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: List deq: EqDecider(T) int_seg: {i..j-} uall: [x:A]. B[x] all: x:A. B[x] exists: x:A. B[x] iff: ⇐⇒ Q not: ¬A and: P ∧ Q apply: a function: x:A ⟶ B[x] natural_number: \$n universe: Type equal: 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: and: P ∧ Q int_seg: {i..j-} uimplies: 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:  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: λ2y.t[x; y] so_apply: x[s1;s2] iff: ⇐⇒ Q rev_implies:  Q deq: EqDecider(T) le: A ≤ B less_than': less_than'(a;b) nat_plus: + true: True uiff: uiff(P;Q) cons: [a b] cand: c∧ B ge: i ≥  eqof: eqof(d) subtract: m subtype_rel: A ⊆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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