### Nuprl Lemma : member_interleaving

`∀[T:Type]. ∀L,L1,L2:T List.  (interleaving(T;L1;L2;L) `` {∀x:T. ((x ∈ L) `⇐⇒` (x ∈ L1) ∨ (x ∈ L2))})`

Proof

Definitions occuring in Statement :  interleaving: `interleaving(T;L1;L2;L)` l_member: `(x ∈ l)` list: `T List` uall: `∀[x:A]. B[x]` guard: `{T}` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` or: `P ∨ Q` universe: `Type`
Definitions unfolded in proof :  guard: `{T}` interleaving: `interleaving(T;L1;L2;L)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` iff: `P `⇐⇒` Q` member: `t ∈ T` prop: `ℙ` rev_implies: `P `` Q` or: `P ∨ Q` subtype_rel: `A ⊆r B` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` exists: `∃x:A. B[x]` l_member: `(x ∈ l)` cand: `A c∧ B` finite': `finite'(T)` ge: `i ≥ j ` decidable: `Dec(P)` false: `False` uiff: `uiff(P;Q)` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` sq_type: `SQType(T)` squash: `↓T` true: `True` surject: `Surj(A;B;f)` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than': `less_than'(a;b)` less_than: `a < b`
Lemmas referenced :  l_member_wf istype-universe nat_wf length_wf_nat set_subtype_base le_wf istype-int int_subtype_base length_wf disjoint_sublists_wf list_wf disjoint_sublists_witness nsub_finite' subtype_base_sq nat_properties decidable__equal_int add-is-int-iff full-omega-unsat intformand_wf intformnot_wf intformeq_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_wf false_wf subtype_rel_self int_seg_wf inject_wf squash_wf true_wf iff_weakening_equal decidable__le intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma less_than_wf int_seg_properties decidable__lt intformless_wf int_formula_prop_less_lemma lelt_wf or_wf int_seg_subtype_nat istype-false select_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma disjoint_sublists_sublist member_sublist
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation_alt lambdaFormation_alt sqequalHypSubstitution productElimination thin independent_pairFormation universeIsType cut introduction extract_by_obid isectElimination hypothesisEquality hypothesis unionIsType productIsType equalityIsType4 applyEquality intEquality lambdaEquality_alt natural_numberEquality independent_isectElimination addEquality because_Cache inhabitedIsType universeEquality dependent_functionElimination independent_functionElimination equalityTransitivity equalitySymmetry instantiate cumulativity setElimination rename applyLambdaEquality unionElimination pointwiseFunctionality promote_hyp baseApply closedConclusion baseClosed approximateComputation dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination functionEquality imageElimination functionIsType imageMemberEquality dependent_set_memberEquality_alt hyp_replacement inlFormation_alt equalityIsType1 productEquality inrFormation_alt

Latex:
\mforall{}[T:Type].  \mforall{}L,L1,L2:T  List.    (interleaving(T;L1;L2;L)  {}\mRightarrow{}  \{\mforall{}x:T.  ((x  \mmember{}  L)  \mLeftarrow{}{}\mRightarrow{}  (x  \mmember{}  L1)  \mvee{}  (x  \mmember{}  L2))\})

Date html generated: 2019_10_15-AM-10_55_21
Last ObjectModification: 2018_10_09-AM-10_18_20

Theory : list!

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