2936
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theory Let
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imports "../Nominal2"
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begin
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atom_decl name
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nominal_datatype trm =
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Var "name"
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| App "trm" "trm"
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2950
0911cb7bf696
changed bind to binds in specifications; bind will cause trouble with Monad_Syntax
Christian Urban <urbanc@in.tum.de>
diff
changeset
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| Lam x::"name" t::"trm" binds x in t
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0911cb7bf696
changed bind to binds in specifications; bind will cause trouble with Monad_Syntax
Christian Urban <urbanc@in.tum.de>
diff
changeset
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| Let as::"assn" t::"trm" binds "bn as" in t
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2936
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and assn =
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ANil
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| ACons "name" "trm" "assn"
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binder
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bn
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where
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"bn ANil = []"
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| "bn (ACons x t as) = (atom x) # (bn as)"
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print_theorems
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thm alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.intros
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thm bn_raw.simps
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thm permute_bn_raw.simps
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thm trm_assn.perm_bn_alpha
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thm trm_assn.permute_bn
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thm trm_assn.fv_defs
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thm trm_assn.eq_iff
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thm trm_assn.bn_defs
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thm trm_assn.bn_inducts
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thm trm_assn.perm_simps
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thm trm_assn.induct
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thm trm_assn.inducts
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thm trm_assn.distinct
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thm trm_assn.supp
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thm trm_assn.fresh
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thm trm_assn.exhaust
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thm trm_assn.strong_exhaust
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thm trm_assn.perm_bn_simps
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lemma alpha_bn_inducts_raw[consumes 1]:
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"\<lbrakk>alpha_bn_raw a b; P3 ANil_raw ANil_raw;
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\<And>trm_raw trm_rawa assn_raw assn_rawa name namea.
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\<lbrakk>alpha_trm_raw trm_raw trm_rawa; alpha_bn_raw assn_raw assn_rawa;
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P3 assn_raw assn_rawa\<rbrakk>
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\<Longrightarrow> P3 (ACons_raw name trm_raw assn_raw)
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(ACons_raw namea trm_rawa assn_rawa)\<rbrakk> \<Longrightarrow> P3 a b"
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by (erule alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.inducts(3)[of _ _ "\<lambda>x y. True" _ "\<lambda>x y. True", simplified]) auto
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lemmas alpha_bn_inducts[consumes 1] = alpha_bn_inducts_raw[quot_lifted]
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lemma alpha_bn_refl: "alpha_bn x x"
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by (induct x rule: trm_assn.inducts(2))
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(rule TrueI, auto simp add: trm_assn.eq_iff)
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lemma alpha_bn_sym: "alpha_bn x y \<Longrightarrow> alpha_bn y x"
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sorry
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lemma alpha_bn_trans: "alpha_bn x y \<Longrightarrow> alpha_bn y z \<Longrightarrow> alpha_bn x z"
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sorry
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lemma bn_inj[rule_format]:
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assumes a: "alpha_bn x y"
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shows "bn x = bn y \<longrightarrow> x = y"
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by (rule alpha_bn_inducts[OF a]) (simp_all add: trm_assn.bn_defs)
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lemma bn_inj2:
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assumes a: "alpha_bn x y"
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shows "\<And>q r. (q \<bullet> bn x) = (r \<bullet> bn y) \<Longrightarrow> permute_bn q x = permute_bn r y"
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using a
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apply(induct rule: alpha_bn_inducts)
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apply(simp add: trm_assn.perm_bn_simps)
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apply(simp add: trm_assn.perm_bn_simps)
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apply(simp add: trm_assn.bn_defs)
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apply(simp add: atom_eqvt)
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done
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lemma Abs_lst_fcb2:
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fixes as bs :: "atom list"
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and x y :: "'b :: fs"
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and c::"'c::fs"
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assumes eq: "[as]lst. x = [bs]lst. y"
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and fcb1: "(set as) \<sharp>* c \<Longrightarrow> (set as) \<sharp>* f as x c"
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and fresh1: "set as \<sharp>* c"
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and fresh2: "set bs \<sharp>* c"
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and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
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and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
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shows "f as x c = f bs y c"
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proof -
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have "supp (as, x, c) supports (f as x c)"
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unfolding supports_def fresh_def[symmetric]
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by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
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then have fin1: "finite (supp (f as x c))"
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by (auto intro: supports_finite simp add: finite_supp)
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have "supp (bs, y, c) supports (f bs y c)"
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unfolding supports_def fresh_def[symmetric]
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by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
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then have fin2: "finite (supp (f bs y c))"
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by (auto intro: supports_finite simp add: finite_supp)
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obtain q::"perm" where
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fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and
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fr2: "supp q \<sharp>* Abs_lst as x" and
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inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"
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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"]
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fin1 fin2
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by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
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have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp
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also have "\<dots> = Abs_lst as x"
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by (simp only: fr2 perm_supp_eq)
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finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp
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then obtain r::perm where
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qq1: "q \<bullet> x = r \<bullet> y" and
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qq2: "q \<bullet> as = r \<bullet> bs" and
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qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"
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apply(drule_tac sym)
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apply(simp only: Abs_eq_iff2 alphas)
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apply(erule exE)
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apply(erule conjE)+
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apply(drule_tac x="p" in meta_spec)
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apply(simp add: set_eqvt)
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apply(blast)
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done
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have "(set as) \<sharp>* f as x c"
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apply(rule fcb1)
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apply(rule fresh1)
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done
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then have "q \<bullet> ((set as) \<sharp>* f as x c)"
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by (simp add: permute_bool_def)
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then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
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apply(simp add: fresh_star_eqvt set_eqvt)
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apply(subst (asm) perm1)
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using inc fresh1 fr1
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apply(auto simp add: fresh_star_def fresh_Pair)
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done
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then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
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then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"
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apply(simp add: fresh_star_eqvt set_eqvt)
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apply(subst (asm) perm2[symmetric])
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using qq3 fresh2 fr1
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apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
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done
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then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)
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have "f as x c = q \<bullet> (f as x c)"
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apply(rule perm_supp_eq[symmetric])
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using inc fcb1[OF fresh1] fr1 by (auto simp add: fresh_star_def)
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also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c"
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apply(rule perm1)
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using inc fresh1 fr1 by (auto simp add: fresh_star_def)
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also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
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also have "\<dots> = r \<bullet> (f bs y c)"
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apply(rule perm2[symmetric])
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using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
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also have "... = f bs y c"
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apply(rule perm_supp_eq)
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using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
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finally show ?thesis by simp
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qed
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lemma Abs_lst1_fcb2:
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fixes a b :: "atom"
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and x y :: "'b :: fs"
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and c::"'c :: fs"
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assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"
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and fcb1: "a \<sharp> c \<Longrightarrow> a \<sharp> f a x c"
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and fresh: "{a, b} \<sharp>* c"
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and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
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and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
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shows "f a x c = f b y c"
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using e
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apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])
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apply(simp_all)
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using fcb1 fresh perm1 perm2
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apply(simp_all add: fresh_star_def)
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done
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function
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apply_assn2 :: "(trm \<Rightarrow> trm) \<Rightarrow> assn \<Rightarrow> assn"
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where
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"apply_assn2 f ANil = ANil"
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| "apply_assn2 f (ACons x t as) = ACons x (f t) (apply_assn2 f as)"
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apply(case_tac x)
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apply(case_tac b rule: trm_assn.exhaust(2))
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apply(simp_all)
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apply(blast)
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done
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termination by lexicographic_order
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lemma apply_assn_eqvt[eqvt]:
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"p \<bullet> (apply_assn2 f a) = apply_assn2 (p \<bullet> f) (p \<bullet> a)"
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apply(induct f a rule: apply_assn2.induct)
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apply simp_all
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apply(perm_simp)
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apply rule
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done
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lemma
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fixes x y :: "'a :: fs"
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shows "[a # as]lst. x = [b # bs]lst. y \<Longrightarrow> [[a]]lst. [as]lst. x = [[b]]lst. [bs]lst. y"
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apply (simp add: Abs_eq_iff)
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apply (elim exE)
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apply (rule_tac x="p" in exI)
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apply (simp add: alphas)
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apply clarify
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apply rule
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apply (simp add: supp_Abs)
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apply blast
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apply (simp add: supp_Abs fresh_star_def)
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apply blast
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done
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lemma
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assumes neq: "a \<noteq> b" "sort_of a = sort_of b"
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shows "[[a]]lst. [[a]]lst. a = [[a]]lst. [[b]]lst. b \<and> [[a, a]]lst. a \<noteq> [[a, b]]lst. b"
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apply (simp add: Abs1_eq_iff)
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apply rule
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apply (simp add: Abs_eq_iff alphas supp_atom fresh_star_def)
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apply (rule_tac x="(a \<rightleftharpoons> b)" in exI)
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apply (simp add: neq)
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apply (simp add: Abs_eq_iff alphas supp_atom fresh_star_def neq)
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done
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nominal_primrec
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subst :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"
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where
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"subst s t (Var x) = (if (s = x) then t else (Var x))"
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| "subst s t (App l r) = App (subst s t l) (subst s t r)"
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| "atom v \<sharp> (s, t) \<Longrightarrow> subst s t (Lam v b) = Lam v (subst s t b)"
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| "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (apply_assn2 (subst s t) as) (subst s t b)"
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apply (simp only: eqvt_def subst_graph_def)
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apply (rule, perm_simp, rule)
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apply (rule TrueI)
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apply (case_tac x)
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apply (rule_tac y="c" and c="(a,b)" in trm_assn.strong_exhaust(1))
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apply (auto simp add: fresh_star_def)[3]
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apply (drule_tac x="assn" in meta_spec)
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apply (simp add: Abs1_eq_iff alpha_bn_refl)
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2956
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apply auto[7]
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prefer 2
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apply(simp)
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thm subst_sumC_def
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thm THE_default_def
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thm theI
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2936
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apply (erule_tac c="(sa, ta)" in Abs_lst1_fcb2)
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apply (simp add: Abs_fresh_iff)
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apply (simp add: fresh_star_def)
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apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def)[2]
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apply (subgoal_tac "apply_assn2 (\<lambda>x2\<Colon>trm. subst_sumC (sa, ta, x2)) asa
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= apply_assn2 (\<lambda>x2\<Colon>trm. subst_sumC (sa, ta, x2)) as")
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prefer 2
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apply (erule alpha_bn_inducts)
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apply simp
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apply (simp only: apply_assn2.simps)
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apply simp
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--"We know nothing about names; not true; but we can apply fcb2"
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defer
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apply (simp only: )
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apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)
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--"We again need induction for fcb assumption; this time true"
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apply (induct_tac as rule: trm_assn.inducts(2))
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apply (rule TrueI)+
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apply (simp_all add: trm_assn.bn_defs fresh_star_def)[2]
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apply (auto simp add: Abs_fresh_iff)[1]
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apply assumption+
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--"But eqvt is not going to be true as well"
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apply (simp add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def trm_assn.fv_bn_eqvt)
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apply (simp add: apply_assn_eqvt)
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apply (drule sym)
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apply (subgoal_tac "p \<bullet> (\<lambda>x2\<Colon>trm. subst_sumC (sa, ta, x2)) = (\<lambda>x2\<Colon>trm. subst_sumC (sa, ta, x2))")
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apply (simp)
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apply (erule alpha_bn_inducts)
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apply simp
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apply simp
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apply (simp add: trm_assn.bn_defs)
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--"Again we cannot relate 'namea' with 'p \<bullet> name'"
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prefer 4
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apply (erule alpha_bn_inducts)
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apply simp_all[2]
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oops
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end
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