Thiserrormessageremindsusthattheuniversevariablefor[T]stillexists,eventhoughitisusuallyhidden.Toapply[id]toitself,thatvariablewouldneedtobelessthanitselfinthetypehierarchy.Universeinconsistencyerrormessagesannouncecaseslikethisonewhereatermcouldonlytype-checkbyviolatinganimpliedconstraintoveruniversevariables.Sucherrorsdemonstratethat[Type]is%\textit{%#<i>#predicative#</i>#%}%,wherethiswordhasaCICmeaningcloselyrelatedtoitsusualmathematicalmeaning.Apredicativesystemenforcestheconstraintthat,foranyobjectofquantifiedtype,noneofthosequantifiersmayeverbeinstantiatedwiththeobjectitself.Impredicativityisassociatedwithpopularparadoxesinsettheory,involvinginconsistentconstructionslike%``%#"#the set of all sets that do not contain themselves.#"#%''%SimilarparadoxeswouldresultfromuncontrolledimpredicativityinCoq.*)
%\index{universeinconsistency}%Thiserrormessageremindsusthattheuniversevariablefor[T]stillexists,eventhoughitisusuallyhidden.Toapply[id]toitself,thatvariablewouldneedtobelessthanitselfinthetypehierarchy.Universeinconsistencyerrormessagesannouncecaseslikethisonewhereatermcouldonlytype-checkbyviolatinganimpliedconstraintoveruniversevariables.Sucherrorsdemonstratethat[Type]is%\textit{%#<i>#predicative#</i>#%}%,wherethiswordhasaCICmeaningcloselyrelatedtoitsusualmathematicalmeaning.Apredicativesystemenforcestheconstraintthat,foranyobjectofquantifiedtype,noneofthosequantifiersmayeverbeinstantiatedwiththeobjectitself.%\index{impredicativity}%Impredicativityisassociatedwithpopularparadoxesinsettheory,involvinginconsistentconstructionslike%``%#"#the set of all sets that do not contain themselves#"#%''%(%\index{Russell'sparadox}%Russell'sparadox).SimilarparadoxeswouldresultfromuncontrolledimpredicativityinCoq.*)
(****InductiveDefinitions*)
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@@ -184,12 +185,13 @@ Inductive exp : Set -> Set :=
Strangely,theuniversevariable[Top.8]onlyappearsinoneplace.Istherenorestrictionimposedonwhichtypesarevalidargumentsto[exp]?Infact,thereisarestriction,butitonlyappearsinaglobalsetofuniverseconstraintsthataremaintained%``%#"#off to the side,#"#%''%notappearingexplicitlyintypes.Wecanprintthecurrentdatabase.*)
Strangely,theuniversevariable[Top.8]onlyappearsinoneplace.Istherenorestrictionimposedonwhichtypesarevalidargumentsto[exp]?Infact,thereisarestriction,butitonlyappearsinaglobalsetofuniverseconstraintsthataremaintained%``%#"#off to the side,#"#%''%notappearingexplicitlyintypes.Wecanprintthecurrentdatabase.%\index{Vernacularcommands!PrintUniverses}%*)
Thisaxiomsaysthatitispermissibletosimplifypatternmatchesoverproofsofequalitieslike[e=e].Theaxiomislogicallyequivalenttosomesimplercorollaries.Inthetheoremnames,%``%#"#UIP#"#%''%standsfor%\index{unicityofidentityproofs}``%#"#unicity of identity proofs#"#%''%,where%``%#"#identity#"#%''%isasynonymfor%``%#"#equality.#"#%''%*)