Nuprl Lemma : firstn_firstn

[L:Top List]. ∀[n:ℤ]. ∀[m:ℕ1].  (firstn(m;firstn(n;L)) firstn(m;L))


Proof




Definitions occuring in Statement :  firstn: firstn(n;as) list: List int_seg: {i..j-} uall: [x:A]. B[x] top: Top add: m natural_number: $n 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 ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: subtype_rel: A ⊆B guard: {T} or: P ∨ Q firstn: firstn(n;as) so_lambda: so_lambda(x,y,z.t[x; y; z]) so_apply: x[s1;s2;s3] cons: [a b] colength: colength(L) so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] decidable: Dec(P) nil: [] it: so_lambda: λ2x.t[x] so_apply: x[s] sq_type: SQType(T) less_than: a < b squash: T less_than': less_than'(a;b) int_seg: {i..j-} lelt: i ≤ j < k bool: 𝔹 unit: Unit btrue: tt uiff: uiff(P;Q) ifthenelse: if then else fi  bfalse: ff
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 int_seg_wf equal-wf-T-base nat_wf colength_wf_list top_wf less_than_transitivity1 less_than_irreflexivity list_wf list-cases list_ind_nil_lemma product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int list_ind_cons_lemma lt_int_wf bool_wf assert_wf le_int_wf bnot_wf int_seg_properties equal-wf-base uiff_transitivity eqtt_to_assert assert_of_lt_int eqff_to_assert assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int subtract-add-cancel decidable__lt lelt_wf
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 dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom addEquality applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality baseClosed instantiate cumulativity imageElimination baseApply closedConclusion equalityElimination

Latex:
\mforall{}[L:Top  List].  \mforall{}[n:\mBbbZ{}].  \mforall{}[m:\mBbbN{}n  +  1].    (firstn(m;firstn(n;L))  \msim{}  firstn(m;L))



Date html generated: 2017_04_17-AM-07_59_01
Last ObjectModification: 2017_02_27-PM-04_29_34

Theory : list_1


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