Nuprl Lemma : taba-property

∀[A,B:Type]. ∀[init:B]. ∀[F:A ⟶ A ⟶ B ⟶ B]. ∀[xs:A List].
  (taba(init;x,x',a.F[x;x';a];xs)
  = accumulate (with value a and list item p):
     let x,x' = p 
     in F[x;x';a]
    over list:
      zip(rev(xs);xs)
    with starting value:
     init)
  ∈ B)


Proof



Definitions occuring in Statement :  taba: taba(init;x,x',a.F[x; x'; a];l),  zip: zip(as;bs),  reverse: rev(as),  list_accum: list_accum,  list: T List,  uall: ∀[x:A]. B[x],  so_apply: x[s1;s2;s3],  function: x:A ⟶ B[x],  spread: spread def,  universe: Type,  equal: s = t ∈ T
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  taba: taba(init;x,x',a.F[x; x'; a];l),  all: ∀x:A. B[x],  decidable: Dec(P),  or: P ∨ Q,  uimplies: b supposing a,  not: ¬A,  implies: P ⇒ Q,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  false: False,  top: Top,  prop: ℙ,  true: True,  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2;s3],  so_apply: x[s1;s2],  pi1: fst(t),  subtype_rel: A ⊆r B,  squash: ↓T,  so_lambda: λ2x.t[x],  so_apply: x[s],  guard: {T},  iff: P ⇐⇒ Q,  and: P ∧ Q,  rev_implies: P ⇐ Q,  nat: ℕ,  ge: i ≥ j ,  so_lambda: so_lambda(x,y,z.t[x; y; z]),  nth_tl: nth_tl(n;as),  le_int: i ≤z j,  lt_int: i <z j,  bnot: ¬bb,  ifthenelse: if b then t else f fi ,  bfalse: ff,  btrue: tt,  cons: [a / b],  le: A ≤ B,  less_than': less_than'(a;b),  colength: colength(L),  nil: [],  it: ⋅,  sq_type: SQType(T),  less_than: a < b,  uiff: uiff(P;Q),  bool: 𝔹,  unit: Unit,  firstn: firstn(n;as),  assert: ↑b,  int_iseg: {i...j},  cand: A c∧ B,  listp: A List+,  pi2: snd(t)
Lemmas referenced :  list_wf,  istype-universe,  decidable__le,  length_wf,  full-omega-unsat,  intformnot_wf,  intformle_wf,  itermVar_wf,  istype-int,  int_formula_prop_not_lemma,  istype-void,  int_formula_prop_le_lemma,  int_term_value_var_lemma,  int_formula_prop_wf,  list_accum_wf,  zip_wf,  reverse_wf,  firstn_all,  subtype_rel_list,  top_wf,  equal_wf,  squash_wf,  true_wf,  pi1_wf_top,  istype-top,  subtype_rel_product,  subtype_rel_self,  iff_weakening_equal,  nat_properties,  intformand_wf,  itermConstant_wf,  intformless_wf,  int_formula_prop_and_lemma,  int_term_value_constant_lemma,  int_formula_prop_less_lemma,  ge_wf,  istype-less_than,  list-cases,  length_of_nil_lemma,  list_ind_nil_lemma,  reverse_nil_lemma,  zip_nil_lemma,  list_accum_nil_lemma,  product_subtype_list,  colength-cons-not-zero,  colength_wf_list,  istype-le,  subtract-1-ge-0,  subtype_base_sq,  intformeq_wf,  int_formula_prop_eq_lemma,  set_subtype_base,  int_subtype_base,  spread_cons_lemma,  decidable__equal_int,  subtract_wf,  itermSubtract_wf,  itermAdd_wf,  int_term_value_subtract_lemma,  int_term_value_add_lemma,  le_wf,  length_of_cons_lemma,  list_ind_cons_lemma,  reverse-cons,  istype-nat,  add-is-int-iff,  false_wf,  le_int_wf,  uiff_transitivity,  equal-wf-T-base,  bool_wf,  assert_wf,  eqtt_to_assert,  assert_of_le_int,  non_neg_length,  lt_int_wf,  less_than_wf,  bnot_wf,  eqff_to_assert,  assert_functionality_wrt_uiff,  bnot_of_le_int,  assert_of_lt_int,  nth_tl_nil,  reduce_tl_nil_lemma,  reduce_tl_cons_lemma,  add-subtract-cancel,  list_decomp,  nth_tl_wf,  cons_wf,  general_length_nth_tl,  length_wf_nat,  decidable__lt,  bool_cases_sqequal,  bool_subtype_base,  assert-bnot,  iff_weakening_uiff,  tl_nth_tl,  firstn_decomp,  select0,  select-nthtl,  istype-false,  firstn_wf,  nil_wf,  hd_wf,  listp_properties,  length_cons,  length_nth_tl,  zip-append,  length-reverse,  length_firstn,  le_weakening2,  zip_cons_cons_lemma,  list_accum_append,  list_accum_cons_lemma,  pi2_wf
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation_alt,  introduction,  cut,  hypothesis,  universeIsType,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  hypothesisEquality,  sqequalRule,  isect_memberEquality_alt,  axiomEquality,  isectIsTypeImplies,  inhabitedIsType,  functionIsType,  because_Cache,  instantiate,  universeEquality,  dependent_functionElimination,  unionElimination,  natural_numberEquality,  independent_isectElimination,  approximateComputation,  independent_functionElimination,  dependent_pairFormation_alt,  lambdaEquality_alt,  int_eqEquality,  voidElimination,  productEquality,  spreadEquality,  applyEquality,  productIsType,  imageElimination,  equalityTransitivity,  equalitySymmetry,  lambdaFormation_alt,  imageMemberEquality,  baseClosed,  productElimination,  setElimination,  rename,  intWeakElimination,  independent_pairFormation,  functionIsTypeImplies,  promote_hyp,  hypothesis_subsumption,  equalityIstype,  dependent_set_memberEquality_alt,  applyLambdaEquality,  baseApply,  closedConclusion,  intEquality,  sqequalBase,  cumulativity,  independent_pairEquality,  lambdaFormation,  pointwiseFunctionality,  addEquality,  equalityElimination,  hyp_replacement,  equalityIsType1,  voidEquality
Latex:
\mforall{}[A,B:Type].  \mforall{}[init:B].  \mforall{}[F:A  {}\mrightarrow{}  A  {}\mrightarrow{}  B  {}\mrightarrow{}  B].  \mforall{}[xs:A  List].
    (taba(init;x,x',a.F[x;x';a];xs)
    =  accumulate  (with  value  a  and  list  item  p):
          let  x,x'  =  p 
          in  F[x;x';a]
        over  list:
            zip(rev(xs);xs)
        with  starting  value:
          init))


Date html generated: 2019_10_15-AM-11_35_24
Last ObjectModification: 2019_06_26-PM-03_42_33

Theory : general


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