### Nuprl Lemma : length-filter-pos

`∀[A:Type]. ∀[P:A ⟶ 𝔹]. ∀[L:A List].  0 < ||filter(P;L)|| supposing (∃x∈L. ↑(P x))`

Proof

Definitions occuring in Statement :  l_exists: `(∃x∈L. P[x])` length: `||as||` filter: `filter(P;l)` list: `T List` assert: `↑b` bool: `𝔹` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` uimplies: `b supposing a` all: `∀x:A. B[x]` prop: `ℙ` so_apply: `x[s]` subtype_rel: `A ⊆r B` implies: `P `` Q` top: `Top` false: `False` iff: `P `⇐⇒` Q` and: `P ∧ Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` assert: `↑b` ifthenelse: `if b then t else f fi ` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb`
Lemmas referenced :  list_induction isect_wf l_exists_wf l_member_wf assert_wf less_than_wf length_wf filter_wf5 subtype_rel_dep_function bool_wf subtype_rel_self set_wf list_wf filter_nil_lemma length_of_nil_lemma false_wf l_exists_nil l_exists_wf_nil filter_cons_lemma eqtt_to_assert length_of_cons_lemma add_nat_plus length_wf_nat nat_plus_wf nat_plus_properties decidable__lt add-is-int-iff satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf itermAdd_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_eq_lemma int_formula_prop_wf equal_wf or_wf true_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot l_exists_cons cons_wf ifthenelse_wf member-less_than
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality cumulativity lambdaFormation hypothesis setElimination rename applyEquality functionExtensionality because_Cache setEquality natural_numberEquality independent_isectElimination independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality addLevel productElimination isectEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry dependent_set_memberEquality independent_pairFormation imageMemberEquality baseClosed applyLambdaEquality pointwiseFunctionality promote_hyp baseApply closedConclusion dependent_pairFormation int_eqEquality intEquality computeAll instantiate functionEquality universeEquality

Latex:
\mforall{}[A:Type].  \mforall{}[P:A  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:A  List].    0  <  ||filter(P;L)||  supposing  (\mexists{}x\mmember{}L.  \muparrow{}(P  x))

Date html generated: 2017_04_17-AM-07_48_04
Last ObjectModification: 2017_02_27-PM-04_22_20

Theory : list_1

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