Nuprl Lemma : filter_trivial

[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List].  filter(P;L) supposing (∀x∈L.↑P[x])


Proof




Definitions occuring in Statement :  l_all: (∀x∈L.P[x]) filter: filter(P;l) list: List assert: b bool: 𝔹 uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s] 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 ≥  guard: {T} uimplies: supposing a prop: so_lambda: λ2x.t[x] so_apply: x[s] subtype_rel: A ⊆B or: P ∨ Q top: Top cons: [a b] colength: colength(L) so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] squash: T sq_stable: SqStable(P) uiff: uiff(P;Q) and: P ∧ Q le: A ≤ B not: ¬A less_than': less_than'(a;b) true: True decidable: Dec(P) iff: ⇐⇒ Q rev_implies:  Q subtract: m nil: [] it: sq_type: SQType(T) less_than: a < b bool: 𝔹 unit: Unit btrue: tt ifthenelse: if then else fi  bfalse: ff exists: x:A. B[x] bnot: ¬bb assert: b
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf l_all_wf assert_wf l_member_wf equal-wf-T-base nat_wf colength_wf_list int_subtype_base list-cases filter_nil_lemma l_all_wf_nil product_subtype_list spread_cons_lemma sq_stable__le le_antisymmetry_iff add_functionality_wrt_le add-associates add-zero zero-add le-add-cancel decidable__le false_wf not-le-2 condition-implies-le minus-add minus-one-mul minus-one-mul-top add-commutes le_wf equal_wf subtract_wf not-ge-2 less-iff-le minus-minus add-swap subtype_base_sq set_subtype_base filter_cons_lemma bool_wf eqtt_to_assert eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot cons_wf list_wf l_all_cons
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination sqequalRule lambdaEquality dependent_functionElimination isect_memberEquality axiomSqEquality applyEquality setEquality equalityTransitivity equalitySymmetry because_Cache unionElimination voidEquality Error :universeIsType,  promote_hyp hypothesis_subsumption productElimination applyLambdaEquality imageMemberEquality baseClosed imageElimination addEquality dependent_set_memberEquality independent_pairFormation minusEquality intEquality instantiate cumulativity equalityElimination dependent_pairFormation Error :functionIsType,  functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:T  List].    filter(P;L)  \msim{}  L  supposing  (\mforall{}x\mmember{}L.\muparrow{}P[x])



Date html generated: 2019_06_20-PM-00_43_39
Last ObjectModification: 2018_09_26-PM-02_16_52

Theory : list_0


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