### Nuprl Lemma : firstn_wf

`∀[A:Type]. ∀[as:A List]. ∀[n:ℤ].  (firstn(n;as) ∈ A List)`

Proof

Definitions occuring in Statement :  firstn: `firstn(n;as)` list: `T List` uall: `∀[x:A]. B[x]` member: `t ∈ T` int: `ℤ` universe: `Type`
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 ` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` subtype_rel: `A ⊆r B` or: `P ∨ Q` firstn: `firstn(n;as)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` squash: `↓T` sq_stable: `SqStable(P)` uiff: `uiff(P;Q)` and: `P ∧ Q` le: `A ≤ B` not: `¬A` less_than': `less_than'(a;b)` true: `True` decidable: `Dec(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` bool: `𝔹` unit: `Unit` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf equal-wf-T-base nat_wf colength_wf_list list-cases list_ind_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 le_wf equal_wf subtract_wf not-ge-2 less-iff-le minus-minus add-swap subtype_base_sq set_subtype_base int_subtype_base list_ind_cons_lemma lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int cons_wf eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot list_wf
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 intEquality cumulativity applyEquality because_Cache unionElimination voidEquality promote_hyp hypothesis_subsumption productElimination applyLambdaEquality imageMemberEquality baseClosed imageElimination addEquality dependent_set_memberEquality independent_pairFormation minusEquality instantiate equalityElimination dependent_pairFormation universeEquality

Latex:
\mforall{}[A:Type].  \mforall{}[as:A  List].  \mforall{}[n:\mBbbZ{}].    (firstn(n;as)  \mmember{}  A  List)

Date html generated: 2017_04_14-AM-08_47_27
Last ObjectModification: 2017_02_27-PM-03_34_43

Theory : list_0

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