--- a/Nominal/Ex/LetRecB.thy Sun Jul 17 11:33:09 2011 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,196 +0,0 @@
-theory LetRecB
-imports "../Nominal2"
-begin
-
-atom_decl name
-
-nominal_datatype let_rec:
- trm =
- Var "name"
-| App "trm" "trm"
-| Lam x::"name" t::"trm" binds x in t
-| Let_Rec bp::"bp" t::"trm" binds "bn bp" in bp t
-and bp =
- Bp "name" "trm"
-binder
- bn::"bp \<Rightarrow> atom list"
-where
- "bn (Bp x t) = [atom x]"
-
-thm let_rec.distinct
-thm let_rec.induct
-thm let_rec.exhaust
-thm let_rec.fv_defs
-thm let_rec.bn_defs
-thm let_rec.perm_simps
-thm let_rec.eq_iff
-thm let_rec.fv_bn_eqvt
-thm let_rec.size_eqvt
-
-
-lemma Abs_lst_fcb2:
- fixes as bs :: "'a :: fs"
- and x y :: "'b :: fs"
- and c::"'c::fs"
- assumes eq: "[ba as]lst. x = [ba bs]lst. y"
- and fcb1: "(set (ba as)) \<sharp>* f as x c"
- and fresh1: "set (ba as) \<sharp>* c"
- and fresh2: "set (ba 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"
- and props: "eqvt ba" "inj ba"
- 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 (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and
- fr2: "supp q \<sharp>* ([ba as]lst. x)" and
- inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))"
- using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)" and x="[ba as]lst. x"]
- fin1 fin2
- by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
- have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp
- also have "\<dots> = [ba as]lst. x"
- by (simp only: fr2 perm_supp_eq)
- finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. y" using eq by simp
- then obtain r::perm where
- qq1: "q \<bullet> x = r \<bullet> y" and
- qq2: "q \<bullet> (ba as) = r \<bullet> (ba bs)" and
- qq3: "supp r \<subseteq> (q \<bullet> (set (ba as))) \<union> set (ba 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 qq4: "q \<bullet> as = r \<bullet> bs" using qq2 props unfolding eqvt_def inj_on_def
- apply(perm_simp)
- apply(simp)
- done
- have "(set (ba as)) \<sharp>* f as x c" by (rule fcb1)
- then have "q \<bullet> ((set (ba as)) \<sharp>* f as x c)"
- by (simp add: permute_bool_def)
- then have "set (q \<bullet> (ba 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> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 qq4
- by simp
- then have "r \<bullet> ((set (ba 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 (ba 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 qq4 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 max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
- by (simp add: permute_pure)
-
-nominal_primrec
- height_trm :: "trm \<Rightarrow> nat"
-and height_bp :: "bp \<Rightarrow> nat"
-where
- "height_trm (Var x) = 1"
-| "height_trm (App l r) = max (height_trm l) (height_trm r)"
-| "height_trm (Lam v b) = 1 + (height_trm b)"
-| "height_trm (Let_Rec bp b) = max (height_bp bp) (height_trm b)"
-| "height_bp (Bp v t) = height_trm t"
- --"eqvt"
- apply (simp only: eqvt_def height_trm_height_bp_graph_def)
- apply (rule, perm_simp, rule, rule TrueI)
- --"completeness"
- apply (case_tac x)
- apply (case_tac a rule: let_rec.exhaust(1))
- apply (auto)[4]
- apply (case_tac b rule: let_rec.exhaust(2))
- apply blast
- apply(simp_all)
- apply (erule_tac c="()" in Abs_lst_fcb2)
- apply (simp_all add: fresh_star_def pure_fresh)[3]
- apply (simp add: eqvt_at_def)
- apply (simp add: eqvt_at_def)
- apply(simp add: eqvt_def)
- apply(perm_simp)
- apply(simp)
- apply(simp add: inj_on_def)
- --"The following could be done by nominal"
- apply (simp add: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])
- apply (simp add: meta_eq_to_obj_eq[OF height_bp_def, symmetric, unfolded fun_eq_iff])
- apply (subgoal_tac "eqvt_at height_bp bp")
- apply (subgoal_tac "eqvt_at height_bp bpa")
- apply (subgoal_tac "eqvt_at height_trm b")
- apply (subgoal_tac "eqvt_at height_trm ba")
- apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inr bp)")
- apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inr bpa)")
- apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inl b)")
- apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inl ba)")
- defer
- apply (simp add: eqvt_at_def height_trm_def)
- apply (simp add: eqvt_at_def height_trm_def)
- apply (simp add: eqvt_at_def height_bp_def)
- apply (simp add: eqvt_at_def height_bp_def)
- apply (subgoal_tac "height_bp bp = height_bp bpa")
- apply (subgoal_tac "height_trm b = height_trm ba")
- apply simp
- apply (subgoal_tac "(\<lambda>as x c. height_trm (snd (bp, b))) as x c = (\<lambda>as x c. height_trm (snd (bpa, ba))) as x c")
- apply simp
- apply (erule_tac c="()" in Abs_lst_fcb2)
- apply (simp add: fresh_star_def pure_fresh)
- apply (simp add: fresh_star_def pure_fresh)
- apply (simp add: fresh_star_def pure_fresh)
- apply (simp add: eqvt_at_def)
- apply (simp add: eqvt_at_def)
- defer defer
- apply (subgoal_tac "(\<lambda>as x c. height_bp (fst (bp, b))) as x c = (\<lambda>as x c. height_bp (fst (bpa, ba))) as x c")
- apply simp
- apply (erule_tac c="()" in Abs_lst_fcb2)
- apply (simp add: fresh_star_def pure_fresh)
- apply (simp add: fresh_star_def pure_fresh)
- apply (simp add: fresh_star_def pure_fresh)
- apply (simp add: fresh_star_def pure_fresh)
- apply (simp add: eqvt_at_def)
- apply (simp add: eqvt_at_def)
---""
- apply(simp_all add: eqvt_def inj_on_def)
- apply(perm_simp)
- apply(simp)
- apply(perm_simp)
- apply(simp)
- done
-
-termination by lexicographic_order
-
-end
-
-
-