Nuprl Lemma : filter_wf3

[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[l:T List].  (filter(λx.P[x];l) ∈ {x:T| ↑P[x]}  List)


Proof




Definitions occuring in Statement :  filter: filter(P;l) list: List assert: b bool: 𝔹 uall: [x:A]. B[x] so_apply: x[s] member: t ∈ T set: {x:A| B[x]}  lambda: λx.A[x] function: x:A ⟶ B[x] universe: Type
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: subtype_rel: A ⊆B or: P ∨ Q top: Top so_apply: x[s] 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: so_lambda: λ2x.t[x] sq_type: SQType(T) less_than: a < b bool: 𝔹 unit: Unit btrue: tt ifthenelse: if then else fi  bfalse: ff
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf equal-wf-T-base nat_wf colength_wf_list list-cases filter_nil_lemma nil_wf assert_wf 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 int_subtype_base filter_cons_lemma bool_wf eqtt_to_assert cons_wf filter_type list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination lambdaEquality dependent_functionElimination axiomEquality equalityTransitivity equalitySymmetry cumulativity applyEquality because_Cache unionElimination isect_memberEquality voidEquality setEquality functionExtensionality promote_hyp hypothesis_subsumption productElimination applyLambdaEquality imageMemberEquality baseClosed imageElimination addEquality dependent_set_memberEquality independent_pairFormation minusEquality intEquality instantiate equalityElimination functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[l:T  List].    (filter(\mlambda{}x.P[x];l)  \mmember{}  \{x:T|  \muparrow{}P[x]\}    List)



Date html generated: 2017_04_14-AM-08_51_42
Last ObjectModification: 2017_02_27-PM-03_37_01

Theory : list_0


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