### Nuprl Lemma : length_filter

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

Proof

Definitions occuring in Statement :  count: `count(P;L)` length: `||as||` filter: `filter(P;l)` list: `T List` nat: `ℕ` bool: `𝔹` uall: `∀[x:A]. B[x]` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` prop: `ℙ` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` count: `count(P;L)` top: `Top` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` not: `¬A` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` guard: `{T}` ge: `i ≥ j ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  list_induction equal_wf nat_wf length_wf_nat filter_wf5 subtype_rel_dep_function bool_wf l_member_wf set_wf count_wf list_wf filter_nil_lemma reduce_nil_lemma length_of_nil_lemma decidable__equal_int satisfiable-full-omega-tt intformnot_wf intformeq_wf itermConstant_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_constant_lemma int_formula_prop_wf false_wf le_wf filter_cons_lemma reduce_cons_lemma eqtt_to_assert length_of_cons_lemma nat_properties length_wf subtype_rel_self intformand_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma int_term_value_add_lemma int_term_value_var_lemma add_nat_wf decidable__le intformle_wf int_formula_prop_le_lemma eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot zero-add reduce_wf ifthenelse_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality hypothesis cumulativity because_Cache applyEquality setEquality independent_isectElimination setElimination rename lambdaFormation functionExtensionality independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality natural_numberEquality unionElimination dependent_pairFormation intEquality computeAll dependent_set_memberEquality equalityTransitivity equalitySymmetry independent_pairFormation equalityElimination productElimination applyLambdaEquality addEquality int_eqEquality promote_hyp instantiate axiomEquality functionEquality universeEquality

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

Date html generated: 2017_04_14-AM-09_31_21
Last ObjectModification: 2017_02_27-PM-04_03_17

Theory : list_1

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