Nuprl Lemma : count-length-filter

[A:Type]. ∀[P:A ⟶ 𝔹]. ∀[L:A List].  (count(P;L) ||filter(P;L)||)


Proof




Definitions occuring in Statement :  count: count(P;L) length: ||as|| filter: filter(P;l) list: List bool: 𝔹 uall: [x:A]. B[x] function: x:A ⟶ B[x] universe: Type sqequal: t
Definitions unfolded in proof :  count: count(P;L) 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 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) bool: 𝔹 unit: Unit btrue: tt uiff: uiff(P;Q) ifthenelse: if then else fi  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 reduce_nil_lemma filter_nil_lemma length_of_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 reduce_cons_lemma filter_cons_lemma bool_wf eqtt_to_assert length_of_cons_lemma length_wf filter_wf5 subtype_rel_dep_function l_member_wf subtype_rel_self set_wf eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot list_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep 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 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 equalityElimination setEquality functionEquality universeEquality

Latex:
\mforall{}[A:Type].  \mforall{}[P:A  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:A  List].    (count(P;L)  \msim{}  ||filter(P;L)||)



Date html generated: 2017_04_17-AM-07_47_53
Last ObjectModification: 2017_02_27-PM-04_22_25

Theory : list_1


Home Index