### Nuprl Lemma : select_append

`∀[T:Type]. ∀[as,bs:T List]. ∀[i:ℕ||as|| + ||bs||].  (as @ bs[i] = if i <z ||as|| then as[i] else bs[i - ||as||] fi  ∈ T)`

Proof

Definitions occuring in Statement :  select: `L[n]` length: `||as||` append: `as @ bs` list: `T List` int_seg: `{i..j-}` ifthenelse: `if b then t else f fi ` lt_int: `i <z j` uall: `∀[x:A]. B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` int_seg: `{i..j-}` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` lelt: `i ≤ j < k` prop: `ℙ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` decidable: `Dec(P)` less_than: `a < b` squash: `↓T` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top`
Lemmas referenced :  lt_int_wf length_wf bool_wf eqtt_to_assert assert_of_lt_int select_append_front lelt_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot not_functionality_wrt_uiff assert_wf less_than_wf select_append_back decidable__le add-is-int-iff full-omega-unsat intformand_wf intformnot_wf intformle_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf false_wf int_seg_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis lambdaFormation unionElimination equalityElimination because_Cache productElimination independent_isectElimination sqequalRule dependent_set_memberEquality independent_pairFormation natural_numberEquality cumulativity dependent_pairFormation equalityTransitivity equalitySymmetry promote_hyp dependent_functionElimination instantiate independent_functionElimination voidElimination pointwiseFunctionality imageElimination baseApply closedConclusion baseClosed approximateComputation lambdaEquality int_eqEquality intEquality isect_memberEquality voidEquality addEquality Error :universeIsType,  axiomEquality Error :inhabitedIsType,  universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[as,bs:T  List].  \mforall{}[i:\mBbbN{}||as||  +  ||bs||].
(as  @  bs[i]  =  if  i  <z  ||as||  then  as[i]  else  bs[i  -  ||as||]  fi  )

Date html generated: 2019_06_20-PM-01_19_45
Last ObjectModification: 2018_09_26-PM-05_20_43

Theory : list_1

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