Nuprl Lemma : bl-all-as-accum

∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹].
  ((∀x∈L.P[x])_b ~ accumulate (with value p and list item x):
                    p ∧b P[x]
                   over list:
                     L
                   with starting value:
                    tt))


Proof



Definitions occuring in Statement :  bl-all: (∀x∈L.P[x])_b,  list_accum: list_accum,  list: T List,  band: p ∧b q,  btrue: tt,  bool: 𝔹,  uall: ∀[x:A]. B[x],  so_apply: x[s],  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,  bl-all: (∀x∈L.P[x])_b,  squash: ↓T,  prop: ℙ,  so_apply: x[s],  all: ∀x:A. B[x],  iff: P ⇐⇒ Q,  and: P ∧ Q,  implies: P ⇒ Q,  rev_implies: P ⇐ Q,  uiff: uiff(P;Q),  so_lambda: λ2x y.t[x; y],  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  band: p ∧b q,  ifthenelse: if b then t else f fi ,  bfalse: ff,  so_apply: x[s1;s2],  true: True,  subtype_rel: A ⊆r B,  guard: {T},  sq_type: SQType(T)
Lemmas referenced :  subtype_base_sq,  bool_wf,  bool_subtype_base,  equal_wf,  squash_wf,  true_wf,  reduce-as-accum,  band_wf,  iff_imp_equal_bool,  assert_wf,  iff_transitivity,  iff_weakening_uiff,  assert_of_band,  iff_wf,  btrue_wf,  list_accum_wf,  eqtt_to_assert,  iff_weakening_equal,  list_wf,  band_commutes
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation,  introduction,  cut,  thin,  instantiate,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  cumulativity,  hypothesis,  independent_isectElimination,  applyEquality,  lambdaEquality,  imageElimination,  hypothesisEquality,  equalityTransitivity,  equalitySymmetry,  because_Cache,  functionExtensionality,  sqequalRule,  lambdaFormation,  independent_pairFormation,  productElimination,  productEquality,  addLevel,  impliesFunctionality,  independent_functionElimination,  unionElimination,  equalityElimination,  dependent_functionElimination,  natural_numberEquality,  imageMemberEquality,  baseClosed,  universeEquality,  sqequalAxiom,  functionEquality,  isect_memberEquality
Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].
    ((\mforall{}x\mmember{}L.P[x])\_b  \msim{}  accumulate  (with  value  p  and  list  item  x):
                                        p  \mwedge{}\msubb{}  P[x]
                                      over  list:
                                          L
                                      with  starting  value:
                                        tt))


Date html generated: 2017_04_17-AM-08_03_18
Last ObjectModification: 2017_02_27-PM-04_33_52

Theory : list_1


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