Nuprl Lemma : filter_append

[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[l1,l2:T List].  (filter(P;l1 l2) filter(P;l1) filter(P;l2))


Proof




Definitions occuring in Statement :  filter: filter(P;l) append: as bs list: List bool: 𝔹 uall: [x:A]. B[x] function: x:A ⟶ B[x] universe: Type sqequal: t
Definitions unfolded in proof :  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 cons: [a b] decidable: Dec(P) colength: colength(L) nil: [] it: guard: {T} so_lambda: λ2x.t[x] so_apply: x[s] sq_type: SQType(T) less_than: a < b squash: T less_than': less_than'(a;b) so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] subtype_rel: A ⊆B
Lemmas referenced :  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 filter_append_sq product_subtype_list colength-cons-not-zero colength_wf_list decidable__le intformnot_wf int_formula_prop_not_lemma istype-le subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf itermSubtract_wf itermAdd_wf int_term_value_subtract_lemma int_term_value_add_lemma le_wf istype-nat list_wf bool_wf istype-universe
Rules used in proof :  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,  axiomSqEquality Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsTypeImplies,  unionElimination because_Cache promote_hyp hypothesis_subsumption productElimination Error :equalityIstype,  Error :dependent_set_memberEquality_alt,  instantiate equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination baseApply closedConclusion baseClosed applyEquality intEquality sqequalBase Error :functionIsType,  universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[l1,l2:T  List].    (filter(P;l1  @  l2)  \msim{}  filter(P;l1)  @  filter(P;l2))



Date html generated: 2019_06_20-PM-01_24_39
Last ObjectModification: 2019_01_15-PM-03_38_43

Theory : list_1


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