
diff -r fb201e383f1b -r da575186d492 Nominal/Ex/Beta.thy
--- a/Nominal/Ex/Beta.thy	Tue Feb 19 05:38:46 2013 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,627 +0,0 @@
-theory Beta
-imports 
-  "../Nominal2"
-begin
-
-
-atom_decl name
-
-
-nominal_datatype lam =
-  Var "name"
-| App "lam" "lam"
-| Lam x::"name" l::"lam"  binds x in l ("Lam [_]. _" [100, 100] 100)
-
-section {* capture-avoiding substitution *}
-
-nominal_primrec
-  subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"  ("_ [_ ::= _]" [90, 90, 90] 90)
-where
-  "(Var x)[y ::= s] = (if x = y then s else (Var x))"
-| "(App t1 t2)[y ::= s] = App (t1[y ::= s]) (t2[y ::= s])"
-| "\<lbrakk>atom x \<sharp> y; atom x \<sharp> s\<rbrakk> \<Longrightarrow> (Lam [x]. t)[y ::= s] = Lam [x].(t[y ::= s])"
-  unfolding eqvt_def subst_graph_def
-  apply (rule, perm_simp, rule)
-  apply(rule TrueI)
-  apply(auto simp add: lam.distinct lam.eq_iff)
-  apply(rule_tac y="a" and c="(aa, b)" in lam.strong_exhaust)
-  apply(blast)+
-  apply(simp_all add: fresh_star_def fresh_Pair_elim)
-  apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2)
-  apply(simp_all add: Abs_fresh_iff)
-  apply(simp add: fresh_star_def fresh_Pair)
-  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
-  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
-done
-
-termination (eqvt)
-  by lexicographic_order
-
-lemma forget:
-  shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"
-  by (nominal_induct t avoiding: x s rule: lam.strong_induct)
-     (auto simp add: lam.fresh fresh_at_base)
-
-lemma fresh_fact:
-  fixes z::"name"
-  assumes a: "atom z \<sharp> s"
-      and b: "z = y \<or> atom z \<sharp> t"
-  shows "atom z \<sharp> t[y ::= s]"
-  using a b
-  by (nominal_induct t avoiding: z y s rule: lam.strong_induct)
-     (auto simp add: lam.fresh fresh_at_base)
-
-lemma substitution_lemma:  
-  assumes a: "x \<noteq> y" "atom x \<sharp> u"
-  shows "t[x ::= s][y ::= u] = t[y ::= u][x ::= s[y ::= u]]"
-using a 
-by (nominal_induct t avoiding: x y s u rule: lam.strong_induct)
-   (auto simp add: fresh_fact forget)
-
-inductive
-  beta :: "lam \<Rightarrow> lam \<Rightarrow> bool" ("_ \<rightarrow> _")
-where
-  beta_beta:  "atom x \<sharp> s \<Longrightarrow> App (Lam [x].t) s \<rightarrow> t[x ::= s]"
-| beta_Lam:   "t \<rightarrow> s \<Longrightarrow> Lam [x].t \<rightarrow> Lam [x].s"
-| beta_App1:  "t \<rightarrow> s \<Longrightarrow> App t u \<rightarrow> App s u"
-| beta_App2:  "t \<rightarrow> s \<Longrightarrow> App u t \<rightarrow> App u s"
-
-equivariance beta
-
-nominal_inductive beta
-  avoids beta_beta: "x" 
-       | beta_Lam: "x"
-by (simp_all add: fresh_star_def fresh_Pair lam.fresh fresh_fact)
-
-inductive
-  equ :: "lam \<Rightarrow> lam \<Rightarrow> bool" ("_ \<approx> _")
-where
-  equ_beta:  "atom x \<sharp> s \<Longrightarrow> App (Lam [x].t) s \<approx> t[x ::= s]"
-| equ_refl:  "t \<approx> t"
-| equ_sym:   "t \<approx> s \<Longrightarrow> s \<approx> t"
-| equ_trans: "\<lbrakk>t1 \<approx> t2; t2 \<approx> t3\<rbrakk> \<Longrightarrow> t1 \<approx> t3"
-| equ_Lam:   "t \<approx> s \<Longrightarrow> Lam [x].t \<approx> Lam [x].s"
-| equ_App1:  "t \<approx> s \<Longrightarrow> App t u \<approx> App s u"
-| equ_App2:  "t \<approx> s \<Longrightarrow> App u t \<approx> App u s"
-
-equivariance equ
-
-nominal_inductive equ
-  avoids equ_beta: "x" 
-       | equ_Lam: "x"
-by (simp_all add: fresh_star_def fresh_Pair lam.fresh fresh_fact)
-
-inductive
-  equ2 :: "lam \<Rightarrow> lam \<Rightarrow> bool" ("_ \<approx>2 _")
-where
-  equ2_beta1:  "\<lbrakk>atom x \<sharp> (t2, s2); s1 \<approx>2 t1; t2 \<approx>2  s2\<rbrakk> \<Longrightarrow> App (Lam [x].t1) t2 \<approx>2 s1[x ::= s2]"
-| equ2_beta2:  "\<lbrakk>atom x \<sharp> (t2, s2); s1 \<approx>2 t1; t2 \<approx>2  s2\<rbrakk> \<Longrightarrow> s1[x ::= s2] \<approx>2 App (Lam [x].t1) t2"
-| equ2_Var:   "Var x \<approx>2 Var x"
-| equ2_Lam:   "t \<approx>2 s \<Longrightarrow> Lam [x].t \<approx>2 Lam [x].s"
-| equ2_App:   "\<lbrakk>t1 \<approx>2 s1; t2 \<approx>2 s2\<rbrakk> \<Longrightarrow> App t1 t2 \<approx>2 App s1 s2"
-
-equivariance equ2
-
-nominal_inductive equ2
-  avoids equ2_beta1: "x"
-       | equ2_beta2: "x" 
-       | equ2_Lam: "x"
-by (simp_all add: fresh_star_def fresh_Pair lam.fresh fresh_fact)
-
-lemma equ2_refl:
-  fixes t::"lam"
-  shows "t \<approx>2 t"
-apply(induct t rule: lam.induct)
-apply(auto intro: equ2.intros)
-done
-
-lemma equ2_symm:
-  fixes t s::"lam"
-  assumes "t \<approx>2 s"
-  shows "s \<approx>2 t"
-using assms
-apply(induct)
-apply(auto intro: equ2.intros)
-done
-
-lemma equ2_trans:
-  fixes t1 t2 t3::"lam"
-  assumes "t1 \<approx>2 t2" "t2 \<approx>2 t3"
-  shows "t1 \<approx>2 t3"
-using assms
-apply(nominal_induct t1 t2 avoiding: t3 rule: equ2.strong_induct)
-defer
-defer
-apply(erule equ2.cases)
-apply(auto)
-apply(rule equ2_beta2)
-apply(simp)
-apply(simp)
-apply(simp)
-apply(rule equ2_refl)
-defer
-apply(rotate_tac 4)
-apply(erule equ2.cases)
-apply(auto)
-oops
-
-lemma "a \<noteq> x \<Longrightarrow> x \<noteq> y \<Longrightarrow> y \<noteq> a \<Longrightarrow> y \<noteq> b \<Longrightarrow> (App (App (Lam[x]. Lam[y]. (App (Var y) (Var x))) (Var a)) (Var b) \<approx> App (Var b) (Var a))"
-  apply (rule equ_trans)
-  apply (rule equ_App1)
-  apply (rule equ_beta)
-  apply (simp add: lam.fresh fresh_at_base)
-  apply (subst subst.simps)
-  apply (simp add: lam.fresh fresh_at_base)+
-  apply (rule equ_trans)
-  apply (rule equ_beta)
-  apply (simp add: lam.fresh fresh_at_base)
-  apply (simp add: equ_refl)
-  done
-
-lemma "\<not> (Var b \<approx>2 Var a) \<Longrightarrow> \<not> (App (App (Lam[x]. Lam[y]. (App (Var y) (Var x))) (Var a)) (Var b) \<approx>2 App (Var b) (Var a))"
-  apply rule
-  apply (erule equ2.cases)
-  apply auto
-  done
-
-lemma [quot_respect]:
-  shows "(op = ===> equ) Var Var"
-  and   "(equ ===> equ ===> equ) App App"
-  and   "(op = ===> equ ===> equ) Beta.Lam Beta.Lam"
-unfolding fun_rel_def
-by (auto intro: equ.intros)
-
-lemma equ_subst1:
-  assumes "t \<approx> s"
-  shows "t[x ::= u] \<approx> s[x ::= u]"
-using assms
-apply(nominal_induct avoiding: x u rule: equ.strong_induct)
-apply(simp)
-apply(rule equ_trans)
-apply(rule equ_beta)
-apply(simp add: fresh_fact)
-apply(subst (2) substitution_lemma)
-apply(simp add: fresh_at_base)
-apply(simp)
-apply(rule equ_refl)
-apply(rule equ_refl)
-apply(auto intro: equ_sym)[1]
-apply(blast intro: equ_trans)
-apply(simp add: equ_Lam)
-apply(simp add: equ_App1)
-apply(simp add: equ_App2)
-done
-
-lemma equ_subst2:
-  assumes "t \<approx> s"
-  shows "u[x ::= t] \<approx> u[x ::= s]"
-using assms
-apply(nominal_induct u avoiding: x t s rule: lam.strong_induct)
-apply(simp add: equ_refl)
-apply(simp)
-apply(smt equ_App1 equ_App2 equ_trans)
-apply(simp)
-apply(metis equ_Lam)
-done
-
-lemma [quot_respect]:
-  shows "(equ ===> op = ===> equ ===> equ) subst subst"
-unfolding fun_rel_def
-by (metis equ_subst1 equ_subst2 equ_trans)
-
-lemma [quot_respect]:
-  shows "(op = ===> equ ===> equ) permute permute"
-unfolding fun_rel_def
-by (auto intro: eqvts)
-
-quotient_type qlam = lam / equ
-apply(rule equivpI)
-apply(rule reflpI)
-apply(simp add: equ_refl)
-apply(rule sympI)
-apply(simp add: equ_sym)
-apply(rule transpI)
-apply(auto intro: equ_trans)
-done
-
-quotient_definition "QVar::name \<Rightarrow> qlam" is Var
-quotient_definition "QApp::qlam \<Rightarrow> qlam \<Rightarrow> qlam" is App
-quotient_definition QLam ("QLam [_]._") 
-  where "QLam::name \<Rightarrow> qlam \<Rightarrow> qlam" is Beta.Lam
-
-lemmas qlam_strong_induct = lam.strong_induct[quot_lifted]
-lemmas qlam_induct = lam.induct[quot_lifted]
-
-instantiation qlam :: pt
-begin
-
-quotient_definition "permute_qlam::perm \<Rightarrow> qlam \<Rightarrow> qlam" 
-  is "permute::perm \<Rightarrow> lam \<Rightarrow> lam"
-
-instance
-apply default
-apply(descending)
-apply(simp)
-apply(rule equ_refl)
-apply(descending)
-apply(simp)
-apply(rule equ_refl)
-done
-
-end
-
-lemma qlam_perm[simp]:
-  shows "p \<bullet> (QVar x) = QVar (p \<bullet> x)"
-  and "p \<bullet> (QApp t s) = QApp (p \<bullet> t) (p \<bullet> s)"
-  and "p \<bullet> (QLam [x].t) = QLam [p \<bullet> x].(p \<bullet> t)"
-apply(descending)
-apply(simp add: equ_refl)
-apply(descending)
-apply(simp add: equ_refl)
-apply(descending)
-apply(simp add: equ_refl)
-done
-
-lemma qlam_supports:
-  shows "{atom x} supports (QVar x)"
-  and   "supp (t, s) supports (QApp t s)"
-  and   "supp (x, t) supports (QLam [x].t)"
-unfolding supports_def fresh_def[symmetric]
-by (auto simp add: swap_fresh_fresh)
-
-lemma qlam_fs:
-  fixes t::"qlam"
-  shows "finite (supp t)"
-apply(induct t rule: qlam_induct)
-apply(rule supports_finite)
-apply(rule qlam_supports)
-apply(simp)
-apply(rule supports_finite)
-apply(rule qlam_supports)
-apply(simp add: supp_Pair)
-apply(rule supports_finite)
-apply(rule qlam_supports)
-apply(simp add: supp_Pair finite_supp)
-done
-
-instantiation qlam :: fs
-begin
-
-instance
-apply(default)
-apply(rule qlam_fs)
-done
-
-end
-
-quotient_definition subst_qlam ("_[_ ::q= _]")
-  where "subst_qlam::qlam \<Rightarrow> name \<Rightarrow> qlam \<Rightarrow> qlam" is subst  
-
-lemma
-  "(QVar x)[y ::q= s] = (if x = y then s else (QVar x))"
-apply(descending)
-apply(simp)
-apply(auto intro: equ_refl)
-done
-
-lemma
-  "(QApp t1 t2)[y ::q= s] = QApp (t1[y ::q= s]) (t2[y ::q= s])"
-apply(descending)
-apply(simp)
-apply(rule equ_refl)
-done
-
-definition
-  "Supp t = \<Inter>{supp s | s. s \<approx> t}"
-
-lemma [quot_respect]:
-  shows "(equ ===> op=) Supp Supp"
-unfolding fun_rel_def
-unfolding Supp_def
-apply(rule allI)+
-apply(rule impI)
-apply(rule_tac f="Inter" in arg_cong)
-apply(auto)
-apply (metis equ_trans)
-by (metis equivp_def qlam_equivp)
-
-quotient_definition "supp_qlam::qlam \<Rightarrow> atom set" 
-  is "Supp::lam \<Rightarrow> atom set"
-
-lemma Supp_supp:
-  "Supp t \<subseteq> supp t"
-unfolding Supp_def
-apply(auto)
-apply(drule_tac x="supp t" in spec)
-apply(auto simp add: equ_refl)
-done
-
-lemma Supp_Lam:
-  "atom a \<notin> Supp (Lam [a].t)"
-proof -
-  have "atom a \<notin> supp (Lam [a].t)" by (simp add: lam.supp)
-  then show ?thesis using Supp_supp
-    by blast
-qed
-
-lemmas s = Supp_Lam[quot_lifted]
-
-definition
-  ssupp :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ ssupp _")
-where
-  "A ssupp x \<equiv> \<forall>p. (\<forall>a \<in> A. (p \<bullet> a) = a) \<longleftrightarrow> (p \<bullet> x) = x"
-
-lemma ssupp_supports:
-  "A ssupp x \<Longrightarrow> A supports x"
-unfolding ssupp_def supports_def
-apply(rule allI)+
-apply(drule_tac x="(a \<rightleftharpoons> b)" in spec)
-apply(auto)
-by (metis swap_atom_simps(3))
-
-lemma ssupp_supp:
-  assumes a: "finite A"
-  and     b: "A ssupp x"
-  shows "supp x = A"
-proof -
-  { assume 0: "A - supp x \<noteq> {}"
-    then obtain a where 1: "a \<in> A - supp x" by auto
-    obtain a' where *: "a' \<in>  UNIV - A" and **: "sort_of a' = sort_of a"
-      by (rule obtain_atom[OF a]) (auto)  
-    have "(a \<rightleftharpoons> a') \<bullet> a = a'" using 1 ** * by (auto)
-    then have w0: "(a \<rightleftharpoons> a') \<bullet> x \<noteq> x"
-      using b unfolding ssupp_def 
-      apply -
-      apply(drule_tac x="(a \<rightleftharpoons> a')" in spec)
-      apply(auto)
-      using 1 *
-      apply(auto)
-      done
-    have w1: "a \<sharp> x" unfolding fresh_def using 1 by auto 
-    have w2: "a' \<sharp> x" using *
-      apply(rule_tac supports_fresh)
-      apply(rule ssupp_supports)
-      apply(rule b)
-      apply(rule a)
-      apply(auto)
-      done
-    have "False" using w0 w1 w2 by (simp add: swap_fresh_fresh)
-    then have "supp x = A" by simp }
-  moreover
-  { assume "A - supp x = {}"
-    moreover
-    have "supp x \<subseteq> A"
-      apply(rule supp_is_subset)
-      apply(rule ssupp_supports)
-      apply(rule b)
-      apply(rule a)
-      done
-    ultimately have "supp x = A"
-      by blast
-  }
-  ultimately show "supp x = A" by blast
-qed
-
-lemma
-  "(supp x) ssupp x"
-unfolding ssupp_def
-apply(auto)
-apply(rule supp_perm_eq)
-apply(simp add: fresh_star_def)
-apply(auto)
-apply(simp add: fresh_perm)
-apply(simp add: fresh_perm[symmetric])
-(*Tzevelekos 2008, section 2.1.2 property 2.5*)
-oops
-
-(* The above is not true. Take p=(a\<leftrightarrow>b) and x={a,b}, then the goal is provably false *)
-lemma
-  "b \<noteq> a \<Longrightarrow> \<not>(supp {a :: name,b}) ssupp {a,b}"
-unfolding ssupp_def
-apply (auto)
-apply (intro exI[of _ "(a\<leftrightarrow>b) :: perm"])
-apply (subgoal_tac "(a \<leftrightarrow> b) \<bullet> {a, b} = {a, b}")
-apply simp
-apply (subgoal_tac "supp {a, b} = {atom a, atom b}")
-apply (simp add: flip_def)
-apply (simp add: supp_of_finite_insert supp_at_base supp_set_empty)
-apply blast
-by (smt empty_eqvt flip_at_simps(1) flip_at_simps(2) insert_commute insert_eqvt)
-
-function
-  qsubst :: "qlam \<Rightarrow> name \<Rightarrow> qlam \<Rightarrow> qlam"  ("_ [_ ::qq= _]" [90, 90, 90] 90)
-where
-  "(QVar x)[y ::qq= s] = (if x = y then s else (QVar x))"
-| "(QApp t1 t2)[y ::qq= s] = QApp (t1[y ::qq= s]) (t2[y ::qq= s])"
-| "\<lbrakk>atom x \<sharp> y; atom x \<sharp> s\<rbrakk> \<Longrightarrow> (QLam [x]. t)[y ::qq= s] = QLam [x].(t[y ::qq= s])"
-apply(simp_all add:)
-oops
-
-
-lemma
-  assumes a: "Lam [x].t \<approx> s"
-  shows "\<exists>x' t'. s = Lam [x']. t' \<and> t \<approx> t'"
-using a
-apply(induct s rule:lam.induct)
-apply(erule equ.cases)
-apply(auto)
-apply(erule equ.cases)
-apply(auto)
-
-
-
-lemma
-  "atom x \<sharp> y \<Longrightarrow> atom x \<notin> supp_qlam s \<Longrightarrow> (QLam [x].t)[y ::q= s] = QLam [x].(t[y ::q= s])"
-apply(descending)
-oops
-
-
-lemma [eqvt]: "(p \<bullet> abs_qlam t) = abs_qlam (p \<bullet> t)"
- apply (subst fun_cong[OF fun_cong[OF meta_eq_to_obj_eq[OF permute_qlam_def]], of p, simplified])
- apply (subst Quotient_rel[OF Quotient_qlam, simplified equivp_reflp[OF qlam_equivp], simplified])
- by (metis Quotient_qlam equ_refl eqvt rep_abs_rsp_left)
-
-lemma supports_abs_qlam:
-  "(supp t) supports (abs_qlam t)"
-unfolding supports_def
-unfolding fresh_def[symmetric]
-apply(auto)
-apply(perm_simp)
-apply(simp add: swap_fresh_fresh)
-done
-
-lemma rep_qlam_inverse:
-  "abs_qlam (rep_qlam t) = t"
-by (metis Quotient_abs_rep Quotient_qlam)
-
-lemma supports_rep_qlam:
-  "supp (rep_qlam t) supports t"
-apply(subst (2) rep_qlam_inverse[symmetric])
-apply(rule supports_abs_qlam)
-done
-
- lemma supports_qvar:
-  "{atom x} supports (QVar x)"
-apply(subgoal_tac "supp (Var x) supports (abs_qlam (Var x))")
-apply(simp add: lam.supp supp_at_base)
-defer
-apply(rule supports_abs_qlam)
-apply(simp add: QVar_def)
-done
-
-section {* Supp *}
-
-lemma s:
-  assumes "s \<approx> t"
-  shows "a \<sharp> (abs_qlam s) \<longleftrightarrow> a \<sharp> (abs_qlam t)"
-using assms
-by (metis Quotient_qlam Quotient_rel_abs)
-
-lemma ss:
-  "Supp t supports (abs_qlam t)"
-apply(simp only: supports_def)
-apply(rule allI)+
-apply(rule impI)
-apply(rule swap_fresh_fresh)
-apply(drule conjunct1)
-apply(auto)[1]
-apply(simp add: Supp_def)
-apply(auto)[1]
-apply(simp add: s[symmetric])
-apply(rule supports_fresh)
-apply(rule supports_abs_qlam)
-apply(simp add: finite_supp)
-apply(simp)
-apply(drule conjunct2)
-apply(auto)[1]
-apply(simp add: Supp_def)
-apply(auto)[1]
-apply(simp add: s[symmetric])
-apply(rule supports_fresh)
-apply(rule supports_abs_qlam)
-apply(simp add: finite_supp)
-apply(simp)
-done
-
-lemma
-  fixes t::"qlam"
-  shows "(supp x) supports (QVar x)"
-apply(rule_tac x="QVar x" in Abs_qlam_cases)
-apply(auto)
-thm QVar_def
-oops
-
-
- 
-section {* Size *}
-
-definition
-  "Size t = Least {size s | s. s \<approx> t}" 
-
-lemma [quot_respect]:
-  shows "(equ ===> op=) Size Size"
-unfolding fun_rel_def
-unfolding Size_def
-apply(auto)
-apply(rule_tac f="Least" in arg_cong)
-apply(auto)
-apply (metis equ_trans)
-by (metis equivp_def qlam_equivp)
-
-instantiation qlam :: size
-begin
-
-quotient_definition "size_qlam::qlam \<Rightarrow> nat" 
-  is "Size::lam \<Rightarrow> nat"
-
-instance
-apply default
-done
-
-end
-
-thm lam.size
-
-lemma
-  "size (QVar x) = 0"
-apply(descending)
-apply(simp add: Size_def)
-apply(rule Least_equality)
-apply(auto)
-apply(simp add: Collect_def)
-apply(rule_tac x="Var x" in exI)
-apply(auto intro: equ_refl)
-done
-
-lemma
-  "size (QLam [x].t) = Suc (size t)"
-apply(descending)
-apply(simp add: Size_def)
-thm Least_Suc
-apply(rule trans)
-apply(rule_tac n="Suc (size t)" in Least_Suc)
-apply(simp add: Collect_def)
-apply(rule_tac x="Lam [x].t" in exI)
-apply(auto intro: equ_refl)[1]
-apply(simp add: Collect_def)
-apply(auto)
-defer
-apply(simp add: Collect_def)
-apply(rule_tac x="t" in exI)
-apply(auto intro: equ_refl)[1]
-apply(simp add: Collect_def)
-defer
-apply(simp add: Collect_def)
-apply(auto)[1]
-
-apply(auto)
-done
-
-term rep_qlam
-lemmas qlam_size_def = Size_def[quot_lifted]
-
-lemma [quot_preserve]:
-  assumes "Quotient equ Abs Rep"
-  shows "(id ---> Rep ---> id) fresh = fresh"
-using assms
-unfolding Quotient_def
-apply(simp add: map_fun_def)
-apply(simp add: comp_def fun_eq_iff)
-
-sorry
-
-lemma [simp]:
-  shows "(QVar x)[y :::= s] = (if x = y then s else (QVar x))"
-  and "(QApp t1 t2)[y :::= s] = QApp (t1[y :::= s]) (t2[y :::= s])"
-  and "\<lbrakk>atom x \<sharp> y; atom x \<sharp> s\<rbrakk> \<Longrightarrow> (QLam [x]. t)[y :::= s] = QLam [x].(t[y :::= s])"
-apply(lifting subst.simps(1))
-apply(lifting subst.simps(2))
-apply(lifting subst.simps(3))
-done
-
-
-
-end
-
-
-




