### Nuprl Lemma : round-robin-list-index

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[L:T List].  (round-robin(L) outl(list-index(eq;L;x))) = x ∈ T supposing (x ∈ L)`

Proof

Definitions occuring in Statement :  round-robin: `round-robin(L)` list-index: `list-index(d;L;x)` l_member: `(x ∈ l)` list: `T List` deq: `EqDecider(T)` outl: `outl(x)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` round-robin: `round-robin(L)` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` nat_plus: `ℕ+` or: `P ∨ Q` false: `False` cons: `[a / b]` top: `Top` guard: `{T}` nat: `ℕ` le: `A ≤ B` cand: `A c∧ B` decidable: `Dec(P)` not: `¬A` uiff: `uiff(P;Q)` subtract: `n - m` less_than': `less_than'(a;b)` true: `True` ge: `i ≥ j ` outl: `outl(x)` int_seg: `{i..j-}` isl: `isl(x)` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` prop: `ℙ` sq_type: `SQType(T)`
Lemmas referenced :  list-index-property subtype_base_sq int_subtype_base rem_base_case outl_wf isl-list-index length_wf list-cases length_of_nil_lemma nil_member product_subtype_list length_of_cons_lemma istype-void length_wf_nat decidable__lt istype-false not-lt-2 condition-implies-le minus-add minus-one-mul zero-add minus-one-mul-top add-commutes add_functionality_wrt_le add-associates add-zero le-add-cancel istype-less_than non_neg_length assert_elim btrue_wf bfalse_wf btrue_neq_bfalse nat_properties int_seg_properties full-omega-unsat intformand_wf intformless_wf itermVar_wf intformnot_wf istype-int int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf l_member_wf list_wf deq_wf istype-universe
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination sqequalRule instantiate cumulativity intEquality because_Cache dependent_functionElimination productElimination independent_functionElimination Error :dependent_set_memberEquality_alt,  unionElimination voidElimination promote_hyp hypothesis_subsumption Error :isect_memberEquality_alt,  Error :inhabitedIsType,  Error :lambdaFormation_alt,  setElimination rename independent_pairFormation natural_numberEquality addEquality minusEquality Error :equalityIstype,  equalityTransitivity equalitySymmetry Error :productIsType,  applyLambdaEquality approximateComputation Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality Error :universeIsType,  universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[x:T].  \mforall{}[L:T  List].
(round-robin(L)  outl(list-index(eq;L;x)))  =  x  supposing  (x  \mmember{}  L)

Date html generated: 2019_06_20-PM-01_56_52
Last ObjectModification: 2019_03_06-AM-10_52_27

Theory : decidable!equality

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