theory Ind_Code
imports "../Base" "../FirstSteps" Simple_Inductive_Package Ind_Prelims
begin

section {* Code *}

subsection {* Definitions *}

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

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

  @{text [display] "def ::= pred \<equiv> \<lambda>zs. \<forall>preds. orules \<longrightarrow> pred zs"}
  
  @{text [display] "ind ::= \<And>zs. pred zs \<Longrightarrow> rules[preds::=Ps] \<Longrightarrow> P zs"}

  @{text [display] "oind ::= \<forall>zs. pred zs \<longrightarrow> orules[preds::=Ps] \<longrightarrow> P zs"}
  
  So we have @{text "pred zs"} and @{text "orules[preds::=Ps]"}; have to show
  @{text "P zs"}. Expanding @{text "pred zs"} gives @{text "\<forall>preds. orules \<longrightarrow> pred zs"}.
  Instantiating the @{text "preds"} with @{text "Ps"} gives
  @{text "orules[preds::=Ps] \<longrightarrow> P zs"}. So we can conclude with @{text "P zs"}.

  We have to show @{text "\<forall>xs. As \<longrightarrow> (\<forall>ys. Bs \<longrightarrow> pred ss)\<^isup>* \<longrightarrow> pred ts"};
  expanding the defs 
  
  @{text [display]
  "\<forall>xs. As \<longrightarrow> (\<forall>ys. Bs \<longrightarrow> (\<forall>preds. orules \<longrightarrow> pred ss))\<^isup>* \<longrightarrow>  (\<forall>preds. orules \<longrightarrow> pred ts"}
  
  so we have @{text "As"}, @{text "(\<forall>ys. Bs \<longrightarrow> (\<forall>preds. orules \<longrightarrow> pred ss))\<^isup>*"},
  @{text "orules"}; and have to show @{text "pred ts"}

  the @{text "orules"} are of the form @{text "\<forall>xs. As \<longrightarrow> (\<forall>ys. Bs \<longrightarrow> pred ss)\<^isup>* \<longrightarrow> pred ts"}.
  
  using the @{text "As"} we ????
*}


text {*
  First we have to produce for each predicate its definitions of the form

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

  We use the following wrapper function to make the definition via
  @{ML LocalTheory.define}. The function takes a predicate name, a syntax
  annotation and a term representing the right-hand side of the definition.
*}

ML %linenosgray{*fun make_defs ((predname, syn), trm) lthy =
let 
  val arg = ((predname, syn), (Attrib.empty_binding, trm))
  val ((_, (_ , thm)), lthy') = LocalTheory.define Thm.internalK arg lthy
in 
  (thm, lthy') 
end*}

text {*
  It returns the definition (as theorem) and the local theory in which this definition has 
  been made. In Line 4 @{ML internalK in Thm} is just a flag attached to the 
  theorem (others possibilities are @{ML definitionK in Thm} or @{ML axiomK in Thm}). 
  These flags just classify theorems and have no significant meaning, except 
  for tools such as finding theorems in the theorem database. We also
  use @{ML empty_binding in Attrib} in Line 3, since the definition does 
  not need any theorem attributes. Note the definition has not yet been 
  stored in the theorem database. So at the moment we can only refer to it
  via the return value. A testcase for this functions is
*}

local_setup %gray {* fn lthy =>
let
  val arg =  ((Binding.name "MyTrue", NoSyn), @{term True})
  val (def, lthy') = make_defs arg lthy 
in
  warning (str_of_thm lthy' def); lthy'
end *}

text {*
  which prints out the theorem @{prop "MyTrue \<equiv> True"}. Since we are
  testing the function inside \isacommand{local\_setup} we have also
  access to theorem associated with this definition.

  \begin{isabelle}
  \isacommand{thm}~@{text "MyTrue_def"}\\
  @{text "> MyTrue \<equiv> True"}
  \end{isabelle}

  The next function constructs the term for the definition, namely 

  @{text [display] "\<lambda>\<^raw:$zs$>. \<forall>preds. orules \<longrightarrow> pred \<^raw:$zs$>"}

  The variables @{text "\<^raw:$zs$>"} need to be chosen so to not occur
  in the @{text orules} and also be distinct from @{text "pred"}. The function
  constructs the term for one particular predicate @{text "pred"}; the number
  of @{text "\<^raw:$zs$>"} is determined by the nunber of types. 
*}

ML %linenosgray{*fun defs_aux lthy orules preds (pred, arg_tys) =
let 
  fun mk_all x P = HOLogic.all_const (fastype_of x) $ lambda x P

  val fresh_args = 
        arg_tys 
        |> map (pair "z")
        |> Variable.variant_frees lthy (preds @ orules) 
        |> map Free
in
  list_comb (pred, fresh_args)
  |> fold_rev (curry HOLogic.mk_imp) orules
  |> fold_rev mk_all preds
  |> fold_rev lambda fresh_args 
end*}

text {*
  The code in Lines 5 to 9 produce the fresh @{text "\<^raw:$zs$>"} with which the 
  predicate is applied. For this it pairs every type with a string @{text [quotes] "z"} 
  (Line 7); then generates variants for all these strings (names) so that they are 
  unique w.r.t.~to the orules and predicates; in Line 9 it generates the corresponding 
  variable terms for the unique names.

  The unique free variables are applied to the predicate (Line 11); then
  the @{text orules} are prefixed (Line 12); in Line 13 we
  quantify over all predicates; and in line 14 we just abstract over all
  the (fresh)  @{text "\<^raw:$zs$>"}, i.e.~the arguments of the predicate.

  A testcase for this function is
*}

local_setup %gray{* fn lthy =>
let
  val orules = [@{prop "even 0"},
                @{prop "\<forall>n::nat. odd n \<longrightarrow> even (Suc n)"},
                @{prop "\<forall>n::nat. even n \<longrightarrow> odd (Suc n)"}] 
  val preds = [@{term "even::nat\<Rightarrow>bool"}, @{term "odd::nat\<Rightarrow>bool"}]
  val pred = @{term "even::nat\<Rightarrow>bool"}
  val arg_tys = [@{typ "nat"}]
  val def = defs_aux lthy orules preds (pred, arg_tys)
in
  warning (Syntax.string_of_term lthy def); lthy
end *}

text {*
  It constructs the term for the predicate @{term "even"}. So we obtain as printout
  the term

  @{text [display] 
"\<lambda>z. \<forall>even odd. (even 0) \<longrightarrow> (\<forall>n. odd n \<longrightarrow> even (Suc n)) 
                         \<longrightarrow> (\<forall>n. even n \<longrightarrow> odd (Suc n)) \<longrightarrow> even z"}

  The main function for the definitions now has to just iterate
  the function @{ML defs_aux} over all predicates. THis is what the
  next function does. 
*}

ML %linenosgray{*fun definitions rules preds prednames syns arg_typss lthy =
let
  val thy = ProofContext.theory_of lthy
  val orules = map (ObjectLogic.atomize_term thy) rules
  val defs = map (defs_aux lthy orules preds) (preds ~~ arg_typss) 
in
  fold_map make_defs (prednames ~~ syns ~~ defs) lthy
end*}

text {*
  The argument @{text "preds"} is the list of predicates as @{ML_type term}s;
  the argument @{text "prednames"} is the list of names of the predicates;
  @{text "arg_tyss"} is the list of argument-type-lists. 
  
  In line 4 we generate the intro rules in the object logic; for this we have to 
  obtain the theory behind the local theory (Line 3); with this we can
  call @{ML defs_aux} to generate the terms for the left-hand sides.
  The actual definitions are made in Line 7.

  A testcase for this function is 
*}

local_setup %gray {* fn lthy =>
let
  val rules = [@{prop "even 0"},
               @{prop "\<And>n::nat. odd n \<Longrightarrow> even (Suc n)"},
               @{prop "\<And>n::nat. even n \<Longrightarrow> odd (Suc n)"}] 
  val preds = [@{term "even::nat\<Rightarrow>bool"}, @{term "odd::nat\<Rightarrow>bool"}]
  val prednames = [Binding.name "even", Binding.name "odd"] 
  val syns = [NoSyn, NoSyn] 
  val arg_tyss = [[@{typ "nat"}], [@{typ "nat"}]]
  val (defs, lthy') = definitions rules preds prednames syns arg_tyss lthy
in
  warning (str_of_thms lthy' defs); lthy
end *}

text {*

  It prints out the two definitions

  @{text [display] 
"even \<equiv> \<lambda>z. \<forall>even odd. (even 0) \<longrightarrow> (\<forall>n. odd n \<longrightarrow> even (Suc n)) 
                                \<longrightarrow> (\<forall>n. even n \<longrightarrow> odd (Suc n)) \<longrightarrow> even z, 
 odd \<equiv> \<lambda>z. \<forall>even odd. (even 0) \<longrightarrow> (\<forall>n. odd n \<longrightarrow> even (Suc n)) 
                                \<longrightarrow> (\<forall>n. even n \<longrightarrow> odd (Suc n)) \<longrightarrow> odd z"}

  This completes the code concerning the definitions. Next comes the code for
  the induction principles. 

  Recall the proof for the induction principle for @{term "even"}: 
*}

lemma 
  assumes prems: "even n"
  shows "P 0 \<Longrightarrow> 
             (\<And>m. Q m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Q (Suc m)) \<Longrightarrow> P n"
apply(atomize (full))
apply(cut_tac prems)
apply(unfold even_def)
apply(drule spec[where x=P])
apply(drule spec[where x=Q])
apply(assumption)
done

text {* 
  To automate this proof we need to be able to instantiate universal 
  quantifiers. For this we use the following helper function. It
  instantiates the @{text "?x"} in @{thm spec} with a @{ML_type cterm}.
*}

ML{*fun inst_spec ctrm = 
 Drule.instantiate' [SOME (ctyp_of_term ctrm)] [NONE, SOME ctrm] @{thm spec}*}

text {*
  For example we can use it to instantiate an assumption:
*}

lemma "\<forall>x1 x2 x3. P (x1::nat) (x2::nat) (x3::nat) \<Longrightarrow> True"
apply (tactic {* 
  let 
    val ctrms = [@{cterm "y1::nat"},@{cterm "y2::nat"},@{cterm "y3::nat"}]
  in 
    EVERY1 (map (dtac o inst_spec) ctrms)
  end *})
txt {* 
  where it produces the goal state

  \begin{minipage}{\textwidth}
  @{subgoals} 
  \end{minipage}*}
(*<*)oops(*>*)

text {*
  Now the tactic for proving the induction rules can be implemented 
  as follows
*}

ML{*fun induction_tac defs prems insts =
  EVERY1 [ObjectLogic.full_atomize_tac,
          cut_facts_tac prems,
          K (rewrite_goals_tac defs),
          EVERY' (map (dtac o inst_spec) insts),
          assume_tac]*}

text {*
  We only have to give it as arguments the premises and the instantiations.
  A testcase for the tactic is
*}

lemma 
  assumes prems: "even n"
  shows "P 0 \<Longrightarrow> 
           (\<And>m. Q m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Q (Suc m)) \<Longrightarrow> P n"
apply(tactic {* 
  let
     val defs = [@{thm even_def}, @{thm odd_def}]
     val insts = [@{cterm "P::nat\<Rightarrow>bool"}, @{cterm "Q::nat\<Rightarrow>bool"}]
  in 
    induction_tac defs @{thms prems} insts 
  end *})
done


text {*
  which indeed proves the lemma. 

  While the generic proof for the induction principle is relatively simple, 
  it is a bit harder to set up the goals just from the given introduction 
  rules. For this we have to construct

  @{text [display] 
  "\<And>\<^raw:$zs$>. pred \<^raw:$zs$> \<Longrightarrow> rules[preds := \<^raw:$Ps$>] \<Longrightarrow> \<^raw:$P$> \<^raw:$zs$>"}

  where the given predicates are replaced by new ones written
  as @{text "\<^raw:$Ps$>"}, and also generate new variables 
  @{text "\<^raw:$zs$>"}.
*}

ML %linenosgray{* fun prove_induction lthy defs rules cnewpreds ((pred, newpred), tys)  =
let
  val zs = replicate (length tys) "z"
  val (newargnames, lthy') = Variable.variant_fixes zs lthy;
  val newargs = map Free (newargnames ~~ tys)
  
  val prem = HOLogic.mk_Trueprop (list_comb (pred, newargs))
  val goal = Logic.list_implies 
         (rules, HOLogic.mk_Trueprop (list_comb (newpred, newargs)))
in
  Goal.prove lthy' [] [prem] goal
  (fn {prems, ...} => induction_tac defs prems cnewpreds)
  |> singleton (ProofContext.export lthy' lthy)
end *}

text {* *}

ML %linenosgray{*fun inductions rules defs preds tyss lthy1  =
let
  val Ps = replicate (length preds) "P"
  val (newprednames, lthy2) = Variable.variant_fixes Ps lthy1
  
  val thy = ProofContext.theory_of lthy2

  val tyss' = map (fn tys => tys ---> HOLogic.boolT) tyss
  val newpreds = map Free (newprednames ~~ tyss')
  val cnewpreds = map (cterm_of thy) newpreds
  val rules' = map (subst_free (preds ~~ newpreds)) rules

in
  map (prove_induction lthy2 defs rules' cnewpreds) 
        (preds ~~ newpreds ~~ tyss)
          |> ProofContext.export lthy2 lthy1
end*}

ML {*
  let 
    val rules = [@{prop "even (0::nat)"},
                 @{prop "\<And>n::nat. odd n \<Longrightarrow> even (Suc n)"},
                 @{prop "\<And>n::nat. even n \<Longrightarrow> odd (Suc n)"}] 
    val defs = [@{thm even_def}, @{thm odd_def}]
    val preds = [@{term "even::nat\<Rightarrow>bool"}, @{term "odd::nat\<Rightarrow>bool"}]
    val tyss = [[@{typ "nat"}], [@{typ "nat"}]]
  in
    inductions rules defs preds tyss @{context}
  end
*}


subsection {* Introduction Rules *}

ML{*val all_elims = fold (fn ct => fn th => th RS inst_spec ct)
val imp_elims = fold (fn th => fn th' => [th', th] MRS @{thm mp})*}

ML{*fun subproof2 prem params2 prems2 =  
 SUBPROOF (fn {prems, ...} =>
   let
     val prem' = prems MRS prem;
     val prem'' = 
       case prop_of prem' of
           _ $ (Const (@{const_name All}, _) $ _) =>
             prem' |> all_elims params2 
                   |> imp_elims prems2
         | _ => prem';
   in 
     rtac prem'' 1 
   end)*}

ML{*fun subproof1 rules preds i = 
 SUBPROOF (fn {params, prems, context = ctxt', ...} =>
   let
     val (prems1, prems2) = chop (length prems - length rules) prems;
     val (params1, params2) = chop (length params - length preds) params;
   in
     rtac (ObjectLogic.rulify (all_elims params1 (nth prems2 i))) 1 
     THEN
     EVERY1 (map (fn prem => subproof2 prem params2 prems2 ctxt') prems1)
   end)*}

ML{*
fun introductions_tac defs rules preds i ctxt =
  EVERY1 [ObjectLogic.rulify_tac,
          K (rewrite_goals_tac defs),
          REPEAT o (resolve_tac [@{thm allI},@{thm impI}]),
          subproof1 rules preds i ctxt]*}

lemma evenS: 
  shows "odd m \<Longrightarrow> even (Suc m)"
apply(tactic {* 
let
  val rules = [@{prop "even (0::nat)"},
                 @{prop "\<And>n::nat. odd n \<Longrightarrow> even (Suc n)"},
                 @{prop "\<And>n::nat. even n \<Longrightarrow> odd (Suc n)"}] 
  val defs = [@{thm even_def}, @{thm odd_def}]
  val preds = [@{term "even::nat\<Rightarrow>bool"}, @{term "odd::nat\<Rightarrow>bool"}]
in
  introductions_tac defs rules preds 1 @{context}
end *})
done

ML{*fun introductions rules preds defs lthy = 
let
  fun prove_intro (i, goal) =
    Goal.prove lthy [] [] goal
      (fn {context, ...} => introductions_tac defs rules preds i context)
in
  map_index prove_intro rules
end*}

text {* main internal function *}

ML %linenosgray{*fun add_inductive_i pred_specs rule_specs lthy =
let
  val syns = map snd pred_specs
  val pred_specs' = map fst pred_specs
  val prednames = map fst pred_specs'
  val preds = map (fn (p, ty) => Free (Binding.name_of p, ty)) pred_specs'

  val tyss = map (binder_types o fastype_of) preds   
  val (attrs, rules) = split_list rule_specs    

  val (defs, lthy') = definitions rules preds prednames syns tyss lthy      
  val ind_rules = inductions rules defs preds tyss lthy' 	
  val intro_rules = introductions rules preds defs lthy'

  val mut_name = space_implode "_" (map Binding.name_of prednames)
  val case_names = map (Binding.name_of o fst) attrs
in
    lthy' 
    |> LocalTheory.notes Thm.theoremK (map (fn (((a, atts), _), th) =>
        ((Binding.qualify false mut_name a, atts), [([th], [])])) (rule_specs ~~ intro_rules)) 
    |-> (fn intross => LocalTheory.note Thm.theoremK
         ((Binding.qualify false mut_name (Binding.name "intros"), []), maps snd intross)) 
    |>> snd 
    ||>> (LocalTheory.notes Thm.theoremK (map (fn (((R, _), _), th) =>
         ((Binding.qualify false (Binding.name_of R) (Binding.name "induct"),
          [Attrib.internal (K (RuleCases.case_names case_names)),
           Attrib.internal (K (RuleCases.consumes 1)),
           Attrib.internal (K (Induct.induct_pred ""))]), [([th], [])]))
          (pred_specs ~~ ind_rules)) #>> maps snd) 
    |> snd
end*}

ML{*fun read_specification' vars specs lthy =
let 
  val specs' = map (fn (a, s) => (a, [s])) specs
  val ((varst, specst), _) = 
                   Specification.read_specification vars specs' lthy
  val specst' = map (apsnd the_single) specst
in   
  (varst, specst')
end*}

ML{*fun add_inductive pred_specs rule_specs lthy =
let
  val (pred_specs', rule_specs') = 
    read_specification' pred_specs rule_specs lthy
in
  add_inductive_i pred_specs' rule_specs' lthy
end*} 

ML{*val spec_parser = 
   OuterParse.opt_target --
   OuterParse.fixes -- 
   Scan.optional 
     (OuterParse.$$$ "where" |--
        OuterParse.!!! 
          (OuterParse.enum1 "|" 
             (SpecParse.opt_thm_name ":" -- OuterParse.prop))) []*}

ML{*val specification =
  spec_parser >>
    (fn ((loc, pred_specs), rule_specs) =>
      Toplevel.local_theory loc (add_inductive pred_specs rule_specs))*}

ML{*val _ = OuterSyntax.command "simple_inductive" "define inductive predicates"
          OuterKeyword.thy_decl specification*}

text {*
  Things to include at the end:

  \begin{itemize}
  \item say something about add-inductive-i to return
  the rules
  \item say that the induction principle is weaker (weaker than
  what the standard inductive package generates)
  \end{itemize}
  
*}

simple_inductive
  Even and Odd
where
  Even0: "Even 0"
| EvenS: "Odd n \<Longrightarrow> Even (Suc n)"
| OddS: "Even n \<Longrightarrow> Odd (Suc n)"

end