### 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: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` top: `Top` add: `n + m` natural_number: `\$n` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r 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: `λ2x y.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 b then t else f 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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