Nuprl Lemma : filter_functionality

[A:Type]. ∀[L:A List]. ∀[f1,f2:A ⟶ 𝔹].  filter(f1;L) filter(f2;L) supposing f1 f2 ∈ (A ⟶ 𝔹)


Proof




Definitions occuring in Statement :  filter: filter(P;l) list: List bool: 𝔹 uimplies: supposing a uall: [x:A]. B[x] function: x:A ⟶ B[x] universe: Type sqequal: t equal: t ∈ 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 satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: subtype_rel: A ⊆B guard: {T} or: P ∨ Q cons: [a b] colength: colength(L) so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] decidable: Dec(P) nil: [] it: 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) iff: ⇐⇒ Q assert: b ifthenelse: if then else fi  btrue: tt true: True rev_implies:  Q
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma 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 less_than_wf equal_wf bool_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases filter_nil_lemma product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int filter_cons_lemma list_wf bool_subtype_base iff_imp_equal_bool and_wf assert_elim assert_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom functionEquality cumulativity functionExtensionality applyEquality equalityTransitivity equalitySymmetry because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination universeEquality addLevel levelHypothesis

Latex:
\mforall{}[A:Type].  \mforall{}[L:A  List].  \mforall{}[f1,f2:A  {}\mrightarrow{}  \mBbbB{}].    filter(f1;L)  \msim{}  filter(f2;L)  supposing  f1  =  f2



Date html generated: 2017_04_14-AM-09_31_36
Last ObjectModification: 2017_02_27-PM-04_03_06

Theory : list_1


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