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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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| Lam x::"name" t::"trm" bind x in t
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| Let as::"assn" t::"trm" bind "bn as" in t
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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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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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5ac9a74d22fd
post-processed eq_iff and supp threormes according to the fv-supp equality
Christian Urban <urbanc@in.tum.de>
diff
changeset
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thm trm_assn.induct
5ac9a74d22fd
post-processed eq_iff and supp threormes according to the fv-supp equality
Christian Urban <urbanc@in.tum.de>
diff
changeset
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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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e44551d067e6
properly exported strong exhaust theorem; cleaned up some examples
Christian Urban <urbanc@in.tum.de>
diff
changeset
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thm trm_assn.exhaust
e44551d067e6
properly exported strong exhaust theorem; cleaned up some examples
Christian Urban <urbanc@in.tum.de>
diff
changeset
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thm trm_assn.strong_exhaust
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lemma alpha_bn_inducts_raw:
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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 = 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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2923
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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 alpha_bn_permute:
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assumes a: "alpha_bn x y"
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and b: "q \<bullet> bn x = r \<bullet> bn y"
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shows "alpha_bn (q \<bullet> x) (r \<bullet> y)"
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proof -
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have "alpha_bn x (permute_bn r y)"
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by (rule alpha_bn_trans[OF a]) (rule trm_assn.perm_bn_alpha)
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then have "alpha_bn (permute_bn r y) x"
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by (rule alpha_bn_sym)
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then have "alpha_bn (permute_bn r y) (permute_bn q x)"
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by (rule alpha_bn_trans) (rule trm_assn.perm_bn_alpha)
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then have "alpha_bn (permute_bn q x) (permute_bn r y)"
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by (rule alpha_bn_sym)
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moreover have "bn (permute_bn q x) = bn (permute_bn r y)"
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using b trm_assn.permute_bn by simp
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ultimately have "permute_bn q x = permute_bn r y"
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using bn_inj by simp
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*)
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lemma lets_bla:
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"x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Let (ACons x (Var y) ANil) (Var x)) \<noteq> (Let (ACons x (Var z) ANil) (Var x))"
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by (simp add: trm_assn.eq_iff)
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lemma lets_ok:
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"(Let (ACons x (Var y) ANil) (Var x)) = (Let (ACons y (Var y) ANil) (Var y))"
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apply (simp add: trm_assn.eq_iff Abs_eq_iff )
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apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
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apply (simp_all add: alphas atom_eqvt supp_at_base fresh_star_def trm_assn.bn_defs trm_assn.supp)
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done
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lemma lets_ok3:
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"x \<noteq> y \<Longrightarrow>
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(Let (ACons x (App (Var y) (Var x)) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>
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(Let (ACons y (App (Var x) (Var y)) (ACons x (Var x) ANil)) (App (Var x) (Var y)))"
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apply (simp add: trm_assn.eq_iff)
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done
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lemma lets_not_ok1:
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"x \<noteq> y \<Longrightarrow>
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(Let (ACons x (Var x) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>
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(Let (ACons y (Var x) (ACons x (Var y) ANil)) (App (Var x) (Var y)))"
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apply (simp add: alphas trm_assn.eq_iff trm_assn.supp fresh_star_def atom_eqvt Abs_eq_iff trm_assn.bn_defs)
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done
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lemma lets_nok:
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"x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>
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(Let (ACons x (App (Var z) (Var z)) (ACons y (Var z) ANil)) (App (Var x) (Var y))) \<noteq>
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(Let (ACons y (Var z) (ACons x (App (Var z) (Var z)) ANil)) (App (Var x) (Var y)))"
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apply (simp add: alphas trm_assn.eq_iff fresh_star_def trm_assn.bn_defs Abs_eq_iff trm_assn.supp trm_assn.distinct)
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done
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lemma
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fixes a b c :: name
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assumes x: "a \<noteq> c" and y: "b \<noteq> c"
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shows "\<exists>p.([atom a], Var c) \<approx>lst (op =) supp p ([atom b], Var c)"
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apply (rule_tac x="(a \<leftrightarrow> b)" in exI)
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apply (simp add: alphas trm_assn.supp supp_at_base x y fresh_star_def atom_eqvt)
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by (metis Rep_name_inverse atom_name_def flip_fresh_fresh fresh_atom fresh_perm x y)
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lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
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by (simp add: permute_pure)
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lemma Abs_lst_fcb2:
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fixes as bs :: "'a :: fs"
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and x y :: "'b :: fs"
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and c::"'c::fs"
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assumes eq: "[ba as]lst. x = [ba bs]lst. y"
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and fcb1: "set (ba as) \<sharp>* f as x c"
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and fresh1: "set (ba as) \<sharp>* c"
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and fresh2: "set (ba 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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(* What we would like in this proof, and lets this proof finish *)
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and bainj: "\<And>q r. q \<bullet> ba as = r \<bullet> ba bs \<Longrightarrow> q \<bullet> as = r \<bullet> bs"
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(* What the user can supply with the help of alpha_bn *)
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(* and bainj: "ba as = ba bs \<Longrightarrow> as = bs"*)
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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 (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and
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fr2: "supp q \<sharp>* ([ba as]lst. x)" and
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inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))"
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using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)"
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and x="[ba as]lst. x"] 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 "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp
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also have "\<dots> = [ba as]lst. x"
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by (simp only: fr2 perm_supp_eq)
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finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. 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> (ba as) = r \<bullet> (ba bs)" and
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qq3: "supp r \<subseteq> (q \<bullet> (set (ba as))) \<union> set (ba 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 (ba as)) \<sharp>* f as x c" by (rule fcb1)
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then have "q \<bullet> ((set (ba 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> (ba 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> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 bainj
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by simp
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then have "r \<bullet> ((set (ba 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 (ba 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 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)
2923
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also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 bainj 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)
2923
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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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2854
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nominal_primrec
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height_trm :: "trm \<Rightarrow> nat"
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and height_assn :: "assn \<Rightarrow> nat"
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where
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"height_trm (Var x) = 1"
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| "height_trm (App l r) = max (height_trm l) (height_trm r)"
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| "height_trm (Lam v b) = 1 + (height_trm b)"
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| "height_trm (Let as b) = max (height_assn as) (height_trm b)"
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| "height_assn ANil = 0"
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| "height_assn (ACons v t as) = max (height_trm t) (height_assn as)"
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apply (simp only: eqvt_def height_trm_height_assn_graph_def)
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apply (rule, perm_simp, rule, rule TrueI)
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apply (case_tac x)
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apply (case_tac a rule: trm_assn.exhaust(1))
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apply (auto)[4]
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apply (drule_tac x="assn" in meta_spec)
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apply (drule_tac x="trm" in meta_spec)
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apply (simp add: alpha_bn_refl)
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apply (case_tac b rule: trm_assn.exhaust(2))
2922
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apply (auto)[2]
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apply(simp_all)
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apply (erule_tac c="()" in Abs_lst_fcb2)
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apply (simp_all add: pure_fresh fresh_star_def)[3]
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apply (simp add: eqvt_at_def)
2854
+ − 230
apply (simp add: eqvt_at_def)
2923
+ − 231
apply assumption
2922
+ − 232
apply(erule conjE)
+ − 233
apply (simp add: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])
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apply (simp add: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff])
+ − 235
apply (subgoal_tac "eqvt_at height_assn as")
+ − 236
apply (subgoal_tac "eqvt_at height_assn asa")
+ − 237
apply (subgoal_tac "eqvt_at height_trm b")
+ − 238
apply (subgoal_tac "eqvt_at height_trm ba")
+ − 239
apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)")
+ − 240
apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr asa)")
+ − 241
apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl b)")
+ − 242
apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl ba)")
+ − 243
defer
+ − 244
apply (simp add: eqvt_at_def height_trm_def)
+ − 245
apply (simp add: eqvt_at_def height_trm_def)
+ − 246
apply (simp add: eqvt_at_def height_assn_def)
+ − 247
apply (simp add: eqvt_at_def height_assn_def)
+ − 248
apply (subgoal_tac "height_assn as = height_assn asa")
+ − 249
apply (subgoal_tac "height_trm b = height_trm ba")
+ − 250
apply simp
+ − 251
apply (erule_tac c="()" in Abs_lst_fcb2)
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apply (simp_all add: pure_fresh fresh_star_def)[3]
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apply (simp_all add: eqvt_at_def)[2]
2923
+ − 254
apply assumption
+ − 255
apply (erule_tac c="()" and ba="bn" in Abs_lst_fcb2)
2922
+ − 256
apply (simp_all add: pure_fresh fresh_star_def)[3]
+ − 257
apply (simp_all add: eqvt_at_def)[2]
2923
+ − 258
apply (rule bn_inj)
+ − 259
prefer 2
+ − 260
apply (simp add: eqvts)
2854
+ − 261
oops
+ − 262
2720
+ − 263
nominal_primrec
+ − 264
subst :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"
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and substa :: "name \<Rightarrow> trm \<Rightarrow> assn \<Rightarrow> assn"
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where
+ − 267
"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)"
+ − 270
| "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (substa s t as) (subst s t b)"
+ − 271
| "substa s t ANil = ANil"
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| "substa s t (ACons v t' as) = ACons v (subst v t t') as"
2842
+ − 273
(*unfolding eqvt_def subst_substa_graph_def
+ − 274
apply (rule, perm_simp)*)
+ − 275
defer
+ − 276
apply (rule TrueI)
+ − 277
apply (case_tac x)
+ − 278
apply (case_tac a)
+ − 279
apply (rule_tac y="c" and c="(aa,b)" in trm_assn.strong_exhaust(1))
+ − 280
apply (auto simp add: fresh_star_def)[3]
+ − 281
apply (drule_tac x="assn" in meta_spec)
+ − 282
apply (simp add: Abs1_eq_iff alpha_bn_refl)
+ − 283
apply (case_tac b)
+ − 284
apply (case_tac c rule: trm_assn.exhaust(2))
+ − 285
apply (auto)[2]
+ − 286
apply blast
+ − 287
apply blast
+ − 288
apply auto
+ − 289
apply (simp_all add: meta_eq_to_obj_eq[OF subst_def, symmetric, unfolded fun_eq_iff])
+ − 290
apply (simp_all add: meta_eq_to_obj_eq[OF substa_def, symmetric, unfolded fun_eq_iff])
2921
+ − 291
prefer 2
2923
+ − 292
apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)
+ − 293
apply (simp_all add: fresh_star_Pair)
+ − 294
prefer 6
+ − 295
apply (erule alpha_bn_inducts)
2921
+ − 296
oops
2720
+ − 297
1600
+ − 298
end
+ − 299
+ − 300
+ − 301