexists{}f1:\mBbbN{}1  {}\mrightarrow{}  \mBbbN{}||L||
            \mexists{}f2:\mBbbN{}||L2||  {}\mrightarrow{}  \mBbbN{}||L||.  (interleaving\_occurence(T;[L[i]];L2;L;f1;f2)  \mwedge{}  ((f1  0)  =  i))


Date html generated: 2020_05_20-AM-07_48_47
Last ObjectModification: 2020_01_25-PM-10_54_34

Theory : list!


Home Index