Nuprl Lemma : last-mapfilter

[T:Type]. ∀[f:Top]. ∀[P:T ⟶ 𝔹]. ∀[L:T List].
  (last(mapfilter(f;P;L)) if null(filter(P;L)) then ⊥ else last(filter(P;L)) fi )


Proof




Definitions occuring in Statement :  mapfilter: mapfilter(f;P;L) last: last(L) null: null(as) filter: filter(P;l) list: List bottom: ifthenelse: if then else fi  bool: 𝔹 uall: [x:A]. B[x] top: Top apply: a function: x:A ⟶ B[x] universe: Type sqequal: t
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: subtype_rel: A ⊆B guard: {T} or: P ∨ Q last: last(L) mapfilter: mapfilter(f;P;L) select: L[n] nil: [] it: so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] ifthenelse: if then else fi  btrue: tt cons: [a b] colength: colength(L) decidable: Dec(P) 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) bool: 𝔹 unit: Unit uiff: uiff(P;Q) bfalse: ff bnot: ¬bb assert: b
Lemmas referenced :  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 filter_nil_lemma map_nil_lemma null_nil_lemma stuck-spread base_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le 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 filter_cons_lemma bool_wf eqtt_to_assert map_cons_lemma null_cons_lemma last-cons mapfilter_wf top_wf assert_wf filter_wf5 subtype_rel_dep_function l_member_wf subtype_rel_self set_wf subtype_rel_list eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot list_wf null_wf assert_of_null
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 dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom cumulativity applyEquality because_Cache unionElimination baseClosed promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality instantiate imageElimination functionExtensionality equalityElimination setEquality functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[f:Top].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:T  List].
    (last(mapfilter(f;P;L))  \msim{}  if  null(filter(P;L))  then  \mbot{}  else  f  last(filter(P;L))  fi  )



Date html generated: 2017_04_17-AM-07_52_32
Last ObjectModification: 2017_02_27-PM-04_25_42

Theory : list_1


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