Nuprl Lemma : ml-accumulate-sq

∀[A,B:Type].
  (∀[f:B ⟶ A ⟶ B]. ∀[l:A List]. ∀[b:B].
     (ml-accumulate(f;b;l) ~ accumulate (with value v and list item a):
                              f v a
                             over list:
                               l
                             with starting value:
                              b))) supposing 
     ((valueall-type(A) ∧ A) and 
     valueall-type(B))


Proof



Definitions occuring in Statement :  ml-accumulate: ml-accumulate(f;b;l),  list_accum: list_accum,  list: T List,  valueall-type: valueall-type(T),  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  and: P ∧ Q,  apply: f a,  function: x:A ⟶ B[x],  universe: Type,  sqequal: s ~ t
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  uimplies: b supposing a,  all: ∀x:A. B[x],  nat: ℕ,  implies: P ⇒ Q,  false: False,  ge: i ≥ j ,  guard: {T},  prop: ℙ,  and: P ∧ Q,  subtype_rel: A ⊆r B,  or: P ∨ Q,  ml-accumulate: ml-accumulate(f;b;l),  top: Top,  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  ml_apply: f(x),  spreadcons: spreadcons,  callbyvalueall: callbyvalueall,  evalall: evalall(t),  nil: [],  it: ⋅,  so_lambda: λ2x.t[x],  so_apply: x[s],  exists: ∃x:A. B[x],  squash: ↓T,  has-value: (a)↓,  has-valueall: has-valueall(a),  ifthenelse: if b then t else f fi ,  btrue: tt,  cons: [a / b],  colength: colength(L),  sq_stable: SqStable(P),  uiff: uiff(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,  sq_type: SQType(T),  less_than: a < b,  bfalse: ff
Lemmas referenced :  nat_properties,  less_than_transitivity1,  less_than_irreflexivity,  ge_wf,  less_than_wf,  equal-wf-T-base,  nat_wf,  colength_wf_list,  list-cases,  list_accum_nil_lemma,  function-value-type,  valueall-type-value-type,  function-valueall-type,  valueall-type-has-valueall,  evalall-reduce,  null_nil_lemma,  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,  int_subtype_base,  list_accum_cons_lemma,  list_wf,  list-valueall-type,  cons_wf,  null_cons_lemma,  valueall-type_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,  cumulativity,  productElimination,  applyEquality,  because_Cache,  unionElimination,  voidEquality,  callbyvalueReduce,  sqleReflexivity,  independent_pairFormation,  imageMemberEquality,  baseClosed,  functionEquality,  promote_hyp,  hypothesis_subsumption,  applyLambdaEquality,  imageElimination,  addEquality,  dependent_set_memberEquality,  minusEquality,  equalityTransitivity,  equalitySymmetry,  intEquality,  instantiate,  functionExtensionality,  productEquality,  universeEquality
Latex:
\mforall{}[A,B:Type].
    (\mforall{}[f:B  {}\mrightarrow{}  A  {}\mrightarrow{}  B].  \mforall{}[l:A  List].  \mforall{}[b:B].
          (ml-accumulate(f;b;l)  \msim{}  accumulate  (with  value  v  and  list  item  a):
                                                            f  v  a
                                                          over  list:
                                                              l
                                                          with  starting  value:
                                                            b)))  supposing 
          ((valueall-type(A)  \mwedge{}  A)  and 
          valueall-type(B))


Date html generated: 2017_09_29-PM-05_51_03
Last ObjectModification: 2017_05_10-PM-06_57_47

Theory : ML


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