### Nuprl Lemma : remove-repeats-fun-as-filter

[A,B:Type]. ∀[eq:EqDecider(B)]. ∀[f:A ⟶ B]. ∀[R:A ⟶ A ⟶ 𝔹]. ∀[L:A List].
(remove-repeats-fun(eq;f;L) filter(λa.(¬b(∃x∈L.R[x;a] ∧b (eq (f x) (f a)))_b);L)) supposing
(sorted-by(λx,y. (↑R[x;y]);L) and
StAntiSym(A;x,y.↑R[x;y]) and
Irrefl(A;x,y.↑R[x;y]))

Proof

Definitions occuring in Statement :  remove-repeats-fun: remove-repeats-fun(eq;f;L) bl-exists: (∃x∈L.P[x])_b sorted-by: sorted-by(R;L) filter: filter(P;l) list: List deq: EqDecider(T) irrefl: Irrefl(T;x,y.E[x; y]) st_anti_sym: StAntiSym(T;x,y.R[x; y]) band: p ∧b q bnot: ¬bb assert: b bool: 𝔹 uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s1;s2] apply: a lambda: λx.A[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 satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: so_apply: x[s1;s2] so_lambda: λ2y.t[x; y] subtype_rel: A ⊆B guard: {T} or: P ∨ Q remove-repeats-fun: remove-repeats-fun(eq;f;L) so_lambda: so_lambda(x,y,z.t[x; y; z]) so_apply: x[s1;s2;s3] cons: [a b] colength: colength(L) 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 deq: EqDecider(T) bool: 𝔹 unit: Unit btrue: tt uiff: uiff(P;Q) eqof: eqof(d) rev_implies:  Q bnot: ¬bb ifthenelse: if then else fi  bfalse: ff assert: b irrefl: Irrefl(T;x,y.E[x; y]) st_anti_sym: StAntiSym(T;x,y.R[x; y]) cand: c∧ B band: p ∧b q label: ...\$L... t
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 sorted-by_wf assert_wf l_member_wf st_anti_sym_wf irrefl_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases filter_nil_lemma list_ind_nil_lemma sorted-by_wf_nil 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_ind_cons_lemma sorted-by-cons filter-filter bl-exists_wf cons_wf band_wf bool_wf eqtt_to_assert assert-bl-exists l_exists_functionality iff_transitivity iff_weakening_uiff assert_of_band safe-assert-deq set_wf eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot l_exists_wf list_wf deq_wf l_exists_iff cons_member and_wf l_all_iff filter-sq bnot_wf eqof_wf not_wf assert_of_bnot l_exists_cons l_all_fwd
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 cumulativity applyEquality functionExtensionality setEquality equalityTransitivity equalitySymmetry because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination equalityElimination productEquality functionEquality universeEquality hyp_replacement addLevel impliesFunctionality levelHypothesis inrFormation inlFormation

Latex:
\mforall{}[A,B:Type].  \mforall{}[eq:EqDecider(B)].  \mforall{}[f:A  {}\mrightarrow{}  B].  \mforall{}[R:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:A  List].
(remove-repeats-fun(eq;f;L)  \msim{}  filter(\mlambda{}a.(\mneg{}\msubb{}(\mexists{}x\mmember{}L.R[x;a]  \mwedge{}\msubb{}  (eq  (f  x)  (f  a)))\_b);L))  supposing
(sorted-by(\mlambda{}x,y.  (\muparrow{}R[x;y]);L)  and
StAntiSym(A;x,y.\muparrow{}R[x;y])  and
Irrefl(A;x,y.\muparrow{}R[x;y]))

Date html generated: 2017_04_17-AM-09_12_33
Last ObjectModification: 2017_02_27-PM-05_20_42

Theory : decidable!equality

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