### Nuprl Lemma : interleaving_split

`∀[T:Type]`
`  ∀L:T List`
`    ∀[P:ℕ||L|| ⟶ ℙ]`
`      ((∀x:ℕ||L||. Dec(P x))`
`      `` (∃L1,L2:T List`
`           ∃f1:ℕ||L1|| ⟶ ℕ||L||`
`            ∃f2:ℕ||L2|| ⟶ ℕ||L||`
`             (interleaving_occurence(T;L1;L2;L;f1;f2)`
`             ∧ ((∀i:ℕ||L1||. (P (f1 i))) ∧ (∀i:ℕ||L2||. (¬(P (f2 i)))))`
`             ∧ (∀i:ℕ||L||`
`                  (((P i) `` (∃j:ℕ||L1||. ((f1 j) = i ∈ ℤ))) ∧ ∃j:ℕ||L2||. ((f2 j) = i ∈ ℤ) supposing ¬(P i))))))`

Proof

Definitions occuring in Statement :  interleaving_occurence: `interleaving_occurence(T;L1;L2;L;f1;f2)` length: `||as||` list: `T List` int_seg: `{i..j-}` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` exists: `∃x:A. B[x]` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` cand: `A c∧ B` subtype_rel: `A ⊆r B` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` sq_type: `SQType(T)` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` sublist_occurence: `sublist_occurence(T;L1;L2;f)` interleaving_occurence: `interleaving_occurence(T;L1;L2;L;f1;f2)` true: `True` label: `...\$L... t` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  increasing_split length_wf_nat all_wf int_seg_wf length_wf decidable_wf list_wf range_sublist subtype_base_sq nat_wf set_subtype_base le_wf int_subtype_base nat_properties decidable__equal_int full-omega-unsat intformand_wf intformnot_wf intformeq_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_formula_prop_wf decidable__le intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma subtype_rel_self interleaving_occurence_wf decidable__lt intformless_wf int_formula_prop_less_lemma lelt_wf not_wf exists_wf equal_wf int_seg_properties disjoint_increasing_onto itermAdd_wf int_term_value_add_lemma iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin isectElimination hypothesisEquality hypothesis independent_functionElimination productElimination natural_numberEquality sqequalRule lambdaEquality applyEquality functionIsType universeIsType universeEquality independent_isectElimination because_Cache dependent_pairFormation instantiate cumulativity intEquality setElimination rename equalityTransitivity equalitySymmetry unionElimination approximateComputation int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation dependent_set_memberEquality functionEquality productEquality functionExtensionality isectEquality promote_hyp lambdaFormation_alt applyLambdaEquality

Latex:
\mforall{}[T:Type]
\mforall{}L:T  List
\mforall{}[P:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}]
((\mforall{}x:\mBbbN{}||L||.  Dec(P  x))
{}\mRightarrow{}  (\mexists{}L1,L2:T  List
\mexists{}f1:\mBbbN{}||L1||  {}\mrightarrow{}  \mBbbN{}||L||
\mexists{}f2:\mBbbN{}||L2||  {}\mrightarrow{}  \mBbbN{}||L||
(interleaving\_occurence(T;L1;L2;L;f1;f2)
\mwedge{}  ((\mforall{}i:\mBbbN{}||L1||.  (P  (f1  i)))  \mwedge{}  (\mforall{}i:\mBbbN{}||L2||.  (\mneg{}(P  (f2  i)))))
\mwedge{}  (\mforall{}i:\mBbbN{}||L||
(((P  i)  {}\mRightarrow{}  (\mexists{}j:\mBbbN{}||L1||.  ((f1  j)  =  i)))
\mwedge{}  \mexists{}j:\mBbbN{}||L2||.  ((f2  j)  =  i)  supposing  \mneg{}(P  i))))))

Date html generated: 2019_10_15-AM-10_57_17
Last ObjectModification: 2018_09_27-AM-09_57_58

Theory : list!

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