### Nuprl Lemma : remove-repeats-fun-map2

`∀[A,B:Type]. ∀[eq:EqDecider(B)]. ∀[L:A List]. ∀[f:{a:A| (a ∈ L)}  ⟶ B].`
`  (map(f;remove-repeats-fun(eq;f;L)) = remove-repeats(eq;map(f;L)) ∈ (B List))`

Proof

Definitions occuring in Statement :  remove-repeats-fun: `remove-repeats-fun(eq;f;L)` remove-repeats: `remove-repeats(eq;L)` l_member: `(x ∈ l)` map: `map(f;as)` list: `T List` deq: `EqDecider(T)` uall: `∀[x:A]. B[x]` set: `{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` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` 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)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` guard: `{T}` 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` rev_implies: `P `` Q` compose: `f o g` deq: `EqDecider(T)` true: `True`
Lemmas referenced :  nat_properties full-omega-unsat 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 l_member_wf equal-wf-T-base nat_wf colength_wf_list int_subtype_base list-cases list_ind_nil_lemma map_nil_lemma remove_repeats_nil_lemma nil_wf 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 decidable__equal_int list_ind_cons_lemma map_cons_lemma remove_repeats_cons_lemma cons_wf cons_member filter-map map_wf filter_wf5 remove-repeats-fun_wf list-set-type subtype_rel_list_set bnot_wf list_wf deq_wf subtype_rel_dep_function subtype_rel_sets set_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 approximateComputation independent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation axiomEquality functionEquality setEquality applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate cumulativity imageElimination functionExtensionality inlFormation inrFormation imageMemberEquality universeEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[eq:EqDecider(B)].  \mforall{}[L:A  List].  \mforall{}[f:\{a:A|  (a  \mmember{}  L)\}    {}\mrightarrow{}  B].
(map(f;remove-repeats-fun(eq;f;L))  =  remove-repeats(eq;map(f;L)))

Date html generated: 2018_05_21-PM-00_51_19
Last ObjectModification: 2018_05_19-AM-06_41_07

Theory : decidable!equality

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