src/Doc/IsarImplementation/Logic.thy
 changeset 52406 1e57c3c4e05c parent 50126 3dec88149176 child 52407 e4662afb3483
     1.1 --- a/src/Doc/IsarImplementation/Logic.thy	Fri Jun 21 16:21:33 2013 -0700
1.2 +++ b/src/Doc/IsarImplementation/Logic.thy	Thu Jun 13 17:40:58 2013 +0200
1.3 @@ -932,6 +932,72 @@
1.4  *}
1.5
1.6
1.7 +subsection {* Sort hypotheses *}
1.8 +
1.9 +text {* Type variables are decorated with sorts, as explained in
1.10 +  \secref{sec:types}.  This constrains type instantiation to certain
1.11 +  ranges of types: variable @{text "\<alpha>\<^sub>s"} may only be assigned to types
1.12 +  @{text "\<tau>"} that belong to sort @{text "s"}.  Within the logic, sort
1.13 +  constraints act like implicit preconditions on the result @{text
1.14 +  "\<lparr>\<alpha>\<^sub>1 : s\<^sub>1\<rparr>, \<dots>, \<lparr>\<alpha>\<^sub>n : s\<^sub>n\<rparr>, \<Gamma> \<turnstile> \<phi>"} where the type variables @{text
1.15 +  "\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n"} cover the propositions @{text "\<Gamma>"}, @{text "\<phi>"}, as
1.16 +  well as the proof of @{text "\<Gamma> \<turnstile> \<phi>"}.
1.17 +
1.18 +  These \emph{sort hypothesis} of a theorem are passed monotonically
1.19 +  through further derivations.  They are redundant, as long as the
1.20 +  statement of a theorem still contains the type variables that are
1.21 +  accounted here.  The logical significance of sort hypotheses is
1.22 +  limited to the boundary case where type variables disappear from the
1.23 +  proposition, e.g.\ @{text "\<lparr>\<alpha>\<^sub>s : s\<rparr> \<turnstile> \<phi>"}.  Since such dangling type
1.24 +  variables can be renamed arbitrarily without changing the
1.25 +  proposition @{text "\<phi>"}, the inference kernel maintains sort
1.26 +  hypotheses in anonymous form @{text "s \<turnstile> \<phi>"}.
1.27 +
1.28 +  In most practical situations, such extra sort hypotheses may be
1.29 +  stripped in a final bookkeeping step, e.g.\ at the end of a proof:
1.30 +  they are typically left over from intermediate reasoning with type
1.31 +  classes that can be satisfied by some concrete type @{text "\<tau>"} of
1.32 +  sort @{text "s"} to replace the hypothetical type variable @{text
1.33 +  "\<alpha>\<^sub>s"}.  *}
1.34 +
1.35 +text %mlref {*
1.36 +  \begin{mldecls}
1.37 +  @{index_ML Thm.extra_shyps: "thm -> sort list"} \\
1.38 +  @{index_ML Thm.strip_shyps: "thm -> thm"} \\
1.39 +  \end{mldecls}
1.40 +
1.41 +  \begin{description}
1.42 +
1.43 +  \item @{ML Thm.extra_shyps}~@{text "thm"} determines the extraneous
1.44 +  sort hypotheses of the given theorem, i.e.\ the sorts that are not
1.45 +  present within type variables of the statement.
1.46 +
1.47 +  \item @{ML Thm.strip_shyps}~@{text "thm"} removes any extraneous
1.48 +  sort hypotheses that can be witnessed from the type signature.
1.49 +
1.50 +  \end{description}
1.51 +*}
1.52 +
1.53 +text %mlex {* The following artificial example demonstrates the
1.54 +  derivation of @{prop False} with a pending sort hypothesis involving
1.55 +  a logically empty sort.  *}
1.56 +
1.57 +class empty =
1.58 +  assumes bad: "\<And>(x::'a) y. x \<noteq> y"
1.59 +
1.60 +theorem (in empty) false: False
1.61 +  using bad by blast
1.62 +
1.63 +ML {*
1.64 +  @{assert} (Thm.extra_shyps @{thm false} = [@{sort empty}])
1.65 +*}
1.66 +
1.67 +text {* Thanks to the inference kernel managing sort hypothesis
1.68 +  according to their logical significance, this example is merely an
1.69 +  instance of \emph{ex falso quodlibet consequitur} --- not a collapse
1.70 +  of the logical framework! *}
1.71 +
1.72 +
1.73  section {* Object-level rules \label{sec:obj-rules} *}
1.74
1.75  text {*