### Nuprl Lemma : list_decomp_member

`∀[T:Type]. ∀L:T List. ∀i:ℕ||L||.  ∃as,bs:T List. (L = (as @ [L[i]] @ bs) ∈ (T List))`

Proof

Definitions occuring in Statement :  select: `L[n]` length: `||as||` append: `as @ bs` cons: `[a / b]` nil: `[]` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` member: `t ∈ T` int_seg: `{i..j-}` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` true: `True` int_iseg: `{i...j}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  firstn_wf nth_tl_wf nth_tl_decomp_eq int_seg_subtype_nat length_wf false_wf int_seg_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_wf list_ind_cons_lemma list_ind_nil_lemma equal_wf list_wf append_wf cons_wf select_wf decidable__le intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma nil_wf exists_wf int_seg_wf subtype_rel_sets lelt_wf le_wf squash_wf true_wf iff_weakening_equal append_firstn_lastn
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation dependent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality setElimination rename hypothesis addEquality natural_numberEquality because_Cache applyEquality independent_isectElimination sqequalRule independent_pairFormation productElimination dependent_functionElimination unionElimination imageElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll universeEquality productEquality setEquality applyLambdaEquality equalityTransitivity equalitySymmetry imageMemberEquality baseClosed independent_functionElimination

Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  \mforall{}i:\mBbbN{}||L||.    \mexists{}as,bs:T  List.  (L  =  (as  @  [L[i]]  @  bs))

Date html generated: 2017_04_14-AM-09_25_33
Last ObjectModification: 2017_02_27-PM-03_59_47

Theory : list_1

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