theory Ind_General_Scheme 
imports Simple_Inductive_Package Ind_Prelims
begin

(*<*)
datatype trm =
  Var "string"
| App "trm" "trm"
| Lam "string" "trm"

simple_inductive 
  fresh :: "string \<Rightarrow> trm \<Rightarrow> bool" 
where
  fresh_var: "a\<noteq>b \<Longrightarrow> fresh a (Var b)"
| fresh_app: "\<lbrakk>fresh a t; fresh a s\<rbrakk> \<Longrightarrow> fresh a (App t s)"
| fresh_lam1: "fresh a (Lam a t)"
| fresh_lam2: "\<lbrakk>a\<noteq>b; fresh a t\<rbrakk> \<Longrightarrow> fresh a (Lam b t)"
(*>*)

section {* The Code in a Nutshell *}

text {*
  (FIXME: perhaps move somewhere else)

  The point of these examples is to get a feeling what the automatic proofs 
  should do in order to solve all inductive definitions we throw at them. For this 
  it is instructive to look at the general construction principle 
  of inductive definitions, which we shall do in the next section.
*}

text {*
  The inductive package will generate the reasoning infrastructure
  for mutually recursive predicates @{text "pred\<^isub>1\<dots>pred\<^isub>n"}. In what
  follows we will have the convention that various, possibly empty collections of 
  ``things'' are indicated either by adding an @{text "s"} or by adding 
  a superscript @{text [quotes] "\<^isup>*"}. The shorthand for the predicates 
  will therefore be @{text "preds"} or @{text "pred\<^sup>*"}. In this case of the
  predicates there must be at least a single one in order to obtain a meaningful
  definition.

  The input for the inductive predicate will be some @{text "preds"} with possible 
  typing and syntax annotations, and also some introduction rules. We call below the 
  introduction rules short as @{text "rules"}. Borrowing some idealised Isabelle 
  notation, one such @{text "rule"} is of the form

  \begin{isabelle}
  @{text "rule ::= 
  \<And>xs. \<^raw:$\underbrace{\mbox{>As\<^raw:}}_{\text{\makebox[0mm]{\rm non-recursive premises}}}$> \<Longrightarrow> 
  \<^raw:$\underbrace{\mbox{>(\<And>ys. Bs \<Longrightarrow> pred ss)\<^isup>*\<^raw:}}_{\text{\rm recursive premises}}$> 
  \<Longrightarrow> pred ts"}
  \end{isabelle}
  
  We actually assume the user will always state rules of this form and we omit
  any code that test this format. So things can go horribly wrong if the
  @{text "rules"} are not of this form.\footnote{FIXME: Exercise to test this
  format.} The @{text As} and @{text Bs} in a @{text "rule"} are formulae not
  involving the inductive predicates @{text "preds"}; the instances @{text
  "pred ss"} and @{text "pred ts"} can stand for different predicates.
  Everything left of @{text [quotes] "\<Longrightarrow> pred ts"} are called the premises of
  the rule. The variables @{text "xs"} are usually omitted in the user's
  input. The variables @{text "ys"} are local with respect to on recursive
  premise. Some examples of @{text "rule"}s the user might write are:


  @{thm [display] fresh_var[no_vars]}

  which has only a single non-recursive premise, whereas

  @{thm [display] evenS[no_vars]}

  has a single recursive premise; the rule

  @{thm [display] accpartI[no_vars]}

  has a recursive premise which has a precondition. In the examples, all 
  rules are stated without the leading meta-quantification @{text "\<And>xs"}.

  The code of the inductive package falls roughly in tree parts involving
  the definitions, the induction principles and the introduction rules, 
  respectively. For the definitions we need to have the @{text rules}
  in a form where the meta-quantifiers and meta-implications are replaced
  by their object logic equivalent. Therefore an @{text "orule"} is of the
  form 

  @{text [display] "orule ::= \<forall>xs. As \<longrightarrow> (\<forall>ys. Bs \<longrightarrow> pred ss)\<^isup>* \<longrightarrow> pred ts"}

  A definition for the predicate @{text "pred"} has then the form

  @{text [display] "def ::= pred \<equiv> \<lambda>zs. \<forall>preds. orules \<longrightarrow> pred zs"}

  The induction principles for the predicate @{text "pred"} are of the
  form

  @{text [display] "ind ::= pred ?zs \<Longrightarrow> rules[preds := ?Ps] \<Longrightarrow> ?P ?zs"}

  where in the @{text "rules"} every @{text pred} is replaced by a new
  (meta)variable @{text "?P"}.

  In order to derive an induction principle for the predicate @{text "pred"}
  we first transform it into the object logic and fix the meta-variables. Hence 
  we have to prove a formula of the form

  @{text [display] "pred zs \<longrightarrow> orules[preds := Ps] \<longrightarrow> P zs"}

  If we assume @{text "pred zs"} and unfold its definition, then we have
  
  @{text [display] "\<forall>preds. orules \<longrightarrow> pred zs"} 

  and must prove

  @{text [display] "orules[preds := Ps] \<longrightarrow> P zs"}

  This can be done by instantiating the @{text "\<forall>preds"} with the @{text "Ps"}. 
  Then we are done since we are left with a simple identity.
  
  The proofs for the introduction rules are more involved. Assuming we want to
  prove the introduction rule 

  @{text [display] "\<And>xs. As \<Longrightarrow> (\<And>ys. Bs \<Longrightarrow> pred ss)\<^isup>* \<Longrightarrow> pred ts"}

  then we can assume

  \begin{isabelle}
  (i)~~@{text "As"}\\
  (ii)~@{text "(\<And>ys. Bs \<Longrightarrow> pred ss)\<^isup>*"}
  \end{isabelle}

  and must show

  @{text [display] "pred ts"}
  
  If we now unfold the definitions for the @{text preds}, we have

  \begin{isabelle}
  (i)~~~@{text "As"}\\
  (ii)~~@{text "(\<And>ys. Bs \<Longrightarrow> \<forall>preds. orules \<longrightarrow> pred ss)\<^isup>*"}\\
  (iii)~@{text "orules"}
  \end{isabelle}

  and need to show

  @{text [display] "pred ts"}

  In the last step we removed some quantifiers and moved the precondition @{text "orules"}  
  into the assumtion. The @{text "orules"} stand for all introduction rules that are given 
  by the user. We apply the one which corresponds to introduction rule we are proving.
  This introduction rule must be of the form 

  @{text [display] "As \<Longrightarrow> (\<And>ys. Bs \<Longrightarrow> pred ss)\<^isup>* \<Longrightarrow> pred ts"}

  When we apply this rule we end up in the goal state where we have to prove

  \begin{isabelle}
  (a)~@{text "As"}\\
  (b)~@{text "(\<And>ys. Bs \<Longrightarrow> pred ss)\<^isup>*"}
  \end{isabelle}

  The goals @{text "As"} we can immediately discharge with the assumption in @{text "(i)"}.
  The goals in @{text "(b)"} we discharge as follows: we assume the @{text "(iv)"} 
  @{text "Bs"} and prove @{text "(c)"} @{text "pred ss"}. We then resolve the 
  @{text "Bs"}  with the assumptions in @{text "(ii)"}. This gives us

  @{text [display] "(\<forall>preds. orules \<longrightarrow> pred ss)\<^isup>*"}

  Instantiating the universal quantifiers and then resolving with the assumptions 
  in @{text "(iii)"} gives us @{text "pred ss"}, which is the goal we are after.

  
  What remains is to implement the reasoning outlined above. 
  For building testcases, we use some shorthands for the definitions 
  of @{text "even/odd"}, @{term "fresh"} and @{term "accpart"}. 
*}


text_raw{*
\begin{figure}[p]
\begin{minipage}{\textwidth}
\begin{isabelle}*}  
ML{*(* even-odd example *)
val eo_defs = [@{thm even_def}, @{thm odd_def}]

val eo_rules =  
  [@{prop "even 0"},
   @{prop "\<And>n. odd n \<Longrightarrow> even (Suc n)"},
   @{prop "\<And>n. even n \<Longrightarrow> odd (Suc n)"}]

val eo_orules =  
  [@{prop "even 0"},
   @{prop "\<forall>n. odd n \<longrightarrow> even (Suc n)"},
   @{prop "\<forall>n. even n \<longrightarrow> odd (Suc n)"}]

val eo_preds =  [@{term "even::nat\<Rightarrow>bool"}, @{term "odd::nat\<Rightarrow>bool"}] 
val eo_prednames = [@{binding "even"}, @{binding "odd"}]
val eo_syns = [NoSyn, NoSyn] 
val eo_arg_tyss = [[@{typ "nat"}], [@{typ "nat"}]] 
val e_pred = @{term "even::nat\<Rightarrow>bool"}
val e_arg_tys = [@{typ "nat"}] 



(* freshness example *)
val fresh_rules =  
  [@{prop "\<And>a b. a \<noteq> b \<Longrightarrow> fresh a (Var b)"},
   @{prop "\<And>a s t. fresh a t \<Longrightarrow> fresh a s \<Longrightarrow> fresh a (App t s)"},
   @{prop "\<And>a t. fresh a (Lam a t)"},
   @{prop "\<And>a b t. a \<noteq> b \<Longrightarrow> fresh a t \<Longrightarrow> fresh a (Lam b t)"}]

val fresh_orules =  
  [@{prop "\<forall>a b. a \<noteq> b \<longrightarrow> fresh a (Var b)"},
   @{prop "\<forall>a s t. fresh a t \<longrightarrow> fresh a s \<longrightarrow> fresh a (App t s)"},
   @{prop "\<forall>a t. fresh a (Lam a t)"},
   @{prop "\<forall>a b t. a \<noteq> b \<longrightarrow> fresh a t \<longrightarrow> fresh a (Lam b t)"}]

val fresh_pred =  @{term "fresh::string\<Rightarrow>trm\<Rightarrow>bool"} 
val fresh_arg_tys = [@{typ "string"}, @{typ "trm"}]



(* accessible-part example *)
val acc_rules = 
     [@{prop "\<And>R x. (\<And>y. R y x \<Longrightarrow> accpart R y) \<Longrightarrow> accpart R x"}]
val acc_pred = @{term "accpart:: ('a \<Rightarrow>'a\<Rightarrow>bool)\<Rightarrow>'a \<Rightarrow>bool"}*}
text_raw{*
\end{isabelle}
\end{minipage}
\caption{Shorthands for the inductive predicates of @{text "even"}-@{text "odd"}, 
  @{text "fresh"} and @{text "accpart"}. The follow the convention @{text "rules"}, 
  @{text "orules"}, @{text "preds"} and so on as used in Section ???. The purpose 
  of these shorthands is to simplify the construction of testcases.}
\end{figure}
*}



end