
diff -r ffb5a181844b -r 8268b277d240 Nominal/Ex/Foo2.thy
--- a/Nominal/Ex/Foo2.thy	Tue Dec 28 00:20:50 2010 +0000
+++ b/Nominal/Ex/Foo2.thy	Tue Dec 28 19:51:25 2010 +0000
@@ -9,6 +9,7 @@
 
 atom_decl name
 
+
 nominal_datatype foo: trm =
   Var "name"
 | App "trm" "trm"
@@ -25,7 +26,6 @@
 | "bn (As_Nil) = []"
 
 
-
 thm foo.bn_defs
 thm foo.permute_bn
 thm foo.perm_bn_alpha
@@ -35,6 +35,7 @@
 thm foo.distinct
 thm foo.induct
 thm foo.inducts
+thm foo.strong_induct
 thm foo.exhaust
 thm foo.strong_exhaust
 thm foo.fv_defs
@@ -47,432 +48,8 @@
 thm foo.fsupp
 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
+thm foo.size
 
-    (* 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
-  assumes a0: "\<And>name c. P1 c (Var name)"
-  and a1: "\<And>trm1 trm2 c. \<lbrakk>\<And>d. P1 d trm1; \<And>d. P1 d trm2\<rbrakk> \<Longrightarrow> P1 c (App trm1 trm2)"
-  and a2: "\<And>name trm c. (\<And>d. P1 d trm) \<Longrightarrow> P1 c (Lam name trm)"
-  and a3: "\<And>a1 t1 a2 t2 c. 
-    \<lbrakk>((set (bn a1)) \<union> (set (bn a2))) \<sharp>* c; \<And>d. P2 c a1; \<And>d. P1 d t1; \<And>d. P2 d a2; \<And>d. P1 d t2\<rbrakk> 
-    \<Longrightarrow> P1 c (Let1 a1 t1 a2 t2)"
-  and a4: "\<And>n1 n2 t1 t2 c. 
-    \<lbrakk>({atom n1} \<union> {atom n2}) \<sharp>* c; \<And>d. P1 d t1; \<And>d. P1 d t2\<rbrakk> \<Longrightarrow> P1 c (Let2 n1 n2 t1 t2)"
-  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"
-  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))
-  apply(simp_all)[2]
-  apply(relation "measure (sum_case (size o snd) (size o snd))")
-  apply(simp_all add: foo.size)
-  done
 
 end
 




