### Nuprl Lemma : decidable__l_exists-proof

`∀[A:Type]. ∀[F:A ⟶ ℙ].  ∀L:A List. ((∀k:A. Dec(F[k])) `` Dec((∃k∈L. F[k])))`

Proof

Definitions occuring in Statement :  l_exists: `(∃x∈L. P[x])` list: `T List` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  l_exists: `(∃x∈L. P[x])` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_seg: `{i..j-}` uimplies: `b supposing a` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` less_than: `a < b` squash: `↓T`
Lemmas referenced :  list_wf decidable_wf all_wf int_seg_wf int_formula_prop_less_lemma intformless_wf decidable__lt int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le int_seg_properties select_wf length_wf decidable__exists_int_seg
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation cut thin instantiate lemma_by_obid sqequalHypSubstitution dependent_functionElimination natural_numberEquality isectElimination hypothesisEquality hypothesis lambdaEquality applyEquality cumulativity setElimination rename independent_isectElimination because_Cache productElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination independent_functionElimination introduction functionEquality universeEquality

Latex:
\mforall{}[A:Type].  \mforall{}[F:A  {}\mrightarrow{}  \mBbbP{}].    \mforall{}L:A  List.  ((\mforall{}k:A.  Dec(F[k]))  {}\mRightarrow{}  Dec((\mexists{}k\mmember{}L.  F[k])))

Date html generated: 2016_05_14-AM-07_47_52
Last ObjectModification: 2016_01_15-AM-08_33_00

Theory : list_1

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