### Nuprl Lemma : select_l_index

`∀[T:Type]. ∀[dT:EqDecider(T)]. ∀[L:T List]. ∀[x:T].  L[index(L;x)] = x ∈ T supposing (x ∈ L)`

Proof

Definitions occuring in Statement :  l_index: `index(L;x)` l_member: `(x ∈ l)` select: `L[n]` list: `T List` deq: `EqDecider(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` prop: `ℙ` l_index: `index(L;x)` deq: `EqDecider(T)` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` all: `∀x:A. B[x]` 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` less_than: `a < b` squash: `↓T` uiff: `uiff(P;Q)` eqof: `eqof(d)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` l_member: `(x ∈ l)` nat: `ℕ` le: `A ≤ B` cand: `A c∧ B` ge: `i ≥ j ` subtype_rel: `A ⊆r B` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  l_member_wf list_wf deq_wf mu-bound-property length_wf_nat select_wf int_seg_properties length_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma int_seg_wf safe-assert-deq assert_wf eqof_wf exists_wf equal_wf lelt_wf nat_properties deq_property l_index_wf non_neg_length squash_wf true_wf iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality sqequalRule isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry universeEquality lambdaEquality applyEquality setElimination rename independent_isectElimination natural_numberEquality productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination addLevel existsFunctionality levelHypothesis existsLevelFunctionality dependent_set_memberEquality applyLambdaEquality independent_functionElimination imageMemberEquality baseClosed

Latex:
\mforall{}[T:Type].  \mforall{}[dT:EqDecider(T)].  \mforall{}[L:T  List].  \mforall{}[x:T].    L[index(L;x)]  =  x  supposing  (x  \mmember{}  L)

Date html generated: 2017_04_17-AM-09_15_59
Last ObjectModification: 2017_02_27-PM-05_21_24

Theory : decidable!equality

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