### Nuprl Lemma : firstn_last

`∀[T:Type]. ∀[L:T List].  L = (firstn(||L|| - 1;L) @ [last(L)]) ∈ (T List) supposing ¬↑null(L)`

Proof

Definitions occuring in Statement :  firstn: `firstn(n;as)` last: `last(L)` length: `||as||` null: `null(as)` append: `as @ bs` cons: `[a / b]` nil: `[]` list: `T List` assert: `↑b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` not: `¬A` subtract: `n - m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ 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` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` 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]` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` append: `as @ bs` true: `True` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` subtype_rel: `A ⊆r B` bfalse: `ff` bool: `𝔹` unit: `Unit` uiff: `uiff(P;Q)` subtract: `n - m` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void 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 intformeq_wf int_formula_prop_eq_lemma list-cases null_nil_lemma list_ind_nil_lemma not_wf true_wf product_subtype_list colength-cons-not-zero colength_wf_list istype-false le_wf assert_wf null_wf subtract-1-ge-0 subtype_base_sq set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le null_cons_lemma length_of_cons_lemma false_wf nat_wf list_wf lt_int_wf length_wf equal-wf-T-base bool_wf list_ind_cons_lemma le_int_wf bnot_wf uiff_transitivity eqtt_to_assert assert_of_lt_int eqff_to_assert assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int cons_wf squash_wf istype-universe equal_wf length_of_nil_lemma append_wf firstn_wf last_wf bfalse_wf assert_elim btrue_neq_bfalse nil_wf subtype_rel_self iff_weakening_equal last_cons add-subtract-cancel last_singleton length_zero non_neg_length
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut thin Error :lambdaFormation_alt,  extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination sqequalRule independent_pairFormation Error :universeIsType,  axiomEquality equalityTransitivity equalitySymmetry applyLambdaEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  unionElimination promote_hyp hypothesis_subsumption productElimination Error :equalityIsType1,  because_Cache Error :dependent_set_memberEquality_alt,  instantiate imageElimination Error :equalityIsType4,  baseApply closedConclusion baseClosed applyEquality intEquality universeEquality addEquality equalityElimination imageMemberEquality hyp_replacement

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].    L  =  (firstn(||L||  -  1;L)  @  [last(L)])  supposing  \mneg{}\muparrow{}null(L)

Date html generated: 2019_06_20-PM-01_34_42
Last ObjectModification: 2018_10_06-AM-11_23_14

Theory : list_1

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