### Nuprl Lemma : apply-alist_wf

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[V:Type]. ∀[L:(T × V) List].  (apply-alist(eq;L;x) ∈ V?)`

Proof

Definitions occuring in Statement :  apply-alist: `apply-alist(eq;L;x)` list: `T List` deq: `EqDecider(T)` uall: `∀[x:A]. B[x]` unit: `Unit` member: `t ∈ T` product: `x:A × B[x]` union: `left + right` 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 ` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` subtype_rel: `A ⊆r B` or: `P ∨ Q` apply-alist: `apply-alist(eq;L;x)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` squash: `↓T` sq_stable: `SqStable(P)` uiff: `uiff(P;Q)` and: `P ∧ Q` le: `A ≤ B` not: `¬A` less_than': `less_than'(a;b)` true: `True` decidable: `Dec(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` deq: `EqDecider(T)` pi1: `fst(t)` exposed-bfalse: `exposed-bfalse` bool: `𝔹` unit: `Unit` btrue: `tt` ifthenelse: `if b then t else f fi ` eqof: `eqof(d)` pi2: `snd(t)` bfalse: `ff` exists: `∃x:A. B[x]` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf equal-wf-T-base nat_wf colength_wf_list list-cases list_ind_nil_lemma it_wf product_subtype_list spread_cons_lemma sq_stable__le le_antisymmetry_iff add_functionality_wrt_le add-associates add-zero zero-add le-add-cancel decidable__le false_wf not-le-2 condition-implies-le minus-add minus-one-mul minus-one-mul-top add-commutes le_wf equal_wf subtract_wf not-ge-2 less-iff-le minus-minus add-swap subtype_base_sq set_subtype_base int_subtype_base list_ind_cons_lemma bool_wf eqtt_to_assert safe-assert-deq unit_wf2 eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot list_ind_wf 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 independent_functionElimination voidElimination lambdaEquality dependent_functionElimination axiomEquality equalityTransitivity equalitySymmetry productEquality cumulativity applyEquality because_Cache unionElimination isect_memberEquality voidEquality inrEquality promote_hyp hypothesis_subsumption productElimination applyLambdaEquality imageMemberEquality baseClosed imageElimination addEquality dependent_set_memberEquality independent_pairFormation minusEquality intEquality instantiate equalityElimination inlEquality dependent_pairFormation unionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[x:T].  \mforall{}[V:Type].  \mforall{}[L:(T  \mtimes{}  V)  List].    (apply-alist(eq;L;x)  \mmember{}  V?)

Date html generated: 2017_04_14-AM-08_46_38
Last ObjectModification: 2017_02_27-PM-03_33_47

Theory : list_0

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