### Nuprl Lemma : append_split2

`∀[T:Type]`
`  ∀L:T List`
`    ∀[P:ℕ||L|| ⟶ ℙ]`
`      ((∀x:ℕ||L||. Dec(P x))`
`      `` (∀i,j:ℕ||L||.  ((P i) `` P j supposing i < j))`
`      `` (∃L_1,L_2:T List. ((L = (L_1 @ L_2) ∈ (T List)) ∧ (∀i:ℕ||L||. (P i `⇐⇒` ||L_1|| ≤ i)))))`

Proof

Definitions occuring in Statement :  length: `||as||` append: `as @ bs` list: `T List` int_seg: `{i..j-}` less_than: `a < b` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` le: `A ≤ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` prop: `ℙ` and: `P ∧ Q` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` so_lambda: `λ2x.t[x]` lelt: `i ≤ j < k` guard: `{T}` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` so_apply: `x[s]` cand: `A c∧ B` int_iseg: `{i...j}` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` le: `A ≤ B` nat: `ℕ` less_than': `less_than'(a;b)` ge: `i ≥ j ` label: `...\$L... t`
Lemmas referenced :  length_wf all_wf int_seg_wf not_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 lelt_wf decidable__exists_int_seg decidable__and2 decidable__all_int_seg decidable__not less_than_wf decidable_wf list_wf firstn_wf nth_tl_wf equal_wf squash_wf true_wf append_firstn_lastn subtype_rel_sets le_wf decidable__le intformle_wf int_formula_prop_le_lemma iff_weakening_equal less_than'_wf append_wf length-append length_firstn exists_wf iff_wf decidable__equal_int itermConstant_wf intformeq_wf int_term_value_constant_lemma int_formula_prop_eq_lemma nil_wf append_back_nil nat_wf false_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma set_wf primrec-wf2 nat_properties
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity natural_numberEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality hypothesis lambdaEquality productEquality applyEquality functionExtensionality because_Cache sqequalRule setElimination rename dependent_set_memberEquality productElimination independent_pairFormation dependent_functionElimination unionElimination imageElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll isect_memberFormation lambdaFormation instantiate independent_functionElimination functionEquality universeEquality isectEquality equalityTransitivity equalitySymmetry setEquality applyLambdaEquality imageMemberEquality baseClosed independent_pairEquality axiomEquality hyp_replacement hypothesis_subsumption

Latex:
\mforall{}[T:Type]
\mforall{}L:T  List
\mforall{}[P:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}]
((\mforall{}x:\mBbbN{}||L||.  Dec(P  x))
{}\mRightarrow{}  (\mforall{}i,j:\mBbbN{}||L||.    ((P  i)  {}\mRightarrow{}  P  j  supposing  i  <  j))
{}\mRightarrow{}  (\mexists{}L\$_{1}\$,L\$_{2}\$:T  List.  ((L  =  (L\$_{1\mbackslash{}\000Cff7d\$  @  L\$_{2}\$))  \mwedge{}  (\mforall{}i:\mBbbN{}||L||.  (P  i  \mLeftarrow{}{}\mRightarrow{}  ||L\$_{1}\$||  \mleq{}  i\000C)))))

Date html generated: 2017_04_14-AM-09_25_30
Last ObjectModification: 2017_02_27-PM-04_00_30

Theory : list_1

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