Nuprl Lemma : list_split_iseg

∀[T:Type]
  ∀f:(T List) ⟶ 𝔹. ∀L1,L2:T List.
    (L1 ≤ L2
    ⇒ let LL1,X = list_split(f;L1) 
       in let LL2,Y = list_split(f;L2) 
          in ((LL1 = LL2 ∈ (T List List)) ∧ X ≤ Y)
             ∨ (∃Z:T List. ∃ZZ:T List List. (((LL1 @ [Z / ZZ]) = LL2 ∈ (T List List)) ∧ X ≤ Z)))


Proof



Definitions occuring in Statement :  list_split: list_split(f;L),  iseg: l1 ≤ l2,  append: as @ bs,  cons: [a / b],  list: T List,  bool: 𝔹,  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  exists: ∃x:A. B[x],  implies: P ⇒ Q,  or: P ∨ Q,  and: P ∧ Q,  function: x:A ⟶ B[x],  spread: spread def,  universe: Type,  equal: s = t ∈ T
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  implies: P ⇒ Q,  member: t ∈ T,  so_lambda: λ2x.t[x],  prop: ℙ,  so_apply: x[s],  squash: ↓T,  iseg: l1 ≤ l2,  exists: ∃x:A. B[x],  true: True,  subtype_rel: A ⊆r B,  uimplies: b supposing a,  guard: {T},  iff: P ⇐⇒ Q,  and: P ∧ Q,  rev_implies: P ⇐ Q,  list_split: list_split(f;L),  top: Top,  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  or: P ∨ Q,  ifthenelse: if b then t else f fi ,  btrue: tt,  cons: [a / b],  bfalse: ff,  bool: 𝔹,  unit: Unit,  it: ⋅,  uiff: uiff(P;Q),  sq_type: SQType(T),  bnot: ¬bb,  assert: ↑b,  false: False,  cand: A c∧ B,  pi1: fst(t),  pi2: snd(t),  not: ¬A,  append: as @ bs,  so_lambda: so_lambda(x,y,z.t[x; y; z]),  so_apply: x[s1;s2;s3]
Lemmas referenced :  list_split_wf,  list_wf,  set_wf,  is_list_splitting_wf,  equal_wf,  iseg_wf,  bool_wf,  squash_wf,  true_wf,  append_wf,  iff_weakening_equal,  list_accum_append,  subtype_rel_list,  top_wf,  list_accum_wf,  list-cases,  null_nil_lemma,  cons_wf,  nil_wf,  product_subtype_list,  null_cons_lemma,  eqtt_to_assert,  eqff_to_assert,  bool_cases_sqequal,  subtype_base_sq,  bool_subtype_base,  assert-bnot,  last_induction,  all_wf,  or_wf,  length_wf,  exists_wf,  length-append,  list_accum_nil_lemma,  iseg_weakening,  and_wf,  pi1_wf_top,  subtype_rel_product,  pi2_wf,  list_accum_cons_lemma,  null_wf3,  equal-wf-T-base,  assert_wf,  bnot_wf,  not_wf,  uiff_transitivity,  assert_of_null,  iff_transitivity,  iff_weakening_uiff,  assert_of_bnot,  iseg_nil,  length_wf_nat,  nat_wf,  nil_iseg,  append_assoc,  list_ind_cons_lemma,  iseg_append
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation,  lambdaFormation,  cut,  introduction,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  cumulativity,  hypothesisEquality,  functionExtensionality,  applyEquality,  hypothesis,  productEquality,  sqequalRule,  lambdaEquality,  spreadEquality,  productElimination,  independent_pairEquality,  setElimination,  rename,  applyLambdaEquality,  imageMemberEquality,  baseClosed,  imageElimination,  equalityTransitivity,  equalitySymmetry,  dependent_functionElimination,  independent_functionElimination,  functionEquality,  universeEquality,  because_Cache,  natural_numberEquality,  setEquality,  independent_isectElimination,  isect_memberEquality,  voidElimination,  voidEquality,  hyp_replacement,  unionElimination,  promote_hyp,  hypothesis_subsumption,  equalityElimination,  dependent_pairFormation,  instantiate,  inlFormation,  dependent_set_memberEquality,  independent_pairFormation,  impliesFunctionality,  inrFormation
Latex:
\mforall{}[T:Type]
    \mforall{}f:(T  List)  {}\mrightarrow{}  \mBbbB{}.  \mforall{}L1,L2:T  List.
        (L1  \mleq{}  L2
        {}\mRightarrow{}  let  LL1,X  =  list\_split(f;L1) 
              in  let  LL2,Y  =  list\_split(f;L2) 
                    in  ((LL1  =  LL2)  \mwedge{}  X  \mleq{}  Y)
                          \mvee{}  (\mexists{}Z:T  List.  \mexists{}ZZ:T  List  List.  (((LL1  @  [Z  /  ZZ])  =  LL2)  \mwedge{}  X  \mleq{}  Z)))


Date html generated: 2018_05_21-PM-08_05_14
Last ObjectModification: 2017_07_26-PM-05_41_11

Theory : general


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