Nuprl Lemma : firstn_decomp

[T:Type]. ∀[j:ℕ]. ∀[l:T List].  (firstn(j 1;l) [l[j 1]] firstn(j;l)) supposing (j 1 < ||l|| and 0 < j)


Definitions occuring in Statement :  firstn: firstn(n;as) select: L[n] length: ||as|| append: as bs cons: [a b] nil: [] list: List nat: less_than: a < b uimplies: supposing a uall: [x:A]. B[x] subtract: m natural_number: $n universe: Type sqequal: t
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T nat: implies:  Q false: False ge: i ≥  guard: {T} uimplies: supposing a prop: less_than: a < b squash: T less_than': less_than'(a;b) and: P ∧ Q all: x:A. B[x] decidable: Dec(P) or: P ∨ Q iff: ⇐⇒ Q not: ¬A rev_implies:  Q uiff: uiff(P;Q) subtract: m subtype_rel: A ⊆B top: Top le: A ≤ B true: True sq_type: SQType(T) firstn: firstn(n;as) so_lambda: so_lambda(x,y,z.t[x; y; z]) so_apply: x[s1;s2;s3] lt_int: i <j select: L[n] cons: [a b] ifthenelse: if then else fi  bfalse: ff btrue: tt append: as bs bool: 𝔹 unit: Unit it: exists: x:A. B[x] bnot: ¬bb assert: b
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf subtract_wf length_wf list_wf decidable__le false_wf not-ge-2 less-iff-le condition-implies-le minus-one-mul zero-add minus-one-mul-top minus-add minus-minus add-associates add-swap add-commutes add_functionality_wrt_le add-zero le-add-cancel nat_wf decidable__equal_int subtype_base_sq int_subtype_base list_decomp decidable__lt not-lt-2 le_antisymmetry_iff list_ind_cons_lemma list_ind_nil_lemma first0 tl_wf subtype_rel_list top_wf not-equal-2 le-add-cancel2 squash_wf true_wf length_tl iff_weakening_equal lt_int_wf bool_wf equal-wf-base assert_wf eqtt_to_assert assert_of_lt_int select-cons-tl eqff_to_assert equal_wf bool_cases_sqequal bool_subtype_base assert-bnot le_int_wf le_wf bnot_wf uiff_transitivity assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination independent_functionElimination voidElimination sqequalRule lambdaEquality dependent_functionElimination isect_memberEquality sqequalAxiom cumulativity equalityTransitivity equalitySymmetry imageElimination productElimination because_Cache unionElimination independent_pairFormation addEquality applyEquality voidEquality intEquality minusEquality universeEquality instantiate imageMemberEquality baseClosed baseApply closedConclusion equalityElimination dependent_pairFormation promote_hyp

\mforall{}[T:Type].  \mforall{}[j:\mBbbN{}].  \mforall{}[l:T  List].
    (firstn(j  -  1;l)  @  [l[j  -  1]]  \msim{}  firstn(j;l))  supposing  (j  -  1  <  ||l||  and  0  <  j)

Date html generated: 2017_04_14-AM-08_48_03
Last ObjectModification: 2017_02_27-PM-03_35_09

Theory : list_0

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