
diff -r 1ddfe28117e9 -r bfb5a70e4319 src/Doc/Corec/Corec.thy
--- a/src/Doc/Corec/Corec.thy	Tue Mar 29 10:57:02 2016 +0200
+++ b/src/Doc/Corec/Corec.thy	Tue Mar 29 17:42:43 2016 +0200
@@ -10,8 +10,10 @@
 
 theory Corec
 imports
+  GCD
   "../Datatypes/Setup"
   "~~/src/HOL/Library/BNF_Corec"
+  "~~/src/HOL/Library/FSet"
 begin
 
 section \<open>Introduction
@@ -20,72 +22,393 @@
 text \<open>
 ...
 \cite{isabelle-datatypes}
+
+* friend
+* up to
+
+BNF
+
+link to papers
 \<close>
 
 
 section \<open>Introductory Examples
   \label{sec:introductory-examples}\<close>
 
+text \<open>
+\<close>
+
 
 subsection \<open>Simple Corecursion
   \label{ssec:simple-corecursion}\<close>
 
+text \<open>
+The case studies by Rutten~\cite{rutten05} and Hinze~\cite{hinze10} on stream
+calculi serve as our starting point. Streams can be defined as follows
+(cf. @{file "~~/src/HOL/Library/Stream.thy"}):
+\<close>
+
+    codatatype (sset: 'a) stream =
+      SCons (shd: 'a) (stl: "'a stream")
+    for
+      map: smap
+
+text \<open>
+The @{command corec} command makes it possible to define functions where the
+corecursive call occurs under two or more constructors:
+\<close>
+
+    corec oneTwos :: "nat stream" where
+      "oneTwos = SCons 1 (SCons 2 oneTwos)"
+
+text \<open>
+\noindent
+This is already beyond the syntactic fragment supported by \keyw{primcorec}.
+
+The following definition of pointwise sum can be performed with either
+\keyw{primcorec} or @{command corec}:
+\<close>
+
+    primcorec ssum :: "('a :: plus) stream \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where
+      "ssum xs ys = SCons (shd xs + shd ys) (ssum (stl xs) (stl ys))"
+
+text \<open>
+\noindent
+Pointwise sum meets the friendliness criterion. We register it as a friend using
+the @{command friend_of_corec} command. The command requires us to give a
+specification of @{const ssum} where a constructor (@{const SCons}) occurs at
+the outermost position on the right-hand side. Here, we can simply reuse the
+\keyw{primcorec} specification above:
+\<close>
+
+    friend_of_corec ssum :: "('a :: plus) stream \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where
+      "ssum xs ys = SCons (shd xs + shd ys) (ssum (stl xs) (stl ys))"
+       apply (rule ssum.code)
+      by transfer_prover
+
+text \<open>
+\noindent
+The command emits two proof goals. The first one corresponds to the equation we
+specified and is trivial to discharge. The second one is a parametricity goal
+and can usually be discharged using the @{text transfer_prover} proof method.
+
+After registering @{const ssum} as a friend, we can use it in the corecursive
+call context, either inside or outside the constructor guard:
+\<close>
+
+    corec fibA :: "nat stream" where
+      "fibA = SCons 0 (ssum (SCons 1 fibA) fibA)"
+
+text \<open>\blankline\<close>
+
+    corec fibB :: "nat stream" where
+      "fibB = ssum (SCons 0 (SCons 1 fibB)) (SCons 0 fibB)"
+
+text \<open>
+Using the @{text "friend"} option, we can simultaneously define a function and
+register it as a friend:
+\<close>
+
+    corec (friend)
+      sprod :: "('a :: {plus,times}) stream \<Rightarrow> 'a stream \<Rightarrow> 'a stream"
+    where
+      "sprod xs ys =
+       SCons (shd xs * shd ys) (ssum (sprod xs (stl ys)) (sprod (stl xs) ys))"
+
+text \<open>\blankline\<close>
+
+    corec (friend) sexp :: "nat stream \<Rightarrow> nat stream" where
+      "sexp xs = SCons (2 ^^ shd xs) (sprod (stl xs) (sexp xs))"
+
+text \<open>
+\noindent
+The parametricity proof goal is given to @{text transfer_prover}.
+
+The @{const sprod} and @{const sexp} functions provide shuffle product and
+exponentiation on streams. We can use them to define the stream of factorial
+numbers in two different ways:
+\<close>
+
+    corec factA :: "nat stream" where
+      "factA = (let zs = SCons 1 factA in sprod zs zs)"
+
+    corec factB :: "nat stream" where
+      "factB = sexp (SCons 0 factB)"
+
+text \<open>
+The arguments of friendly operations can be of complex types involving the
+target codatatype. The following example defines the supremum of a finite set of
+streams by primitive corecursion and registers it as friendly:
+\<close>
+
+    corec (friend) sfsup :: "nat stream fset \<Rightarrow> nat stream" where
+      "sfsup X = SCons (Sup (fset (fimage shd X))) (sfsup (fimage stl X))"
+
+text \<open>
+\noindent
+In general, the arguments may be any BNF, with the restriction that the target
+codatatype (@{typ "nat stream"}) may occur only in a live position of the BNF.
+For this reason, the following operation, on unbounded sets, cannot be
+registered as a friend:
+\<close>
+
+    corec ssup :: "nat stream set \<Rightarrow> nat stream" where
+      "ssup X = SCons (Sup (image shd X)) (ssup (image stl X))"
+
 
 subsection \<open>Nested Corecursion
   \label{ssec:nested-corecursion}\<close>
 
+text \<open>
+The package generally supports arbitrary codatatypes with multiple constructors
+and nesting through other type constructors (BNFs). Consider the following type
+of finitely branching Rose trees of potentially infinite depth:
+\<close>
 
-subsection \<open>Polymorphism
-  \label{ssec:polymorphism}\<close>
+    codatatype 'a tree =
+      Node (lab: 'a) (sub: "'a tree list")
+
+text \<open>
+We first define the pointwise sum of two trees analogously to @{const ssum}:
+\<close>
+
+    corec (friend) tplus :: "('a :: plus) tree \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
+      "tplus t u =
+       Node (lab t + lab u) (map (\<lambda>(t', u'). tplus t' u') (zip (sub t) (sub u)))"
+
+text \<open>
+\noindent
+Here, @{const map} is the standard map function on lists, and @{const zip}
+converts two parallel lists into a list of pairs. The @{const tplus} function is
+primitively corecursive. Instead of @{text "corec (friend)"}, we could also have
+used \keyw{primcorec} and @{command friend_of_corec}, as we did for
+@{const ssum}.
+
+Once @{const tplus} is registered as friendly, we can use it in the corecursive
+call context:
+\<close>
+
+    corec (friend) ttimes :: "('a :: {plus,times}) tree \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
+      "ttimes t u = Node (lab t * lab u)
+         (map (\<lambda>(t', u'). tplus (ttimes t u') (ttimes t' u)) (zip (sub t) (sub u)))"
+
+
+subsection \<open>Mixed Recursion--Corecursion
+  \label{ssec:mixed-recursion-corecursion}\<close>
+
+text \<open>
+It is often convenient to let a corecursive function perform some finite
+computation before producing a constructor. With mixed recursion--corecursion, a
+finite number of unguarded recursive calls perform this calculation before
+reaching a guarded corecursive call.
+
+Intuitively, the unguarded recursive call can be unfolded to arbitrary finite
+depth, ultimately yielding a purely corecursive definition. An example is the
+@{term primes} function from Di Gianantonio and Miculan
+\cite{di-gianantonio-miculan-2003}:
+\<close>
+
+    corecursive primes :: "nat \<Rightarrow> nat \<Rightarrow> nat stream" where
+      "primes m n =
+       (if (m = 0 \<and> n > 1) \<or> coprime m n then
+          SCons n (primes (m * n) (n + 1))
+        else
+          primes m (n + 1))"
+      apply (relation "measure (\<lambda>(m, n).
+        if n = 0 then 1 else if coprime m n then 0 else m - n mod m)")
+       apply (auto simp: mod_Suc intro: Suc_lessI)
+       apply (metis One_nat_def coprime_Suc_nat gcd.commute gcd_red_nat)
+      by (metis diff_less_mono2 lessI mod_less_divisor)
+
+text \<open>
+\noindent
+The @{command corecursive} command is a variant of @{command corec} that allows
+us to specify a termination argument for any unguarded self-call.
+
+When called with @{term "m = 1"} and @{term "n = 2"}, the @{const primes}
+function computes the stream of prime numbers. The unguarded call in the
+@{text else} branch increments @{term n} until it is coprime to the first
+argument @{term m} (i.e., the greatest common divisor of @{term m} and
+@{term n} is @{term 1}).
+
+For any positive integers @{term m} and @{term n}, the numbers @{term m} and
+@{term "m * n + 1"} are coprime, yielding an upper bound on the number of times
+@{term n} is increased. Hence, the function will take the @{text else} branch at
+most finitely often before taking the then branch and producing one constructor.
+There is a slight complication when @{term "m = 0 \<and> n > 1"}: Without the first
+disjunct in the @{text "if"} condition, the function could stall. (This corner
+case was overlooked in the original example \cite{di-gianantonio-miculan-2003}.)
+
+In the following example, which defines the stream of Catalan numbers,
+termination is discharged automatically using @{text lexicographic_order}:
+\<close>
+
+    corec catalan :: "nat \<Rightarrow> nat stream" where
+      "catalan n =
+       (if n > 0 then ssum (catalan (n - 1)) (SCons 0 (catalan (n + 1)))
+        else SCons 1 (catalan 1))"
+
+text \<open>
+A more elaborate case study, revolving around the filter function on lazy lists,
+is presented in @{file "~~/src/HOL/Corec_Examples/LFilter.thy"}.
+\<close>
+
+
+subsection \<open>Self-Friendship
+  \label{ssec:self-friendship}\<close>
+
+text \<open>
+Paradoxically, the package allows us to simultaneously define a function and use
+it as its own friend, as in the following definition of a ``skewed product'':
+\<close>
+
+    corec (friend)
+      sskew :: "('a :: {plus,times}) stream \<Rightarrow> 'a stream \<Rightarrow> 'a stream"
+    where
+      "sskew xs ys =
+       SCons (shd xs * shd ys) (sskew (sskew xs (stl ys)) (sskew (stl xs) ys))"
+
+text \<open>
+\noindent
+Such definitions, with nested self-calls on the right-hand side, cannot be
+separated into a @{command corec} part and a @{command friend_of_corec} part.
+\<close>
 
 
 subsection \<open>Coinduction
   \label{ssec:coinduction}\<close>
 
+text \<open>
+Once a corecursive specification has been accepted, we normally want to reason
+about it. The @{text codatatype} command generates a structural coinduction
+principle that matches primitively corecursive functions. For nonprimitive
+specifications, our package provides the more advanced proof principle of
+\emph{coinduction up to congruence}---or simply \emph{coinduction up-to}.
+
+The structural coinduction principle for @{typ "'a stream"}, called
+@{thm [source] stream.coinduct}, is as follows:
+%
+\[@{thm stream.coinduct[no_vars]}\]
+%
+Coinduction allows us to prove an equality @{text "l = r"} on streams by
+providing a relation @{text R} that relates @{text l} and @{text r} (first
+premise) and that constitutes a bisimulation (second premise). Streams that are
+related by a bisimulation cannot be distinguished by taking observations (via
+the selectors @{const shd} and @{const stl}); hence they must be equal.
+
+The coinduction up-to principle after registering @{const sskew} as friendly is
+available as @{thm [source] sskew.coinduct} or
+@{thm [source] stream.coinduct_upto(2)}:
+%
+\[@{thm sfsup.coinduct[no_vars]}\]
+%
+This rule is almost identical to structural coinduction, except that the
+corecursive application of @{term R} is replaced by
+@{term "stream.v5.congclp R"}. The @{const stream.v5.congclp} predicate is
+equipped with the following introduction rules:
+\<close>
+
+(* FIXME:
+thm
+  sskew.cong_base
+  sskew.cong_refl
+  sskew.cong_intros
+
+thm stream.coinduct_upto(2)
+  sskew.coinduct
+
+thm stream.cong_intros
+thm sfsup.cong_intros
+*)
+
 
 subsection \<open>Uniqueness Reasoning
   \label{ssec:uniqueness-reasoning}\<close>
 
+text \<open>
+...
+\<close>
+
 
 section \<open>Command Syntax
   \label{sec:command-syntax}\<close>
 
+text \<open>
+...
+\<close>
 
-subsection \<open>corec and corecursive
+
+subsection \<open>\keyw{corec} and \keyw{corecursive}
   \label{ssec:corec-and-corecursive}\<close>
 
+text \<open>
+...
+\<close>
+
 
-subsection \<open>friend_of_corec
+subsection \<open>\keyw{friend_of_corec}
   \label{ssec:friend-of-corec}\<close>
 
+text \<open>
+...
+\<close>
 
-subsection \<open>coinduction_upto
+
+subsection \<open>\keyw{coinduction_upto}
   \label{ssec:coinduction-upto}\<close>
 
+text \<open>
+...
+\<close>
+
 
 section \<open>Generated Theorems
   \label{sec:generated-theorems}\<close>
 
+text \<open>
+...
+\<close>
 
-subsection \<open>corec and corecursive
+
+subsection \<open>\keyw{corec} and \keyw{corecursive}
   \label{ssec:corec-and-corecursive}\<close>
 
+text \<open>
+...
+\<close>
+
 
-subsection \<open>friend_of_corec
+subsection \<open>\keyw{friend_of_corec}
   \label{ssec:friend-of-corec}\<close>
 
+text \<open>
+...
+\<close>
 
-subsection \<open>coinduction_upto
+
+subsection \<open>\keyw{coinduction_upto}
   \label{ssec:coinduction-upto}\<close>
 
+text \<open>
+...
+\<close>
+
 
 section \<open>Proof Method
   \label{sec:proof-method}\<close>
 
+text \<open>
+...
+\<close>
 
-subsection \<open>corec_unique
+
+subsection \<open>\textit{corec_unique}
   \label{ssec:corec-unique}\<close>
 
+text \<open>
+...
+\<close>
+
 
 section \<open>Known Bugs and Limitations
   \label{sec:known-bugs-and-limitations}\<close>
@@ -104,6 +427,7 @@
   * limitation on type of friend
   * unfinished tactics
   * polymorphism of corecUU_transfer
+  * alternative views
 
 \end{enumerate}
 \<close>




