### Nuprl Lemma : assert-exists_sublist

`∀[T:Type]. ∀L:T List. ∀P:(T List) ⟶ 𝔹.  (↑exists_sublist(L;P) `⇐⇒` ∃LL:T List. (LL ⊆ L ∧ (↑(P LL))))`

Proof

Definitions occuring in Statement :  exists_sublist: `exists_sublist(L;P)` sublist: `L1 ⊆ L2` list: `T List` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` prop: `ℙ` and: `P ∧ Q` so_apply: `x[s]` implies: `P `` Q` exists_sublist: `exists_sublist(L;P)` ifthenelse: `if b then t else f fi ` btrue: `tt` top: `Top` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` bfalse: `ff` iff: `P `⇐⇒` Q` exists: `∃x:A. B[x]` cand: `A c∧ B` rev_implies: `P `` Q` uimplies: `b supposing a` guard: `{T}` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` or: `P ∨ Q` cons: `[a / b]` sq_type: `SQType(T)` assert: `↑b` true: `True`
Lemmas referenced :  list_induction all_wf list_wf bool_wf iff_wf assert_wf exists_sublist_wf exists_wf sublist_wf null_nil_lemma null_cons_lemma spread_cons_lemma nil_wf nil-sublist assert_witness sublist_nil assert_functionality_wrt_uiff or_wf cons_wf assert_of_bor bor_wf sublist_tl2 cons_sublist_cons list-cases product_subtype_list and_wf equal_wf assert_elim subtype_base_sq bool_subtype_base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality functionEquality cumulativity hypothesis functionExtensionality applyEquality productEquality independent_functionElimination rename dependent_functionElimination isect_memberEquality voidElimination voidEquality because_Cache universeEquality independent_pairFormation dependent_pairFormation productElimination independent_isectElimination equalitySymmetry unionElimination addLevel allFunctionality impliesFunctionality orFunctionality orLevelFunctionality inlFormation promote_hyp hypothesis_subsumption inrFormation dependent_set_memberEquality applyLambdaEquality setElimination equalityTransitivity levelHypothesis instantiate natural_numberEquality

Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  \mforall{}P:(T  List)  {}\mrightarrow{}  \mBbbB{}.    (\muparrow{}exists\_sublist(L;P)  \mLeftarrow{}{}\mRightarrow{}  \mexists{}LL:T  List.  (LL  \msubseteq{}  L  \mwedge{}  (\muparrow{}(P  LL))))

Date html generated: 2018_05_21-PM-00_34_15
Last ObjectModification: 2017_10_12-AM-10_13_20

Theory : list_1

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