Nuprl Lemma : l-ind_wf

[A,B:Type]. ∀[L:A List]. ∀[x:B]. ∀[F:A ⟶ (A List) ⟶ B ⟶ B].  (l-ind(L;x;h,t,r.F[h;t;r]) ∈ B)


Proof




Definitions occuring in Statement :  l-ind: l-ind(L;nilcase;h,t,r.F[h; t; r]) list: List uall: [x:A]. B[x] so_apply: x[s1;s2;s3] member: t ∈ T function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  guard: {T} uimplies: supposing a prop: subtype_rel: A ⊆B or: P ∨ Q 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: λ2y.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: ⇐⇒ Q rev_implies:  Q subtract: m nil: [] it: so_lambda: λ2x.t[x] so_apply: x[s] sq_type: SQType(T) less_than: a < 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 l_ind_nil_lemma 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 l_ind_cons_lemma list_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 cumulativity applyEquality because_Cache unionElimination isect_memberEquality voidEquality promote_hyp hypothesis_subsumption productElimination applyLambdaEquality imageMemberEquality baseClosed imageElimination addEquality dependent_set_memberEquality independent_pairFormation minusEquality intEquality instantiate functionExtensionality functionEquality universeEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[L:A  List].  \mforall{}[x:B].  \mforall{}[F:A  {}\mrightarrow{}  (A  List)  {}\mrightarrow{}  B  {}\mrightarrow{}  B].    (l-ind(L;x;h,t,r.F[h;t;r])  \mmember{}  B)



Date html generated: 2017_04_14-AM-08_34_06
Last ObjectModification: 2017_02_27-PM-03_21_38

Theory : list_0


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