### Nuprl Lemma : imax-list-filter-member

`∀L:ℤ List. ∀P:ℤ ⟶ 𝔹.  (imax-list(filter(P;L)) ∈ L) supposing ¬(filter(P;L) = [] ∈ (ℤ List))`

Proof

Definitions occuring in Statement :  imax-list: `imax-list(L)` l_member: `(x ∈ l)` filter: `filter(P;l)` nil: `[]` list: `T List` bool: `𝔹` uimplies: `b supposing a` all: `∀x:A. B[x]` not: `¬A` function: `x:A ⟶ B[x]` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` not: `¬A` implies: `P `` Q` false: `False` uall: `∀[x:A]. B[x]` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` prop: `ℙ` or: `P ∨ Q` cons: `[a / b]` top: `Top` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` and: `P ∧ Q` guard: `{T}` decidable: `Dec(P)` uiff: `uiff(P;Q)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q`
Lemmas referenced :  equal-wf-T-base list_wf filter_wf5 subtype_rel_dep_function bool_wf l_member_wf subtype_rel_self set_wf list-cases length_of_nil_lemma not_wf equal-wf-base product_subtype_list length_of_cons_lemma add_nat_plus length_wf_nat less_than_wf nat_plus_wf nat_plus_properties decidable__lt add-is-int-iff satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf itermAdd_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_eq_lemma int_formula_prop_wf false_wf equal_wf list_subtype_base int_subtype_base imax-list-member filter_is_sublist member_sublist member_filter_2 imax-list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut introduction sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality voidElimination extract_by_obid isectElimination intEquality hypothesis applyEquality setEquality independent_isectElimination setElimination rename because_Cache baseClosed unionElimination independent_functionElimination equalityTransitivity equalitySymmetry promote_hyp hypothesis_subsumption productElimination isect_memberEquality voidEquality dependent_set_memberEquality natural_numberEquality independent_pairFormation imageMemberEquality applyLambdaEquality pointwiseFunctionality baseApply closedConclusion dependent_pairFormation int_eqEquality computeAll functionEquality

Latex:
\mforall{}L:\mBbbZ{}  List.  \mforall{}P:\mBbbZ{}  {}\mrightarrow{}  \mBbbB{}.    (imax-list(filter(P;L))  \mmember{}  L)  supposing  \mneg{}(filter(P;L)  =  [])

Date html generated: 2017_04_17-AM-07_40_17
Last ObjectModification: 2017_02_27-PM-04_14_08

Theory : list_1

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