theory LetSimple1+
−
imports "../Nominal2" +
−
begin+
−
+
−
lemma Abs_lst_fcb2:+
−
  fixes as bs :: "atom list"+
−
    and x y :: "'b :: fs"+
−
    and c::"'c::fs"+
−
  assumes eq: "[as]lst. x = [bs]lst. y"+
−
  and fcb1: "(set as) \<sharp>* f as x c"+
−
  and fresh1: "set as \<sharp>* c"+
−
  and fresh2: "set bs \<sharp>* c"+
−
  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"+
−
  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"+
−
  shows "f as x c = f bs y c"+
−
proof -+
−
  have "supp (as, x, c) supports (f as x c)"+
−
    unfolding  supports_def fresh_def[symmetric]+
−
    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)+
−
  then have fin1: "finite (supp (f as x c))"+
−
    by (auto intro: supports_finite simp add: finite_supp)+
−
  have "supp (bs, y, c) supports (f bs y c)"+
−
    unfolding  supports_def fresh_def[symmetric]+
−
    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)+
−
  then have fin2: "finite (supp (f bs y c))"+
−
    by (auto intro: supports_finite simp add: finite_supp)+
−
  obtain q::"perm" where +
−
    fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and +
−
    fr2: "supp q \<sharp>* Abs_lst as x" and +
−
    inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"+
−
    using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"]  +
−
      fin1 fin2+
−
    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)+
−
  have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp+
−
  also have "\<dots> = Abs_lst as x"+
−
    by (simp only: fr2 perm_supp_eq)+
−
  finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp+
−
  then obtain r::perm where +
−
    qq1: "q \<bullet> x = r \<bullet> y" and +
−
    qq2: "q \<bullet> as = r \<bullet> bs" and +
−
    qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"+
−
    apply(drule_tac sym)+
−
    apply(simp only: Abs_eq_iff2 alphas)+
−
    apply(erule exE)+
−
    apply(erule conjE)++
−
    apply(drule_tac x="p" in meta_spec)+
−
    apply(simp add: set_eqvt)+
−
    apply(blast)+
−
    done+
−
  have "(set as) \<sharp>* f as x c" by (rule fcb1)+
−
  then have "q \<bullet> ((set as) \<sharp>* f as x c)"+
−
    by (simp add: permute_bool_def)+
−
  then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"+
−
    apply(simp add: fresh_star_eqvt set_eqvt)+
−
    apply(subst (asm) perm1)+
−
    using inc fresh1 fr1+
−
    apply(auto simp add: fresh_star_def fresh_Pair)+
−
    done+
−
  then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp+
−
  then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"+
−
    apply(simp add: fresh_star_eqvt set_eqvt)+
−
    apply(subst (asm) perm2[symmetric])+
−
    using qq3 fresh2 fr1+
−
    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)+
−
    done+
−
  then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)+
−
  have "f as x c = q \<bullet> (f as x c)"+
−
    apply(rule perm_supp_eq[symmetric])+
−
    using inc fcb1 fr1 by (auto simp add: fresh_star_def)+
−
  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" +
−
    apply(rule perm1)+
−
    using inc fresh1 fr1 by (auto simp add: fresh_star_def)+
−
  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp+
−
  also have "\<dots> = r \<bullet> (f bs y c)"+
−
    apply(rule perm2[symmetric])+
−
    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)+
−
  also have "... = f bs y c"+
−
    apply(rule perm_supp_eq)+
−
    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)+
−
  finally show ?thesis by simp+
−
qed+
−
+
−
lemma Abs_lst1_fcb2:+
−
  fixes a b :: "atom"+
−
    and x y :: "'b :: fs"+
−
    and c::"'c :: fs"+
−
  assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"+
−
  and fcb1: "a \<sharp> f a x c"+
−
  and fresh: "{a, b} \<sharp>* c"+
−
  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"+
−
  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"+
−
  shows "f a x c = f b y c"+
−
using e+
−
apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])+
−
apply(simp_all)+
−
using fcb1 fresh perm1 perm2+
−
apply(simp_all add: fresh_star_def)+
−
done+
−
+
−
+
−
atom_decl name+
−
+
−
nominal_datatype trm =+
−
  Var "name"+
−
| App "trm" "trm"+
−
| Let x::"name" "trm" t::"trm"   bind x in t+
−
+
−
print_theorems+
−
+
−
thm trm.fv_defs+
−
thm trm.eq_iff +
−
thm trm.bn_defs+
−
thm trm.bn_inducts+
−
thm trm.perm_simps+
−
thm trm.induct+
−
thm trm.inducts+
−
thm trm.distinct+
−
thm trm.supp+
−
thm trm.fresh+
−
thm trm.exhaust+
−
thm trm.strong_exhaust+
−
thm trm.perm_bn_simps+
−
+
−
nominal_primrec+
−
    height_trm :: "trm \<Rightarrow> nat"+
−
where+
−
  "height_trm (Var x) = 1"+
−
| "height_trm (App l r) = max (height_trm l) (height_trm r)"+
−
| "height_trm (Let x t s) = max (height_trm t) (height_trm s)"+
−
  apply (simp only: eqvt_def height_trm_graph_def)+
−
  apply (rule, perm_simp, rule, rule TrueI)+
−
  apply (case_tac x rule: trm.exhaust(1))+
−
  apply (auto)[3]+
−
  apply(simp_all)[5]+
−
  apply (subgoal_tac "height_trm_sumC t = height_trm_sumC ta")+
−
  apply (subgoal_tac "height_trm_sumC s = height_trm_sumC sa")+
−
  apply simp+
−
  apply(simp)+
−
  apply(erule conjE)+
−
  apply(erule_tac c="()" in Abs_lst1_fcb2)+
−
  apply(simp_all add: fresh_star_def pure_fresh)[2]+
−
  apply(simp_all add: eqvt_at_def)[2]+
−
  apply(simp)+
−
  done+
−
+
−
termination+
−
  by lexicographic_order+
−
+
−
+
−
nominal_primrec (invariant "\<lambda>x (y::atom set). finite y")+
−
  frees_set :: "trm \<Rightarrow> atom set"+
−
where+
−
  "frees_set (Var x) = {atom x}"+
−
| "frees_set (App t1 t2) = frees_set t1 \<union> frees_set t2"+
−
| "frees_set (Let x t s) = (frees_set s) - {atom x} \<union> (frees_set t)"+
−
  apply(simp add: eqvt_def frees_set_graph_def)+
−
  apply(rule, perm_simp, rule)+
−
  apply(erule frees_set_graph.induct)+
−
  apply(auto)[3]+
−
  apply(rule_tac y="x" in trm.exhaust)+
−
  apply(auto)[3]+
−
  apply(simp_all)[5]+
−
  apply(simp)+
−
  apply(erule conjE)+
−
  apply(subgoal_tac "frees_set_sumC s - {atom x} = frees_set_sumC sa - {atom xa}")+
−
  apply(simp)+
−
  apply(erule_tac c="()" in Abs_lst1_fcb2)+
−
  apply(simp add: fresh_minus_atom_set)+
−
  apply(simp add: fresh_star_def fresh_Unit)+
−
  apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl)+
−
  apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl)+
−
  done+
−
+
−
termination+
−
  by lexicographic_order+
−
+
−
+
−
nominal_primrec+
−
  subst :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm"  ("_ [_ ::= _]" [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])"+
−
| "atom x \<sharp> (y, s) \<Longrightarrow> (Let x t t')[y ::= s] = Let x (t[y ::= s]) (t'[y ::= s])"+
−
  apply(simp add: eqvt_def subst_graph_def)+
−
  apply (rule, perm_simp, rule)+
−
  apply(rule TrueI)+
−
  apply(auto)[1]+
−
  apply(rule_tac y="a" and c="(aa, b)" in trm.strong_exhaust)+
−
  apply(blast)++
−
  apply(simp_all add: fresh_star_def fresh_Pair_elim)[1]+
−
  apply(blast)+
−
  apply(simp_all)[5]+
−
  apply(simp (no_asm_use))+
−
  apply(simp)+
−
  apply(erule conjE)++
−
  apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2)+
−
  apply(simp 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+
−
  by lexicographic_order+
−
+
−
+
−
end+
−