### Nuprl Lemma : member_filter_2

`∀[T:Type]. ∀L:T List. ∀P:{x:T| (x ∈ L)}  ⟶ 𝔹. ∀x:T.  ((x ∈ filter(P;L)) `⇐⇒` (x ∈ L) ∧ (↑(P x)))`

Proof

Definitions occuring in Statement :  l_member: `(x ∈ l)` filter: `filter(P;l)` list: `T List` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` prop: `ℙ` and: `P ∧ Q` so_apply: `x[s]` implies: `P `` Q` top: `Top` iff: `P `⇐⇒` Q` uimplies: `b supposing a` not: `¬A` false: `False` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` squash: `↓T` true: `True` cand: `A c∧ B`
Lemmas referenced :  list_induction all_wf l_member_wf bool_wf iff_wf filter_wf5 assert_wf list_wf filter_nil_lemma null_nil_lemma btrue_wf member-implies-null-eq-bfalse nil_wf btrue_neq_bfalse assert_witness subtype_rel_dep_function cons_wf subtype_rel_sets cons_member equal_wf subtype_rel_self set_wf filter_cons_lemma eqtt_to_assert eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot or_wf assert_elim not_assert_elim
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality functionEquality setEquality cumulativity because_Cache hypothesis setElimination rename functionExtensionality applyEquality productEquality dependent_set_memberEquality independent_functionElimination dependent_functionElimination universeEquality isect_memberEquality voidElimination voidEquality independent_pairFormation independent_isectElimination equalityTransitivity equalitySymmetry productElimination inrFormation comment inlFormation unionElimination equalityElimination dependent_pairFormation promote_hyp instantiate addLevel impliesFunctionality andLevelFunctionality hyp_replacement applyLambdaEquality imageMemberEquality baseClosed imageElimination natural_numberEquality levelHypothesis

Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  \mforall{}P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}.  \mforall{}x:T.    ((x  \mmember{}  filter(P;L))  \mLeftarrow{}{}\mRightarrow{}  (x  \mmember{}  L)  \mwedge{}  (\muparrow{}(P  x)))

Date html generated: 2017_04_14-AM-08_53_08
Last ObjectModification: 2017_02_27-PM-03_38_37

Theory : list_0

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