### Nuprl Lemma : member-filter

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

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]` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` 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]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` prop: `ℙ` so_apply: `x[s]` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` uimplies: `b supposing a` not: `¬A` false: `False` rev_implies: `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]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` squash: `↓T` true: `True`
Lemmas referenced :  list_induction assert_wf iff_wf l_member_wf filter_type list_wf filter_nil_lemma null_nil_lemma btrue_wf member-implies-null-eq-bfalse nil_wf istype-assert btrue_neq_bfalse filter_cons_lemma eqtt_to_assert eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot istype-universe equal_functionality_wrt_subtype_rel2 subtype_rel_set_simple cons_member cons_wf assert_elim equal_wf and_wf not_assert_elim or_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt lambdaFormation_alt cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality_alt functionEquality setEquality applyEquality hypothesis universeIsType setElimination rename independent_functionElimination dependent_functionElimination Error :memTop,  independent_pairFormation dependent_set_memberEquality_alt independent_isectElimination equalityTransitivity equalitySymmetry voidElimination because_Cache setIsType inhabitedIsType unionElimination equalityElimination productElimination dependent_pairFormation_alt equalityIstype promote_hyp instantiate cumulativity functionIsType productIsType universeEquality inlFormation_alt inrFormation_alt unionIsType applyLambdaEquality imageMemberEquality baseClosed imageElimination inlFormation natural_numberEquality dependent_set_memberEquality inrFormation lambdaFormation lambdaEquality

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

Date html generated: 2020_05_19-PM-09_37_46
Last ObjectModification: 2020_01_04-PM-07_57_53

Theory : list_0

Home Index