diff -r 16ffbc8442ca -r ffb5a181844b Nominal/Ex/Foo2.thy
--- a/Nominal/Ex/Foo2.thy	Sun Dec 26 16:35:16 2010 +0000
+++ b/Nominal/Ex/Foo2.thy	Tue Dec 28 00:20:50 2010 +0000
@@ -26,7 +26,6 @@
 
 
 
-
 thm foo.bn_defs
 thm foo.permute_bn
 thm foo.perm_bn_alpha
@@ -49,6 +48,409 @@
 thm foo.supp
 thm foo.fresh
 
+ML {*
+
+open Function_Lib
+
+type rec_call_info = int * (string * typ) list * term list * term list
+
+datatype scheme_case = SchemeCase of
+ {bidx : int,
+  qs: (string * typ) list,
+  oqnames: string list,
+  gs: term list,
+  lhs: term list,
+  rs: rec_call_info list}
+
+datatype scheme_branch = SchemeBranch of
+ {P : term,
+  xs: (string * typ) list,
+  ws: (string * typ) list,
+  Cs: term list}
+
+datatype ind_scheme = IndScheme of
+ {T: typ, (* sum of products *)
+  branches: scheme_branch list,
+  cases: scheme_case list}
+
+val ind_atomize = Raw_Simplifier.rewrite true @{thms induct_atomize}
+val ind_rulify = Raw_Simplifier.rewrite true @{thms induct_rulify}
+
+fun meta thm = thm RS eq_reflection
+
+val sum_prod_conv = Raw_Simplifier.rewrite true
+  (map meta (@{thm split_conv} :: @{thms sum.cases}))
+
+fun term_conv thy cv t =
+  cv (cterm_of thy t)
+  |> prop_of |> Logic.dest_equals |> snd
+
+fun mk_relT T = HOLogic.mk_setT (HOLogic.mk_prodT (T, T))
+
+fun dest_hhf ctxt t =
+  let
+    val ((vars, imp), ctxt') = Function_Lib.focus_term t ctxt
+  in
+    (ctxt', vars, Logic.strip_imp_prems imp, Logic.strip_imp_concl imp)
+  end
+
+fun mk_scheme' ctxt cases concl =
+  let
+    fun mk_branch concl =
+      let
+        val _ = tracing ("concl:\n" ^ Syntax.string_of_term ctxt concl)
+        val (_, ws, Cs, _ $ Pxs) = dest_hhf ctxt concl
+        val (P, xs) = strip_comb Pxs
+        val _ = tracing ("xs: " ^ commas (map @{make_string} xs)) 
+        val _ =  map dest_Free xs
+        val _ = tracing ("done")
+      in
+        SchemeBranch { P=P, xs=map dest_Free xs, ws=ws, Cs=Cs }
+      end
+
+    val (branches, cases') = (* correction *)
+      case Logic.dest_conjunctions concl of
+        [conc] =>
+        let
+          val _ $ Pxs = Logic.strip_assums_concl conc
+          val (P, _) = strip_comb Pxs
+          val (cases', conds) =
+            take_prefix (Term.exists_subterm (curry op aconv P)) cases
+          val concl' = fold_rev (curry Logic.mk_implies) conds conc
+        in
+          ([mk_branch concl'], cases')
+        end
+      | concls => (map mk_branch concls, cases)
+
+    fun mk_case premise =
+      let
+        val (ctxt', qs, prems, _ $ Plhs) = dest_hhf ctxt premise
+        val (P, lhs) = strip_comb Plhs
+
+        fun bidx Q =
+          find_index (fn SchemeBranch {P=P',...} => Q aconv P') branches
+
+        fun mk_rcinfo pr =
+          let
+            val (_, Gvs, Gas, _ $ Phyp) = dest_hhf ctxt' pr
+            val (P', rcs) = strip_comb Phyp
+          in
+            (bidx P', Gvs, Gas, rcs)
+          end
+
+        fun is_pred v = exists (fn SchemeBranch {P,...} => v aconv P) branches
+
+        val (gs, rcprs) =
+          take_prefix (not o Term.exists_subterm is_pred) prems
+      in
+        SchemeCase {bidx=bidx P, qs=qs, oqnames=map fst qs(*FIXME*),
+          gs=gs, lhs=lhs, rs=map mk_rcinfo rcprs}
+      end
+
+    fun PT_of (SchemeBranch { xs, ...}) =
+      foldr1 HOLogic.mk_prodT (map snd xs)
+
+    val ST = Balanced_Tree.make (uncurry SumTree.mk_sumT) (map PT_of branches)
+  in
+    IndScheme {T=ST, cases=map mk_case cases', branches=branches }
+  end
+
+fun mk_completeness ctxt (IndScheme {cases, branches, ...}) bidx =
+  let
+    val SchemeBranch { xs, ws, Cs, ... } = nth branches bidx
+    val relevant_cases = filter (fn SchemeCase {bidx=bidx', ...} => bidx' = bidx) cases
+
+    val allqnames = fold (fn SchemeCase {qs, ...} => fold (insert (op =) o Free) qs) relevant_cases []
+    val (Pbool :: xs') = map Free (Variable.variant_frees ctxt allqnames (("P", HOLogic.boolT) :: xs))
+    val Cs' = map (Pattern.rewrite_term (ProofContext.theory_of ctxt) (filter_out (op aconv) (map Free xs ~~ xs')) []) Cs
+
+    fun mk_case (SchemeCase {qs, oqnames, gs, lhs, ...}) =
+      HOLogic.mk_Trueprop Pbool
+      |> fold_rev (fn x_l => curry Logic.mk_implies (HOLogic.mk_Trueprop(HOLogic.mk_eq x_l)))
+           (xs' ~~ lhs)
+      |> fold_rev (curry Logic.mk_implies) gs
+      |> fold_rev mk_forall_rename (oqnames ~~ map Free qs)
+  in
+    HOLogic.mk_Trueprop Pbool
+    |> fold_rev (curry Logic.mk_implies o mk_case) relevant_cases
+    |> fold_rev (curry Logic.mk_implies) Cs'
+    |> fold_rev (Logic.all o Free) ws
+    |> fold_rev mk_forall_rename (map fst xs ~~ xs')
+    |> mk_forall_rename ("P", Pbool)
+  end
+
+fun mk_wf R (IndScheme {T, ...}) =
+  HOLogic.Trueprop $ (Const (@{const_name wf}, mk_relT T --> HOLogic.boolT) $ R)
+
+fun mk_ineqs R (IndScheme {T, cases, branches}) =
+  let
+    fun inject i ts =
+       SumTree.mk_inj T (length branches) (i + 1) (foldr1 HOLogic.mk_prod ts)
+
+    val thesis = Free ("thesis", HOLogic.boolT) (* FIXME *)
+
+    fun mk_pres bdx args =
+      let
+        val SchemeBranch { xs, ws, Cs, ... } = nth branches bdx
+        fun replace (x, v) t = betapply (lambda (Free x) t, v)
+        val Cs' = map (fold replace (xs ~~ args)) Cs
+        val cse =
+          HOLogic.mk_Trueprop thesis
+          |> fold_rev (curry Logic.mk_implies) Cs'
+          |> fold_rev (Logic.all o Free) ws
+      in
+        Logic.mk_implies (cse, HOLogic.mk_Trueprop thesis)
+      end
+
+    fun f (SchemeCase {bidx, qs, oqnames, gs, lhs, rs, ...}) =
+      let
+        fun g (bidx', Gvs, Gas, rcarg) =
+          let val export =
+            fold_rev (curry Logic.mk_implies) Gas
+            #> fold_rev (curry Logic.mk_implies) gs
+            #> fold_rev (Logic.all o Free) Gvs
+            #> fold_rev mk_forall_rename (oqnames ~~ map Free qs)
+          in
+            (HOLogic.mk_mem (HOLogic.mk_prod (inject bidx' rcarg, inject bidx lhs), R)
+             |> HOLogic.mk_Trueprop
+             |> export,
+             mk_pres bidx' rcarg
+             |> export
+             |> Logic.all thesis)
+          end
+      in
+        map g rs
+      end
+  in
+    map f cases
+  end
+
+
+fun mk_ind_goal thy branches =
+  let
+    fun brnch (SchemeBranch { P, xs, ws, Cs, ... }) =
+      HOLogic.mk_Trueprop (list_comb (P, map Free xs))
+      |> fold_rev (curry Logic.mk_implies) Cs
+      |> fold_rev (Logic.all o Free) ws
+      |> term_conv thy ind_atomize
+      |> Object_Logic.drop_judgment thy
+      |> HOLogic.tupled_lambda (foldr1 HOLogic.mk_prod (map Free xs))
+  in
+    SumTree.mk_sumcases HOLogic.boolT (map brnch branches)
+  end
+
+fun mk_induct_rule ctxt R x complete_thms wf_thm ineqss
+  (IndScheme {T, cases=scases, branches}) =
+  let
+    val n = length branches
+    val scases_idx = map_index I scases
+
+    fun inject i ts =
+      SumTree.mk_inj T n (i + 1) (foldr1 HOLogic.mk_prod ts)
+    val P_of = nth (map (fn (SchemeBranch { P, ... }) => P) branches)
+
+    val thy = ProofContext.theory_of ctxt
+    val cert = cterm_of thy
+
+    val P_comp = mk_ind_goal thy branches
+
+    (* Inductive Hypothesis: !!z. (z,x):R ==> P z *)
+    val ihyp = Term.all T $ Abs ("z", T,
+      Logic.mk_implies
+        (HOLogic.mk_Trueprop (
+          Const (@{const_name Set.member}, HOLogic.mk_prodT (T, T) --> mk_relT T --> HOLogic.boolT) 
+          $ (HOLogic.pair_const T T $ Bound 0 $ x)
+          $ R),
+         HOLogic.mk_Trueprop (P_comp $ Bound 0)))
+      |> cert
+
+    val aihyp = Thm.assume ihyp
+
+    (* Rule for case splitting along the sum types *)
+    val xss = map (fn (SchemeBranch { xs, ... }) => map Free xs) branches
+    val pats = map_index (uncurry inject) xss
+    val sum_split_rule =
+      Pat_Completeness.prove_completeness thy [x] (P_comp $ x) xss (map single pats)
+
+    fun prove_branch (bidx, (SchemeBranch { P, xs, ws, Cs, ... }, (complete_thm, pat))) =
+      let
+        val fxs = map Free xs
+        val branch_hyp = Thm.assume (cert (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, pat))))
+
+        val C_hyps = map (cert #> Thm.assume) Cs
+
+        val (relevant_cases, ineqss') =
+          (scases_idx ~~ ineqss)
+          |> filter (fn ((_, SchemeCase {bidx=bidx', ...}), _) => bidx' = bidx)
+          |> split_list
+
+        fun prove_case (cidx, SchemeCase {qs, gs, lhs, rs, ...}) ineq_press =
+          let
+            val case_hyps =
+              map (Thm.assume o cert o HOLogic.mk_Trueprop o HOLogic.mk_eq) (fxs ~~ lhs)
+
+            val cqs = map (cert o Free) qs
+            val ags = map (Thm.assume o cert) gs
+
+            val replace_x_ss = HOL_basic_ss addsimps (branch_hyp :: case_hyps)
+            val sih = full_simplify replace_x_ss aihyp
+
+            fun mk_Prec (idx, Gvs, Gas, rcargs) (ineq, pres) =
+              let
+                val cGas = map (Thm.assume o cert) Gas
+                val cGvs = map (cert o Free) Gvs
+                val import = fold Thm.forall_elim (cqs @ cGvs)
+                  #> fold Thm.elim_implies (ags @ cGas)
+                val ipres = pres
+                  |> Thm.forall_elim (cert (list_comb (P_of idx, rcargs)))
+                  |> import
+              in
+                sih
+                |> Thm.forall_elim (cert (inject idx rcargs))
+                |> Thm.elim_implies (import ineq) (* Psum rcargs *)
+                |> Conv.fconv_rule sum_prod_conv
+                |> Conv.fconv_rule ind_rulify
+                |> (fn th => th COMP ipres) (* P rs *)
+                |> fold_rev (Thm.implies_intr o cprop_of) cGas
+                |> fold_rev Thm.forall_intr cGvs
+              end
+
+            val P_recs = map2 mk_Prec rs ineq_press   (*  [P rec1, P rec2, ... ]  *)
+
+            val step = HOLogic.mk_Trueprop (list_comb (P, lhs))
+              |> fold_rev (curry Logic.mk_implies o prop_of) P_recs
+              |> fold_rev (curry Logic.mk_implies) gs
+              |> fold_rev (Logic.all o Free) qs
+              |> cert
+
+            val Plhs_to_Pxs_conv =
+              foldl1 (uncurry Conv.combination_conv)
+                (Conv.all_conv :: map (fn ch => K (Thm.symmetric (ch RS eq_reflection))) case_hyps)
+
+            val res = Thm.assume step
+              |> fold Thm.forall_elim cqs
+              |> fold Thm.elim_implies ags
+              |> fold Thm.elim_implies P_recs (* P lhs *)
+              |> Conv.fconv_rule (Conv.arg_conv Plhs_to_Pxs_conv) (* P xs *)
+              |> fold_rev (Thm.implies_intr o cprop_of) (ags @ case_hyps)
+              |> fold_rev Thm.forall_intr cqs (* !!qs. Gas ==> xs = lhss ==> P xs *)
+          in
+            (res, (cidx, step))
+          end
+
+        val (cases, steps) = split_list (map2 prove_case relevant_cases ineqss')
+
+        val bstep = complete_thm
+          |> Thm.forall_elim (cert (list_comb (P, fxs)))
+          |> fold (Thm.forall_elim o cert) (fxs @ map Free ws)
+          |> fold Thm.elim_implies C_hyps
+          |> fold Thm.elim_implies cases (* P xs *)
+          |> fold_rev (Thm.implies_intr o cprop_of) C_hyps
+          |> fold_rev (Thm.forall_intr o cert o Free) ws
+
+        val Pxs = cert (HOLogic.mk_Trueprop (P_comp $ x))
+          |> Goal.init
+          |> (Simplifier.rewrite_goals_tac (map meta (branch_hyp :: @{thm split_conv} :: @{thms sum.cases}))
+              THEN CONVERSION ind_rulify 1)
+          |> Seq.hd
+          |> Thm.elim_implies (Conv.fconv_rule Drule.beta_eta_conversion bstep)
+          |> Goal.finish ctxt
+          |> Thm.implies_intr (cprop_of branch_hyp)
+          |> fold_rev (Thm.forall_intr o cert) fxs
+      in
+        (Pxs, steps)
+      end
+
+    val (branches, steps) =
+      map_index prove_branch (branches ~~ (complete_thms ~~ pats))
+      |> split_list |> apsnd flat
+
+    val istep = sum_split_rule
+      |> fold (fn b => fn th => Drule.compose_single (b, 1, th)) branches
+      |> Thm.implies_intr ihyp
+      |> Thm.forall_intr (cert x) (* "!!x. (!!y<x. P y) ==> P x" *)
+
+    val induct_rule =
+      @{thm "wf_induct_rule"}
+      |> (curry op COMP) wf_thm
+      |> (curry op COMP) istep
+
+    val steps_sorted = map snd (sort (int_ord o pairself fst) steps)
+  in
+    (steps_sorted, induct_rule)
+  end
+
+
+fun mk_ind_tac comp_tac pres_tac term_tac ctxt facts =
+  (ALLGOALS (Method.insert_tac facts)) THEN HEADGOAL (SUBGOAL (fn (t, i) =>
+  let
+    val (ctxt', _, cases, concl) = dest_hhf ctxt t
+    val scheme as IndScheme {T=ST, branches, ...} = mk_scheme' ctxt' cases concl
+    val ([Rn,xn], ctxt'') = Variable.variant_fixes ["R","x"] ctxt'
+    val R = Free (Rn, mk_relT ST)
+    val x = Free (xn, ST)
+    val cert = cterm_of (ProofContext.theory_of ctxt)
+
+    val ineqss = mk_ineqs R scheme
+      |> map (map (pairself (Thm.assume o cert)))
+    val complete =
+      map_range (mk_completeness ctxt scheme #> cert #> Thm.assume) (length branches)
+    val wf_thm = mk_wf R scheme |> cert |> Thm.assume
+
+    val (descent, pres) = split_list (flat ineqss)
+    val newgoals = complete @ pres @ wf_thm :: descent
+
+    val (steps, indthm) =
+      mk_induct_rule ctxt'' R x complete wf_thm ineqss scheme
+
+    fun project (i, SchemeBranch {xs, ...}) =
+      let
+        val inst = (foldr1 HOLogic.mk_prod (map Free xs))
+          |> SumTree.mk_inj ST (length branches) (i + 1)
+          |> cert
+      in
+        indthm
+        |> Drule.instantiate' [] [SOME inst]
+        |> simplify SumTree.sumcase_split_ss
+        |> Conv.fconv_rule ind_rulify
+      end
+
+    val res = Conjunction.intr_balanced (map_index project branches)
+      |> fold_rev Thm.implies_intr (map cprop_of newgoals @ steps)
+      |> Drule.generalize ([], [Rn])
+
+    val nbranches = length branches
+    val npres = length pres
+  in
+    Thm.compose_no_flatten false (res, length newgoals) i
+    THEN term_tac (i + nbranches + npres)
+    THEN (EVERY (map (TRY o pres_tac) ((i + nbranches + npres - 1) downto (i + nbranches))))
+    THEN (EVERY (map (TRY o comp_tac) ((i + nbranches - 1) downto i)))
+  end))
+
+
+fun induction_schema_tac ctxt =
+  mk_ind_tac (K all_tac) (assume_tac APPEND' Goal.assume_rule_tac ctxt) (K all_tac) ctxt;
+
+*}
+
+ML {*
+val trm1 = @{prop "P1 &&& P2 &&& P3"}
+val trm2 = @{prop "(P1 &&& P2) &&& P3 &&& P4"}
+*}
+
+ML {*
+  Logic.dest_conjunctions trm2
+*}
+
+lemma
+  shows "P1" "P2" "P4"
+oops
+
+lemma 
+  shows "P1" "P2" "P3" "P4"
+oops
+
 lemma strong_induct:
   fixes c :: "'a :: fs"
   and assg :: assg and trm :: trm
@@ -63,8 +465,7 @@
   and a5: "\<And>c. P2 c As_Nil"
   and a6: "\<And>name1 name2 trm assg c. \<lbrakk>\<And>d. P1 d trm; \<And>d. P2 d assg\<rbrakk> \<Longrightarrow> P2 c (As name1 name2 trm assg)"
   shows "P1 c trm" "P2 c assg"
-  using assms
-  apply(induction_schema)
+  apply(raw_tactic {* induction_schema_tac @{context} @{thms assms} *})
   apply(rule_tac y="trm" and c="c" in foo.strong_exhaust(1))
   apply(simp_all)[5]
   apply(rule_tac ya="assg" in foo.strong_exhaust(2))