Nuprl Lemma : lastn-as-accum

∀[A:Type]. ∀[n:ℤ]. ∀[L:A List].
  (lastn(n;L) ~ accumulate (with value b and list item x):
                 if ||b|| <z n then b @ [x] else tl(b @ [x]) fi 
                over list:
                  L
                with starting value:
                 []))


Proof



Definitions occuring in Statement :  lastn: lastn(n;L),  length: ||as||,  append: as @ bs,  tl: tl(l),  list_accum: list_accum,  cons: [a / b],  nil: [],  list: T List,  ifthenelse: if b then t else f fi ,  lt_int: i <z j,  uall: ∀[x:A]. B[x],  int: ℤ,  universe: Type,  sqequal: s ~ t
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  all: ∀x:A. B[x],  decidable: Dec(P),  or: P ∨ Q,  subtype_rel: A ⊆r B,  uimplies: b supposing a,  top: Top,  nat: ℕ,  implies: P ⇒ Q,  false: False,  ge: i ≥ j ,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  not: ¬A,  and: P ∧ Q,  prop: ℙ,  guard: {T},  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  cons: [a / b],  colength: colength(L),  nil: [],  it: ⋅,  so_lambda: λ2x.t[x],  so_apply: x[s],  sq_type: SQType(T),  less_than: a < b,  squash: ↓T,  less_than': less_than'(a;b),  append: as @ bs,  so_lambda: so_lambda(x,y,z.t[x; y; z]),  so_apply: x[s1;s2;s3],  bool: 𝔹,  unit: Unit,  btrue: tt,  uiff: uiff(P;Q),  ifthenelse: if b then t else f fi ,  bfalse: ff,  true: True,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  bnot: ¬bb,  assert: ↑b,  nat_plus: ℕ+,  le: A ≤ B,  listp: A List+
Lemmas referenced :  decidable__le,  list_wf,  lastn-0,  subtype_rel_list,  top_wf,  nat_properties,  satisfiable-full-omega-tt,  intformand_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  intformless_wf,  int_formula_prop_and_lemma,  int_formula_prop_le_lemma,  int_term_value_constant_lemma,  int_term_value_var_lemma,  int_formula_prop_less_lemma,  int_formula_prop_wf,  ge_wf,  less_than_wf,  equal-wf-T-base,  nat_wf,  colength_wf_list,  less_than_transitivity1,  less_than_irreflexivity,  list-cases,  list_accum_nil_lemma,  product_subtype_list,  spread_cons_lemma,  intformeq_wf,  itermAdd_wf,  int_formula_prop_eq_lemma,  int_term_value_add_lemma,  intformnot_wf,  int_formula_prop_not_lemma,  le_wf,  equal_wf,  subtract_wf,  itermSubtract_wf,  int_term_value_subtract_lemma,  subtype_base_sq,  set_subtype_base,  int_subtype_base,  decidable__equal_int,  list_accum_cons_lemma,  length_of_nil_lemma,  list_ind_nil_lemma,  reduce_tl_cons_lemma,  lt_int_wf,  bool_wf,  equal-wf-base,  assert_wf,  le_int_wf,  bnot_wf,  uiff_transitivity,  eqtt_to_assert,  assert_of_lt_int,  eqff_to_assert,  assert_functionality_wrt_uiff,  bnot_of_lt_int,  assert_of_le_int,  lastn-nil,  length_wf,  append_nil_sq,  lastn-cases,  bnot_of_le_int,  ifthenelse_wf,  append_wf,  cons_wf,  nil_wf,  tl_wf,  squash_wf,  true_wf,  length_append,  iff_weakening_equal,  length-singleton,  bool_cases_sqequal,  bool_subtype_base,  assert-bnot,  length_tl,  listp_properties,  length-append,  length_of_cons_lemma,  add_nat_plus,  length_wf_nat,  nat_plus_wf,  nat_plus_properties,  decidable__lt,  list_ind_cons_lemma,  non_neg_length
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation,  introduction,  cut,  extract_by_obid,  sqequalHypSubstitution,  dependent_functionElimination,  thin,  hypothesisEquality,  natural_numberEquality,  hypothesis,  unionElimination,  sqequalAxiom,  isectElimination,  cumulativity,  sqequalRule,  isect_memberEquality,  because_Cache,  intEquality,  universeEquality,  applyEquality,  independent_isectElimination,  lambdaEquality,  voidElimination,  voidEquality,  lambdaFormation,  setElimination,  rename,  intWeakElimination,  dependent_pairFormation,  int_eqEquality,  independent_pairFormation,  computeAll,  independent_functionElimination,  promote_hyp,  hypothesis_subsumption,  productElimination,  equalityTransitivity,  equalitySymmetry,  applyLambdaEquality,  dependent_set_memberEquality,  addEquality,  baseClosed,  instantiate,  imageElimination,  baseApply,  closedConclusion,  equalityElimination,  imageMemberEquality
Latex:
\mforall{}[A:Type].  \mforall{}[n:\mBbbZ{}].  \mforall{}[L:A  List].
    (lastn(n;L)  \msim{}  accumulate  (with  value  b  and  list  item  x):
                                  if  ||b||  <z  n  then  b  @  [x]  else  tl(b  @  [x])  fi 
                                over  list:
                                    L
                                with  starting  value:
                                  []))


Date html generated: 2018_05_21-PM-06_42_55
Last ObjectModification: 2017_07_26-PM-04_54_30

Theory : general


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