### 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: `T 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: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` apply: `f a` lambda: `λx.A[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` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]` subtype_rel: `A ⊆r 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: `P `⇐⇒` Q` deq: `EqDecider(T)` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` eqof: `eqof(d)` rev_implies: `P `` Q` bnot: `¬bb` ifthenelse: `if b then t else f fi ` bfalse: `ff` assert: `↑b` irrefl: `Irrefl(T;x,y.E[x; y])` st_anti_sym: `StAntiSym(T;x,y.R[x; y])` cand: `A 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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