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

`∀[A,B:Type].`
`  ∀eq:EqDecider(B). ∀f:A ⟶ B. ∀L:A List. ∀a:A.`
`    ((a ∈ remove-repeats-fun(eq;f;L)) `⇐⇒` ∃i:ℕ||L||. ((L[i] = a ∈ A) ∧ (∀j:ℕi. (¬((f L[j]) = (f a) ∈ B)))))`

Proof

Definitions occuring in Statement :  remove-repeats-fun: `remove-repeats-fun(eq;f;L)` l_member: `(x ∈ l)` select: `L[n]` length: `||as||` list: `T List` deq: `EqDecider(T)` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` not: `¬A` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` prop: `ℙ` and: `P ∧ Q` int_seg: `{i..j-}` uimplies: `b supposing a` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` less_than: `a < b` squash: `↓T` so_apply: `x[s]` 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]` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` deq: `EqDecider(T)` le: `A ≤ B` less_than': `less_than'(a;b)` nat_plus: `ℕ+` true: `True` uiff: `uiff(P;Q)` cons: `[a / b]` cand: `A c∧ B` ge: `i ≥ j ` eqof: `eqof(d)` subtract: `n - m` subtype_rel: `A ⊆r B`
Lemmas referenced :  list_induction all_wf iff_wf l_member_wf remove-repeats-fun_wf exists_wf int_seg_wf length_wf equal_wf select_wf int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma not_wf list_wf list_ind_nil_lemma length_of_nil_lemma stuck-spread base_wf null_nil_lemma btrue_wf member-implies-null-eq-bfalse nil_wf btrue_neq_bfalse equal-wf-base-T list_ind_cons_lemma length_of_cons_lemma cons_wf filter_wf5 bnot_wf deq_wf cons_member false_wf add_nat_plus length_wf_nat less_than_wf nat_plus_wf nat_plus_properties add-is-int-iff itermAdd_wf intformeq_wf int_term_value_add_lemma int_formula_prop_eq_lemma lelt_wf non_neg_length member_filter_2 assert_wf eqof_wf iff_transitivity iff_weakening_uiff assert_of_bnot safe-assert-deq add-member-int_seg2 subtract_wf itermSubtract_wf int_term_value_subtract_lemma select-cons-tl add-subtract-cancel decidable__equal_int squash_wf le_wf iff_weakening_equal select_cons_tl true_wf member_filter
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality cumulativity because_Cache functionExtensionality applyEquality hypothesis natural_numberEquality productEquality setElimination rename independent_isectElimination productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination independent_functionElimination baseClosed equalityTransitivity equalitySymmetry setEquality functionEquality universeEquality dependent_set_memberEquality imageMemberEquality applyLambdaEquality pointwiseFunctionality promote_hyp baseApply closedConclusion addEquality impliesFunctionality inlFormation inrFormation

Latex:
\mforall{}[A,B:Type].
\mforall{}eq:EqDecider(B).  \mforall{}f:A  {}\mrightarrow{}  B.  \mforall{}L:A  List.  \mforall{}a:A.
((a  \mmember{}  remove-repeats-fun(eq;f;L))  \mLeftarrow{}{}\mRightarrow{}  \mexists{}i:\mBbbN{}||L||.  ((L[i]  =  a)  \mwedge{}  (\mforall{}j:\mBbbN{}i.  (\mneg{}((f  L[j])  =  (f  a))))))

Date html generated: 2017_04_17-AM-09_12_07
Last ObjectModification: 2017_02_27-PM-05_20_47

Theory : decidable!equality

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