### Nuprl Lemma : list-to-set-filter

`∀[T:Type]. ∀eq:EqDecider(T). ∀P:T ⟶ 𝔹. ∀L:T List.  (list-to-set(eq;filter(P;L)) ~ filter(P;list-to-set(eq;L)))`

Proof

Definitions occuring in Statement :  list-to-set: `list-to-set(eq;L)` filter: `filter(P;l)` list: `T List` deq: `EqDecider(T)` bool: `𝔹` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` function: `x:A ⟶ B[x]` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.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)` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` iff: `P `⇐⇒` Q` bfalse: `ff` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` cand: `A c∧ B`
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-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases filter_nil_lemma list_to_set_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 equal_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-to-set-cons bool_wf eqtt_to_assert deq-member_wf list-to-set_wf assert-deq-member filter_wf5 subtype_rel_dep_function l_member_wf subtype_rel_self set_wf eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot list_wf deq_wf member_filter
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation thin 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 cumulativity applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination functionExtensionality equalityElimination setEquality functionEquality universeEquality

Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T).  \mforall{}P:T  {}\mrightarrow{}  \mBbbB{}.  \mforall{}L:T  List.
(list-to-set(eq;filter(P;L))  \msim{}  filter(P;list-to-set(eq;L)))

Date html generated: 2017_04_17-AM-09_10_07
Last ObjectModification: 2017_02_27-PM-05_18_30

Theory : decidable!equality

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