Nuprl Lemma : select_wf

[A:Type]. ∀[l:A List]. ∀[n:ℤ].  (l[n] ∈ A) supposing (n < ||l|| and (0 ≤ n))


Definitions occuring in Statement :  select: L[n] length: ||as|| list: List less_than: a < b uimplies: supposing a uall: [x:A]. B[x] le: A ≤ B member: t ∈ T natural_number: $n int: universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  guard: {T} uimplies: supposing a prop: subtype_rel: A ⊆B or: P ∨ Q le: A ≤ B and: P ∧ Q cons: [a b] colength: colength(L) so_lambda: λ2y.t[x; y] top: Top so_apply: x[s1;s2] squash: T sq_stable: SqStable(P) uiff: uiff(P;Q) not: ¬A less_than': less_than'(a;b) true: True decidable: Dec(P) iff: ⇐⇒ Q rev_implies:  Q subtract: m nil: [] it: so_lambda: λ2x.t[x] so_apply: x[s] sq_type: SQType(T) less_than: a < b select: L[n] bool: 𝔹 unit: Unit btrue: tt bfalse: ff exists: x:A. B[x] bnot: ¬bb ifthenelse: if then else fi  assert: b has-value: (a)↓ nat_plus: +
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf length_wf le_wf equal-wf-T-base nat_wf colength_wf_list list-cases length_of_nil_lemma nil_wf product_subtype_list spread_cons_lemma sq_stable__le le_antisymmetry_iff add_functionality_wrt_le add-associates add-zero zero-add le-add-cancel decidable__le false_wf not-le-2 condition-implies-le minus-add minus-one-mul minus-one-mul-top add-commutes equal_wf subtract_wf not-ge-2 less-iff-le minus-minus add-swap subtype_base_sq set_subtype_base int_subtype_base lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot value-type-has-value int-value-type not-lt-2 length_of_cons_lemma non_neg_length length_wf_nat cons_wf list_wf le_reflexive one-mul add-mul-special two-mul mul-distributes-right zero-mul omega-shadow mul-distributes mul-associates mul-commutes le-add-cancel-alt decidable__lt
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination lambdaEquality dependent_functionElimination isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry cumulativity intEquality applyEquality because_Cache unionElimination productElimination voidEquality promote_hyp hypothesis_subsumption applyLambdaEquality imageMemberEquality baseClosed imageElimination addEquality dependent_set_memberEquality independent_pairFormation minusEquality instantiate equalityElimination lessCases sqequalAxiom dependent_pairFormation callbyvalueReduce sqequalIntensionalEquality universeEquality multiplyEquality

\mforall{}[A:Type].  \mforall{}[l:A  List].  \mforall{}[n:\mBbbZ{}].    (l[n]  \mmember{}  A)  supposing  (n  <  ||l||  and  (0  \mleq{}  n))

Date html generated: 2017_04_14-AM-08_36_30
Last ObjectModification: 2017_02_27-PM-03_28_36

Theory : list_0

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