Nuprl Lemma : cbv_list_accum-is-list_accum

∀[T,T':Type]. ∀[l:T List]. ∀[y:T']. ∀[f:T' ⟶ T ⟶ T'].
  cbv_list_accum(x,a.f[x;a];y;l) ~ accumulate (with value x and list item a):
                                    f[x;a]
                                   over list:
                                     l
                                   with starting value:
                                    y) 
  supposing value-type(T')


Proof



Definitions occuring in Statement :  cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),  list_accum: list_accum,  list: T List,  value-type: value-type(T),  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  so_apply: x[s1;s2],  function: x:A ⟶ B[x],  universe: Type,  sqequal: s ~ t
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  all: ∀x:A. B[x],  nat: ℕ,  implies: P ⇒ Q,  false: False,  ge: i ≥ j ,  guard: {T},  uimplies: b supposing a,  prop: ℙ,  subtype_rel: A ⊆r B,  or: P ∨ Q,  cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),  list_accum: list_accum,  nil: [],  it: ⋅,  cons: [a / b],  colength: colength(L),  so_lambda: λ2x y.t[x; y],  top: Top,  so_apply: x[s1;s2],  squash: ↓T,  sq_stable: SqStable(P),  uiff: uiff(P;Q),  and: P ∧ Q,  le: A ≤ B,  not: ¬A,  less_than': less_than'(a;b),  true: True,  decidable: Dec(P),  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  subtract: n - m,  so_lambda: λ2x.t[x],  so_apply: x[s],  sq_type: SQType(T),  less_than: a < b,  has-value: (a)↓
Lemmas referenced :  nat_properties,  less_than_transitivity1,  less_than_irreflexivity,  ge_wf,  less_than_wf,  value-type_wf,  equal-wf-T-base,  nat_wf,  colength_wf_list,  int_subtype_base,  list-cases,  product_subtype_list,  spread_cons_lemma,  sq_stable__le,  le_antisymmetry_iff,  add_functionality_wrt_le,  add-associates,  add-zero,  zero-add,  le-add-cancel,  decidable__le,  false_wf,  not-le-2,  condition-implies-le,  minus-add,  minus-one-mul,  minus-one-mul-top,  add-commutes,  le_wf,  equal_wf,  subtract_wf,  not-ge-2,  less-iff-le,  minus-minus,  add-swap,  subtype_base_sq,  set_subtype_base,  value-type-has-value,  list_wf
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation,  introduction,  cut,  thin,  lambdaFormation,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  hypothesisEquality,  hypothesis,  setElimination,  rename,  intWeakElimination,  natural_numberEquality,  independent_isectElimination,  independent_functionElimination,  voidElimination,  sqequalRule,  lambdaEquality,  dependent_functionElimination,  isect_memberEquality,  sqequalAxiom,  equalityTransitivity,  equalitySymmetry,  functionEquality,  applyEquality,  because_Cache,  unionElimination,  callbyvalueReduce,  sqleReflexivity,  promote_hyp,  hypothesis_subsumption,  productElimination,  voidEquality,  applyLambdaEquality,  imageMemberEquality,  baseClosed,  imageElimination,  addEquality,  dependent_set_memberEquality,  independent_pairFormation,  minusEquality,  intEquality,  instantiate,  cumulativity,  universeEquality
Latex:
\mforall{}[T,T':Type].  \mforall{}[l:T  List].  \mforall{}[y:T'].  \mforall{}[f:T'  {}\mrightarrow{}  T  {}\mrightarrow{}  T'].
    cbv\_list\_accum(x,a.f[x;a];y;l)  \msim{}  accumulate  (with  value  x  and  list  item  a):
                                                                        f[x;a]
                                                                      over  list:
                                                                          l
                                                                      with  starting  value:
                                                                        y) 
    supposing  value-type(T')


Date html generated: 2018_05_21-PM-00_23_39
Last ObjectModification: 2018_05_19-AM-06_55_53

Theory : omega


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