Nuprl Lemma : append_firstn_lastn_sq

[L:Top List]. ∀[n:ℕ||L|| 1].  (L firstn(n;L) nth_tl(n;L))


Proof




Definitions occuring in Statement :  firstn: firstn(n;as) length: ||as|| nth_tl: nth_tl(n;as) append: as bs list: List int_seg: {i..j-} uall: [x:A]. B[x] top: Top add: m natural_number: $n 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 int_seg: {i..j-} firstn: firstn(n;as) so_lambda: so_lambda(x,y,z.t[x; y; z]) so_apply: x[s1;s2;s3] append: as bs 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) bool: 𝔹 unit: Unit btrue: tt uiff: uiff(P;Q) ifthenelse: if then else fi  bfalse: ff bnot: ¬bb assert: b lelt: i ≤ j < k le: A ≤ B nth_tl: nth_tl(n;as)
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 length_wf top_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list_wf list-cases nth_tl_nil list_ind_nil_lemma nil_wf 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 eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot cons_wf length_of_cons_lemma int_seg_properties non_neg_length decidable__lt lelt_wf le_int_wf assert_wf bnot_wf reduce_tl_cons_lemma uiff_transitivity assert_of_le_int assert_functionality_wrt_uiff bnot_of_le_int
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 equalityElimination

Latex:
\mforall{}[L:Top  List].  \mforall{}[n:\mBbbN{}||L||  +  1].    (L  \msim{}  firstn(n;L)  @  nth\_tl(n;L))



Date html generated: 2017_04_14-AM-09_25_23
Last ObjectModification: 2017_02_27-PM-04_00_02

Theory : list_1


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