theory Ind_General_Scheme 

imports  "../Base" Simple_Inductive_Package

begin

(*<*)

simple_inductive

  trcl :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"

where

  base: "trcl R x x"

| step: "trcl R x y \<Longrightarrow> R y z \<Longrightarrow> trcl R x z"

simple_inductive

  even and odd

where

  even0: "even 0"

| evenS: "odd n \<Longrightarrow> even (Suc n)"

| oddS: "even n \<Longrightarrow> odd (Suc n)"

simple_inductive

  accpart :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"

where

  accpartI: "(\<And>y. R y x \<Longrightarrow> accpart R y) \<Longrightarrow> accpart R x"

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\label{sec:nutshell} *}

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'' (lists, nested implications and so on) are indicated either by

  adding an @{text [quotes] "s"} or by adding a superscript @{text [quotes]

  "\<^isup>*"}. The shorthand for the predicates will therefore be @{text

  "preds"} or @{text "pred\<^sup>*"}. In the case of the predicates there must

  be, of course, at least a single one in order to obtain a meaningful

  definition.

  The input for the inductive package 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 assumed to be 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}

  For the purposes here, we will assume the @{text rules} have this format and

  omit any code that actually tests this. Therefore ``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"}

  stand for formulae not involving the inductive predicates @{text "preds"};

  the instances @{text "pred ss"} and @{text "pred ts"} can stand for

  different predicates, like @{text "pred\<^isub>1 ss"} and @{text

  "pred\<^isub>2 ts"}, in case mutual recursive predicates are defined; the

  terms @{text ss} and @{text ts} are the arguments of these predicates. Every

  formula left of @{text [quotes] "\<Longrightarrow> pred ts"} is a premise of the rule. The

  outermost quantified variables @{text "xs"} are usually omitted in the

  user's input. The quantification for the variables @{text "ys"} is local

  with respect to one recursive premise and must be given. Some examples of

  @{text "rule"}s 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 single recursive premise that has a precondition. As usual all 

  rules are stated without the leading meta-quantification @{text "\<And>xs"}.

  The output of the inductive package will be definitions for the predicates, 

  induction principles and introduction rules.  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 equivalents. 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 every 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 fresh

  meta-variable @{text "?P"}.

  In order to derive an induction principle for the predicate @{text "pred"},

  we first transform @{text ind} 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 an

  assumption

  @{text [display] "\<forall>preds. orules \<longrightarrow> pred zs"} 

  and must prove the goal

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

  This can be done by instantiating the @{text "\<forall>preds"}-quantification 

  with the @{text "Ps"}. Then we are done since we are left with a simple 

  identity.

  Although the user declares the introduction rules @{text rules}, they must 

  also be derived from the @{text defs}. These derivations are a bit 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 have assumptions of the form

  \begin{isabelle}

  (i)~~@{text "As"}\\

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

  \end{isabelle}

  and must show the goal

  @{text [display] "pred ts"}

  If we now unfold the definitions for the @{text preds}, we have assumptions

  \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 assumption. The @{text "orules"} stand for all introduction rules that are given 

  by the user. We apply the @{text orule} that corresponds to introduction rule we are 

  proving. After transforming the object connectives into meta-connectives, this introduction 

  rule must necessarily 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

  goals of the form

  \begin{isabelle}

  (a)~@{text "As"}\\

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

  \end{isabelle}

  We can immediately discharge the goals @{text "As"} using the assumptions in 

  @{text "(i)"}. The goals in @{text "(b)"} can be discharged as follows: we 

  assume the @{text "Bs"} and prove @{text "pred ss"}. For this we resolve the 

  @{text "Bs"}  with the assumptions in @{text "(ii)"}. This gives us the 

  assumptions

  @{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.

  This completes the proof for introduction rules.

  What remains is to implement in Isabelle the reasoning outlined in this section. 

  We will describe the code in the next section. For building testcases, we use the shorthands for 

  @{text "even/odd"}, @{term "fresh"} and @{term "accpart"}

  defined in Figure~\ref{fig:shorthands}.

*}

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_mxs = [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 @{text "even"}-@{text "odd"}, 

  @{text "fresh"} and @{text "accpart"}. The names of these shorthands follow 

  the convention @{text "rules"}, @{text "orules"}, @{text "preds"} and so on. 

  The purpose of these shorthands is to simplify the construction of testcases

  in Section~\ref{sec:code}.\label{fig:shorthands}}

\end{figure}

*}

end