
diff -r 95aac598a526 -r e838f7d90f81 Nominal/Ex/Lambda.thy
--- a/Nominal/Ex/Lambda.thy	Mon May 17 17:54:07 2010 +0100
+++ b/Nominal/Ex/Lambda.thy	Fri May 21 10:44:07 2010 +0200
@@ -1,5 +1,5 @@
 theory Lambda
-imports "../NewParser"
+imports "../NewParser" Quotient_Option
 begin
 
 atom_decl name
@@ -472,6 +472,321 @@
 nominal_inductive typing
 *)
 
+(* Substitution *)
+
+definition new where
+  "new s = (THE x. \<forall>a \<in> s. atom x \<noteq> a)"
+
+primrec match_Var_raw where
+  "match_Var_raw (Var_raw x) = Some x"
+| "match_Var_raw (App_raw x y) = None"
+| "match_Var_raw (Lam_raw n t) = None"
+
+quotient_definition
+  "match_Var :: lam \<Rightarrow> name option"
+is match_Var_raw
+
+lemma [quot_respect]: "(alpha_lam_raw ===> op =) match_Var_raw match_Var_raw"
+  apply rule
+  apply (induct_tac a b rule: alpha_lam_raw.induct)
+  apply simp_all
+  done
+
+lemmas match_Var_simps = match_Var_raw.simps[quot_lifted]
+
+primrec match_App_raw where
+  "match_App_raw (Var_raw x) = None"
+| "match_App_raw (App_raw x y) = Some (x, y)"
+| "match_App_raw (Lam_raw n t) = None"
+
+quotient_definition
+  "match_App :: lam \<Rightarrow> (lam \<times> lam) option"
+is match_App_raw
+
+lemma [quot_respect]:
+  "(alpha_lam_raw ===> option_rel (prod_rel alpha_lam_raw alpha_lam_raw)) match_App_raw match_App_raw"
+  apply (intro fun_relI)
+  apply (induct_tac a b rule: alpha_lam_raw.induct)
+  apply simp_all
+  done
+
+lemmas match_App_simps = match_App_raw.simps[quot_lifted]
+
+primrec match_Lam_raw where
+  "match_Lam_raw (S :: atom set) (Var_raw x) = None"
+| "match_Lam_raw S (App_raw x y) = None"
+| "match_Lam_raw S (Lam_raw n t) = (let z = new (S \<union> (fv_lam_raw t - {atom n})) in Some (z, (n \<leftrightarrow> z) \<bullet> t))"
+
+quotient_definition
+  "match_Lam :: (atom set) \<Rightarrow> lam \<Rightarrow> (name \<times> lam) option"
+is match_Lam_raw
+
+lemma [quot_respect]:
+  "(op = ===> alpha_lam_raw ===> option_rel (prod_rel op = alpha_lam_raw)) match_Lam_raw match_Lam_raw"
+  proof (intro fun_relI, clarify)
+    fix S t s
+    assume a: "alpha_lam_raw t s"
+    show "option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S t) (match_Lam_raw S s)"
+      using a proof (induct t s rule: alpha_lam_raw.induct)
+      case goal1 show ?case by simp
+    next
+      case goal2 show ?case by simp
+    next
+      case (goal3 x t y s)
+      then obtain p where "({atom x}, t) \<approx>gen (\<lambda>x1 x2. alpha_lam_raw x1 x2 \<and>
+                              option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S x1)
+                               (match_Lam_raw S x2)) fv_lam_raw p ({atom y}, s)" ..
+      then have
+        c: "fv_lam_raw t - {atom x} = fv_lam_raw s - {atom y}" and
+        d: "(fv_lam_raw t - {atom x}) \<sharp>* p" and
+        e: "alpha_lam_raw (p \<bullet> t) s" and
+        f: "option_rel (prod_rel op = alpha_lam_raw) (match_Lam_raw S (p \<bullet> t)) (match_Lam_raw S s)" and
+        g: "p \<bullet> {atom x} = {atom y}" unfolding alphas(1) by - (elim conjE, assumption)+
+      let ?z = "new (S \<union> (fv_lam_raw t - {atom x}))"
+      have h: "?z = new (S \<union> (fv_lam_raw s - {atom y}))" using c by simp
+      show ?case
+        unfolding match_Lam_raw.simps Let_def option_rel.simps prod_rel.simps split_conv
+      proof
+        show "?z = new (S \<union> (fv_lam_raw s - {atom y}))" by (fact h)
+      next
+        have "atom y \<sharp> p" sorry
+        have "fv_lam_raw t \<sharp>* ((x \<leftrightarrow> y) \<bullet> p)" sorry
+        then have "alpha_lam_raw (((x \<leftrightarrow> y) \<bullet> p) \<bullet> t) t" sorry
+        have "alpha_lam_raw (p \<bullet> t) ((x \<leftrightarrow> y) \<bullet> t)" sorry
+        have "alpha_lam_raw t ((x \<leftrightarrow> y) \<bullet> s)" sorry
+        then have "alpha_lam_raw ((x \<leftrightarrow> ?z) \<bullet> t) ((y \<leftrightarrow> ?z) \<bullet> s)" using eqvts(15) sorry
+        then show "alpha_lam_raw ((x \<leftrightarrow> new (S \<union> (fv_lam_raw t - {atom x}))) \<bullet> t)
+                  ((y \<leftrightarrow> new (S \<union> (fv_lam_raw s - {atom y}))) \<bullet> s)" unfolding h .
+      qed
+    qed
+  qed
+
+lemmas match_Lam_simps = match_Lam_raw.simps[quot_lifted]
+
+lemma app_some: "match_App x = Some (a, b) \<Longrightarrow> x = App a b"
+by (induct x rule: lam.induct) (simp_all add: match_App_simps)
+
+lemma lam_some: "match_Lam S x = Some (z, s) \<Longrightarrow> x = Lam z s \<and> atom z \<sharp> S"
+  apply (induct x rule: lam.induct)
+  apply (simp_all add: match_Lam_simps)
+  apply (simp add: Let_def)
+  apply (erule conjE)
+  apply (thin_tac "match_Lam S lam = Some (z, s) \<Longrightarrow> lam = Lam z s \<and> atom z \<sharp> S")
+  apply (rule conjI)
+  apply (simp add: lam.eq_iff)
+  apply (rule_tac x="(name \<leftrightarrow> z)" in exI)
+  apply (simp add: alphas)
+  apply (simp add: eqvts)
+  apply (simp only: lam.fv(3)[symmetric])
+  apply (subgoal_tac "Lam name lam = Lam z s")
+  apply (simp del: lam.fv)
+  prefer 3
+  apply (thin_tac "(name \<leftrightarrow> new (S \<union> (fv_lam lam - {atom name}))) \<bullet> lam = s")
+  apply (simp only: new_def)
+  apply (subgoal_tac "\<forall>a \<in> S. atom z \<noteq> a")
+  apply (simp only: fresh_def)
+  
+  thm new_def
+  apply simp
+
+
+function subst where
+"subst v s t = (
+  case match_Var t of Some n \<Rightarrow> if n = v then s else Var n | None \<Rightarrow>
+  case match_App t of Some (l, r) \<Rightarrow> App (subst v s l) (subst v s r) | None \<Rightarrow>
+  case match_Lam (supp (v,s)) t of Some (n, t) \<Rightarrow> Lam n (subst v s t) | None \<Rightarrow> undefined)"
+by pat_completeness auto
+
+termination apply (relation "measure (\<lambda>(_, _, t). size t)")
+apply auto[1]
+defer
+apply (case_tac a) apply simp
+apply (frule app_some) apply simp
+apply (case_tac a) apply simp
+apply (frule app_some) apply simp
+apply (case_tac a) apply simp
+apply (frule lam_some)
+ apply simp
+done
+
+lemmas lam_exhaust = lam_raw.exhaust[quot_lifted]
+
+lemma subst_eqvt:
+  "p \<bullet> (subst v s t) = subst (p \<bullet> v) (p \<bullet> s) (p \<bullet> t)"
+  proof (induct v s t rule: subst.induct)
+    case (1 v s t)
+    show ?case proof (cases t rule: lam_exhaust)
+      fix n
+      assume "t = Var n"
+      then show ?thesis by (simp add: match_Var_simps)
+    next
+      fix l r
+      assume "t = App l r"
+      then show ?thesis
+        apply (simp only:)
+        apply (subst subst.simps)
+        apply (subst match_Var_simps)
+        apply (simp only: option.cases)
+        apply (subst match_App_simps)
+        apply (simp only: option.cases)
+        apply (simp only: prod.cases)
+        apply (simp only: lam.perm)
+        apply (subst (3) subst.simps)
+        apply (subst match_Var_simps)
+       apply (simp only: option.cases)
+        apply (subst match_App_simps)
+        apply (simp only: option.cases)
+        apply (simp only: prod.cases)
+        apply (subst 1(2)[of "(l, r)" "l" "r"])
+        apply (simp add: match_Var_simps)
+        apply (simp add: match_App_simps)
+        apply (rule refl)
+        apply (subst 1(3)[of "(l, r)" "l" "r"])
+        apply (simp add: match_Var_simps)
+        apply (simp add: match_App_simps)
+        apply (rule refl)
+        apply (rule refl)
+        done
+    next
+      fix n t'
+      assume "t = Lam n t'"
+      then show ?thesis
+        apply (simp only: )
+        apply (simp only: lam.perm)
+        apply (subst subst.simps)
+        apply (subst match_Var_simps)
+        apply (simp only: option.cases)
+        apply (subst match_App_simps)
+        apply (simp only: option.cases)
+        apply (subst match_Lam_simps)
+        apply (simp only: Let_def)
+        apply (simp only: option.cases)
+        apply (simp only: prod.cases)
+        apply (subst (2) subst.simps)
+        apply (subst match_Var_simps)
+        apply (simp only: option.cases)
+        apply (subst match_App_simps)
+        apply (simp only: option.cases)
+        apply (subst match_Lam_simps)
+        apply (simp only: Let_def)
+        apply (simp only: option.cases)
+        apply (simp only: prod.cases)
+        apply (simp only: lam.perm)
+        apply (simp only: lam.eq_iff)
+        sorry
+    qed
+  qed
+
+lemma size_no_change: "size (p \<bullet> (t :: lam_raw)) = size t"
+  by (induct t) simp_all
+
+function
+  subst_raw :: "lam_raw \<Rightarrow> name \<Rightarrow> lam_raw \<Rightarrow> lam_raw"
+where
+  "subst_raw (Var_raw x) y s = (if x=y then s else (Var_raw x))"
+| "subst_raw (App_raw l r) y s = App_raw (subst_raw l y s) (subst_raw r y s)"
+| "subst_raw (Lam_raw x t) y s =
+      Lam_raw (new ({atom y} \<union> fv_lam_raw s \<union> fv_lam_raw t - {atom x}))
+       (subst_raw ((x \<leftrightarrow> (new ({atom y} \<union> fv_lam_raw s \<union> fv_lam_raw t - {atom x}))) \<bullet> t) y s)"
+  by (pat_completeness, auto)
+termination
+  apply (relation "measure (\<lambda>(t, y, s). (size t))")
+  apply (auto simp add: size_no_change)
+  done
+
+lemma fv_subst[simp]: "fv_lam_raw (subst_raw t y s) =
+  (if (atom y \<in> fv_lam_raw t) then fv_lam_raw s \<union> (fv_lam_raw t - {atom y}) else fv_lam_raw t)"
+  apply (induct t arbitrary: s)
+  apply (auto simp add: supp_at_base)[1]
+  apply (auto simp add: supp_at_base)[1]
+  apply (simp only: fv_lam_raw.simps)
+  apply simp
+  apply (rule conjI)
+  apply clarify
+  oops
+
+lemma new_eqvt[eqvt]: "p \<bullet> (new s) = new (p \<bullet> s)"
+  oops
+
+lemma subst_var_raw_eqvt[eqvt]: "p \<bullet> (subst_raw t y s) = subst_raw (p \<bullet> t) (p \<bullet> y) (p \<bullet> s)"
+  apply (induct t arbitrary: p y s)
+  apply simp_all
+  apply(perm_simp)
+  oops
+
+lemma subst_id: "alpha_lam_raw (subst_raw x d (Var_raw d)) x"
+  apply (induct x arbitrary: d)
+  apply (simp_all add: alpha_lam_raw.intros)
+  apply (rule alpha_lam_raw.intros)
+  apply (rule_tac x="(name \<leftrightarrow> new (insert (atom d) (supp d)))" in exI)
+  apply (simp add: alphas)
+  oops
+
+quotient_definition
+  subst2 ("_ [ _ ::= _ ]" [100,100,100] 100)
+where
+  "subst2 :: lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" is "subst_raw"
+
+lemmas fv_rsp = quot_respect(10)[simplified]
+
+lemma subst_rsp_pre1:
+  assumes a: "alpha_lam_raw a b"
+  shows "alpha_lam_raw (subst_raw a y c) (subst_raw b y c)"
+  using a
+  apply (induct a b arbitrary: c y rule: alpha_lam_raw.induct)
+  apply (simp add: equivp_reflp[OF lam_equivp])
+  apply (simp add: alpha_lam_raw.intros)
+  apply (simp only: alphas)
+  apply clarify
+  apply (simp only: subst_raw.simps)
+  apply (rule alpha_lam_raw.intros)
+  apply (simp only: alphas)
+  sorry
+
+lemma subst_rsp_pre2:
+  assumes a: "alpha_lam_raw a b"
+  shows "alpha_lam_raw (subst_raw c y a) (subst_raw c y b)"
+  using a
+  apply (induct c arbitrary: a b y)
+  apply (simp add: equivp_reflp[OF lam_equivp])
+  apply (simp add: alpha_lam_raw.intros)
+  apply simp
+  apply (rule alpha_lam_raw.intros)
+  apply (rule_tac x="((new (insert (atom y) (fv_lam_raw a \<union> fv_lam_raw c) -
+                       {atom name}))\<leftrightarrow>(new (insert (atom y) (fv_lam_raw b \<union> fv_lam_raw c) -
+                        {atom name})))" in exI)
+  apply (simp only: alphas)
+  apply (tactic {* split_conj_tac 1 *})
+  prefer 3
+  sorry
+
+lemma [quot_respect]:
+  "(alpha_lam_raw ===> op = ===> alpha_lam_raw ===> alpha_lam_raw) subst_raw subst_raw"
+  proof (intro fun_relI, simp)
+    fix a b c d :: lam_raw
+    fix y :: name
+    assume a: "alpha_lam_raw a b"
+    assume b: "alpha_lam_raw c d"
+    have c: "alpha_lam_raw (subst_raw a y c) (subst_raw b y c)" using subst_rsp_pre1 a by simp
+    then have d: "alpha_lam_raw (subst_raw b y c) (subst_raw b y d)" using subst_rsp_pre2 b by simp
+    show "alpha_lam_raw (subst_raw a y c) (subst_raw b y d)"
+      using c d equivp_transp[OF lam_equivp] by blast
+  qed
+
+lemma simp3:
+  "x \<noteq> y \<Longrightarrow> atom x \<notin> fv_lam_raw s \<Longrightarrow> alpha_lam_raw (subst_raw (Lam_raw x t) y s) (Lam_raw x (subst_raw t y s))"
+  apply simp
+  apply (rule alpha_lam_raw.intros)
+  apply (rule_tac x ="(x \<leftrightarrow> (new (insert (atom y) (fv_lam_raw s \<union> fv_lam_raw t) -
+                    {atom x})))" in exI)
+  apply (simp only: alphas)
+  sorry
+
+lemmas subst_simps = subst_raw.simps(1-2)[quot_lifted,no_vars]
+  simp3[quot_lifted,simplified lam.supp,simplified fresh_def[symmetric], no_vars]
+
+
+thm subst_raw.simps(3)[quot_lifted,no_vars]
 
 end
 




