Nuprl Lemma : map-filter

[T,A:Type]. ∀[f:T ⟶ A]. ∀[P:T ⟶ 𝔹]. ∀[Q:A ⟶ 𝔹].
  ∀[L:T List]. (map(f;filter(P;L)) filter(Q;map(f;L))) supposing ∀x:T. (f x) x


Proof




Definitions occuring in Statement :  filter: filter(P;l) map: map(f;as) list: List bool: 𝔹 uimplies: supposing a uall: [x:A]. B[x] all: x:A. B[x] apply: a function: x:A ⟶ B[x] universe: Type sqequal: t equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  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 cons: [a b] colength: colength(L) so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] decidable: Dec(P) nil: [] it: 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) uiff: uiff(P;Q) bool: 𝔹 unit: Unit btrue: tt ifthenelse: if then else fi  bfalse: ff
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 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 map_cons_lemma list_wf all_wf bool_wf assert_wf bnot_wf not_wf assert_functionality_wrt_uiff eqtt_to_assert uiff_transitivity eqff_to_assert assert_of_bnot
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 promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination functionExtensionality functionEquality universeEquality equalityElimination

Latex:
\mforall{}[T,A:Type].  \mforall{}[f:T  {}\mrightarrow{}  A].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[Q:A  {}\mrightarrow{}  \mBbbB{}].
    \mforall{}[L:T  List].  (map(f;filter(P;L))  \msim{}  filter(Q;map(f;L)))  supposing  \mforall{}x:T.  Q  (f  x)  =  P  x



Date html generated: 2017_04_17-AM-08_37_14
Last ObjectModification: 2017_02_27-PM-04_55_21

Theory : list_1


Home Index