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+(*  Title:      HOL/Examples/Adhoc_Overloading.thy
+    Author:     Christian Sternagel
+*)
+
+section \<open>Ad Hoc Overloading\<close>
+
+theory Adhoc_Overloading
+imports
+  Main
+  "HOL-Library.Infinite_Set"
+begin
+
+text \<open>Adhoc overloading allows to overload a constant depending on
+its type. Typically this involves to introduce an uninterpreted
+constant (used for input and output) and then add some variants (used
+internally).\<close>
+
+subsection \<open>Plain Ad Hoc Overloading\<close>
+
+text \<open>Consider the type of first-order terms.\<close>
+datatype ('a, 'b) "term" =
+  Var 'b |
+  Fun 'a "('a, 'b) term list"
+
+text \<open>The set of variables of a term might be computed as follows.\<close>
+fun term_vars :: "('a, 'b) term \<Rightarrow> 'b set" where
+  "term_vars (Var x) = {x}" |
+  "term_vars (Fun f ts) = \<Union>(set (map term_vars ts))"
+
+text \<open>However, also for \emph{rules} (i.e., pairs of terms) and term
+rewrite systems (i.e., sets of rules), the set of variables makes
+sense. Thus we introduce an unspecified constant \<open>vars\<close>.\<close>
+
+consts vars :: "'a \<Rightarrow> 'b set"
+
+text \<open>Which is then overloaded with variants for terms, rules, and TRSs.\<close>
+adhoc_overloading
+  vars term_vars
+
+value [nbe] "vars (Fun ''f'' [Var 0, Var 1])"
+
+fun rule_vars :: "('a, 'b) term \<times> ('a, 'b) term \<Rightarrow> 'b set" where
+  "rule_vars (l, r) = vars l \<union> vars r"
+
+adhoc_overloading
+  vars rule_vars
+
+value [nbe] "vars (Var 1, Var 0)"
+
+definition trs_vars :: "(('a, 'b) term \<times> ('a, 'b) term) set \<Rightarrow> 'b set" where
+  "trs_vars R = \<Union>(rule_vars ` R)"
+
+adhoc_overloading
+  vars trs_vars
+
+value [nbe] "vars {(Var 1, Var 0)}"
+
+text \<open>Sometimes it is necessary to add explicit type constraints
+before a variant can be determined.\<close>
+(*value "vars R" (*has multiple instances*)*)
+value "vars (R :: (('a, 'b) term \<times> ('a, 'b) term) set)"
+
+text \<open>It is also possible to remove variants.\<close>
+no_adhoc_overloading
+  vars term_vars rule_vars 
+
+(*value "vars (Var 1)" (*does not have an instance*)*)
+
+text \<open>As stated earlier, the overloaded constant is only used for
+input and output. Internally, always a variant is used, as can be
+observed by the configuration option \<open>show_variants\<close>.\<close>
+
+adhoc_overloading
+  vars term_vars
+
+declare [[show_variants]]
+
+term "vars (Var 1)" (*which yields: "term_vars (Var 1)"*)
+
+
+subsection \<open>Adhoc Overloading inside Locales\<close>
+
+text \<open>As example we use permutations that are parametrized over an
+atom type \<^typ>\<open>'a\<close>.\<close>
+
+definition perms :: "('a \<Rightarrow> 'a) set" where
+  "perms = {f. bij f \<and> finite {x. f x \<noteq> x}}"
+
+typedef 'a perm = "perms :: ('a \<Rightarrow> 'a) set"
+  by standard (auto simp: perms_def)
+
+text \<open>First we need some auxiliary lemmas.\<close>
+lemma permsI [Pure.intro]:
+  assumes "bij f" and "MOST x. f x = x"
+  shows "f \<in> perms"
+  using assms by (auto simp: perms_def) (metis MOST_iff_finiteNeg)
+
+lemma perms_imp_bij:
+  "f \<in> perms \<Longrightarrow> bij f"
+  by (simp add: perms_def)
+
+lemma perms_imp_MOST_eq:
+  "f \<in> perms \<Longrightarrow> MOST x. f x = x"
+  by (simp add: perms_def) (metis MOST_iff_finiteNeg)
+
+lemma id_perms [simp]:
+  "id \<in> perms"
+  "(\<lambda>x. x) \<in> perms"
+  by (auto simp: perms_def bij_def)
+
+lemma perms_comp [simp]:
+  assumes f: "f \<in> perms" and g: "g \<in> perms"
+  shows "(f \<circ> g) \<in> perms"
+  apply (intro permsI bij_comp)
+  apply (rule perms_imp_bij [OF g])
+  apply (rule perms_imp_bij [OF f])
+  apply (rule MOST_rev_mp [OF perms_imp_MOST_eq [OF g]])
+  apply (rule MOST_rev_mp [OF perms_imp_MOST_eq [OF f]])
+  by simp
+
+lemma perms_inv:
+  assumes f: "f \<in> perms"
+  shows "inv f \<in> perms"
+  apply (rule permsI)
+  apply (rule bij_imp_bij_inv)
+  apply (rule perms_imp_bij [OF f])
+  apply (rule MOST_mono [OF perms_imp_MOST_eq [OF f]])
+  apply (erule subst, rule inv_f_f)
+  apply (rule bij_is_inj [OF perms_imp_bij [OF f]])
+  done
+
+lemma bij_Rep_perm: "bij (Rep_perm p)"
+  using Rep_perm [of p] unfolding perms_def by simp
+
+instantiation perm :: (type) group_add
+begin
+
+definition "0 = Abs_perm id"
+definition "- p = Abs_perm (inv (Rep_perm p))"
+definition "p + q = Abs_perm (Rep_perm p \<circ> Rep_perm q)"
+definition "(p1::'a perm) - p2 = p1 + - p2"
+
+lemma Rep_perm_0: "Rep_perm 0 = id"
+  unfolding zero_perm_def by (simp add: Abs_perm_inverse)
+
+lemma Rep_perm_add:
+  "Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2"
+  unfolding plus_perm_def by (simp add: Abs_perm_inverse Rep_perm)
+
+lemma Rep_perm_uminus:
+  "Rep_perm (- p) = inv (Rep_perm p)"
+  unfolding uminus_perm_def by (simp add: Abs_perm_inverse perms_inv Rep_perm)
+
+instance
+  apply standard
+  unfolding Rep_perm_inject [symmetric]
+  unfolding minus_perm_def
+  unfolding Rep_perm_add
+  unfolding Rep_perm_uminus
+  unfolding Rep_perm_0
+  apply (simp_all add: o_assoc inv_o_cancel [OF bij_is_inj [OF bij_Rep_perm]])
+  done
+
+end
+
+lemmas Rep_perm_simps =
+  Rep_perm_0
+  Rep_perm_add
+  Rep_perm_uminus
+
+
+section \<open>Permutation Types\<close>
+
+text \<open>We want to be able to apply permutations to arbitrary types. To
+this end we introduce a constant \<open>PERMUTE\<close> together with
+convenient infix syntax.\<close>
+
+consts PERMUTE :: "'a perm \<Rightarrow> 'b \<Rightarrow> 'b" (infixr \<open>\<bullet>\<close> 75)
+
+text \<open>Then we add a locale for types \<^typ>\<open>'b\<close> that support
+appliciation of permutations.\<close>
+locale permute =
+  fixes permute :: "'a perm \<Rightarrow> 'b \<Rightarrow> 'b"
+  assumes permute_zero [simp]: "permute 0 x = x"
+    and permute_plus [simp]: "permute (p + q) x = permute p (permute q x)"
+begin
+
+adhoc_overloading
+  PERMUTE permute
+
+end
+
+text \<open>Permuting atoms.\<close>
+definition permute_atom :: "'a perm \<Rightarrow> 'a \<Rightarrow> 'a" where
+  "permute_atom p a = (Rep_perm p) a"
+
+adhoc_overloading
+  PERMUTE permute_atom
+
+interpretation atom_permute: permute permute_atom
+  by standard (simp_all add: permute_atom_def Rep_perm_simps)
+
+text \<open>Permuting permutations.\<close>
+definition permute_perm :: "'a perm \<Rightarrow> 'a perm \<Rightarrow> 'a perm" where
+  "permute_perm p q = p + q - p"
+
+adhoc_overloading
+  PERMUTE permute_perm
+
+interpretation perm_permute: permute permute_perm
+  apply standard
+  unfolding permute_perm_def
+  apply simp
+  apply (simp only: diff_conv_add_uminus minus_add add.assoc)
+  done
+
+text \<open>Permuting functions.\<close>
+locale fun_permute =
+  dom: permute perm1 + ran: permute perm2
+  for perm1 :: "'a perm \<Rightarrow> 'b \<Rightarrow> 'b"
+  and perm2 :: "'a perm \<Rightarrow> 'c \<Rightarrow> 'c"
+begin
+
+adhoc_overloading
+  PERMUTE perm1 perm2
+
+definition permute_fun :: "'a perm \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'c)" where
+  "permute_fun p f = (\<lambda>x. p \<bullet> (f (-p \<bullet> x)))"
+
+adhoc_overloading
+  PERMUTE permute_fun
+
+end
+
+sublocale fun_permute \<subseteq> permute permute_fun
+  by (unfold_locales, auto simp: permute_fun_def)
+     (metis dom.permute_plus minus_add)
+
+lemma "(Abs_perm id :: nat perm) \<bullet> Suc 0 = Suc 0"
+  unfolding permute_atom_def
+  by (metis Rep_perm_0 id_apply zero_perm_def)
+
+interpretation atom_fun_permute: fun_permute permute_atom permute_atom
+  by (unfold_locales)
+
+adhoc_overloading
+  PERMUTE atom_fun_permute.permute_fun
+
+lemma "(Abs_perm id :: 'a perm) \<bullet> id = id"
+  unfolding atom_fun_permute.permute_fun_def
+  unfolding permute_atom_def
+  by (metis Rep_perm_0 id_def inj_imp_inv_eq inj_on_id uminus_perm_def zero_perm_def)
+
+end