Nuprl Lemma : filter-equal

[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L1,L2:T List].
  (filter(P;L1) filter(P;L2) ∈ (T List)) supposing 
     ((∀i:ℕ||L1||. ((L1[i] L2[i] ∈ T) ∨ ((¬↑(P L1[i])) ∧ (¬↑(P L2[i]))))) and 
     (||L1|| ||L2|| ∈ ℤ))


Definitions occuring in Statement :  select: L[n] length: ||as|| filter: filter(P;l) list: List int_seg: {i..j-} assert: b bool: 𝔹 uimplies: supposing a uall: [x:A]. B[x] all: x:A. B[x] not: ¬A or: P ∨ Q and: P ∧ Q apply: a function: x:A ⟶ B[x] natural_number: $n int: universe: Type equal: t ∈ T
Definitions unfolded in proof :  true: True assert: b iff: ⇐⇒ Q rev_implies:  Q bfalse: ff ifthenelse: if then else fi  btrue: tt unit: Unit bool: 𝔹 uall: [x:A]. B[x] member: t ∈ T all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  uimplies: supposing a not: ¬A satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] top: Top and: P ∧ Q prop: or: P ∨ Q select: L[n] nil: [] it: so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] int_seg: {i..j-} lelt: i ≤ j < k cons: [a b] colength: colength(L) guard: {T} so_lambda: λ2x.t[x] so_apply: x[s] sq_type: SQType(T) le: A ≤ B subtype_rel: A ⊆B decidable: Dec(P) less_than: a < b squash: T less_than': less_than'(a;b) uiff: uiff(P;Q) nat_plus: +
Lemmas referenced :  select-cons squash_wf true_wf add-subtract-cancel bool_subtype_base iff_imp_equal_bool le_int_wf bfalse_wf iff_functionality_wrt_iff iff_weakening_uiff assert_of_le_int iff_weakening_equal assert_of_bnot eqff_to_assert uiff_transitivity eqtt_to_assert nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf istype-less_than list-cases length_of_nil_lemma stuck-spread istype-base filter_nil_lemma nil_wf int_seg_wf int_seg_properties product_subtype_list colength-cons-not-zero subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma length_of_cons_lemma filter_cons_lemma non_neg_length itermAdd_wf int_term_value_add_lemma length_wf colength_wf_list decidable__le intformnot_wf int_formula_prop_not_lemma istype-le decidable__equal_int subtract_wf itermSubtract_wf int_term_value_subtract_lemma le_wf add-is-int-iff false_wf select_wf cons_wf decidable__lt istype-assert istype-nat list_wf bool_wf istype-universe istype-false add_nat_plus length_wf_nat nat_plus_properties equal-wf-T-base assert_wf bnot_wf not_wf assert_elim not_assert_elim btrue_neq_bfalse
Rules used in proof :  imageMemberEquality cumulativity equalityElimination sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut thin Error :lambdaFormation_alt,  extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination sqequalRule independent_pairFormation Error :universeIsType,  axiomEquality Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsTypeImplies,  unionElimination baseClosed because_Cache Error :functionIsType,  equalityTransitivity equalitySymmetry productElimination Error :equalityIstype,  sqequalBase promote_hyp hypothesis_subsumption instantiate applyLambdaEquality addEquality applyEquality Error :dependent_set_memberEquality_alt,  imageElimination baseApply closedConclusion intEquality pointwiseFunctionality Error :unionIsType,  Error :productIsType,  universeEquality

\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L1,L2:T  List].
    (filter(P;L1)  =  filter(P;L2))  supposing 
          ((\mforall{}i:\mBbbN{}||L1||.  ((L1[i]  =  L2[i])  \mvee{}  ((\mneg{}\muparrow{}(P  L1[i]))  \mwedge{}  (\mneg{}\muparrow{}(P  L2[i])))))  and 
          (||L1||  =  ||L2||))

Date html generated: 2019_06_20-PM-02_13_14
Last ObjectModification: 2019_06_20-PM-02_08_13

Theory : list_1

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