### Nuprl Lemma : list-index-property

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

Proof

Definitions occuring in Statement :  list-index: `list-index(d;L;x)` l_member: `(x ∈ l)` select: `L[n]` list: `T List` deq: `EqDecider(T)` outl: `outl(x)` 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` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` nat: `ℕ` false: `False` ge: `i ≥ j ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` prop: `ℙ` or: `P ∨ Q` list-index: `list-index(d;L;x)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` isl: `isl(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` decidable: `Dec(P)` subtype_rel: `A ⊆r B` outl: `outl(x)` btrue: `tt` true: `True` int_seg: `{i..j-}` lelt: `i ≤ j < k` subtract: `n - m` bool: `𝔹` unit: `Unit` eqof: `eqof(d)` deq: `EqDecider(T)` uiff: `uiff(P;Q)` bnot: `¬bb`
Lemmas referenced :  isl-list-index nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf istype-less_than list-cases list_ind_nil_lemma stuck-spread istype-base product_subtype_list colength-cons-not-zero colength_wf_list istype-le subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le le_wf list_ind_cons_lemma list-index_wf istype-nat l_member_wf list_wf deq_wf istype-universe select-cons-tl int_seg_properties decidable__lt add-associates add-swap add-commutes zero-add istype-true select_wf length_wf eqof_wf eqtt_to_assert safe-assert-deq eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality dependent_functionElimination productElimination independent_functionElimination hypothesis lambdaFormation_alt setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation dependent_pairFormation_alt lambdaEquality_alt int_eqEquality isect_memberEquality_alt voidElimination sqequalRule independent_pairFormation universeIsType axiomEquality functionIsTypeImplies inhabitedIsType unionElimination baseClosed promote_hyp hypothesis_subsumption equalityIstype because_Cache dependent_set_memberEquality_alt instantiate equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination baseApply closedConclusion applyEquality intEquality sqequalBase isectIsTypeImplies universeEquality addEquality functionIsType equalityElimination cumulativity

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

Date html generated: 2019_10_15-AM-10_24_24
Last ObjectModification: 2019_08_05-PM-02_04_47

Theory : decidable!equality

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