Nuprl Lemma : firstn_all

[L:Top List]. ∀[n:ℤ].  firstn(n;L) supposing ||L|| ≤ n


Proof




Definitions occuring in Statement :  firstn: firstn(n;as) length: ||as|| list: List uimplies: supposing a uall: [x:A]. B[x] top: Top le: A ≤ B int: sqequal: t
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 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: λ2y.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: ⇐⇒ 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 bool: 𝔹 unit: Unit btrue: tt ifthenelse: if then else fi  bfalse: ff exists: x:A. B[x] bnot: ¬bb assert: b nat_plus: +
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf le_wf length_wf top_wf equal-wf-T-base nat_wf colength_wf_list list_wf list-cases length_of_nil_lemma list_ind_nil_lemma 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 length_of_cons_lemma list_ind_cons_lemma lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot non_neg_length length_wf_nat subtype_rel-equal base_wf not-lt-2 add-is-int-iff le_reflexive one-mul add-mul-special two-mul mul-distributes-right zero-mul omega-shadow mul-distributes mul-commutes mul-associates minus-zero
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 independent_functionElimination voidElimination sqequalRule lambdaEquality dependent_functionElimination isect_memberEquality sqequalAxiom equalityTransitivity equalitySymmetry intEquality applyEquality because_Cache unionElimination voidEquality promote_hyp hypothesis_subsumption productElimination applyLambdaEquality imageMemberEquality baseClosed imageElimination addEquality dependent_set_memberEquality independent_pairFormation minusEquality instantiate cumulativity equalityElimination dependent_pairFormation sqequalIntensionalEquality baseApply closedConclusion multiplyEquality

Latex:
\mforall{}[L:Top  List].  \mforall{}[n:\mBbbZ{}].    firstn(n;L)  \msim{}  L  supposing  ||L||  \mleq{}  n



Date html generated: 2018_05_21-PM-00_20_36
Last ObjectModification: 2017_10_18-PM-00_43_25

Theory : list_0


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