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theory Lambda
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imports
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"../Nominal2"
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begin
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atom_decl name
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nominal_datatype lam =
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Var "name"
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| App "lam" "lam"
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| Lam x::"name" l::"lam" binds x in l ("Lam [_]. _" [100, 100] 100)
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nominal_datatype sem =
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L e::"env" x::"name" l::"lam" binds x "bn e" in l
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| N "neu"
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and neu =
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V "name"
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| A "neu" "sem"
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and env =
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ENil
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| ECons "env" "name" "sem"
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binder
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bn
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where
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"bn ENil = []"
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| "bn (ECons env x v) = (atom x) # (bn env)"
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thm sem_neu_env.supp
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lemma [simp]:
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shows "finite (fv_bn env)"
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apply(induct env rule: sem_neu_env.inducts(3))
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apply(rule TrueI)+
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apply(auto simp add: sem_neu_env.supp finite_supp)
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done
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lemma test1:
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shows "supp p \<sharp>* (fv_bn env) \<Longrightarrow> (p \<bullet> env) = permute_bn p env"
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apply(induct env rule: sem_neu_env.inducts(3))
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apply(rule TrueI)+
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apply(auto simp add: sem_neu_env.perm_bn_simps sem_neu_env.supp)
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apply(drule meta_mp)
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apply(drule fresh_star_supp_conv)
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apply(subst (asm) supp_finite_atom_set)
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apply(simp add: finite_supp)
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apply(simp add: fresh_star_Un)
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apply(rule fresh_star_supp_conv)
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apply(subst supp_finite_atom_set)
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apply(simp)
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apply(simp)
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apply(simp)
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apply(rule perm_supp_eq)
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apply(drule fresh_star_supp_conv)
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apply(subst (asm) supp_finite_atom_set)
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apply(simp add: finite_supp)
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apply(simp add: fresh_star_Un)
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apply(rule fresh_star_supp_conv)
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apply(simp)
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done
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thm alpha_sem_raw_alpha_neu_raw_alpha_env_raw_alpha_bn_raw.inducts(4)[no_vars]
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lemma alpha_bn_inducts_raw[consumes 1]:
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"\<lbrakk>alpha_bn_raw x7 x8;
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P4 ENil_raw ENil_raw;
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\<And>env_raw env_rawa sem_raw sem_rawa name namea.
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\<lbrakk>alpha_bn_raw env_raw env_rawa; P4 env_raw env_rawa; alpha_sem_raw sem_raw sem_rawa\<rbrakk>
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\<Longrightarrow> P4 (ECons_raw env_raw name sem_raw) (ECons_raw env_rawa namea sem_rawa)\<rbrakk>
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\<Longrightarrow> P4 x7 x8"
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apply(induct rule: alpha_sem_raw_alpha_neu_raw_alpha_env_raw_alpha_bn_raw.inducts(4))
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apply(rule TrueI)+
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apply(auto)
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done
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lemmas alpha_bn_inducts[consumes 1] = alpha_bn_inducts_raw[quot_lifted]
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lemma test2:
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shows "alpha_bn env1 env2 \<Longrightarrow> p \<bullet> (bn env1) = q \<bullet> (bn env2) \<Longrightarrow> permute_bn p env1 = permute_bn q env2"
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apply(induct rule: alpha_bn_inducts)
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apply(auto simp add: sem_neu_env.perm_bn_simps sem_neu_env.bn_defs)
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apply(simp add: atom_eqvt)
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done
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lemma fresh_star_Union:
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assumes "as \<subseteq> bs" "bs \<sharp>* x"
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shows "as \<sharp>* x"
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using assms by (auto simp add: fresh_star_def)
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nominal_primrec (invariant "\<lambda>x y. case x of Inl (x1, y1) \<Rightarrow>
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supp y \<subseteq> (supp y1 - set (bn x1)) \<union> (fv_bn x1) | Inr (x2, y2) \<Rightarrow> supp y \<subseteq> supp x2 \<union> supp y2")
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evals :: "env \<Rightarrow> lam \<Rightarrow> sem" and
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evals_aux :: "sem \<Rightarrow> sem \<Rightarrow> sem"
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where
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"evals ENil (Var x) = N (V x)"
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| "evals (ECons tail y v) (Var x) = (if x = y then v else evals tail (Var x))"
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| "atom x \<sharp> env \<Longrightarrow> evals env (Lam [x]. t) = L env x t"
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| "evals env (App t1 t2) = evals_aux (evals env t1) (evals env t2)"
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| "set (atom x # bn env) \<sharp>* (fv_bn env, t') \<Longrightarrow> evals_aux (L env x t) t' = evals (ECons env x t') t"
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| "evals_aux (N n) t' = N (A n t')"
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apply(simp add: eqvt_def evals_evals_aux_graph_def)
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apply(perm_simp)
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apply(simp)
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apply(erule evals_evals_aux_graph.induct)
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apply(simp add: sem_neu_env.supp lam.supp sem_neu_env.bn_defs)
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apply(simp add: sem_neu_env.supp lam.supp sem_neu_env.bn_defs)
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apply(rule conjI)
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apply(rule impI)
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apply(blast)
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apply(rule impI)
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apply(simp add: supp_at_base)
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apply(blast)
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apply(simp add: sem_neu_env.supp lam.supp sem_neu_env.bn_defs)
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apply(blast)
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apply(simp add: sem_neu_env.supp lam.supp sem_neu_env.bn_defs)
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apply(blast)
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apply(simp add: sem_neu_env.supp lam.supp sem_neu_env.bn_defs)
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apply(blast)
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apply(simp add: sem_neu_env.supp lam.supp sem_neu_env.bn_defs)
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--"completeness"
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apply(case_tac x)
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apply(simp)
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apply(case_tac a)
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apply(simp)
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apply(case_tac aa rule: sem_neu_env.exhaust(3))
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apply(simp add: sem_neu_env.fresh)
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apply(case_tac b rule: lam.exhaust)
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apply(metis)+
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apply(case_tac aa rule: sem_neu_env.exhaust(3))
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apply(rule_tac y="b" and c="env" in lam.strong_exhaust)
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apply(metis)+
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apply(simp add: fresh_star_def)
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apply(simp)
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apply(rule_tac y="b" and c="ECons env name sem" in lam.strong_exhaust)
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apply(metis)+
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apply(simp add: fresh_star_def)
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apply(simp)
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apply(case_tac b)
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apply(simp)
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apply(rule_tac y="a" and c="(a, ba)" in sem_neu_env.strong_exhaust(1))
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apply(simp)
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apply(rotate_tac 4)
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apply(drule_tac x="name" in meta_spec)
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apply(drule_tac x="env" in meta_spec)
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apply(drule_tac x="ba" in meta_spec)
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apply(drule_tac x="lam" in meta_spec)
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apply(simp add: sem_neu_env.alpha_refl)
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apply(rotate_tac 9)
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apply(drule_tac meta_mp)
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apply(simp (no_asm_use) add: fresh_star_def sem_neu_env.fresh fresh_Pair)
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apply(simp)
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apply(subst fresh_finite_atom_set)
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defer
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apply(simp)
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apply(subst fresh_finite_atom_set)
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defer
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apply(simp)
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apply(metis)+
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--"compatibility"
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apply(all_trivials)
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apply(simp)
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apply(simp)
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defer
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apply(simp)
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apply(simp)
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apply (simp add: meta_eq_to_obj_eq[OF evals_def, symmetric, unfolded fun_eq_iff])
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apply (subgoal_tac "eqvt_at (\<lambda>(a, b). evals a b) (ECons env x t'a, t)")
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apply (subgoal_tac "eqvt_at (\<lambda>(a, b). evals a b) (ECons enva xa t'a, ta)")
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apply (thin_tac "eqvt_at evals_evals_aux_sumC (Inl (ECons env x t'a, t))")
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apply (thin_tac "eqvt_at evals_evals_aux_sumC (Inl (ECons enva xa t'a, ta))")
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apply(erule conjE)+
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defer
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apply (simp_all add: eqvt_at_def evals_def)[3]
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defer
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defer
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apply(simp add: sem_neu_env.alpha_refl)
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apply(erule conjE)+
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apply(erule_tac c="(env, enva)" 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(perm_simp)
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apply(simp add: fresh_star_Pair perm_supp_eq)
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apply(perm_simp)
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apply(simp add: fresh_star_Pair perm_supp_eq)
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apply(simp add: sem_neu_env.bn_defs sem_neu_env.supp)
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using at_set_avoiding3
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apply -
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apply(drule_tac x="set (atom x # bn env)" in meta_spec)
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apply(drule_tac x="(fv_bn env, fv_bn enva, env, enva, x, xa, t, ta, t'a)" in meta_spec)
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apply(drule_tac x="[atom x # bn env]lst. t" in meta_spec)
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apply(simp (no_asm_use) add: finite_supp Abs_fresh_star_iff)
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apply(drule meta_mp)
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apply(simp add: supp_Pair finite_supp supp_finite_atom_set)
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apply(drule meta_mp)
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apply(simp add: fresh_star_def)
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apply(erule exE)
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apply(erule conjE)+
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apply(rule trans)
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apply(rule sym)
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apply(rule_tac p="p" in perm_supp_eq)
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apply(simp)
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apply(perm_simp)
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apply(simp add: fresh_star_Un fresh_star_insert)
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apply(rule conjI)
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apply(simp (no_asm_use) add: fresh_star_def fresh_Pair)
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apply(simp add: fresh_def)
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apply(simp add: supp_finite_atom_set)
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apply(blast)
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apply(rule conjI)
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apply(simp (no_asm_use) add: fresh_star_def fresh_Pair)
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apply(simp add: fresh_def)
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apply(simp add: supp_finite_atom_set)
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apply(blast)
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apply(rule conjI)
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apply(simp (no_asm_use) add: fresh_star_def fresh_Pair)
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apply(simp add: fresh_def)
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apply(simp add: supp_finite_atom_set)
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apply(blast)
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apply(simp (no_asm_use) add: fresh_star_def fresh_Pair)
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apply(simp add: fresh_def)
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apply(simp add: supp_finite_atom_set)
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apply(blast)
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apply(simp add: eqvt_at_def perm_supp_eq)
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apply(subst (3) perm_supp_eq)
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apply(simp)
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apply(simp add: fresh_star_def fresh_Pair)
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apply(auto)[1]
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apply(rotate_tac 6)
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apply(drule sym)
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apply(simp)
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apply(rotate_tac 11)
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apply(drule trans)
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apply(rule sym)
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apply(rule_tac p="p" in supp_perm_eq)
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apply(assumption)
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apply(rotate_tac 11)
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apply(rule sym)
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apply(simp add: atom_eqvt)
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apply(simp (no_asm_use) add: Abs_eq_iff2 alphas)
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apply(erule conjE | erule exE)+
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apply(rule trans)
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apply(rule sym)
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apply(rule_tac p="pa" in perm_supp_eq)
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apply(erule fresh_star_Union)
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apply(simp (no_asm_use) add: fresh_star_insert fresh_star_Un)
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apply(rule conjI)
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apply(perm_simp)
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apply(simp add: fresh_star_insert fresh_star_Un)
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apply(simp add: fresh_Pair)
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apply(simp add: fresh_def)
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apply(simp add: supp_finite_atom_set)
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apply(blast)
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apply(rule conjI)
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apply(perm_simp)
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apply(simp add: fresh_star_insert fresh_star_Un)
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apply(simp add: fresh_Pair)
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apply(simp add: fresh_def)
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apply(simp add: supp_finite_atom_set)
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apply(blast)
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apply(rule conjI)
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apply(perm_simp)
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defer
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apply(perm_simp)
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apply(simp add: fresh_star_insert fresh_star_Un)
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apply(simp add: fresh_star_Pair)
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apply(simp add: fresh_star_def fresh_def)
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apply(simp add: supp_finite_atom_set)
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apply(blast)
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apply(simp)
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apply(perm_simp)
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apply(subst (3) perm_supp_eq)
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apply(erule fresh_star_Union)
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apply(simp add: fresh_star_insert fresh_star_Un)
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apply(simp add: fresh_star_def fresh_Pair)
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apply(subgoal_tac "pa \<bullet> enva = p \<bullet> env")
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apply(simp)
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defer
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apply(simp (no_asm_use) add: fresh_star_insert fresh_star_Un)
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apply(simp (no_asm_use) add: fresh_star_def)
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apply(rule ballI)
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apply(subgoal_tac "a \<notin> supp ta - insert (atom xa) (set (bn enva)) \<union> (fv_bn enva \<union> supp t'a)")
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apply(simp only: fresh_def)
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apply(blast)
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apply(simp (no_asm_use))
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apply(rule conjI)
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apply(blast)
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apply(simp add: fresh_Pair)
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apply(simp add: fresh_star_def fresh_def)
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apply(simp add: supp_finite_atom_set)
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apply(subst test1)
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apply(erule fresh_star_Union)
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apply(simp add: fresh_star_insert fresh_star_Un)
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apply(simp add: fresh_star_def fresh_Pair)
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apply(subst test1)
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apply(simp)
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297 |
apply(simp add: fresh_star_insert fresh_star_Un)
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298 |
apply(simp add: fresh_star_def fresh_Pair)
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2968
|
299 |
apply(rule sym)
|
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2969
|
300 |
apply(erule test2)
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2968
|
301 |
apply(perm_simp)
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|
302 |
apply(simp)
|
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2969
|
303 |
done
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304 |
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305 |
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2968
|
306 |
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2952
|
307 |
|
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2969
|
308 |
text {* can probably not proved by a trivial size argument *}
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309 |
termination (* apply(lexicographic_order) *)
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2968
|
310 |
sorry
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2952
|
311 |
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2955
|
312 |
lemma [eqvt]:
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313 |
shows "(p \<bullet> evals env t) = evals (p \<bullet> env) (p \<bullet> t)"
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2967
|
314 |
and "(p \<bullet> evals_aux v s) = evals_aux (p \<bullet> v) (p \<bullet> s)"
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2955
|
315 |
sorry
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316 |
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|
317 |
(* fixme: should be a provided lemma *)
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318 |
lemma fv_bn_finite:
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|
319 |
shows "finite (fv_bn env)"
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|
320 |
apply(induct env rule: sem_neu_env.inducts(3))
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|
321 |
apply(auto simp add: sem_neu_env.supp finite_supp)
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|
322 |
done
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323 |
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|
324 |
lemma test:
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2956
|
325 |
fixes env::"env"
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2955
|
326 |
shows "supp (evals env t) \<subseteq> (supp t - set (bn env)) \<union> (fv_bn env)"
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2967
|
327 |
and "supp (evals_aux s v) \<subseteq> (supp s) \<union> (supp v)"
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|
328 |
apply(induct env t and s v rule: evals_evals_aux.induct)
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2955
|
329 |
apply(simp add: sem_neu_env.supp lam.supp supp_Nil sem_neu_env.bn_defs)
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|
330 |
apply(simp add: sem_neu_env.supp lam.supp supp_Nil supp_Cons sem_neu_env.bn_defs)
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|
331 |
apply(rule conjI)
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|
332 |
apply(auto)[1]
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|
333 |
apply(rule impI)
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|
334 |
apply(simp)
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2956
|
335 |
apply(simp add: supp_at_base)
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|
336 |
apply(blast)
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2955
|
337 |
apply(simp)
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|
338 |
apply(subst sem_neu_env.supp)
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|
339 |
apply(simp add: sem_neu_env.supp lam.supp)
|
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|
340 |
apply(auto)[1]
|
|
2956
|
341 |
apply(simp add: lam.supp sem_neu_env.supp)
|
|
|
342 |
apply(blast)
|
|
|
343 |
apply(simp add: sem_neu_env.supp sem_neu_env.bn_defs)
|
|
2967
|
344 |
apply(blast)
|
|
2956
|
345 |
apply(simp add: sem_neu_env.supp)
|
|
|
346 |
done
|
|
|
347 |
|
|
2955
|
348 |
|
|
2954
|
349 |
nominal_primrec
|
|
|
350 |
reify :: "sem \<Rightarrow> lam" and
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|
351 |
reifyn :: "neu \<Rightarrow> lam"
|
|
|
352 |
where
|
|
2955
|
353 |
"atom x \<sharp> (env, y, t) \<Longrightarrow> reify (L env y t) = Lam [x]. (reify (evals (ECons env y (N (V x))) t))"
|
|
2954
|
354 |
| "reify (N n) = reifyn n"
|
|
|
355 |
| "reifyn (V x) = Var x"
|
|
|
356 |
| "reifyn (A n d) = App (reifyn n) (reify d)"
|
|
2955
|
357 |
apply(subgoal_tac "\<And>p x y. reify_reifyn_graph x y \<Longrightarrow> reify_reifyn_graph (p \<bullet> x) (p \<bullet> y)")
|
|
|
358 |
apply(simp add: eqvt_def)
|
|
|
359 |
apply(simp add: permute_fun_def)
|
|
|
360 |
apply(rule allI)
|
|
|
361 |
apply(rule ext)
|
|
|
362 |
apply(rule ext)
|
|
|
363 |
apply(rule iffI)
|
|
|
364 |
apply(drule_tac x="p" in meta_spec)
|
|
|
365 |
apply(drule_tac x="- p \<bullet> x" in meta_spec)
|
|
|
366 |
apply(drule_tac x="- p \<bullet> xa" in meta_spec)
|
|
|
367 |
apply(simp add: permute_bool_def)
|
|
|
368 |
apply(simp add: permute_bool_def)
|
|
|
369 |
apply(erule reify_reifyn_graph.induct)
|
|
|
370 |
apply(perm_simp)
|
|
|
371 |
apply(rule reify_reifyn_graph.intros)
|
|
|
372 |
apply(rule_tac p="-p" in permute_boolE)
|
|
|
373 |
apply(perm_simp add: permute_minus_cancel)
|
|
|
374 |
apply(simp)
|
|
|
375 |
apply(simp)
|
|
|
376 |
apply(perm_simp)
|
|
|
377 |
apply(rule reify_reifyn_graph.intros)
|
|
|
378 |
apply(simp)
|
|
|
379 |
apply(perm_simp)
|
|
|
380 |
apply(rule reify_reifyn_graph.intros)
|
|
|
381 |
apply(perm_simp)
|
|
|
382 |
apply(rule reify_reifyn_graph.intros)
|
|
|
383 |
apply(simp)
|
|
|
384 |
apply(simp)
|
|
|
385 |
apply(rule TrueI)
|
|
2954
|
386 |
--"completeness"
|
|
|
387 |
apply(case_tac x)
|
|
|
388 |
apply(simp)
|
|
|
389 |
apply(case_tac a rule: sem_neu_env.exhaust(1))
|
|
2955
|
390 |
apply(subgoal_tac "\<exists>x::name. atom x \<sharp> (env, name, lam)")
|
|
|
391 |
apply(metis)
|
|
|
392 |
apply(rule obtain_fresh)
|
|
|
393 |
apply(blast)
|
|
|
394 |
apply(blast)
|
|
2954
|
395 |
apply(case_tac b rule: sem_neu_env.exhaust(2))
|
|
|
396 |
apply(simp)
|
|
|
397 |
apply(simp)
|
|
|
398 |
apply(metis)
|
|
|
399 |
--"compatibility"
|
|
|
400 |
apply(all_trivials)
|
|
|
401 |
defer
|
|
|
402 |
apply(simp)
|
|
|
403 |
apply(simp)
|
|
2955
|
404 |
apply(simp)
|
|
|
405 |
apply(erule conjE)
|
|
|
406 |
apply (simp add: meta_eq_to_obj_eq[OF reify_def, symmetric, unfolded fun_eq_iff])
|
|
2956
|
407 |
apply (subgoal_tac "eqvt_at (\<lambda>t. reify t) (evals (ECons env y (N (V x))) t)")
|
|
2955
|
408 |
apply (subgoal_tac "eqvt_at (\<lambda>t. reify t) (evals (ECons enva ya (N (V xa))) ta)")
|
|
2956
|
409 |
apply (thin_tac "eqvt_at reify_reifyn_sumC (Inl (evals (ECons env y (N (V x))) t))")
|
|
2955
|
410 |
apply (thin_tac "eqvt_at reify_reifyn_sumC (Inl (evals (ECons enva ya (N (V xa))) ta))")
|
|
|
411 |
defer
|
|
|
412 |
apply (simp_all add: eqvt_at_def reify_def)[2]
|
|
2956
|
413 |
apply(subgoal_tac "\<exists>c::name. atom c \<sharp> (x, xa, env, enva, y, ya, t, ta)")
|
|
2955
|
414 |
prefer 2
|
|
|
415 |
apply(rule obtain_fresh)
|
|
|
416 |
apply(blast)
|
|
|
417 |
apply(erule exE)
|
|
|
418 |
apply(rule trans)
|
|
|
419 |
apply(rule sym)
|
|
|
420 |
apply(rule_tac a="x" and b="c" in flip_fresh_fresh)
|
|
|
421 |
apply(simp add: Abs_fresh_iff)
|
|
|
422 |
apply(simp add: Abs_fresh_iff fresh_Pair)
|
|
|
423 |
apply(auto)[1]
|
|
|
424 |
apply(rule fresh_eqvt_at)
|
|
|
425 |
back
|
|
|
426 |
apply(assumption)
|
|
|
427 |
apply(simp add: finite_supp)
|
|
2956
|
428 |
apply(rule_tac S="supp (env, y, x, t)" in supports_fresh)
|
|
2955
|
429 |
apply(simp add: supports_def fresh_def[symmetric])
|
|
|
430 |
apply(perm_simp)
|
|
|
431 |
apply(simp add: swap_fresh_fresh fresh_Pair)
|
|
|
432 |
apply(simp add: finite_supp)
|
|
|
433 |
apply(simp add: fresh_def[symmetric])
|
|
|
434 |
apply(simp add: eqvt_at_def)
|
|
|
435 |
apply(simp add: eqvt_at_def[symmetric])
|
|
|
436 |
apply(perm_simp)
|
|
|
437 |
apply(simp add: flip_fresh_fresh)
|
|
|
438 |
apply(rule sym)
|
|
|
439 |
apply(rule trans)
|
|
|
440 |
apply(rule sym)
|
|
|
441 |
apply(rule_tac a="xa" and b="c" in flip_fresh_fresh)
|
|
|
442 |
apply(simp add: Abs_fresh_iff)
|
|
|
443 |
apply(simp add: Abs_fresh_iff fresh_Pair)
|
|
|
444 |
apply(auto)[1]
|
|
|
445 |
apply(rule fresh_eqvt_at)
|
|
|
446 |
back
|
|
|
447 |
apply(assumption)
|
|
|
448 |
apply(simp add: finite_supp)
|
|
|
449 |
apply(rule_tac S="supp (enva, ya, xa, ta)" in supports_fresh)
|
|
|
450 |
apply(simp add: supports_def fresh_def[symmetric])
|
|
|
451 |
apply(perm_simp)
|
|
|
452 |
apply(simp add: swap_fresh_fresh fresh_Pair)
|
|
|
453 |
apply(simp add: finite_supp)
|
|
|
454 |
apply(simp add: fresh_def[symmetric])
|
|
|
455 |
apply(simp add: eqvt_at_def)
|
|
|
456 |
apply(simp add: eqvt_at_def[symmetric])
|
|
|
457 |
apply(perm_simp)
|
|
|
458 |
apply(simp add: flip_fresh_fresh)
|
|
|
459 |
apply(simp (no_asm) add: Abs1_eq_iff)
|
|
2969
|
460 |
(* HERE *)
|
|
2956
|
461 |
thm at_set_avoiding3
|
|
|
462 |
using at_set_avoiding3
|
|
|
463 |
apply -
|
|
|
464 |
apply(drule_tac x="set (atom y # bn env)" in meta_spec)
|
|
|
465 |
apply(drule_tac x="(env, enva)" in meta_spec)
|
|
|
466 |
apply(drule_tac x="[atom y # bn env]lst. t" in meta_spec)
|
|
|
467 |
apply(simp (no_asm_use) add: finite_supp)
|
|
|
468 |
apply(drule meta_mp)
|
|
|
469 |
apply(rule Abs_fresh_star)
|
|
|
470 |
apply(auto)[1]
|
|
|
471 |
apply(erule exE)
|
|
|
472 |
apply(erule conjE)+
|
|
|
473 |
apply(drule_tac q="(x \<leftrightarrow> c)" in eqvt_at_perm)
|
|
|
474 |
apply(perm_simp)
|
|
|
475 |
apply(simp add: flip_fresh_fresh fresh_Pair)
|
|
|
476 |
apply(drule_tac q="(xa \<leftrightarrow> c)" in eqvt_at_perm)
|
|
|
477 |
apply(perm_simp)
|
|
|
478 |
apply(simp add: flip_fresh_fresh fresh_Pair)
|
|
|
479 |
apply(drule sym)
|
|
2967
|
480 |
(* HERE *)
|
|
|
481 |
apply(rotate_tac 9)
|
|
|
482 |
apply(drule sym)
|
|
2956
|
483 |
apply(rotate_tac 9)
|
|
|
484 |
apply(drule trans)
|
|
|
485 |
apply(rule sym)
|
|
|
486 |
apply(rule_tac p="p" in supp_perm_eq)
|
|
2967
|
487 |
apply(assumption)
|
|
2956
|
488 |
apply(simp)
|
|
|
489 |
apply(perm_simp)
|
|
2967
|
490 |
apply(simp (no_asm_use) add: Abs_eq_iff2 alphas)
|
|
|
491 |
apply(erule conjE | erule exE)+
|
|
2956
|
492 |
apply(clarify)
|
|
|
493 |
apply(rule trans)
|
|
|
494 |
apply(rule sym)
|
|
|
495 |
apply(rule_tac p="pa" in perm_supp_eq)
|
|
|
496 |
defer
|
|
|
497 |
apply(rule sym)
|
|
|
498 |
apply(rule trans)
|
|
|
499 |
apply(rule sym)
|
|
|
500 |
apply(rule_tac p="p" in perm_supp_eq)
|
|
|
501 |
defer
|
|
|
502 |
apply(simp add: atom_eqvt)
|
|
|
503 |
apply(drule_tac q="(x \<leftrightarrow> c)" in eqvt_at_perm)
|
|
|
504 |
apply(perm_simp)
|
|
|
505 |
apply(simp add: flip_fresh_fresh fresh_Pair)
|
|
|
506 |
|
|
2955
|
507 |
apply(rule sym)
|
|
|
508 |
apply(erule_tac Abs_lst1_fcb2')
|
|
|
509 |
apply(rule fresh_eqvt_at)
|
|
|
510 |
back
|
|
|
511 |
apply(drule_tac q="(c \<leftrightarrow> x)" in eqvt_at_perm)
|
|
|
512 |
apply(perm_simp)
|
|
|
513 |
apply(simp add: flip_fresh_fresh)
|
|
|
514 |
apply(simp add: finite_supp)
|
|
|
515 |
apply(rule supports_fresh)
|
|
|
516 |
apply(rule_tac S="supp (enva, ya, xa, ta)" in supports_fresh)
|
|
|
517 |
apply(simp add: supports_def fresh_def[symmetric])
|
|
|
518 |
apply(perm_simp)
|
|
|
519 |
apply(simp add: swap_fresh_fresh fresh_Pair)
|
|
|
520 |
apply(simp add: finite_supp)
|
|
|
521 |
apply(simp add: fresh_def[symmetric])
|
|
|
522 |
apply(simp add: eqvt_at_def)
|
|
|
523 |
apply(simp add: eqvt_at_def[symmetric])
|
|
|
524 |
apply(perm_simp)
|
|
|
525 |
apply(rule fresh_eqvt_at)
|
|
|
526 |
back
|
|
|
527 |
apply(drule_tac q="(c \<leftrightarrow> x)" in eqvt_at_perm)
|
|
|
528 |
apply(perm_simp)
|
|
|
529 |
apply(simp add: flip_fresh_fresh)
|
|
|
530 |
apply(assumption)
|
|
|
531 |
apply(simp add: finite_supp)
|
|
2954
|
532 |
sorry
|
|
2952
|
533 |
|
|
2954
|
534 |
termination sorry
|
|
|
535 |
|
|
|
536 |
definition
|
|
|
537 |
eval :: "lam \<Rightarrow> sem"
|
|
|
538 |
where
|
|
|
539 |
"eval t = evals ENil t"
|
|
|
540 |
|
|
|
541 |
definition
|
|
|
542 |
normalize :: "lam \<Rightarrow> lam"
|
|
|
543 |
where
|
|
|
544 |
"normalize t = reify (eval t)"
|
|
|
545 |
|
|
|
546 |
end |