### Nuprl Lemma : member_filter2

`∀[T:Type]. ∀L:T List. ∀P:ℕ||L|| ⟶ 𝔹. ∀x:T.  ((x ∈ filter2(P;L)) `⇐⇒` ∃i:ℕ||L||. ((x = L[i] ∈ T) ∧ (↑(P i))))`

Proof

Definitions occuring in Statement :  filter2: `filter2(P;L)` l_member: `(x ∈ l)` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
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` int_seg: `{i..j-}` uimplies: `b supposing a` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` less_than: `a < b` squash: `↓T` so_apply: `x[s]` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` l_member: `(x ∈ l)` cand: `A c∧ B` nat: `ℕ` ge: `i ≥ j ` le: `A ≤ B` less_than': `less_than'(a;b)` nat_plus: `ℕ+` true: `True` uiff: `uiff(P;Q)` bool: `𝔹` unit: `Unit` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` subtype_rel: `A ⊆r B` sq_type: `SQType(T)` subtract: `n - m` cons: `[a / b]`
Lemmas referenced :  list_induction all_wf int_seg_wf length_wf bool_wf iff_wf l_member_wf filter2_wf exists_wf equal_wf select_wf int_seg_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma assert_wf istype-universe length_of_nil_lemma filter2_nil_lemma stuck-spread istype-base length_of_cons_lemma list_wf nat_properties nat_wf less_than_wf istype-false add_nat_plus length_wf_nat nat_plus_properties add-is-int-iff itermAdd_wf intformeq_wf int_term_value_add_lemma int_formula_prop_eq_lemma false_wf le_wf equal-wf-T-base bnot_wf not_wf eqtt_to_assert uiff_transitivity eqff_to_assert assert_of_bnot squash_wf true_wf cons_filter2 subtype_rel_self iff_weakening_equal cons_wf non_neg_length decidable__equal_int subtype_base_sq int_subtype_base add-member-int_seg2 subtract_wf itermSubtract_wf int_term_value_subtract_lemma select_cons_tl select-cons-tl add-subtract-cancel select-cons-hd subtract-add-cancel
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 natural_numberEquality hypothesis because_Cache productEquality setElimination rename independent_isectElimination productElimination dependent_functionElimination unionElimination approximateComputation independent_functionElimination dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination independent_pairFormation universeIsType imageElimination applyEquality functionIsType baseClosed inhabitedIsType productIsType equalityIsType1 universeEquality dependent_set_memberEquality_alt imageMemberEquality equalityTransitivity equalitySymmetry applyLambdaEquality pointwiseFunctionality promote_hyp baseApply closedConclusion addEquality equalityElimination instantiate cumulativity intEquality

Latex:
\mforall{}[T:Type]
\mforall{}L:T  List.  \mforall{}P:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbB{}.  \mforall{}x:T.    ((x  \mmember{}  filter2(P;L))  \mLeftarrow{}{}\mRightarrow{}  \mexists{}i:\mBbbN{}||L||.  ((x  =  L[i])  \mwedge{}  (\muparrow{}(P  i))))

Date html generated: 2019_10_15-AM-10_55_05
Last ObjectModification: 2018_10_09-AM-10_21_29

Theory : list!

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