### Nuprl Lemma : list-index_wf

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[L:T List].  (list-index(eq;L;x) ∈ ℕ||L|| + Top)`

Proof

Definitions occuring in Statement :  list-index: `list-index(d;L;x)` length: `||as||` list: `T List` deq: `EqDecider(T)` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` top: `Top` member: `t ∈ T` union: `left + right` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` 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]` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` uiff: `uiff(P;Q)` int_seg: `{i..j-}` lelt: `i ≤ j < k` subtract: `n - m` bool: `𝔹` unit: `Unit` btrue: `tt` ifthenelse: `if b then t else f fi ` le: `A ≤ B` nat_plus: `ℕ+` true: `True` bfalse: `ff` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma 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 less_than_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases length_of_nil_lemma list_ind_nil_lemma int_seg_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int length_of_cons_lemma list_ind_cons_lemma add-member-int_seg2 decidable__lt length_wf eqof_wf bool_wf eqtt_to_assert safe-assert-deq false_wf add_nat_plus length_wf_nat nat_plus_wf nat_plus_properties add-is-int-iff lelt_wf top_wf eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot subtype_rel_union int_seg_subtype list_wf deq_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry cumulativity applyEquality because_Cache unionElimination inrEquality promote_hyp hypothesis_subsumption productElimination applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination inlEquality equalityElimination imageMemberEquality pointwiseFunctionality baseApply closedConclusion universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[x:T].  \mforall{}[L:T  List].    (list-index(eq;L;x)  \mmember{}  \mBbbN{}||L||  +  Top)

Date html generated: 2017_04_17-AM-09_15_02
Last ObjectModification: 2017_02_27-PM-05_20_37

Theory : decidable!equality

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