# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1294071672 0
# Node ID 0865caafbfe6072e420261b7da9ae8da39e9b0cd
# Parent  64b4cb2c2bf805f4266546e2a2c05c5e2cf22f3e
file with most of the strong rule induction development

diff -r 64b4cb2c2bf8 -r 0865caafbfe6 Nominal/Ex/Typing.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/Typing.thy	Mon Jan 03 16:21:12 2011 +0000
@@ -0,0 +1,540 @@
+theory Lambda
+imports "../Nominal2" 
+begin
+
+
+atom_decl name
+
+nominal_datatype lam =
+  Var "name"
+| App "lam" "lam"
+| Lam x::"name" l::"lam"  bind x in l
+
+thm lam.distinct
+thm lam.induct
+thm lam.exhaust lam.strong_exhaust
+thm lam.fv_defs
+thm lam.bn_defs
+thm lam.perm_simps
+thm lam.eq_iff
+thm lam.fv_bn_eqvt
+thm lam.size_eqvt
+
+ML {*
+fun mk_cminus p = Thm.capply @{cterm "uminus::perm \<Rightarrow> perm"} p 
+
+fun minus_permute_intro_tac p = 
+  rtac (Drule.instantiate' [] [SOME (mk_cminus p)] @{thm permute_boolE})
+
+fun minus_permute_elim p thm = 
+  thm RS (Drule.instantiate' [] [NONE, SOME (mk_cminus p)] @{thm permute_boolI})
+*}
+
+ML {*
+fun real_head_of (@{term Trueprop} $ t) = real_head_of t
+  | real_head_of (Const ("==>", _) $ _ $ t) = real_head_of t
+  | real_head_of (Const (@{const_name all}, _) $ Abs (_, _, t)) = real_head_of t
+  | real_head_of (Const (@{const_name All}, _) $ Abs (_, _, t)) = real_head_of t
+  | real_head_of (Const ("HOL.induct_forall", _) $ Abs (_, _, t)) = real_head_of t
+  | real_head_of t = head_of t  
+*}
+
+ML {* 
+fun mk_vc_compat (avoid, avoid_trm) prems concl_args params =
+  let
+    fun aux arg = arg
+      |> mk_fresh_star avoid_trm 
+      |> HOLogic.mk_Trueprop
+      |> (curry Logic.list_implies) prems
+      |> (curry list_all_free) params
+  in 
+    if null avoid then [] else map aux concl_args
+  end
+*}
+
+ML {*
+fun map_term prop f trm =
+  if prop trm 
+  then f trm
+  else case trm of
+    (t1 $ t2) => map_term prop f t1 $ map_term prop f t2
+  | Abs (x, T, t) => Abs (x, T, map_term prop f t)
+  | _ => trm
+*}
+
+ML {*
+fun add_p_c p (c, c_ty) trm =
+  let
+    val (P, args) = strip_comb trm
+    val (P_name, P_ty) = dest_Free P
+    val (ty_args, bool) = strip_type P_ty
+    val args' = map (mk_perm p) args
+  in
+    list_comb (Free (P_name, (c_ty :: ty_args) ---> bool),  c :: args')
+    |> (fn t => HOLogic.all_const c_ty $ lambda c t )
+    |> (fn t => HOLogic.all_const @{typ perm} $  lambda p t)
+  end
+*}
+
+ML {*
+fun induct_forall_const T = Const ("HOL.induct_forall", (T --> @{typ bool}) --> @{typ bool})
+fun mk_induct_forall (a, T) t =  induct_forall_const T $ Abs (a, T, t)
+*}
+
+ML {*
+fun add_c_prop qnt Ps (c, c_name, c_ty) trm =
+  let
+    fun add t = 
+      let
+        val (P, args) = strip_comb t
+        val (P_name, P_ty) = dest_Free P
+        val (ty_args, bool) = strip_type P_ty
+        val args' = args
+          |> qnt ? map (incr_boundvars 1)
+      in
+        list_comb (Free (P_name, (c_ty :: ty_args) ---> bool), c :: args')
+        |> qnt ? mk_induct_forall (c_name, c_ty)
+      end
+  in
+    map_term (member (op =) Ps o head_of) add trm
+  end
+*}
+
+ML {*
+fun prep_prem Ps c_name c_ty (avoid, avoid_trm) (params, prems, concl) =
+  let
+    val prems' = prems
+      |> map (incr_boundvars 1) 
+      |> map (add_c_prop true Ps (Bound 0, c_name, c_ty))
+
+    val avoid_trm' = avoid_trm
+      |> (curry list_abs_free) (params @ [(c_name, c_ty)])
+      |> strip_abs_body
+      |> (fn t => mk_fresh_star_ty c_ty t (Bound 0))
+      |> HOLogic.mk_Trueprop
+
+    val prems'' = 
+      if null avoid 
+      then prems' 
+      else avoid_trm' :: prems'
+
+    val concl' = concl
+      |> incr_boundvars 1 
+      |> add_c_prop false Ps (Bound 0, c_name, c_ty)  
+  in
+    mk_full_horn (params @ [(c_name, c_ty)]) prems'' concl'
+  end
+*}
+
+
+ML {*
+fun same_name (Free (a1, _), Free (a2, _)) = (a1 = a2)
+  | same_name (Var (a1, _), Var (a2, _)) = (a1 = a2)
+  | same_name (Const (a1, _), Const (a2, _)) = (a1 = a2)
+  | same_name _ = false
+*}
+
+ML {*
+(* local abbreviations *)
+fun eqvt_stac ctxt = Nominal_Permeq.eqvt_strict_tac ctxt @{thms permute_minus_cancel} []  
+fun eqvt_srule ctxt = Nominal_Permeq.eqvt_strict_rule ctxt @{thms permute_minus_cancel} []  
+*}
+
+ML {*
+val all_elims = 
+  let
+     fun spec' ct = Drule.instantiate' [SOME (ctyp_of_term ct)] [NONE, SOME ct] @{thm spec}
+  in
+    fold (fn ct => fn th => th RS spec' ct)
+  end
+*}
+
+ML {*
+fun helper_tac flag prm p ctxt =
+  Subgoal.SUBPROOF (fn {context, prems, ...} =>
+    let
+      val prems' = prems
+        |> map (minus_permute_elim p)
+        |> map (eqvt_srule context)
+     
+      val prm' = (prems' MRS prm)
+        |> flag ? (all_elims [p])
+        |> flag ? (simplify (HOL_basic_ss addsimps @{thms permute_minus_cancel}))
+    in
+      simp_tac (HOL_ss addsimps (prm' :: @{thms induct_forall_def })) 1
+    end) ctxt
+*}
+
+ML {*
+fun non_binder_tac prem intr_cvars Ps ctxt = 
+  Subgoal.SUBPROOF (fn {context, params, prems, ...} =>
+    let
+      val thy = ProofContext.theory_of context
+      val (prms, p, _) = split_last2 (map snd params)
+      val prm_tys = map (fastype_of o term_of) prms
+      val cperms = map (cterm_of thy o perm_const) prm_tys
+      val p_prms = map2 (fn ct1 => fn ct2 => Thm.mk_binop ct1 p ct2) cperms prms 
+      val prem' = cterm_instantiate (intr_cvars ~~ p_prms) prem
+
+      (* for inductive-premises*)
+      fun tac1 prm = helper_tac true prm p context
+
+      (* for non-inductive premises *)   
+      fun tac2 prm =  
+        EVERY' [ minus_permute_intro_tac p, 
+                 eqvt_stac context, 
+                 helper_tac false prm p context ]
+
+      fun select prm (t, i) =
+        (if member same_name Ps (real_head_of t) then tac1 prm else tac2 prm) i
+    in
+      EVERY1 [rtac prem', RANGE (map (SUBGOAL o select) prems) ]
+    end) ctxt
+*}
+
+ML {*
+fun binder_tac prem intr_cvars Ps fresh_thms avoid avoid_trms ctxt = 
+  Subgoal.FOCUS (fn {context, params, ...} =>
+    let
+      val thy = ProofContext.theory_of context
+      val (prms, p, _) = split_last2 (map snd params)
+      val prm_tys = map (fastype_of o term_of) prms
+      val cperms = map (cterm_of thy o perm_const) prm_tys
+      val p_prms = map2 (fn ct1 => fn ct2 => Thm.mk_binop ct1 p ct2) cperms prms 
+      val prem' = cterm_instantiate (intr_cvars ~~ p_prms) prem
+    in
+      Skip_Proof.cheat_tac thy
+      (* EVERY1 [rtac prem'] *)  
+    end) ctxt
+*}
+
+ML {*
+fun case_tac ctxt fresh_thms Ps avoid avoid_trm intr_cvars prem =
+  let
+    val tac1 = non_binder_tac prem intr_cvars Ps ctxt
+    val tac2 = binder_tac prem intr_cvars Ps fresh_thms avoid avoid_trm ctxt
+  in 
+    EVERY' [ rtac @{thm allI}, rtac @{thm allI}, eqvt_stac ctxt,
+             if null avoid then tac1 else tac2 ]
+  end
+*}
+
+ML {*
+fun tac raw_induct fresh_thms Ps avoids avoid_trms intr_cvars {prems, context} =
+  let
+    val cases_tac = map4 (case_tac context fresh_thms Ps) avoids avoid_trms intr_cvars prems
+  in 
+    EVERY1 [ DETERM o rtac raw_induct, RANGE cases_tac ]
+  end
+*}
+
+ML {*
+val normalise = @{lemma "(Q --> (!p c. P p c)) ==> (\<And>c. Q ==> P (0::perm) c)" by simp}
+*}
+
+ML {*
+fun prove_strong_inductive rule_names avoids raw_induct intrs ctxt =
+  let
+    val thy = ProofContext.theory_of ctxt
+    val ((_, [raw_induct']), ctxt') = Variable.import true [raw_induct] ctxt
+
+    val (ind_prems, ind_concl) = raw_induct'
+      |> prop_of
+      |> Logic.strip_horn
+      |>> map strip_full_horn
+    val params = map (fn (x, _, _) => x) ind_prems
+    val param_trms = (map o map) Free params  
+
+    val intr_vars_tys = map (fn t => rev (Term.add_vars (prop_of t) [])) intrs
+    val intr_vars = (map o map) fst intr_vars_tys
+    val intr_vars_substs = map2 (curry (op ~~)) intr_vars param_trms
+    val intr_cvars = (map o map) (cterm_of thy o Var) intr_vars_tys
+
+    val (intr_prems, intr_concls) = intrs
+      |> map prop_of
+      |> map2 subst_Vars intr_vars_substs
+      |> map Logic.strip_horn
+      |> split_list
+
+    val intr_concls_args = map (snd o strip_comb o HOLogic.dest_Trueprop) intr_concls 
+      
+    val avoid_trms = avoids
+      |> (map o map) (setify ctxt') 
+      |> map fold_union
+
+    val vc_compat_goals = 
+      map4 mk_vc_compat (avoids ~~ avoid_trms) intr_prems intr_concls_args params
+
+    val ([c_name, a, p], ctxt'') = Variable.variant_fixes ["c", "'a", "p"] ctxt'
+    val c_ty = TFree (a, @{sort fs})
+    val c = Free (c_name, c_ty)
+    val p = Free (p, @{typ perm})
+
+    val (preconds, ind_concls) = ind_concl
+      |> HOLogic.dest_Trueprop
+      |> HOLogic.dest_conj 
+      |> map HOLogic.dest_imp
+      |> split_list
+
+    val Ps = map (fst o strip_comb) ind_concls
+
+    val ind_concl' = ind_concls
+      |> map (add_p_c p (c, c_ty))
+      |> (curry (op ~~)) preconds  
+      |> map HOLogic.mk_imp
+      |> fold_conj
+      |> HOLogic.mk_Trueprop
+
+    val ind_prems' = ind_prems
+      |> map2 (prep_prem Ps c_name c_ty) (avoids ~~ avoid_trms)   
+
+    fun after_qed ctxt_outside fresh_thms ctxt = 
+      let
+        val thms = Goal.prove ctxt [] ind_prems' ind_concl' 
+          (tac raw_induct fresh_thms Ps avoids avoid_trms intr_cvars) 
+          |> singleton (ProofContext.export ctxt ctxt_outside)
+          |> Datatype_Aux.split_conj_thm
+          |> map (fn thm => thm RS normalise)
+          |> map (asm_full_simplify (HOL_basic_ss addsimps @{thms permute_zero induct_rulify})) 
+          |> map (Drule.rotate_prems (length ind_prems'))
+          |> map zero_var_indexes
+
+        val _ = tracing ("RESULTS\n" ^ cat_lines (map (Syntax.string_of_term ctxt o prop_of) thms))
+      in
+        ctxt
+      end 
+  in
+    Proof.theorem NONE (after_qed ctxt) ((map o map) (rpair []) vc_compat_goals) ctxt''
+  end
+*}
+
+ML {*
+fun prove_strong_inductive_cmd (pred_name, avoids) ctxt =
+  let
+    val thy = ProofContext.theory_of ctxt;
+    val ({names, ...}, {raw_induct, intrs, ...}) =
+      Inductive.the_inductive ctxt (Sign.intern_const thy pred_name);
+
+    val rule_names = 
+      hd names
+      |> the o Induct.lookup_inductP ctxt
+      |> fst o Rule_Cases.get
+      |> map fst
+
+    val _ = (case duplicates (op = o pairself fst) avoids of
+        [] => ()
+      | xs => error ("Duplicate case names: " ^ commas_quote (map fst xs)))
+
+    val _ = (case subtract (op =) rule_names (map fst avoids) of
+        [] => ()
+      | xs => error ("No such case(s) in inductive definition: " ^ commas_quote xs))
+
+    val avoids_ordered = order_default (op =) [] rule_names avoids
+      
+    fun read_avoids avoid_trms intr =
+      let
+        (* fixme hack *)
+        val (((_, ctrms), _), ctxt') = Variable.import true [intr] ctxt
+        val trms = map (term_of o snd) ctrms
+        val ctxt'' = fold Variable.declare_term trms ctxt' 
+      in
+        map (Syntax.read_term ctxt'') avoid_trms 
+      end 
+
+    val avoid_trms = map2 read_avoids avoids_ordered intrs
+  in
+    prove_strong_inductive rule_names avoid_trms raw_induct intrs ctxt
+  end
+*}
+
+ML {*
+(* outer syntax *)
+local
+  structure P = Parse;
+  structure S = Scan
+  
+  val _ = Keyword.keyword "avoids"
+
+  val single_avoid_parser = 
+    P.name -- (P.$$$ ":" |-- P.and_list1 P.term)
+
+  val avoids_parser = 
+    S.optional (P.$$$ "avoids" |-- P.enum1 "|" single_avoid_parser) []
+
+  val main_parser = P.xname -- avoids_parser
+in
+  val _ =
+  Outer_Syntax.local_theory_to_proof "nominal_inductive"
+    "prove strong induction theorem for inductive predicate involving nominal datatypes"
+      Keyword.thy_goal (main_parser >> prove_strong_inductive_cmd)
+end
+*}
+
+inductive
+  Acc :: "('a::pt \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
+where
+  AccI: "(\<And>y. R y x \<Longrightarrow> Acc R y) \<Longrightarrow> Acc R x"
+
+(*
+equivariance Acc
+*)
+
+lemma Acc_eqvt [eqvt]:
+  fixes p::"perm"
+  assumes a: "Acc R x"
+  shows "Acc (p \<bullet> R) (p \<bullet> x)"
+using a
+apply(induct)
+apply(rule AccI)
+apply(rotate_tac 1)
+apply(drule_tac x="-p \<bullet> y" in meta_spec)
+apply(simp)
+apply(drule meta_mp)
+apply(rule_tac p="p" in permute_boolE)
+apply(perm_simp add: permute_minus_cancel)
+apply(assumption)
+apply(assumption)
+done
+ 
+
+nominal_inductive Acc .
+
+section {* Typing *}
+
+nominal_datatype ty =
+  TVar string
+| TFun ty ty ("_ \<rightarrow> _") 
+
+lemma ty_fresh:
+  fixes x::"name"
+  and   T::"ty"
+  shows "atom x \<sharp> T"
+apply (nominal_induct T rule: ty.strong_induct)
+apply (simp_all add: ty.fresh pure_fresh)
+done
+
+
+
+inductive
+  valid :: "(name \<times> ty) list \<Rightarrow> bool"
+where
+  "valid []"
+| "\<lbrakk>atom x \<sharp> Gamma; valid Gamma\<rbrakk> \<Longrightarrow> valid ((x, T)#Gamma)"
+
+inductive
+  typing :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ : _" [60,60,60] 60) 
+where
+    t_Var[intro]: "\<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Var x : T"
+  | t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : T2"
+  | t_Lam[intro]: "\<lbrakk>atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam x t : T1 \<rightarrow> T2"
+
+thm typing.intros
+thm typing.induct
+
+
+
+equivariance valid
+equivariance typing
+
+
+nominal_inductive typing
+  avoids t_Lam: "x"
+  apply -
+  apply(simp_all add: fresh_star_def ty_fresh lam.fresh)?
+  done
+
+
+
+lemma
+  fixes c::"'a::fs"
+  assumes a: "\<Gamma> \<turnstile> t : T" 
+  and a1: "\<And>\<Gamma> x T c. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P c \<Gamma> (Var x) T"
+  and a2: "\<And>\<Gamma> t1 T1 T2 t2 c. \<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<And>d. P d \<Gamma> t1 T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1; \<And>d. P d \<Gamma> t2 T1\<rbrakk> 
+           \<Longrightarrow> P c \<Gamma> (App t1 t2) T2"
+  and a3: "\<And>x \<Gamma> T1 t T2 c. \<lbrakk>{atom x} \<sharp>* c; atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2; \<And>d. P d ((x, T1) # \<Gamma>) t T2\<rbrakk> 
+           \<Longrightarrow> P c \<Gamma> (Lam x t) T1 \<rightarrow> T2"
+  shows "P c \<Gamma> t T"
+proof -
+  from a have "\<And>p c. P c (p \<bullet> \<Gamma>) (p \<bullet> t) (p \<bullet> T)"
+  proof (induct)
+    case (t_Var \<Gamma> x T p c)
+    then show ?case
+      apply -
+      apply(perm_strict_simp)
+      thm a1
+      apply(rule a1)
+      apply(drule_tac p="p" in permute_boolI)
+      apply(perm_strict_simp add: permute_minus_cancel)
+      apply(assumption)
+      apply(rotate_tac 1)
+      apply(drule_tac p="p" in permute_boolI)
+      apply(perm_strict_simp add: permute_minus_cancel)
+      apply(assumption)
+      done
+  next
+    case (t_App \<Gamma> t1 T1 T2 t2 p c)
+    then show ?case
+      apply -
+      apply(perm_strict_simp)
+      apply(rule a2)
+      apply(drule_tac p="p" in permute_boolI)
+      apply(perm_strict_simp add: permute_minus_cancel)
+      apply(assumption)
+      apply(assumption)
+      apply(rotate_tac 2)
+      apply(drule_tac p="p" in permute_boolI)
+      apply(perm_strict_simp add: permute_minus_cancel)
+      apply(assumption)
+      apply(assumption)
+      done
+  next
+    case (t_Lam x \<Gamma> T1 t T2 p c)
+    then show ?case
+      apply -
+      apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom x}) \<sharp>* c \<and> 
+        supp (p \<bullet> \<Gamma>, p \<bullet> Lam x t, p \<bullet> (T1 \<rightarrow> T2)) \<sharp>* q")
+      apply(erule exE)
+      apply(rule_tac t="p \<bullet> \<Gamma>" and  s="(q + p) \<bullet> \<Gamma>" in subst)
+      apply(simp only: permute_plus)
+      apply(rule supp_perm_eq)
+      apply(simp add: supp_Pair fresh_star_Un)
+      apply(rule_tac t="p \<bullet> Lam x t" and  s="(q + p) \<bullet> Lam x t" in subst)
+      apply(simp only: permute_plus)
+      apply(rule supp_perm_eq)
+      apply(simp add: supp_Pair fresh_star_Un)
+      apply(rule_tac t="p \<bullet> (T1 \<rightarrow> T2)" and  s="(q + p) \<bullet> (T1 \<rightarrow> T2)" in subst)
+      apply(simp only: permute_plus)
+      apply(rule supp_perm_eq)
+      apply(simp add: supp_Pair fresh_star_Un)
+      apply(simp (no_asm) only: eqvts)
+      apply(rule a3)
+      apply(simp only: eqvts permute_plus)
+      apply(rule_tac p="- (q + p)" in permute_boolE)
+      apply(perm_strict_simp add: permute_minus_cancel)
+      apply(assumption)
+       apply(rule_tac p="- (q + p)" in permute_boolE)
+      apply(perm_strict_simp add: permute_minus_cancel)
+      apply(assumption)
+      apply(perm_strict_simp)
+      apply(simp only:)
+      apply(rule at_set_avoiding2)
+      apply(simp add: finite_supp)
+      apply(simp add: finite_supp)
+      apply(simp add: finite_supp)
+      apply(rule_tac p="-p" in permute_boolE)
+      apply(perm_strict_simp add: permute_minus_cancel)
+	--"supplied by the user"
+      apply(simp add: fresh_star_Pair)
+      sorry
+  qed
+  then have "P c (0 \<bullet> \<Gamma>) (0 \<bullet> t) (0 \<bullet> T)" .
+  then show "P c \<Gamma> t T" by simp
+qed
+
+*)
+
+
+end
+
+
+