### Nuprl Lemma : remove-repeats-filter

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:T ⟶ 𝔹]. ∀[L:T List].`
`  (remove-repeats(eq;filter(P;L)) = filter(P;remove-repeats(eq;L)) ∈ (T List))`

Proof

Definitions occuring in Statement :  remove-repeats: `remove-repeats(eq;L)` filter: `filter(P;l)` list: `T List` deq: `EqDecider(T)` bool: `𝔹` uall: `∀[x:A]. B[x]` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` prop: `ℙ` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` top: `Top` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` ifthenelse: `if b then t else f fi ` squash: `↓T` true: `True` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` deq: `EqDecider(T)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` not: `¬A` eqof: `eqof(d)` band: `p ∧b q`
Lemmas referenced :  list_induction equal_wf list_wf remove-repeats_wf filter_wf5 subtype_rel_dep_function bool_wf l_member_wf set_wf subtype_rel_self filter_nil_lemma remove_repeats_nil_lemma nil_wf filter_cons_lemma remove_repeats_cons_lemma eqtt_to_assert cons_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot filter-filter band_wf bnot_wf iff_weakening_equal iff_imp_equal_bool assert_elim and_wf not_assert_elim btrue_neq_bfalse assert_wf not_wf iff_transitivity eqof_wf iff_weakening_uiff assert_of_band assert_of_bnot safe-assert-deq iff_wf deq_wf band_commutes
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality cumulativity hypothesis because_Cache applyEquality setEquality independent_isectElimination setElimination rename lambdaFormation independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality functionExtensionality unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination imageElimination natural_numberEquality imageMemberEquality baseClosed dependent_pairFormation promote_hyp instantiate independent_pairFormation addLevel levelHypothesis dependent_set_memberEquality applyLambdaEquality productEquality impliesFunctionality andLevelFunctionality impliesLevelFunctionality axiomEquality functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:T  List].
(remove-repeats(eq;filter(P;L))  =  filter(P;remove-repeats(eq;L)))

Date html generated: 2017_04_17-AM-09_10_35
Last ObjectModification: 2017_02_27-PM-05_19_19

Theory : decidable!equality

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