src/HOL/ex/Classpackage.thy
author haftmann
Sun Jul 23 07:21:41 2006 +0200 (2006-07-23)
changeset 20187 af47971ea304
parent 20178 e56fa3c8b1f1
child 20383 58f65fc90cf4
permissions -rw-r--r--
small adjustments
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(*  ID:         $Id$
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    Author:     Florian Haftmann, TU Muenchen
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*)
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header {* Test and Examples for Pure/Tools/class_package.ML *}
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theory Classpackage
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imports Main
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begin
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class semigroup =
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  fixes mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>\<otimes>" 70)
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  assumes assoc: "x \<^loc>\<otimes> y \<^loc>\<otimes> z = x \<^loc>\<otimes> (y \<^loc>\<otimes> z)"
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instance nat :: semigroup
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  "m \<otimes> n \<equiv> m + n"
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proof
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  fix m n q :: nat 
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  from semigroup_nat_def show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)" by simp
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qed
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instance int :: semigroup
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  "k \<otimes> l \<equiv> k + l"
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proof
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  fix k l j :: int
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  from semigroup_int_def show "k \<otimes> l \<otimes> j = k \<otimes> (l \<otimes> j)" by simp
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qed
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instance (type) list :: semigroup
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  "xs \<otimes> ys \<equiv> xs @ ys"
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proof
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  fix xs ys zs :: "'a list"
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  show "xs \<otimes> ys \<otimes> zs = xs \<otimes> (ys \<otimes> zs)"
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  proof -
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    from semigroup_list_def have "\<And>xs ys\<Colon>'a list. xs \<otimes> ys \<equiv> xs @ ys" .
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    thus ?thesis by simp
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  qed
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qed
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class monoidl = semigroup +
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  fixes one :: 'a ("\<^loc>\<one>")
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  assumes neutl: "\<^loc>\<one> \<^loc>\<otimes> x = x"
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instance monoidl_num_def: nat :: monoidl and int :: monoidl
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  "\<one> \<equiv> 0"
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  "\<one> \<equiv> 0"
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proof
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  fix n :: nat
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  from monoidl_num_def show "\<one> \<otimes> n = n" by simp
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next
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  fix k :: int
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  from monoidl_num_def show "\<one> \<otimes> k = k" by simp
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qed
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instance (type) list :: monoidl
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  "\<one> \<equiv> []"
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proof
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  fix xs :: "'a list"
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  show "\<one> \<otimes> xs = xs"
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  proof -
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    from mult_list_def have "\<And>xs ys\<Colon>'a list. xs \<otimes> ys \<equiv> xs @ ys" .
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    moreover from monoidl_list_def have "\<one> \<equiv> []\<Colon>'a list" by simp
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    ultimately show ?thesis by simp
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  qed
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qed  
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class monoid = monoidl +
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  assumes neutr: "x \<^loc>\<otimes> \<^loc>\<one> = x"
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instance monoid_list_def: (type) list :: monoid
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proof
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  fix xs :: "'a list"
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  show "xs \<otimes> \<one> = xs"
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  proof -
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    from mult_list_def have "\<And>xs ys\<Colon>'a list. xs \<otimes> ys \<equiv> xs @ ys" .
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    moreover from monoid_list_def have "\<one> \<equiv> []\<Colon>'a list" by simp
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    ultimately show ?thesis by simp
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  qed
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qed  
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class monoid_comm = monoid +
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  assumes comm: "x \<^loc>\<otimes> y = y \<^loc>\<otimes> x"
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instance monoid_comm_num_def: nat :: monoid_comm and int :: monoid_comm
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proof
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  fix n :: nat
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  from monoid_comm_num_def show "n \<otimes> \<one> = n" by simp
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next
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  fix n m :: nat
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  from monoid_comm_num_def show "n \<otimes> m = m \<otimes> n" by simp
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next
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  fix k :: int
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  from monoid_comm_num_def show "k \<otimes> \<one> = k" by simp
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next
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  fix k l :: int
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  from monoid_comm_num_def show "k \<otimes> l = l \<otimes> k" by simp
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qed
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definition (in monoid)
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  units :: "'a set"
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  units_def: "units = { y. \<exists>x. x \<^loc>\<otimes> y = \<^loc>\<one> \<and> y \<^loc>\<otimes> x = \<^loc>\<one> }"
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lemma (in monoid) inv_obtain:
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  assumes ass: "x \<in> units"
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  obtains y where "y \<^loc>\<otimes> x = \<^loc>\<one>" and "x \<^loc>\<otimes> y = \<^loc>\<one>"
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proof -
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  from ass units_def obtain y
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    where "y \<^loc>\<otimes> x = \<^loc>\<one>" and "x \<^loc>\<otimes> y = \<^loc>\<one>" by auto
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  thus ?thesis ..
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qed
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lemma (in monoid) inv_unique:
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  assumes eq: "y \<^loc>\<otimes> x = \<^loc>\<one>" "x \<^loc>\<otimes> y' = \<^loc>\<one>"
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  shows "y = y'"
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proof -
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  from eq neutr have "y = y \<^loc>\<otimes> (x \<^loc>\<otimes> y')" by simp
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  also with assoc have "... = (y \<^loc>\<otimes> x) \<^loc>\<otimes> y'" by simp
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  also with eq neutl have "... = y'" by simp
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  finally show ?thesis .
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qed
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lemma (in monoid) units_inv_comm:
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  assumes inv: "x \<^loc>\<otimes> y = \<^loc>\<one>"
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    and G: "x \<in> units"
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  shows "y \<^loc>\<otimes> x = \<^loc>\<one>"
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proof -
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  from G inv_obtain obtain z
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    where z_choice: "z \<^loc>\<otimes> x = \<^loc>\<one>" by blast
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  from inv neutl neutr have "x \<^loc>\<otimes> y \<^loc>\<otimes> x = x \<^loc>\<otimes> \<^loc>\<one>" by simp
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  with assoc have "z \<^loc>\<otimes> x \<^loc>\<otimes> y \<^loc>\<otimes> x = z \<^loc>\<otimes> x \<^loc>\<otimes> \<^loc>\<one>" by simp
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  with neutl z_choice show ?thesis by simp
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qed
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consts
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  reduce :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"
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primrec
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  "reduce f g 0 x = g"
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  "reduce f g (Suc n) x = f x (reduce f g n x)"
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definition (in monoid)
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  npow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
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  npow_def_prim: "npow n x = reduce (op \<^loc>\<otimes>) \<^loc>\<one> n x"
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abbreviation (in monoid)
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  abbrev_npow :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infix "\<^loc>\<up>" 75)
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  "x \<^loc>\<up> n \<equiv> npow n x"
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lemma (in monoid) npow_def:
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  "x \<^loc>\<up> 0 = \<^loc>\<one>"
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  "x \<^loc>\<up> Suc n = x \<^loc>\<otimes> x \<^loc>\<up> n"
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using npow_def_prim by simp_all
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lemma (in monoid) nat_pow_one:
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  "\<^loc>\<one> \<^loc>\<up> n = \<^loc>\<one>"
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using npow_def neutl by (induct n) simp_all
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lemma (in monoid) nat_pow_mult:
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  "npow n x \<^loc>\<otimes> npow m x = npow (n + m) x"
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proof (induct n)
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  case 0 with neutl npow_def show ?case by simp
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next
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  case (Suc n) with Suc.hyps assoc npow_def show ?case by simp
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qed
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lemma (in monoid) nat_pow_pow:
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  "npow n (npow m x) = npow (n * m) x"
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using npow_def nat_pow_mult by (induct n) simp_all
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class group = monoidl +
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  fixes inv :: "'a \<Rightarrow> 'a" ("\<^loc>\<div> _" [81] 80)
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  assumes invl: "\<^loc>\<div> x \<^loc>\<otimes> x = \<^loc>\<one>"
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class group_comm = group + monoid_comm
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instance group_comm_int_def: int :: group_comm
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  "\<div> k \<equiv> - (k\<Colon>int)"
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proof
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  fix k :: int
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  from group_comm_int_def show "\<div> k \<otimes> k = \<one>" by simp
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qed
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lemma (in group) cancel:
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  "(x \<^loc>\<otimes> y = x \<^loc>\<otimes> z) = (y = z)"
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proof
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  fix x y z :: 'a
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  assume eq: "x \<^loc>\<otimes> y = x \<^loc>\<otimes> z"
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  hence "\<^loc>\<div> x \<^loc>\<otimes> (x \<^loc>\<otimes> y) = \<^loc>\<div> x \<^loc>\<otimes> (x \<^loc>\<otimes> z)" by simp
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  with assoc have "\<^loc>\<div> x \<^loc>\<otimes> x \<^loc>\<otimes> y = \<^loc>\<div> x \<^loc>\<otimes> x \<^loc>\<otimes> z" by simp
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  with neutl invl show "y = z" by simp
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next
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  fix x y z :: 'a
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  assume eq: "y = z"
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  thus "x \<^loc>\<otimes> y = x \<^loc>\<otimes> z" by simp
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qed
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lemma (in group) neutr:
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  "x \<^loc>\<otimes> \<^loc>\<one> = x"
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proof -
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  from invl have "\<^loc>\<div> x \<^loc>\<otimes> x = \<^loc>\<one>" by simp
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  with assoc [symmetric] neutl invl have "\<^loc>\<div> x \<^loc>\<otimes> (x \<^loc>\<otimes> \<^loc>\<one>) = \<^loc>\<div> x \<^loc>\<otimes> x" by simp
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  with cancel show ?thesis by simp
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qed
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lemma (in group) invr:
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  "x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<one>"
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proof -
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  from neutl invl have "\<^loc>\<div> x \<^loc>\<otimes> x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<div> x" by simp
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  with neutr have "\<^loc>\<div> x \<^loc>\<otimes> x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<div> x \<^loc>\<otimes> \<^loc>\<one> " by simp
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  with assoc have "\<^loc>\<div> x \<^loc>\<otimes> (x \<^loc>\<otimes> \<^loc>\<div> x) = \<^loc>\<div> x \<^loc>\<otimes> \<^loc>\<one> " by simp
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  with cancel show ?thesis ..
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qed
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interpretation group < monoid
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proof -
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  fix x :: "'a"
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  from neutr show "x \<^loc>\<otimes> \<^loc>\<one> = x" .
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qed
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instance group < monoid
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proof
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  fix x :: "'a\<Colon>group"
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  from group.neutr show "x \<otimes> \<one> = x" .
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qed
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lemma (in group) all_inv [intro]:
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  "(x\<Colon>'a) \<in> monoid.units (op \<^loc>\<otimes>) \<^loc>\<one>"
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  unfolding units_def
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proof -
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  fix x :: "'a"
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  from invl invr have "\<^loc>\<div> x \<^loc>\<otimes> x = \<^loc>\<one>" and "x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<one>" . 
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  then obtain y where "y \<^loc>\<otimes> x = \<^loc>\<one>" and "x \<^loc>\<otimes> y = \<^loc>\<one>" ..
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  hence "\<exists>y\<Colon>'a. y \<^loc>\<otimes> x = \<^loc>\<one> \<and> x \<^loc>\<otimes> y = \<^loc>\<one>" by blast
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  thus "x \<in> {y\<Colon>'a. \<exists>x\<Colon>'a. x \<^loc>\<otimes> y = \<^loc>\<one> \<and> y \<^loc>\<otimes> x = \<^loc>\<one>}" by simp
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qed
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lemma (in group) cancer:
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  "(y \<^loc>\<otimes> x = z \<^loc>\<otimes> x) = (y = z)"
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proof
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  assume eq: "y \<^loc>\<otimes> x = z \<^loc>\<otimes> x"
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  with assoc [symmetric] have "y \<^loc>\<otimes> (x \<^loc>\<otimes> \<^loc>\<div> x) = z \<^loc>\<otimes> (x \<^loc>\<otimes> \<^loc>\<div> x)" by (simp del: invr)
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  with invr neutr show "y = z" by simp
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next
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  assume eq: "y = z"
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  thus "y \<^loc>\<otimes> x = z \<^loc>\<otimes> x" by simp
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qed
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lemma (in group) inv_one:
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  "\<^loc>\<div> \<^loc>\<one> = \<^loc>\<one>"
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proof -
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  from neutl have "\<^loc>\<div> \<^loc>\<one> = \<^loc>\<one> \<^loc>\<otimes> (\<^loc>\<div> \<^loc>\<one>)" ..
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  moreover from invr have "... = \<^loc>\<one>" by simp
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  finally show ?thesis .
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qed
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lemma (in group) inv_inv:
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  "\<^loc>\<div> (\<^loc>\<div> x) = x"
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proof -
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  from invl invr neutr
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    have "\<^loc>\<div> (\<^loc>\<div> x) \<^loc>\<otimes> \<^loc>\<div> x \<^loc>\<otimes> x = x \<^loc>\<otimes> \<^loc>\<div> x \<^loc>\<otimes> x" by simp
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  with assoc [symmetric]
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    have "\<^loc>\<div> (\<^loc>\<div> x) \<^loc>\<otimes> (\<^loc>\<div> x \<^loc>\<otimes> x) = x \<^loc>\<otimes> (\<^loc>\<div> x \<^loc>\<otimes> x)" by simp
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  with invl neutr show ?thesis by simp
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qed
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lemma (in group) inv_mult_group:
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  "\<^loc>\<div> (x \<^loc>\<otimes> y) = \<^loc>\<div> y \<^loc>\<otimes> \<^loc>\<div> x"
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proof -
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  from invl have "\<^loc>\<div> (x \<^loc>\<otimes> y) \<^loc>\<otimes> (x \<^loc>\<otimes> y) = \<^loc>\<one>" by simp
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  with assoc have "\<^loc>\<div> (x \<^loc>\<otimes> y) \<^loc>\<otimes> x \<^loc>\<otimes> y = \<^loc>\<one>" by simp
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  with neutl have "\<^loc>\<div> (x \<^loc>\<otimes> y) \<^loc>\<otimes> x \<^loc>\<otimes> y \<^loc>\<otimes> \<^loc>\<div> y \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<div> y \<^loc>\<otimes> \<^loc>\<div> x" by simp
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  with assoc have "\<^loc>\<div> (x \<^loc>\<otimes> y) \<^loc>\<otimes> (x \<^loc>\<otimes> (y \<^loc>\<otimes> \<^loc>\<div> y) \<^loc>\<otimes> \<^loc>\<div> x) = \<^loc>\<div> y \<^loc>\<otimes> \<^loc>\<div> x" by simp
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  with invr neutr show ?thesis by simp
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qed
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lemma (in group) inv_comm:
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  "x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<div> x \<^loc>\<otimes> x"
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using invr invl by simp
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definition (in group)
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  pow :: "int \<Rightarrow> 'a \<Rightarrow> 'a"
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  pow_def: "pow k x = (if k < 0 then \<^loc>\<div> (monoid.npow (op \<^loc>\<otimes>) \<^loc>\<one> (nat (-k)) x)
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    else (monoid.npow (op \<^loc>\<otimes>) \<^loc>\<one> (nat k) x))"
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abbreviation (in group)
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  abbrev_pow :: "'a \<Rightarrow> int \<Rightarrow> 'a" (infix "\<^loc>\<up>" 75)
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  "x \<^loc>\<up> k \<equiv> pow k x"
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lemma (in group) int_pow_zero:
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  "x \<^loc>\<up> (0\<Colon>int) = \<^loc>\<one>"
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using npow_def pow_def by simp
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lemma (in group) int_pow_one:
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  "\<^loc>\<one> \<^loc>\<up> (k\<Colon>int) = \<^loc>\<one>"
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using pow_def nat_pow_one inv_one by simp
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instance group_prod_def: (group, group) * :: group
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  mult_prod_def: "x \<otimes> y \<equiv> let (x1, x2) = x; (y1, y2) = y in
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              (x1 \<otimes> y1, x2 \<otimes> y2)"
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  mult_one_def: "\<one> \<equiv> (\<one>, \<one>)"
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  mult_inv_def: "\<div> x \<equiv> let (x1, x2) = x in (\<div> x1, \<div> x2)"
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by default (simp_all add: split_paired_all group_prod_def assoc neutl invl)
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instance group_comm_prod_def: (group_comm, group_comm) * :: group_comm
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by default (simp_all add: split_paired_all group_prod_def assoc neutl invl comm)
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definition
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  "x = ((2\<Colon>nat) \<otimes> \<one> \<otimes> 3, (2\<Colon>int) \<otimes> \<one> \<otimes> \<div> 3, [1\<Colon>nat, 2] \<otimes> \<one> \<otimes> [1, 2, 3])"
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  "y = (2 \<Colon> int, \<div> 2 \<Colon> int) \<otimes> \<one> \<otimes> (3, \<div> 3)"
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code_generate "op \<otimes>" \<one> inv
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code_generate (ml, haskell) x
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code_generate (ml, haskell) y
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code_serialize ml (-)
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end