Nuprl Lemma : firstn_decomp2

[T:Type]. ∀[j:ℕ]. ∀[l:T List].  (firstn(j 1;l) [l[j 1]] firstn(j;l)) supposing ((j ≤ ||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] le: A ≤ B 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 ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A all: x:A. B[x] top: Top and: P ∧ Q prop: less_than: a < b squash: T less_than': less_than'(a;b) decidable: Dec(P) or: P ∨ Q sq_type: SQType(T) guard: {T} subtract: m le: A ≤ B 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 subtype_rel: A ⊆B true: True iff: ⇐⇒ Q rev_implies:  Q bool: 𝔹 unit: Unit it: uiff: uiff(P;Q) bnot: ¬bb assert: b
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf le_wf length_wf list_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_wf decidable__equal_int subtype_base_sq int_subtype_base list_decomp decidable__lt intformeq_wf int_formula_prop_eq_lemma list_ind_cons_lemma list_ind_nil_lemma first0 tl_wf subtype_rel_list top_wf 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 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 dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom cumulativity equalityTransitivity equalitySymmetry imageElimination productElimination because_Cache unionElimination universeEquality instantiate applyEquality imageMemberEquality baseClosed baseApply closedConclusion equalityElimination 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  \mleq{}  ||l||)  and  0  <  j)

Date html generated: 2017_04_17-AM-07_52_08
Last ObjectModification: 2017_02_27-PM-04_25_09

Theory : list_1

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