Nuprl Lemma : filter-equals

∀[T:Type]
  ∀P:T ⟶ 𝔹. ∀L1,L2:T List.
    (filter(P;L1) = L2 ∈ (T List)
       ⇐⇒ (∀x:T. ((x ∈ L2) ⇐⇒ (x ∈ L1) ∧ (↑(P x)))) ∧ (∀x,y:T.  (x before y ∈ L2 ⇒ x before y ∈ L1))) supposing 
       (no_repeats(T;L2) and 
       no_repeats(T;L1))


Proof



Definitions occuring in Statement :  l_before: x before y ∈ l,  no_repeats: no_repeats(T;l),  l_member: (x ∈ l),  filter: filter(P;l),  list: T List,  assert: ↑b,  bool: 𝔹,  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  iff: P ⇐⇒ Q,  implies: P ⇒ Q,  and: P ∧ Q,  apply: f a,  function: x:A ⟶ B[x],  universe: Type,  equal: s = t ∈ T
Definitions unfolded in proof :  implies: P ⇒ Q,  rev_implies: P ⇐ Q,  iff: P ⇐⇒ Q,  and: P ∧ Q,  istype: istype(T),  so_apply: x[s],  subtype_rel: A ⊆r B,  uimplies: b supposing a,  prop: ℙ,  so_lambda: λ2x.t[x],  member: t ∈ T,  all: ∀x:A. B[x],  uall: ∀[x:A]. B[x],  false: False,  not: ¬A,  top: Top,  ge: i ≥ j ,  uiff: uiff(P;Q),  guard: {T},  satisfiable_int_formula: satisfiable_int_formula(fmla),  or: P ∨ Q,  decidable: Dec(P),  nat_plus: ℕ+,  cand: A c∧ B,  cons: [a / b],  select: L[n],  less_than': less_than'(a;b),  le: A ≤ B,  nat: ℕ,  exists: ∃x:A. B[x],  l_member: (x ∈ l),  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  ifthenelse: if b then t else f fi ,  bfalse: ff,  sq_type: SQType(T),  assert: ↑b,  true: True,  squash: ↓T
Lemmas referenced :  istype-universe,  istype-assert,  length_wf,  l_before_wf,  assert_wf,  l_member_wf,  bool_wf,  subtype_rel_dep_function,  filter_wf5,  equal_wf,  iff_wf,  no_repeats_wf,  list_wf,  list_induction,  length_of_nil_lemma,  bfalse_wf,  null_cons_lemma,  null_wf,  cons_wf,  assert_witness,  btrue_neq_bfalse,  member-implies-null-eq-bfalse,  btrue_wf,  null_nil_lemma,  no_repeats_witness,  equal-wf-base-T,  nil_wf,  istype-void,  filter_nil_lemma,  int_formula_prop_le_lemma,  intformle_wf,  decidable__le,  nat_properties,  select_wf,  false_wf,  int_formula_prop_eq_lemma,  int_term_value_add_lemma,  int_term_value_var_lemma,  int_formula_prop_and_lemma,  intformeq_wf,  itermAdd_wf,  itermVar_wf,  intformand_wf,  add-is-int-iff,  nat_plus_properties,  istype-less_than,  int_formula_prop_wf,  int_term_value_constant_lemma,  int_formula_prop_less_lemma,  int_formula_prop_not_lemma,  istype-int,  itermConstant_wf,  intformless_wf,  intformnot_wf,  full-omega-unsat,  decidable__lt,  length_wf_nat,  add_nat_plus,  length_of_cons_lemma,  istype-le,  not_wf,  bnot_wf,  equal-wf-T-base,  filter_cons_lemma,  eqtt_to_assert,  uiff_transitivity,  eqff_to_assert,  assert_of_bnot,  product_subtype_list,  list-cases,  no_repeats_cons,  cons_one_one,  cons_member,  assert_elim,  subtype_base_sq,  bool_subtype_base,  cons_before,  l_before_member,  true_wf,  squash_wf,  l_before_member2,  not_assert_elim
Rules used in proof :  universeEquality,  instantiate,  dependent_functionElimination,  applyLambdaEquality,  inhabitedIsType,  equalityIstype,  productIsType,  isectIsType,  functionIsType,  independent_functionElimination,  productEquality,  rename,  setElimination,  independent_isectElimination,  setIsType,  setEquality,  universeIsType,  because_Cache,  applyEquality,  isectEquality,  hypothesis,  functionEquality,  lambdaEquality_alt,  sqequalRule,  hypothesisEquality,  isectElimination,  sqequalHypSubstitution,  extract_by_obid,  introduction,  thin,  cut,  lambdaFormation_alt,  isect_memberFormation_alt,  sqequalReflexivity,  computationStep,  sqequalTransitivity,  sqequalSubstitution,  dependent_set_memberEquality_alt,  sqequalBase,  productElimination,  equalitySymmetry,  equalityTransitivity,  independent_pairFormation,  baseClosed,  voidElimination,  isect_memberEquality_alt,  int_eqEquality,  closedConclusion,  baseApply,  promote_hyp,  pointwiseFunctionality,  approximateComputation,  unionElimination,  natural_numberEquality,  dependent_pairFormation_alt,  equalityElimination,  hypothesis_subsumption,  inlFormation_alt,  inrFormation_alt,  unionIsType,  hyp_replacement,  unionEquality,  cumulativity,  imageMemberEquality,  imageElimination
Latex:
\mforall{}[T:Type]
    \mforall{}P:T  {}\mrightarrow{}  \mBbbB{}.  \mforall{}L1,L2:T  List.
        (filter(P;L1)  =  L2
              \mLeftarrow{}{}\mRightarrow{}  (\mforall{}x:T.  ((x  \mmember{}  L2)  \mLeftarrow{}{}\mRightarrow{}  (x  \mmember{}  L1)  \mwedge{}  (\muparrow{}(P  x))))
                      \mwedge{}  (\mforall{}x,y:T.    (x  before  y  \mmember{}  L2  {}\mRightarrow{}  x  before  y  \mmember{}  L1)))  supposing 
              (no\_repeats(T;L2)  and 
              no\_repeats(T;L1))


Date html generated: 2019_10_15-AM-10_23_04
Last ObjectModification: 2019_08_05-PM-02_01_26

Theory : list_1


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