### Nuprl Lemma : first_index-positive

`∀[T:Type]. ∀P:T ⟶ 𝔹. ∀L:T List.  (0 < index-of-first x in L.P[x] `⇐⇒` (∃x∈L. ↑P[x]))`

Proof

Definitions occuring in Statement :  first_index: `index-of-first x in L.P[x]` l_exists: `(∃x∈L. P[x])` list: `T List` assert: `↑b` bool: `𝔹` less_than: `a < b` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` first_index: `index-of-first x in L.P[x]` member: `t ∈ T` 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` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` less_than: `a < b` squash: `↓T` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` l_exists: `(∃x∈L. P[x])`
Lemmas referenced :  bool_wf list_wf l_member_wf l_exists_wf assert_wf exists_wf iff_wf search_wf less_than_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 length_wf int_seg_properties select_wf length_wf_nat search_property
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut lemma_by_obid sqequalHypSubstitution dependent_functionElimination thin isectElimination hypothesisEquality hypothesis lambdaEquality applyEquality cumulativity setElimination rename independent_isectElimination natural_numberEquality productElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll because_Cache imageElimination addLevel impliesFunctionality independent_functionElimination setEquality functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}P:T  {}\mrightarrow{}  \mBbbB{}.  \mforall{}L:T  List.    (0  <  index-of-first  x  in  L.P[x]  \mLeftarrow{}{}\mRightarrow{}  (\mexists{}x\mmember{}L.  \muparrow{}P[x]))

Date html generated: 2016_05_15-PM-03_24_59
Last ObjectModification: 2016_01_16-AM-10_48_21

Theory : general

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