author paulson Wed, 03 Apr 2019 16:38:59 +0100 changeset 70043 350acd367028 parent 70042 45787384ff86 child 70044 da5857dbcbb9
theory Poly_Mapping
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Poly_Mapping.thy	Wed Apr 03 16:38:59 2019 +0100
@@ -0,0 +1,1878 @@
+(* Author: Andreas Lochbihler, ETH Zurich
+   Author: Florian Haftmann, TU Muenchen
+   with some material ported from HOL Light by LCP
+*)
+
+section \<open>Polynomial mapping: combination of almost everywhere zero functions with an algebraic view\<close>
+
+theory Poly_Mapping
+imports Groups_Big_Fun Fun_Lexorder More_List
+begin
+
+subsection \<open>Preliminary: auxiliary operations for \qt{almost everywhere zero}\<close>
+
+text \<open>
+  A central notion for polynomials are functions being \qt{almost everywhere zero}.
+  For these we provide some auxiliary definitions and lemmas.
+\<close>
+
+lemma finite_mult_not_eq_zero_leftI:
+  fixes f :: "'b \<Rightarrow> 'a :: mult_zero"
+  assumes "finite {a. f a \<noteq> 0}"
+  shows "finite {a. g a * f a \<noteq> 0}"
+proof -
+  have "{a. g a * f a \<noteq> 0} \<subseteq> {a. f a \<noteq> 0}" by auto
+  then show ?thesis using assms by (rule finite_subset)
+qed
+
+lemma finite_mult_not_eq_zero_rightI:
+  fixes f :: "'b \<Rightarrow> 'a :: mult_zero"
+  assumes "finite {a. f a \<noteq> 0}"
+  shows "finite {a. f a * g a \<noteq> 0}"
+proof -
+  have "{a. f a * g a \<noteq> 0} \<subseteq> {a. f a \<noteq> 0}" by auto
+  then show ?thesis using assms by (rule finite_subset)
+qed
+
+lemma finite_mult_not_eq_zero_prodI:
+  fixes f g :: "'a \<Rightarrow> 'b::semiring_0"
+  assumes "finite {a. f a \<noteq> 0}" (is "finite ?F")
+  assumes "finite {b. g b \<noteq> 0}" (is "finite ?G")
+  shows "finite {(a, b). f a * g b \<noteq> 0}"
+proof -
+  from assms have "finite (?F \<times> ?G)"
+    by blast
+  then have "finite {(a, b). f a \<noteq> 0 \<and> g b \<noteq> 0}"
+    by simp
+  then show ?thesis
+    by (rule rev_finite_subset) auto
+qed
+
+lemma finite_not_eq_zero_sumI:
+  fixes f g :: "'a::monoid_add \<Rightarrow> 'b::semiring_0"
+  assumes "finite {a. f a \<noteq> 0}" (is "finite ?F")
+  assumes "finite {b. g b \<noteq> 0}" (is "finite ?G")
+  shows "finite {a + b | a b. f a \<noteq> 0 \<and> g b \<noteq> 0}" (is "finite ?FG")
+proof -
+  from assms have "finite (?F \<times> ?G)"
+  then have "finite (case_prod plus  (?F \<times> ?G))"
+    by (rule finite_imageI)
+  also have "case_prod plus  (?F \<times> ?G) = ?FG"
+    by auto
+  finally show ?thesis
+    by simp
+qed
+
+lemma finite_mult_not_eq_zero_sumI:
+  fixes f g :: "'a::monoid_add \<Rightarrow> 'b::semiring_0"
+  assumes "finite {a. f a \<noteq> 0}"
+  assumes "finite {b. g b \<noteq> 0}"
+  shows "finite {a + b | a b. f a * g b \<noteq> 0}"
+proof -
+  from assms
+  have "finite {a + b | a b. f a \<noteq> 0 \<and> g b \<noteq> 0}"
+    by (rule finite_not_eq_zero_sumI)
+  then show ?thesis
+    by (rule rev_finite_subset) (auto dest: mult_not_zero)
+qed
+
+lemma finite_Sum_any_not_eq_zero_weakenI:
+  assumes "finite {a. \<exists>b. f a b \<noteq> 0}"
+  shows "finite {a. Sum_any (f a) \<noteq> 0}"
+proof -
+  have "{a. Sum_any (f a) \<noteq> 0} \<subseteq> {a. \<exists>b. f a b \<noteq> 0}"
+    by (auto elim: Sum_any.not_neutral_obtains_not_neutral)
+  then show ?thesis using assms by (rule finite_subset)
+qed
+
+context zero
+begin
+
+definition "when" :: "'a \<Rightarrow> bool \<Rightarrow> 'a" (infixl "when" 20)
+where
+  "(a when P) = (if P then a else 0)"
+
+text \<open>
+  Case distinctions always complicate matters, particularly when
+  nested.  The @{const "when"} operation allows to minimise these
+  if @{term 0} is the false-case value and makes proof obligations
+\<close>
+
+lemma "when" [simp]:
+  "P \<Longrightarrow> (a when P) = a"
+  "\<not> P \<Longrightarrow> (a when P) = 0"
+
+lemma when_simps [simp]:
+  "(a when True) = a"
+  "(a when False) = 0"
+  by simp_all
+
+lemma when_cong:
+  assumes "P \<longleftrightarrow> Q"
+    and "Q \<Longrightarrow> a = b"
+  shows "(a when P) = (b when Q)"
+  using assms by (simp add: when_def)
+
+lemma zero_when [simp]:
+  "(0 when P) = 0"
+
+lemma when_when:
+  "(a when P when Q) = (a when P \<and> Q)"
+  by (cases Q) simp_all
+
+lemma when_commute:
+  "(a when Q when P) = (a when P when Q)"
+  by (simp add: when_when conj_commute)
+
+lemma when_neq_zero [simp]:
+  "(a when P) \<noteq> 0 \<longleftrightarrow> P \<and> a \<noteq> 0"
+  by (cases P) simp_all
+
+end
+
+begin
+
+  "(a + b when P) = (a when P) + (b when P)"
+
+end
+
+context semiring_1
+begin
+
+lemma zero_power_eq:
+  "0 ^ n = (1 when n = 0)"
+
+end
+
+begin
+
+lemma Sum_any_when_equal [simp]:
+  "(\<Sum>a. (f a when a = b)) = f b"
+
+lemma Sum_any_when_equal' [simp]:
+  "(\<Sum>a. (f a when b = a)) = f b"
+
+lemma Sum_any_when_independent:
+  "(\<Sum>a. g a when P) = ((\<Sum>a. g a) when P)"
+  by (cases P) simp_all
+
+lemma Sum_any_when_dependent_prod_right:
+  "(\<Sum>(a, b). g a when b = h a) = (\<Sum>a. g a)"
+proof -
+  have "inj_on (\<lambda>a. (a, h a)) {a. g a \<noteq> 0}"
+    by (rule inj_onI) auto
+  then show ?thesis unfolding Sum_any.expand_set
+    by (rule sum.reindex_cong) auto
+qed
+
+lemma Sum_any_when_dependent_prod_left:
+  "(\<Sum>(a, b). g b when a = h b) = (\<Sum>b. g b)"
+proof -
+  have "(\<Sum>(a, b). g b when a = h b) = (\<Sum>(b, a). g b when a = h b)"
+    by (rule Sum_any.reindex_cong [of prod.swap]) (simp_all add: fun_eq_iff)
+  then show ?thesis by (simp add: Sum_any_when_dependent_prod_right)
+qed
+
+end
+
+begin
+
+lemma when_diff_distrib:
+  "(a - b when P) = (a when P) - (b when P)"
+
+end
+
+begin
+
+lemma when_uminus_distrib:
+  "(- a when P) = - (a when P)"
+
+end
+
+context mult_zero
+begin
+
+lemma mult_when:
+  "a * (b when P) = (a * b when P)"
+  by (cases P) simp_all
+
+lemma when_mult:
+  "(a when P) * b = (a * b when P)"
+  by (cases P) simp_all
+
+end
+
+
+subsection \<open>Type definition\<close>
+
+text \<open>
+  The following type is of central importance:
+\<close>
+
+typedef (overloaded) ('a, 'b) poly_mapping ("(_ \<Rightarrow>\<^sub>0 /_)" [1, 0] 0) =
+  "{f :: 'a \<Rightarrow> 'b::zero. finite {x. f x \<noteq> 0}}"
+  morphisms lookup Abs_poly_mapping
+proof -
+  have "(\<lambda>_::'a. (0 :: 'b)) \<in> ?poly_mapping" by simp
+  then show ?thesis by (blast intro!: exI)
+qed
+
+declare lookup_inverse [simp]
+declare lookup_inject [simp]
+
+lemma lookup_Abs_poly_mapping [simp]:
+  "finite {x. f x \<noteq> 0} \<Longrightarrow> lookup (Abs_poly_mapping f) = f"
+  using Abs_poly_mapping_inverse [of f] by simp
+
+lemma finite_lookup [simp]:
+  "finite {k. lookup f k \<noteq> 0}"
+  using poly_mapping.lookup [of f] by simp
+
+lemma finite_lookup_nat [simp]:
+  fixes f :: "'a \<Rightarrow>\<^sub>0 nat"
+  shows "finite {k. 0 < lookup f k}"
+  using poly_mapping.lookup [of f] by simp
+
+lemma poly_mapping_eqI:
+  assumes "\<And>k. lookup f k = lookup g k"
+  shows "f = g"
+  using assms unfolding poly_mapping.lookup_inject [symmetric]
+  by blast
+
+lemma poly_mapping_eq_iff: "a = b \<longleftrightarrow> lookup a = lookup b"
+  by auto
+
+text \<open>
+  We model the universe of functions being \qt{almost everywhere zero}
+  by means of a separate type @{typ "('a, 'b) poly_mapping"}.
+  For convenience we provide a suggestive infix syntax which
+  is a variant of the usual function space syntax.  Conversion between both types
+  happens through the morphisms
+  \begin{quote}
+    @{term_type lookup}
+  \end{quote}
+  \begin{quote}
+    @{term_type Abs_poly_mapping}
+  \end{quote}
+  satisfying
+  \begin{quote}
+    @{thm lookup_inverse}
+  \end{quote}
+  \begin{quote}
+    @{thm lookup_Abs_poly_mapping}
+  \end{quote}
+  Luckily, we have rarely to deal with those low-level morphisms explicitly
+  but rely on Isabelle's \emph{lifting} package with its method \<open>transfer\<close>
+  and its specification tool \<open>lift_definition\<close>.
+\<close>
+
+setup_lifting type_definition_poly_mapping
+code_datatype Abs_poly_mapping\<comment>\<open>FIXME? workaround for preventing \<open>code_abstype\<close> setup\<close>
+
+text \<open>
+  @{typ "'a \<Rightarrow>\<^sub>0 'b"} serves distinctive purposes:
+  \begin{enumerate}
+    \item A clever nesting as @{typ "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"}
+      later in theory \<open>MPoly\<close> gives a suitable
+      Interpreting @{typ "nat \<Rightarrow>\<^sub>0 nat"} as a mapping from variable identifiers
+      to exponents yields monomials, and the whole type maps monomials
+      to coefficients.  Lets call this the \emph{ultimate interpretation}.
+    \item A further more specialised type isomorphic to @{typ "nat \<Rightarrow>\<^sub>0 'a"}
+      is apt to direct implementation using code generation
+      \cite{Haftmann-Nipkow:2010:code}.
+  \end{enumerate}
+  Note that despite the names \qt{mapping} and \qt{lookup} suggest something
+  implementation-near, it is best to keep @{typ "'a \<Rightarrow>\<^sub>0 'b"} as an abstract
+  \emph{algebraic} type
+  providing operations like \qt{addition}, \qt{multiplication} without any notion
+  of key-order, data structures etc.  This implementations-specific notions are
+  easily introduced later for particular implementations but do not provide any
+  gain for specifying logical properties of polynomials.
+\<close>
+
+
+text \<open>
+  The additive structure covers the usual operations \<open>0\<close>, \<open>+\<close> and
+  (unary and binary) \<open>-\<close>.  Recalling the ultimate interpretation, it
+  is obvious that these have just lift the corresponding operations on values
+  to mappings.
+
+  Isabelle has a rich hierarchy of algebraic type classes, and in such situations
+  of pointwise lifting a typical pattern is to have instantiations for a considerable
+  number of type classes.
+
+  The operations themselves are specified using \<open>lift_definition\<close>, where
+  the proofs of the \qt{almost everywhere zero} property can be significantly involved.
+
+  The @{const lookup} operation is supposed to be usable explicitly (unless in
+  other situations where the morphisms between types are somehow internal
+  to the \emph{lifting} package).  Hence it is good style to provide explicit rewrite
+  rules how @{const lookup} acts on operations immediately.
+\<close>
+
+instantiation poly_mapping :: (type, zero) zero
+begin
+
+lift_definition zero_poly_mapping :: "'a \<Rightarrow>\<^sub>0 'b"
+  is "\<lambda>k. 0"
+  by simp
+
+instance ..
+
+end
+
+lemma Abs_poly_mapping [simp]: "Abs_poly_mapping (\<lambda>k. 0) = 0"
+
+lemma lookup_zero [simp]: "lookup 0 k = 0"
+  by transfer rule
+
+begin
+
+lift_definition plus_poly_mapping ::
+    "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b"
+  is "\<lambda>f1 f2 k. f1 k + f2 k"
+proof -
+  fix f1 f2 :: "'a \<Rightarrow> 'b"
+  assume "finite {k. f1 k \<noteq> 0}"
+    and "finite {k. f2 k \<noteq> 0}"
+  then have "finite ({k. f1 k \<noteq> 0} \<union> {k. f2 k \<noteq> 0})" by auto
+  moreover have "{x. f1 x + f2 x \<noteq> 0} \<subseteq> {k. f1 k \<noteq> 0} \<union> {k. f2 k \<noteq> 0}"
+    by auto
+  ultimately show "finite {x. f1 x + f2 x \<noteq> 0}"
+    by (blast intro: finite_subset)
+qed
+
+instance
+  by intro_classes (transfer, simp add: fun_eq_iff ac_simps)+
+
+end
+
+  "lookup (f + g) k = lookup f k + lookup g k"
+  by transfer rule
+
+  by intro_classes (transfer, simp add: fun_eq_iff ac_simps)+
+
+lemma lookup_sum: "lookup (sum pp X) i = sum (\<lambda>x. lookup (pp x) i) X"
+  by (induction rule: infinite_finite_induct) (auto simp: lookup_add)
+
+  by intro_classes (transfer, simp add: fun_eq_iff)+*)
+
+begin
+
+lift_definition minus_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b"
+  is "\<lambda>f1 f2 k. f1 k - f2 k"
+proof -
+  fix f1 f2 :: "'a \<Rightarrow> 'b"
+  assume "finite {k. f1 k \<noteq> 0}"
+    and "finite {k. f2 k \<noteq> 0}"
+  then have "finite ({k. f1 k \<noteq> 0} \<union> {k. f2 k \<noteq> 0})" by auto
+  moreover have "{x. f1 x - f2 x \<noteq> 0} \<subseteq> {k. f1 k \<noteq> 0} \<union> {k. f2 k \<noteq> 0}"
+    by auto
+  ultimately show "finite {x. f1 x - f2 x \<noteq> 0}" by (blast intro: finite_subset)
+qed
+
+instance
+
+end
+
+begin
+
+lift_definition uminus_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b"
+  is uminus
+  by simp
+
+
+instance
+  by intro_classes (transfer, simp add: fun_eq_iff ac_simps)+
+
+end
+
+lemma lookup_uminus [simp]:
+  "lookup (- f) k = - lookup f k"
+  by transfer simp
+
+lemma lookup_minus:
+  "lookup (f - g) k = lookup f k - lookup g k"
+  by transfer rule
+
+
+subsection \<open>Multiplicative structure\<close>
+
+instantiation poly_mapping :: (zero, zero_neq_one) zero_neq_one
+begin
+
+lift_definition one_poly_mapping :: "'a \<Rightarrow>\<^sub>0 'b"
+  is "\<lambda>k. 1 when k = 0"
+  by simp
+
+instance
+  by intro_classes (transfer, simp add: fun_eq_iff)
+
+end
+
+lemma lookup_one:
+  "lookup 1 k = (1 when k = 0)"
+  by transfer rule
+
+lemma lookup_one_zero [simp]:
+  "lookup 1 0 = 1"
+  by transfer simp
+
+definition prod_fun :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a::monoid_add \<Rightarrow> 'b::semiring_0"
+where
+  "prod_fun f1 f2 k = (\<Sum>l. f1 l * (\<Sum>q. (f2 q when k = l + q)))"
+
+lemma prod_fun_unfold_prod:
+  fixes f g :: "'a :: monoid_add \<Rightarrow> 'b::semiring_0"
+  assumes fin_f: "finite {a. f a \<noteq> 0}"
+  assumes fin_g: "finite {b. g b \<noteq> 0}"
+  shows "prod_fun f g k = (\<Sum>(a, b). f a * g b when k = a + b)"
+proof -
+  let ?C = "{a. f a \<noteq> 0} \<times> {b. g b \<noteq> 0}"
+  from fin_f fin_g have "finite ?C" by blast
+  moreover have "{a. \<exists>b. (f a * g b when k = a + b) \<noteq> 0} \<times>
+    {b. \<exists>a. (f a * g b when k = a + b) \<noteq> 0} \<subseteq> {a. f a \<noteq> 0} \<times> {b. g b \<noteq> 0}"
+    by auto
+  ultimately show ?thesis using fin_g
+    by (auto simp add: prod_fun_def
+      Sum_any.cartesian_product [of "{a. f a \<noteq> 0} \<times> {b. g b \<noteq> 0}"] Sum_any_right_distrib mult_when)
+qed
+
+lemma finite_prod_fun:
+  fixes f1 f2 :: "'a :: monoid_add \<Rightarrow> 'b :: semiring_0"
+  assumes fin1: "finite {l. f1 l \<noteq> 0}"
+  and fin2: "finite {q. f2 q \<noteq> 0}"
+  shows "finite {k. prod_fun f1 f2 k \<noteq> 0}"
+proof -
+  have *: "finite {k. (\<exists>l. f1 l \<noteq> 0 \<and> (\<exists>q. f2 q \<noteq> 0 \<and> k = l + q))}"
+    using assms by simp
+  { fix k l
+    have "{q. (f2 q when k = l + q) \<noteq> 0} \<subseteq> {q. f2 q \<noteq> 0 \<and> k = l + q}" by auto
+    with fin2 have "sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q} = (\<Sum>q. (f2 q when k = l + q))"
+      by (simp add: Sum_any.expand_superset [of "{q. f2 q \<noteq> 0 \<and> k = l + q}"]) }
+  note aux = this
+  have "{k. (\<Sum>l. f1 l * sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q}) \<noteq> 0}
+    \<subseteq> {k. (\<exists>l. f1 l * sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q} \<noteq> 0)}"
+    by (auto elim!: Sum_any.not_neutral_obtains_not_neutral)
+  also have "\<dots> \<subseteq> {k. (\<exists>l. f1 l \<noteq> 0 \<and> sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q} \<noteq> 0)}"
+    by (auto dest: mult_not_zero)
+  also have "\<dots> \<subseteq> {k. (\<exists>l. f1 l \<noteq> 0 \<and> (\<exists>q. f2 q \<noteq> 0 \<and> k = l + q))}"
+    by (auto elim!: sum.not_neutral_contains_not_neutral)
+  finally have "finite {k. (\<Sum>l. f1 l * sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q}) \<noteq> 0}"
+    using * by (rule finite_subset)
+  with aux have "finite {k. (\<Sum>l. f1 l * (\<Sum>q. (f2 q when k = l + q))) \<noteq> 0}"
+    by simp
+  with fin2 show ?thesis
+qed
+
+instantiation poly_mapping :: (monoid_add, semiring_0) semiring_0
+begin
+
+lift_definition times_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b"
+  is prod_fun
+by(rule finite_prod_fun)
+
+instance
+proof
+  fix a b c :: "'a \<Rightarrow>\<^sub>0 'b"
+  show "a * b * c = a * (b * c)"
+  proof transfer
+    fix f g h :: "'a \<Rightarrow> 'b"
+    assume fin_f: "finite {a. f a \<noteq> 0}" (is "finite ?F")
+    assume fin_g: "finite {b. g b \<noteq> 0}" (is "finite ?G")
+    assume fin_h: "finite {c. h c \<noteq> 0}" (is "finite ?H")
+    from fin_f fin_g have fin_fg: "finite {(a, b). f a * g b \<noteq> 0}" (is "finite ?FG")
+      by (rule finite_mult_not_eq_zero_prodI)
+    from fin_g fin_h have fin_gh: "finite {(b, c). g b * h c \<noteq> 0}" (is "finite ?GH")
+      by (rule finite_mult_not_eq_zero_prodI)
+    from fin_f fin_g have fin_fg': "finite {a + b | a b. f a * g b \<noteq> 0}" (is "finite ?FG'")
+      by (rule finite_mult_not_eq_zero_sumI)
+    then have fin_fg'': "finite {d. (\<Sum>(a, b). f a * g b when d = a + b) \<noteq> 0}"
+      by (auto intro: finite_Sum_any_not_eq_zero_weakenI)
+    from fin_g fin_h have fin_gh': "finite {b + c | b c. g b * h c \<noteq> 0}" (is "finite ?GH'")
+      by (rule finite_mult_not_eq_zero_sumI)
+    then have fin_gh'': "finite {d. (\<Sum>(b, c). g b * h c when d = b + c) \<noteq> 0}"
+      by (auto intro: finite_Sum_any_not_eq_zero_weakenI)
+    show "prod_fun (prod_fun f g) h = prod_fun f (prod_fun g h)" (is "?lhs = ?rhs")
+    proof
+      fix k
+      from fin_f fin_g fin_h fin_fg''
+      have "?lhs k = (\<Sum>(ab, c). (\<Sum>(a, b). f a * g b when ab = a + b) * h c when k = ab + c)"
+      also have "\<dots> = (\<Sum>(ab, c). (\<Sum>(a, b). f a * g b * h c when k = ab + c when ab = a + b))"
+        apply (subst Sum_any_left_distrib)
+        using fin_fg apply (simp add: split_def)
+        apply (subst Sum_any_when_independent [symmetric])
+        apply (simp add: when_when when_mult mult_when split_def conj_commute)
+        done
+      also have "\<dots> = (\<Sum>(ab, c, a, b). f a * g b * h c when k = ab + c when ab = a + b)"
+        apply (subst Sum_any.cartesian_product2 [of "(?FG' \<times> ?H) \<times> ?FG"])
+        apply (auto simp add: finite_cartesian_product_iff fin_fg fin_fg' fin_h dest: mult_not_zero)
+        done
+      also have "\<dots> = (\<Sum>(ab, c, a, b). f a * g b * h c when k = a + b + c when ab = a + b)"
+        by (rule Sum_any.cong) (simp add: split_def when_def)
+      also have "\<dots> = (\<Sum>(ab, cab). (case cab of (c, a, b) \<Rightarrow> f a * g b * h c when k = a + b + c)
+        when ab = (case cab of (c, a, b) \<Rightarrow> a + b))"
+      also have "\<dots> = (\<Sum>(c, a, b). f a * g b * h c when k = a + b + c)"
+      also have "\<dots> = (\<Sum>(bc, cab). (case cab of (c, a, b) \<Rightarrow> f a * g b * h c when k = a + b + c)
+        when bc = (case cab of (c, a, b) \<Rightarrow> b + c))"
+      also have "\<dots> = (\<Sum>(bc, c, a, b). f a * g b * h c when k = a + b + c when bc = b + c)"
+      also have "\<dots> = (\<Sum>(bc, c, a, b). f a * g b * h c when bc = b + c when k = a + bc)"
+        by (rule Sum_any.cong) (simp add: split_def when_def ac_simps)
+      also have "\<dots> = (\<Sum>(a, bc, b, c). f a * g b * h c when bc = b + c when k = a + bc)"
+      proof -
+        have "bij (\<lambda>(a, d, b, c). (d, c, a, b))"
+          by (auto intro!: bijI injI surjI [of _ "\<lambda>(d, c, a, b). (a, d, b, c)"] simp add: split_def)
+        then show ?thesis
+          by (rule Sum_any.reindex_cong) auto
+      qed
+      also have "\<dots> = (\<Sum>(a, bc). (\<Sum>(b, c). f a * g b * h c when bc = b + c when k = a + bc))"
+        apply (subst Sum_any.cartesian_product2 [of "(?F \<times> ?GH') \<times> ?GH"])
+        apply (auto simp add: finite_cartesian_product_iff fin_f fin_gh fin_gh' ac_simps dest: mult_not_zero)
+        done
+      also have "\<dots> = (\<Sum>(a, bc). f a * (\<Sum>(b, c). g b * h c when bc = b + c) when k = a + bc)"
+        apply (subst Sum_any_right_distrib)
+        using fin_gh apply (simp add: split_def)
+        apply (subst Sum_any_when_independent [symmetric])
+        apply (simp add: when_when when_mult mult_when split_def ac_simps)
+        done
+      also from fin_f fin_g fin_h fin_gh''
+      have "\<dots> = ?rhs k"
+      finally show "?lhs k = ?rhs k" .
+    qed
+  qed
+  show "(a + b) * c = a * c + b * c"
+  proof transfer
+    fix f g h :: "'a \<Rightarrow> 'b"
+    assume fin_f: "finite {k. f k \<noteq> 0}"
+    assume fin_g: "finite {k. g k \<noteq> 0}"
+    assume fin_h: "finite {k. h k \<noteq> 0}"
+    show "prod_fun (\<lambda>k. f k + g k) h = (\<lambda>k. prod_fun f h k + prod_fun g h k)"
+      apply (rule ext)
+      apply (auto simp add: prod_fun_def algebra_simps)
+      apply (subst Sum_any.distrib)
+      using fin_f fin_g apply (auto intro: finite_mult_not_eq_zero_rightI)
+      done
+  qed
+  show "a * (b + c) = a * b + a * c"
+  proof transfer
+    fix f g h :: "'a \<Rightarrow> 'b"
+    assume fin_f: "finite {k. f k \<noteq> 0}"
+    assume fin_g: "finite {k. g k \<noteq> 0}"
+    assume fin_h: "finite {k. h k \<noteq> 0}"
+    show "prod_fun f (\<lambda>k. g k + h k) = (\<lambda>k. prod_fun f g k + prod_fun f h k)"
+      apply (rule ext)
+      apply (subst Sum_any.distrib)
+      apply (auto intro: fin_g fin_h)
+      apply (subst Sum_any.distrib)
+      using fin_f apply (rule finite_mult_not_eq_zero_rightI)
+      using fin_f apply (rule finite_mult_not_eq_zero_rightI)
+      done
+  qed
+  show "0 * a = 0"
+    by transfer (simp add: prod_fun_def [abs_def])
+  show "a * 0 = 0"
+    by transfer (simp add: prod_fun_def [abs_def])
+qed
+
+end
+
+lemma lookup_mult:
+  "lookup (f * g) k = (\<Sum>l. lookup f l * (\<Sum>q. lookup g q when k = l + q))"
+  by transfer (simp add: prod_fun_def)
+
+instance poly_mapping :: (comm_monoid_add, comm_semiring_0) comm_semiring_0
+proof
+  fix a b c :: "'a \<Rightarrow>\<^sub>0 'b"
+  show "a * b = b * a"
+  proof transfer
+    fix f g :: "'a \<Rightarrow> 'b"
+    assume fin_f: "finite {a. f a \<noteq> 0}"
+    assume fin_g: "finite {b. g b \<noteq> 0}"
+    show "prod_fun f g = prod_fun g f"
+    proof
+      fix k
+      have fin1: "\<And>l. finite {a. (f a when k = l + a) \<noteq> 0}"
+        using fin_f by auto
+      have fin2: "\<And>l. finite {b. (g b when k = l + b) \<noteq> 0}"
+        using fin_g by auto
+      from fin_f fin_g have "finite {(a, b). f a \<noteq> 0 \<and> g b \<noteq> 0}" (is "finite ?AB")
+        by simp
+      show "prod_fun f g k = prod_fun g f k"
+        apply (subst Sum_any_right_distrib)
+        apply (rule fin2)
+        apply (subst Sum_any_right_distrib)
+         apply (rule fin1)
+        apply (subst Sum_any.swap [of ?AB])
+        apply (fact \<open>finite ?AB\<close>)
+        apply (auto simp add: mult_when ac_simps)
+        done
+    qed
+  qed
+  show "(a + b) * c = a * c + b * c"
+  proof transfer
+    fix f g h :: "'a \<Rightarrow> 'b"
+    assume fin_f: "finite {k. f k \<noteq> 0}"
+    assume fin_g: "finite {k. g k \<noteq> 0}"
+    assume fin_h: "finite {k. h k \<noteq> 0}"
+    show "prod_fun (\<lambda>k. f k + g k) h = (\<lambda>k. prod_fun f h k + prod_fun g h k)"
+      apply (auto simp add: prod_fun_def fun_eq_iff algebra_simps)
+      apply (subst Sum_any.distrib)
+      using fin_f apply (rule finite_mult_not_eq_zero_rightI)
+      using fin_g apply (rule finite_mult_not_eq_zero_rightI)
+      apply simp_all
+      done
+  qed
+qed
+
+instance poly_mapping :: (monoid_add, semiring_0_cancel) semiring_0_cancel
+  ..
+
+instance poly_mapping :: (comm_monoid_add, comm_semiring_0_cancel) comm_semiring_0_cancel
+  ..
+
+instance poly_mapping :: (monoid_add, semiring_1) semiring_1
+proof
+  fix a :: "'a \<Rightarrow>\<^sub>0 'b"
+  show "1 * a = a"
+    by transfer (simp add: prod_fun_def [abs_def] when_mult)
+  show "a * 1 = a"
+    apply transfer
+    apply (simp add: prod_fun_def [abs_def] Sum_any_right_distrib Sum_any_left_distrib mult_when)
+    apply (subst when_commute)
+    apply simp
+    done
+qed
+
+instance poly_mapping :: (comm_monoid_add, comm_semiring_1) comm_semiring_1
+proof
+  fix a :: "'a \<Rightarrow>\<^sub>0 'b"
+  show "1 * a = a"
+    by transfer (simp add: prod_fun_def [abs_def])
+qed
+
+instance poly_mapping :: (monoid_add, semiring_1_cancel) semiring_1_cancel
+  ..
+
+instance poly_mapping :: (monoid_add, ring) ring
+  ..
+
+instance poly_mapping :: (comm_monoid_add, comm_ring) comm_ring
+  ..
+
+instance poly_mapping :: (monoid_add, ring_1) ring_1
+  ..
+
+instance poly_mapping :: (comm_monoid_add, comm_ring_1) comm_ring_1
+  ..
+
+
+subsection \<open>Single-point mappings\<close>
+
+lift_definition single :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b::zero"
+  is "\<lambda>k v k'. (v when k = k')"
+  by simp
+
+lemma inj_single [iff]:
+  "inj (single k)"
+proof (rule injI, transfer)
+  fix k :: 'b and a b :: "'a::zero"
+  assume "(\<lambda>k'. a when k = k') = (\<lambda>k'. b when k = k')"
+  then have "(\<lambda>k'. a when k = k') k = (\<lambda>k'. b when k = k') k"
+    by (rule arg_cong)
+  then show "a = b" by simp
+qed
+
+lemma lookup_single:
+  "lookup (single k v) k' = (v when k = k')"
+  by transfer rule
+
+lemma lookup_single_eq [simp]:
+  "lookup (single k v) k = v"
+  by transfer simp
+
+lemma lookup_single_not_eq:
+  "k \<noteq> k' \<Longrightarrow> lookup (single k v) k' = 0"
+  by transfer simp
+
+lemma single_zero [simp]:
+  "single k 0 = 0"
+  by transfer simp
+
+lemma single_one [simp]:
+  "single 0 1 = 1"
+  by transfer simp
+
+  "single k (a + b) = single k a + single k b"
+
+lemma single_uminus:
+  "single k (- a) = - single k a"
+  by transfer (simp add: fun_eq_iff when_uminus_distrib)
+
+lemma single_diff:
+  "single k (a - b) = single k a - single k b"
+  by transfer (simp add: fun_eq_iff when_diff_distrib)
+
+lemma single_numeral [simp]:
+  "single 0 (numeral n) = numeral n"
+
+lemma lookup_numeral:
+  "lookup (numeral n) k = (numeral n when k = 0)"
+proof -
+  have "lookup (numeral n) k = lookup (single 0 (numeral n)) k"
+    by simp
+  then show ?thesis unfolding lookup_single by simp
+qed
+
+lemma single_of_nat [simp]:
+  "single 0 (of_nat n) = of_nat n"
+
+lemma lookup_of_nat:
+  "lookup (of_nat n) k = (of_nat n when k = 0)"
+proof -
+  have "lookup (of_nat n) k = lookup (single 0 (of_nat n)) k"
+    by simp
+  then show ?thesis unfolding lookup_single by simp
+qed
+
+lemma of_nat_single:
+  "of_nat = single 0 \<circ> of_nat"
+
+lemma mult_single:
+  "single k a * single l b = single (k + l) (a * b)"
+proof transfer
+  fix k l :: 'a and a b :: 'b
+  show "prod_fun (\<lambda>k'. a when k = k') (\<lambda>k'. b when l = k') = (\<lambda>k'. a * b when k + l = k')"
+  proof
+    fix k'
+    have "prod_fun (\<lambda>k'. a when k = k') (\<lambda>k'. b when l = k') k' = (\<Sum>n. a * b when l = n when k' = k + n)"
+      by (simp add: prod_fun_def Sum_any_right_distrib mult_when when_mult)
+    also have "\<dots> = (\<Sum>n. a * b when k' = k + n when l = n)"
+      by (simp add: when_when conj_commute)
+    also have "\<dots> = (a * b when k' = k + l)"
+      by simp
+    also have "\<dots> = (a * b when k + l = k')"
+    finally show "prod_fun (\<lambda>k'. a when k = k') (\<lambda>k'. b when l = k') k' =
+      (\<lambda>k'. a * b when k + l = k') k'" .
+  qed
+qed
+
+instance poly_mapping :: (monoid_add, semiring_char_0) semiring_char_0
+  by intro_classes (auto intro: inj_compose inj_of_nat simp add: of_nat_single)
+
+instance poly_mapping :: (monoid_add, ring_char_0) ring_char_0
+  ..
+
+lemma single_of_int [simp]:
+  "single 0 (of_int k) = of_int k"
+  by (cases k) (simp_all add: single_diff single_uminus)
+
+lemma lookup_of_int:
+  "lookup (of_int l) k = (of_int l when k = 0)"
+proof -
+  have "lookup (of_int l) k = lookup (single 0 (of_int l)) k"
+    by simp
+  then show ?thesis unfolding lookup_single by simp
+qed
+
+
+subsection \<open>Integral domains\<close>
+
+instance poly_mapping :: ("{ordered_cancel_comm_monoid_add, linorder}", semiring_no_zero_divisors) semiring_no_zero_divisors
+  text \<open>The @{class "linorder"} constraint is a pragmatic device for the proof --- maybe it can be dropped\<close>
+proof
+  fix f g :: "'a \<Rightarrow>\<^sub>0 'b"
+  assume "f \<noteq> 0" and "g \<noteq> 0"
+  then show "f * g \<noteq> 0"
+  proof transfer
+    fix f g :: "'a \<Rightarrow> 'b"
+    define F where "F = {a. f a \<noteq> 0}"
+    moreover define G where "G = {a. g a \<noteq> 0}"
+    ultimately have [simp]:
+      "\<And>a. f a \<noteq> 0 \<longleftrightarrow> a \<in> F"
+      "\<And>b. g b \<noteq> 0 \<longleftrightarrow> b \<in> G"
+      by simp_all
+    assume "finite {a. f a \<noteq> 0}"
+    then have [simp]: "finite F"
+      by simp
+    assume "finite {a. g a \<noteq> 0}"
+    then have [simp]: "finite G"
+      by simp
+    assume "f \<noteq> (\<lambda>a. 0)"
+    then obtain a where "f a \<noteq> 0"
+      by (auto simp add: fun_eq_iff)
+    assume "g \<noteq> (\<lambda>b. 0)"
+    then obtain b where "g b \<noteq> 0"
+      by (auto simp add: fun_eq_iff)
+    from \<open>f a \<noteq> 0\<close> and \<open>g b \<noteq> 0\<close> have "F \<noteq> {}" and "G \<noteq> {}"
+      by auto
+    note Max_F = \<open>finite F\<close> \<open>F \<noteq> {}\<close>
+    note Max_G = \<open>finite G\<close> \<open>G \<noteq> {}\<close>
+    from Max_F and Max_G have [simp]:
+      "Max F \<in> F"
+      "Max G \<in> G"
+      by auto
+    from Max_F Max_G have [dest!]:
+      "\<And>a. a \<in> F \<Longrightarrow> a \<le> Max F"
+      "\<And>b. b \<in> G \<Longrightarrow> b \<le> Max G"
+      by auto
+    define q where "q = Max F + Max G"
+    have "(\<Sum>(a, b). f a * g b when q = a + b) =
+      (\<Sum>(a, b). f a * g b when q = a + b when a \<in> F \<and> b \<in> G)"
+      by (rule Sum_any.cong) (auto simp add: split_def when_def q_def intro: ccontr)
+    also have "\<dots> =
+      (\<Sum>(a, b). f a * g b when (Max F, Max G) = (a, b))"
+    proof (rule Sum_any.cong)
+      fix ab :: "'a \<times> 'a"
+      obtain a b where [simp]: "ab = (a, b)"
+        by (cases ab) simp_all
+      have [dest!]:
+        "a \<le> Max F \<Longrightarrow> Max F \<noteq> a \<Longrightarrow> a < Max F"
+        "b \<le> Max G \<Longrightarrow> Max G \<noteq> b \<Longrightarrow> b < Max G"
+        by auto
+      show "(case ab of (a, b) \<Rightarrow> f a * g b when q = a + b when a \<in> F \<and> b \<in> G) =
+         (case ab of (a, b) \<Rightarrow> f a * g b when (Max F, Max G) = (a, b))"
+        by (auto simp add: split_def when_def q_def dest: add_strict_mono [of a "Max F" b "Max G"])
+    qed
+    also have "\<dots> = (\<Sum>ab. (case ab of (a, b) \<Rightarrow> f a * g b) when
+      (Max F, Max G) = ab)"
+      unfolding split_def when_def by auto
+    also have "\<dots> \<noteq> 0"
+      by simp
+    finally have "prod_fun f g q \<noteq> 0"
+    then show "prod_fun f g \<noteq> (\<lambda>k. 0)"
+      by (auto simp add: fun_eq_iff)
+  qed
+qed
+
+instance poly_mapping :: ("{ordered_cancel_comm_monoid_add, linorder}", ring_no_zero_divisors) ring_no_zero_divisors
+  ..
+
+instance poly_mapping :: ("{ordered_cancel_comm_monoid_add, linorder}", ring_1_no_zero_divisors) ring_1_no_zero_divisors
+  ..
+
+instance poly_mapping :: ("{ordered_cancel_comm_monoid_add, linorder}", idom) idom
+  ..
+
+
+subsection \<open>Mapping order\<close>
+
+instantiation poly_mapping :: (linorder, "{zero, linorder}") linorder
+begin
+
+lift_definition less_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"
+  is less_fun
+  .
+
+lift_definition less_eq_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"
+  is "\<lambda>f g. less_fun f g \<or> f = g"
+  .
+
+instance proof (rule class.Orderings.linorder.of_class.intro)
+  show "class.linorder (less_eq :: (_ \<Rightarrow>\<^sub>0 _) \<Rightarrow> _) less"
+  proof (rule linorder_strictI, rule order_strictI)
+    fix f g h :: "'a \<Rightarrow>\<^sub>0 'b"
+    show "f \<le> g \<longleftrightarrow> f < g \<or> f = g"
+      by transfer (rule refl)
+    show "\<not> f < f"
+      by transfer (rule less_fun_irrefl)
+    show "f < g \<or> f = g \<or> g < f"
+    proof transfer
+      fix f g :: "'a \<Rightarrow> 'b"
+      assume "finite {k. f k \<noteq> 0}" and "finite {k. g k \<noteq> 0}"
+      then have "finite ({k. f k \<noteq> 0} \<union> {k. g k \<noteq> 0})"
+        by simp
+      moreover have "{k. f k \<noteq> g k} \<subseteq> {k. f k \<noteq> 0} \<union> {k. g k \<noteq> 0}"
+        by auto
+      ultimately have "finite {k. f k \<noteq> g k}"
+        by (rule rev_finite_subset)
+      then show "less_fun f g \<or> f = g \<or> less_fun g f"
+        by (rule less_fun_trichotomy)
+    qed
+    assume "f < g" then show "\<not> g < f"
+      by transfer (rule less_fun_asym)
+    note \<open>f < g\<close> moreover assume "g < h"
+      ultimately show "f < h"
+      by transfer (rule less_fun_trans)
+  qed
+qed
+
+end
+
+proof (intro_classes, transfer)
+  fix f g h :: "'a \<Rightarrow> 'b"
+  assume *: "less_fun f g \<or> f = g"
+  { assume "less_fun f g"
+    then obtain k where "f k < g k" "(\<And>k'. k' < k \<Longrightarrow> f k' = g k')"
+      by (blast elim!: less_funE)
+    then have "h k + f k < h k + g k" "(\<And>k'. k' < k \<Longrightarrow> h k' + f k' = h k' + g k')"
+      by simp_all
+    then have "less_fun (\<lambda>k. h k + f k) (\<lambda>k. h k + g k)"
+      by (blast intro: less_funI)
+  }
+  with * show "less_fun (\<lambda>k. h k + f k) (\<lambda>k. h k + g k) \<or> (\<lambda>k. h k + f k) = (\<lambda>k. h k + g k)"
+    by (auto simp add: fun_eq_iff)
+qed
+
+  ..
+
+  ..
+
+  ..
+
+  ..
+
+text \<open>
+  For pragmatism we leave out the final elements in the hierarchy:
+  @{class linordered_ring}, @{class linordered_ring_strict}, @{class linordered_idom};
+  remember that the order instance is a mere technical device, not a deeper algebraic
+  property.
+\<close>
+
+
+subsection \<open>Fundamental mapping notions\<close>
+
+lift_definition keys :: "('a \<Rightarrow>\<^sub>0 'b::zero) \<Rightarrow> 'a set"
+  is "\<lambda>f. {k. f k \<noteq> 0}" .
+
+lift_definition range :: "('a \<Rightarrow>\<^sub>0 'b::zero) \<Rightarrow> 'b set"
+  is "\<lambda>f :: 'a \<Rightarrow> 'b. Set.range f - {0}" .
+
+lemma finite_keys [simp]:
+  "finite (keys f)"
+  by transfer
+
+lemma not_in_keys_iff_lookup_eq_zero:
+  "k \<notin> keys f \<longleftrightarrow> lookup f k = 0"
+  by transfer simp
+
+lemma lookup_not_eq_zero_eq_in_keys:
+  "lookup f k \<noteq> 0 \<longleftrightarrow> k \<in> keys f"
+  by transfer simp
+
+  "lookup f k = 0 \<Longrightarrow> \<not> k \<in> keys f"
+
+lemma finite_range [simp]: "finite (Poly_Mapping.range p)"
+proof transfer
+  fix f :: "'b \<Rightarrow> 'a"
+  assume *: "finite {x. f x \<noteq> 0}"
+  have "Set.range f - {0} \<subseteq> f  {x. f x \<noteq> 0}"
+    by auto
+  thus "finite (Set.range f - {0})"
+    by(rule finite_subset)(rule finite_imageI[OF *])
+qed
+
+lemma in_keys_lookup_in_range [simp]:
+  "k \<in> keys f \<Longrightarrow> lookup f k \<in> range f"
+  by transfer simp
+
+lemma in_keys_iff: "x \<in> (keys s) = (lookup s x \<noteq> 0)"
+  by (transfer, simp)
+
+lemma keys_zero [simp]:
+  "keys 0 = {}"
+  by transfer simp
+
+lemma range_zero [simp]:
+  "range 0 = {}"
+  by transfer auto
+
+  "keys (f + g) \<subseteq> keys f \<union> keys g"
+  by transfer auto
+
+lemma keys_one [simp]:
+  "keys 1 = {0}"
+  by transfer simp
+
+lemma range_one [simp]:
+  "range 1 = {1}"
+  by transfer (auto simp add: when_def)
+
+lemma keys_single [simp]:
+  "keys (single k v) = (if v = 0 then {} else {k})"
+  by transfer simp
+
+lemma range_single [simp]:
+  "range (single k v) = (if v = 0 then {} else {v})"
+  by transfer (auto simp add: when_def)
+
+lemma keys_mult:
+  "keys (f * g) \<subseteq> {a + b | a b. a \<in> keys f \<and> b \<in> keys g}"
+  apply transfer
+  apply (auto simp add: prod_fun_def dest!: mult_not_zero elim!: Sum_any.not_neutral_obtains_not_neutral)
+  apply blast
+  done
+
+lemma setsum_keys_plus_distrib:
+  assumes hom_0: "\<And>k. f k 0 = 0"
+  and hom_plus: "\<And>k. k \<in> Poly_Mapping.keys p \<union> Poly_Mapping.keys q \<Longrightarrow> f k (Poly_Mapping.lookup p k + Poly_Mapping.lookup q k) = f k (Poly_Mapping.lookup p k) + f k (Poly_Mapping.lookup q k)"
+  shows
+  "(\<Sum>k\<in>Poly_Mapping.keys (p + q). f k (Poly_Mapping.lookup (p + q) k)) =
+   (\<Sum>k\<in>Poly_Mapping.keys p. f k (Poly_Mapping.lookup p k)) +
+   (\<Sum>k\<in>Poly_Mapping.keys q. f k (Poly_Mapping.lookup q k))"
+  (is "?lhs = ?p + ?q")
+proof -
+  let ?A = "Poly_Mapping.keys p \<union> Poly_Mapping.keys q"
+  have "?lhs = (\<Sum>k\<in>?A. f k (Poly_Mapping.lookup p k + Poly_Mapping.lookup q k))"
+    apply(rule sum.mono_neutral_cong_left)
+     apply(transfer fixing: f)
+    apply(transfer fixing: f)
+    done
+  also have "\<dots> = (\<Sum>k\<in>?A. f k (Poly_Mapping.lookup p k) + f k (Poly_Mapping.lookup q k))"
+  also have "\<dots> = (\<Sum>k\<in>?A. f k (Poly_Mapping.lookup p k)) + (\<Sum>k\<in>?A. f k (Poly_Mapping.lookup q k))"
+    (is "_ = ?p' + ?q'")
+  also have "?p' = ?p"
+    by (simp add: hom_0 in_keys_iff sum.mono_neutral_cong_right)
+  also have "?q' = ?q"
+    by (simp add: hom_0 in_keys_iff sum.mono_neutral_cong_right)
+  finally show ?thesis .
+qed
+
+subsection \<open>Degree\<close>
+
+definition degree :: "(nat \<Rightarrow>\<^sub>0 'a::zero) \<Rightarrow> nat"
+where
+  "degree f = Max (insert 0 (Suc  keys f))"
+
+lemma degree_zero [simp]:
+  "degree 0 = 0"
+  unfolding degree_def by transfer simp
+
+lemma degree_one [simp]:
+  "degree 1 = 1"
+  unfolding degree_def by transfer simp
+
+lemma degree_single_zero [simp]:
+  "degree (single k 0) = 0"
+  unfolding degree_def by transfer simp
+
+lemma degree_single_not_zero [simp]:
+  "v \<noteq> 0 \<Longrightarrow> degree (single k v) = Suc k"
+  unfolding degree_def by transfer simp
+
+lemma degree_zero_iff [simp]:
+  "degree f = 0 \<longleftrightarrow> f = 0"
+unfolding degree_def proof transfer
+  fix f :: "nat \<Rightarrow> 'a"
+  assume "finite {n. f n \<noteq> 0}"
+  then have fin: "finite (insert 0 (Suc  {n. f n \<noteq> 0}))" by auto
+  show "Max (insert 0 (Suc  {n. f n \<noteq> 0})) = 0 \<longleftrightarrow> f = (\<lambda>n. 0)" (is "?P \<longleftrightarrow> ?Q")
+  proof
+    assume ?P
+    have "{n. f n \<noteq> 0} = {}"
+    proof (rule ccontr)
+      assume "{n. f n \<noteq> 0} \<noteq> {}"
+      then obtain n where "n \<in> {n. f n \<noteq> 0}" by blast
+      then have "{n. f n \<noteq> 0} = insert n {n. f n \<noteq> 0}" by auto
+      then have "Suc  {n. f n \<noteq> 0} = insert (Suc n) (Suc  {n. f n \<noteq> 0})" by auto
+      with \<open>?P\<close> have "Max (insert 0 (insert (Suc n) (Suc  {n. f n \<noteq> 0}))) = 0" by simp
+      then have "Max (insert (Suc n) (insert 0 (Suc  {n. f n \<noteq> 0}))) = 0"
+      with fin have "max (Suc n) (Max (insert 0 (Suc  {n. f n \<noteq> 0}))) = 0"
+        by simp
+      then show False by simp
+    qed
+    then show ?Q by (simp add: fun_eq_iff)
+  next
+    assume ?Q then show ?P by simp
+  qed
+qed
+
+lemma degree_greater_zero_in_keys:
+  assumes "0 < degree f"
+  shows "degree f - 1 \<in> keys f"
+proof -
+  from assms have "keys f \<noteq> {}"
+    by (auto simp add: degree_def)
+  then show ?thesis unfolding degree_def
+    by (simp add: mono_Max_commute [symmetric] mono_Suc)
+qed
+
+lemma in_keys_less_degree:
+  "n \<in> keys f \<Longrightarrow> n < degree f"
+unfolding degree_def by transfer (auto simp add: Max_gr_iff)
+
+lemma beyond_degree_lookup_zero:
+  "degree f \<le> n \<Longrightarrow> lookup f n = 0"
+  unfolding degree_def by transfer auto
+
+  "degree (f + g) \<le> max (degree f) (Poly_Mapping.degree g)"
+unfolding degree_def proof transfer
+  fix f g :: "nat \<Rightarrow> 'a"
+  assume f: "finite {x. f x \<noteq> 0}"
+  assume g: "finite {x. g x \<noteq> 0}"
+  let ?f = "Max (insert 0 (Suc  {k. f k \<noteq> 0}))"
+  let ?g = "Max (insert 0 (Suc  {k. g k \<noteq> 0}))"
+  have "Max (insert 0 (Suc  {k. f k + g k \<noteq> 0})) \<le> Max (insert 0 (Suc  ({k. f k \<noteq> 0} \<union> {k. g k \<noteq> 0})))"
+    by (rule Max.subset_imp) (insert f g, auto)
+  also have "\<dots> = max ?f ?g"
+    using f g by (simp_all add: image_Un Max_Un [symmetric])
+  finally show "Max (insert 0 (Suc  {k. f k + g k \<noteq> 0}))
+    \<le> max (Max (insert 0 (Suc  {k. f k \<noteq> 0}))) (Max (insert 0 (Suc  {k. g k \<noteq> 0})))"
+    .
+qed
+
+lemma sorted_list_of_set_keys:
+  "sorted_list_of_set (keys f) = filter (\<lambda>k. k \<in> keys f) [0..<degree f]" (is "_ = ?r")
+proof -
+  have "keys f = set ?r"
+    by (auto dest: in_keys_less_degree)
+  moreover have "sorted_list_of_set (set ?r) = ?r"
+    unfolding sorted_list_of_set_sort_remdups
+    by (simp add: remdups_filter filter_sort [symmetric])
+  ultimately show ?thesis by simp
+qed
+
+
+subsection \<open>Inductive structure\<close>
+
+lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b::zero) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b"
+  is "\<lambda>k v f. f(k := v)"
+proof -
+  fix f :: "'a \<Rightarrow> 'b" and k' v
+  assume "finite {k. f k \<noteq> 0}"
+  then have "finite (insert k' {k. f k \<noteq> 0})"
+    by simp
+  then show "finite {k. (f(k' := v)) k \<noteq> 0}"
+    by (rule rev_finite_subset) auto
+qed
+
+lemma update_induct [case_names const update]:
+  assumes const': "P 0"
+  assumes update': "\<And>f a b. a \<notin> keys f \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> P f \<Longrightarrow> P (update a b f)"
+  shows "P f"
+proof -
+  obtain g where "f = Abs_poly_mapping g" and "finite {a. g a \<noteq> 0}"
+    by (cases f) simp_all
+  define Q where "Q g = P (Abs_poly_mapping g)" for g
+  from \<open>finite {a. g a \<noteq> 0}\<close> have "Q g"
+  proof (induct g rule: finite_update_induct)
+    case const with const' Q_def show ?case
+      by simp
+  next
+    case (update a b g)
+    from \<open>finite {a. g a \<noteq> 0}\<close> \<open>g a = 0\<close> have "a \<notin> keys (Abs_poly_mapping g)"
+      by (simp add: Abs_poly_mapping_inverse keys.rep_eq)
+    moreover note \<open>b \<noteq> 0\<close>
+    moreover from \<open>Q g\<close> have "P (Abs_poly_mapping g)"
+    ultimately have "P (update a b (Abs_poly_mapping g))"
+      by (rule update')
+    also from \<open>finite {a. g a \<noteq> 0}\<close>
+    have "update a b (Abs_poly_mapping g) = Abs_poly_mapping (g(a := b))"
+      by (simp add: update.abs_eq eq_onp_same_args)
+    finally show ?case
+      by (simp add: Q_def fun_upd_def)
+  qed
+  then show ?thesis by (simp add: Q_def \<open>f = Abs_poly_mapping g\<close>)
+qed
+
+lemma lookup_update:
+  "lookup (update k v f) k' = (if k = k' then v else lookup f k')"
+  by transfer simp
+
+lemma keys_update:
+  "keys (update k v f) = (if v = 0 then keys f - {k} else insert k (keys f))"
+  by transfer auto
+
+
+subsection \<open>Quasi-functorial structure\<close>
+
+lift_definition map :: "('b::zero \<Rightarrow> 'c::zero)
+  \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c::zero)"
+  is "\<lambda>g f k. g (f k) when f k \<noteq> 0"
+  by simp
+
+context
+  fixes f :: "'b \<Rightarrow> 'a"
+  assumes inj_f: "inj f"
+begin
+
+lift_definition map_key :: "('a \<Rightarrow>\<^sub>0 'c::zero) \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'c"
+  is "\<lambda>p. p \<circ> f"
+proof -
+  fix g :: "'c \<Rightarrow> 'd" and p :: "'a \<Rightarrow> 'c"
+  assume "finite {x. p x \<noteq> 0}"
+  hence "finite (f  {y. p (f y) \<noteq> 0})"
+    by(rule finite_subset[rotated]) auto
+  thus "finite {x. (p \<circ> f) x \<noteq> 0}" unfolding o_def
+    by(rule finite_imageD)(rule subset_inj_on[OF inj_f], simp)
+qed
+
+end
+
+lemma map_key_compose:
+  assumes [transfer_rule]: "inj f" "inj g"
+  shows "map_key f (map_key g p) = map_key (g \<circ> f) p"
+proof -
+  from assms have [transfer_rule]: "inj (g \<circ> f)"
+  show ?thesis by transfer(simp add: o_assoc)
+qed
+
+lemma map_key_id:
+  "map_key (\<lambda>x. x) p = p"
+proof -
+  have [transfer_rule]: "inj (\<lambda>x. x)" by simp
+  show ?thesis by transfer(simp add: o_def)
+qed
+
+context
+  fixes f :: "'a \<Rightarrow> 'b"
+  assumes inj_f [transfer_rule]: "inj f"
+begin
+
+lemma map_key_map:
+  "map_key f (map g p) = map g (map_key f p)"
+  by transfer (simp add: fun_eq_iff)
+
+lemma map_key_plus:
+  "map_key f (p + q) = map_key f p + map_key f q"
+  by transfer (simp add: fun_eq_iff)
+
+lemma keys_map_key:
+  "keys (map_key f p) = f - keys p"
+  by transfer auto
+
+lemma map_key_zero [simp]:
+  "map_key f 0 = 0"
+  by transfer (simp add: fun_eq_iff)
+
+lemma map_key_single [simp]:
+  "map_key f (single (f k) v) = single k v"
+  by transfer (simp add: fun_eq_iff inj_onD [OF inj_f] when_def)
+
+end
+
+lemma mult_map_scale_conv_mult: "map ((*) s) p = single 0 s * p"
+proof(transfer fixing: s)
+  fix p :: "'a \<Rightarrow> 'b"
+  assume *: "finite {x. p x \<noteq> 0}"
+  { fix x
+    have "prod_fun (\<lambda>k'. s when 0 = k') p x =
+          (\<Sum>l :: 'a. if l = 0 then s * (\<Sum>q. p q when x = q) else 0)"
+      by(auto simp add: prod_fun_def when_def intro: Sum_any.cong simp del: Sum_any.delta)
+    also have "\<dots> = (\<lambda>k. s * p k when p k \<noteq> 0) x" by(simp add: when_def)
+    also note calculation }
+  then show "(\<lambda>k. s * p k when p k \<noteq> 0) = prod_fun (\<lambda>k'. s when 0 = k') p"
+qed
+
+lemma map_single [simp]:
+  "(c = 0 \<Longrightarrow> f 0 = 0) \<Longrightarrow> map f (single x c) = single x (f c)"
+by transfer(auto simp add: fun_eq_iff when_def)
+
+lemma map_eq_zero_iff: "map f p = 0 \<longleftrightarrow> (\<forall>k \<in> keys p. f (lookup p k) = 0)"
+by transfer(auto simp add: fun_eq_iff when_def)
+
+subsection \<open>Canonical dense representation of @{typ "nat \<Rightarrow>\<^sub>0 'a"}\<close>
+
+abbreviation no_trailing_zeros :: "'a :: zero list \<Rightarrow> bool"
+where
+  "no_trailing_zeros \<equiv> no_trailing ((=) 0)"
+
+lift_definition "nth" :: "'a list \<Rightarrow> (nat \<Rightarrow>\<^sub>0 'a::zero)"
+  is "nth_default 0"
+  by (fact finite_nth_default_neq_default)
+
+text \<open>
+  The opposite direction is directly specified on (later)
+  type \<open>nat_mapping\<close>.
+\<close>
+
+lemma nth_Nil [simp]:
+  "nth [] = 0"
+  by transfer (simp add: fun_eq_iff)
+
+lemma nth_singleton [simp]:
+  "nth [v] = single 0 v"
+proof (transfer, rule ext)
+  fix n :: nat and v :: 'a
+  show "nth_default 0 [v] n = (v when 0 = n)"
+    by (auto simp add: nth_default_def nth_append)
+qed
+
+lemma nth_replicate [simp]:
+  "nth (replicate n 0 @ [v]) = single n v"
+proof (transfer, rule ext)
+  fix m n :: nat and v :: 'a
+  show "nth_default 0 (replicate n 0 @ [v]) m = (v when n = m)"
+    by (auto simp add: nth_default_def nth_append)
+qed
+
+lemma nth_strip_while [simp]:
+  "nth (strip_while ((=) 0) xs) = nth xs"
+  by transfer (fact nth_default_strip_while_dflt)
+
+lemma nth_strip_while' [simp]:
+  "nth (strip_while (\<lambda>k. k = 0) xs) = nth xs"
+  by (subst eq_commute) (fact nth_strip_while)
+
+lemma nth_eq_iff:
+  "nth xs = nth ys \<longleftrightarrow> strip_while (HOL.eq 0) xs = strip_while (HOL.eq 0) ys"
+  by transfer (simp add: nth_default_eq_iff)
+
+lemma lookup_nth [simp]:
+  "lookup (nth xs) = nth_default 0 xs"
+  by (fact nth.rep_eq)
+
+lemma keys_nth [simp]:
+  "keys (nth xs) =  fst  {(n, v) \<in> set (enumerate 0 xs). v \<noteq> 0}"
+proof transfer
+  fix xs :: "'a list"
+  { fix n
+    assume "nth_default 0 xs n \<noteq> 0"
+    then have "n < length xs" and "xs ! n \<noteq> 0"
+      by (auto simp add: nth_default_def split: if_splits)
+    then have "(n, xs ! n) \<in> {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}" (is "?x \<in> ?A")
+      by (auto simp add: in_set_conv_nth enumerate_eq_zip)
+    then have "fst ?x \<in> fst  ?A"
+      by blast
+    then have "n \<in> fst  {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}"
+      by simp
+  }
+  then show "{k. nth_default 0 xs k \<noteq> 0} = fst  {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}"
+    by (auto simp add: in_enumerate_iff_nth_default_eq)
+qed
+
+lemma range_nth [simp]:
+  "range (nth xs) = set xs - {0}"
+  by transfer simp
+
+lemma degree_nth:
+  "no_trailing_zeros xs \<Longrightarrow> degree (nth xs) = length xs"
+unfolding degree_def proof transfer
+  fix xs :: "'a list"
+  assume *: "no_trailing_zeros xs"
+  let ?A = "{n. nth_default 0 xs n \<noteq> 0}"
+  let ?f = "nth_default 0 xs"
+  let ?bound = "Max (insert 0 (Suc  {n. ?f n \<noteq> 0}))"
+  show "?bound = length xs"
+  proof (cases "xs = []")
+    case False
+    with * obtain n where n: "n < length xs" "xs ! n \<noteq> 0"
+      by (fastforce simp add: no_trailing_unfold last_conv_nth neq_Nil_conv)
+    then have "?bound = Max (Suc  {k. (k < length xs \<longrightarrow> xs ! k \<noteq> (0::'a)) \<and> k < length xs})"
+      by (subst Max_insert) (auto simp add: nth_default_def)
+    also let ?A = "{k. k < length xs \<and> xs ! k \<noteq> 0}"
+    have "{k. (k < length xs \<longrightarrow> xs ! k \<noteq> (0::'a)) \<and> k < length xs} = ?A" by auto
+    also have "Max (Suc  ?A) = Suc (Max ?A)" using n
+      by (subst mono_Max_commute [where f = Suc, symmetric]) (auto simp add: mono_Suc)
+    also {
+      have "Max ?A \<in> ?A" using n Max_in [of ?A] by fastforce
+      hence "Suc (Max ?A) \<le> length xs" by simp
+      moreover from * False have "length xs - 1 \<in> ?A"
+        by(auto simp add: no_trailing_unfold last_conv_nth)
+      hence "length xs - 1 \<le> Max ?A" using Max_ge[of ?A "length xs - 1"] by auto
+      hence "length xs \<le> Suc (Max ?A)" by simp
+      ultimately have "Suc (Max ?A) = length xs" by simp }
+    finally show ?thesis .
+  qed simp
+qed
+
+lemma nth_trailing_zeros [simp]:
+  "nth (xs @ replicate n 0) = nth xs"
+  by transfer simp
+
+lemma nth_idem:
+  "nth (List.map (lookup f) [0..<degree f]) = f"
+  unfolding degree_def by transfer
+    (auto simp add: nth_default_def fun_eq_iff not_less)
+
+lemma nth_idem_bound:
+  assumes "degree f \<le> n"
+  shows "nth (List.map (lookup f) [0..<n]) = f"
+proof -
+  from assms obtain m where "n = degree f + m"
+    by (blast dest: le_Suc_ex)
+  then have "[0..<n] = [0..<degree f] @ [degree f..<degree f + m]"
+  moreover have "List.map (lookup f) [degree f..<degree f + m] = replicate m 0"
+    by (rule replicate_eqI) (auto simp add: beyond_degree_lookup_zero)
+  ultimately show ?thesis by (simp add: nth_idem)
+qed
+
+
+subsection \<open>Canonical sparse representation of @{typ "'a \<Rightarrow>\<^sub>0 'b"}\<close>
+
+lift_definition the_value :: "('a \<times> 'b) list \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b::zero"
+  is "\<lambda>xs k. case map_of xs k of None \<Rightarrow> 0 | Some v \<Rightarrow> v"
+proof -
+  fix xs :: "('a \<times> 'b) list"
+  have fin: "finite {k. \<exists>v. map_of xs k = Some v}"
+    using finite_dom_map_of [of xs] unfolding dom_def by auto
+  then show "finite {k. (case map_of xs k of None \<Rightarrow> 0 | Some v \<Rightarrow> v) \<noteq> 0}"
+    using fin by (simp split: option.split)
+qed
+
+definition items :: "('a::linorder \<Rightarrow>\<^sub>0 'b::zero) \<Rightarrow> ('a \<times> 'b) list"
+where
+  "items f = List.map (\<lambda>k. (k, lookup f k)) (sorted_list_of_set (keys f))"
+
+text \<open>
+  For the canonical sparse representation we provide both
+  directions of morphisms since the specification of
+  ordered association lists in theory \<open>OAList\<close>
+  will support arbitrary linear orders @{class linorder}
+  as keys, not just natural numbers @{typ nat}.
+\<close>
+
+lemma the_value_items [simp]:
+  "the_value (items f) = f"
+  unfolding items_def
+  by transfer (simp add: fun_eq_iff map_of_map_restrict restrict_map_def)
+
+lemma lookup_the_value:
+  "lookup (the_value xs) k = (case map_of xs k of None \<Rightarrow> 0 | Some v \<Rightarrow> v)"
+  by transfer rule
+
+lemma items_the_value:
+  assumes "sorted (List.map fst xs)" and "distinct (List.map fst xs)" and "0 \<notin> snd  set xs"
+  shows "items (the_value xs) = xs"
+proof -
+  from assms have "sorted_list_of_set (set (List.map fst xs)) = List.map fst xs"
+    unfolding sorted_list_of_set_sort_remdups by (simp add: distinct_remdups_id sorted_sort_id)
+  moreover from assms have "keys (the_value xs) = fst  set xs"
+    by transfer (auto simp add: image_def split: option.split dest: set_map_of_compr)
+  ultimately show ?thesis
+    unfolding items_def using assms
+    by (auto simp add: lookup_the_value intro: map_idI)
+qed
+
+lemma the_value_Nil [simp]:
+  "the_value [] = 0"
+  by transfer (simp add: fun_eq_iff)
+
+lemma the_value_Cons [simp]:
+  "the_value (x # xs) = update (fst x) (snd x) (the_value xs)"
+  by transfer (simp add: fun_eq_iff)
+
+lemma items_zero [simp]:
+  "items 0 = []"
+  unfolding items_def by simp
+
+lemma items_one [simp]:
+  "items 1 = [(0, 1)]"
+  unfolding items_def by transfer simp
+
+lemma items_single [simp]:
+  "items (single k v) = (if v = 0 then [] else [(k, v)])"
+  unfolding items_def by simp
+
+lemma in_set_items_iff [simp]:
+  "(k, v) \<in> set (items f) \<longleftrightarrow> k \<in> keys f \<and> lookup f k = v"
+  unfolding items_def by transfer auto
+
+
+subsection \<open>Size estimation\<close>
+
+context
+  fixes f :: "'a \<Rightarrow> nat"
+  and g :: "'b :: zero \<Rightarrow> nat"
+begin
+
+definition poly_mapping_size :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> nat"
+where
+  "poly_mapping_size m = g 0 + (\<Sum>k \<in> keys m. Suc (f k + g (lookup m k)))"
+
+lemma poly_mapping_size_0 [simp]:
+  "poly_mapping_size 0 = g 0"
+
+lemma poly_mapping_size_single [simp]:
+  "poly_mapping_size (single k v) = (if v = 0 then g 0 else g 0 + f k + g v + 1)"
+  unfolding poly_mapping_size_def by transfer simp
+
+lemma keys_less_poly_mapping_size:
+  "k \<in> keys m \<Longrightarrow> f k + g (lookup m k) < poly_mapping_size m"
+unfolding poly_mapping_size_def
+proof transfer
+  fix k :: 'a and m :: "'a \<Rightarrow> 'b" and f :: "'a \<Rightarrow> nat" and g
+  let ?keys = "{k. m k \<noteq> 0}"
+  assume *: "finite ?keys" "k \<in> ?keys"
+  then have "f k + g (m k) = (\<Sum>k' \<in> ?keys. f k' + g (m k') when k' = k)"
+    by (simp add: sum.delta when_def)
+  also have "\<dots> < (\<Sum>k' \<in> ?keys. Suc (f k' + g (m k')))" using *
+    by (intro sum_strict_mono) (auto simp add: when_def)
+  also have "\<dots> \<le> g 0 + \<dots>" by simp
+  finally have "f k + g (m k) < \<dots>" .
+  then show "f k + g (m k) < g 0 + (\<Sum>k | m k \<noteq> 0. Suc (f k + g (m k)))"
+    by simp
+qed
+
+lemma lookup_le_poly_mapping_size:
+  "g (lookup m k) \<le> poly_mapping_size m"
+proof (cases "k \<in> keys m")
+  case True
+  with keys_less_poly_mapping_size [of k m]
+  show ?thesis by simp
+next
+  case False
+  then show ?thesis
+    by (simp add: Poly_Mapping.poly_mapping_size_def in_keys_iff)
+qed
+
+lemma poly_mapping_size_estimation:
+  "k \<in> keys m \<Longrightarrow> y \<le> f k + g (lookup m k) \<Longrightarrow> y < poly_mapping_size m"
+  using keys_less_poly_mapping_size by (auto intro: le_less_trans)
+
+lemma poly_mapping_size_estimation2:
+  assumes "v \<in> range m" and "y \<le> g v"
+  shows "y < poly_mapping_size m"
+proof -
+  from assms obtain k where *: "lookup m k = v" "v \<noteq> 0"
+    by transfer blast
+  from * have "k \<in> keys m"
+  then show ?thesis
+  proof (rule poly_mapping_size_estimation)
+    from assms * have "y \<le> g (lookup m k)"
+      by simp
+    then show "y \<le> f k + g (lookup m k)"
+      by simp
+  qed
+qed
+
+end
+
+lemma poly_mapping_size_one [simp]:
+  "poly_mapping_size f g 1 = g 0 + f 0 + g 1 + 1"
+  unfolding poly_mapping_size_def by transfer simp
+
+lemma poly_mapping_size_cong [fundef_cong]:
+  "m = m' \<Longrightarrow> g 0 = g' 0 \<Longrightarrow> (\<And>k. k \<in> keys m' \<Longrightarrow> f k = f' k)
+    \<Longrightarrow> (\<And>v. v \<in> range m' \<Longrightarrow> g v = g' v)
+    \<Longrightarrow> poly_mapping_size f g m = poly_mapping_size f' g' m'"
+  by (auto simp add: poly_mapping_size_def intro!: sum.cong)
+
+instantiation poly_mapping :: (type, zero) size
+begin
+
+definition "size = poly_mapping_size (\<lambda>_. 0) (\<lambda>_. 0)"
+
+instance ..
+
+end
+
+
+subsection \<open>Further mapping operations and properties\<close>
+
+text \<open>It is like in algebra: there are many definitions, some are also used\<close>
+
+lift_definition mapp ::
+  "('a \<Rightarrow> 'b :: zero \<Rightarrow> 'c :: zero) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c)"
+  is "\<lambda>f p k. (if k \<in> keys p then f k (lookup p k) else 0)"
+  by simp
+
+lemma mapp_cong [fundef_cong]:
+  "\<lbrakk> m = m'; \<And>k. k \<in> keys m' \<Longrightarrow> f k (lookup m' k) = f' k (lookup m' k) \<rbrakk>
+  \<Longrightarrow> mapp f m = mapp f' m'"
+  by transfer (auto simp add: fun_eq_iff)
+
+lemma lookup_mapp:
+  "lookup (mapp f p) k = (f k (lookup p k) when k \<in> keys p)"
+
+lemma keys_mapp_subset: "keys (mapp f p) \<subseteq> keys p"
+  by (meson in_keys_iff mapp.rep_eq subsetI)
+
+subsection\<open>Free Abelian Groups Over a Type\<close>
+
+abbreviation frag_of :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>0 int"
+  where "frag_of c \<equiv> Poly_Mapping.single c (1::int)"
+
+lemma lookup_frag_of [simp]:
+   "Poly_Mapping.lookup(frag_of c) = (\<lambda>x. if x = c then 1 else 0)"
+  by (force simp add: lookup_single_not_eq)
+
+lemma frag_of_nonzero [simp]: "frag_of a \<noteq> 0"
+proof -
+  let ?f = "\<lambda>x. if x = a then 1 else (0::int)"
+  have "?f \<noteq> (\<lambda>x. 0::int)"
+    by (auto simp: fun_eq_iff)
+  then have "Poly_Mapping.lookup (Abs_poly_mapping ?f) \<noteq> Poly_Mapping.lookup (Abs_poly_mapping (\<lambda>x. 0))"
+    by fastforce
+  then show ?thesis
+    by (metis lookup_single_eq lookup_zero)
+qed
+
+definition frag_cmul :: "int \<Rightarrow> ('a \<Rightarrow>\<^sub>0 int) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 int)"
+  where "frag_cmul c a = Abs_poly_mapping (\<lambda>x. c * Poly_Mapping.lookup a x)"
+
+lemma frag_cmul_zero [simp]: "frag_cmul 0 x = 0"
+
+lemma frag_cmul_zero2 [simp]: "frag_cmul c 0 = 0"
+
+lemma frag_cmul_one [simp]: "frag_cmul 1 x = x"
+  by (auto simp: frag_cmul_def Poly_Mapping.poly_mapping.lookup_inverse)
+
+lemma frag_cmul_minus_one [simp]: "frag_cmul (-1) x = -x"
+  by (simp add: frag_cmul_def uminus_poly_mapping_def poly_mapping_eqI)
+
+lemma frag_cmul_cmul [simp]: "frag_cmul c (frag_cmul d x) = frag_cmul (c*d) x"
+  by (simp add: frag_cmul_def mult_ac)
+
+lemma lookup_frag_cmul [simp]: "poly_mapping.lookup (frag_cmul c x) i = c * poly_mapping.lookup x i"
+
+lemma minus_frag_cmul [simp]: "- frag_cmul k x = frag_cmul (-k) x"
+
+lemma keys_frag_of: "Poly_Mapping.keys(frag_of a) = {a}"
+  by simp
+
+lemma finite_cmul_nonzero: "finite {x. c * Poly_Mapping.lookup a x \<noteq> (0::int)}"
+  by simp
+
+lemma keys_cmul: "Poly_Mapping.keys(frag_cmul c a) \<subseteq> Poly_Mapping.keys a"
+  using finite_cmul_nonzero [of c a]
+  by (metis lookup_frag_cmul mult_zero_right not_in_keys_iff_lookup_eq_zero subsetI)
+
+
+lemma keys_cmul_iff [iff]: "i \<in> Poly_Mapping.keys (frag_cmul c x) \<longleftrightarrow> i \<in> Poly_Mapping.keys x \<and> c \<noteq> 0"
+  apply (auto simp: )
+  apply (meson subsetD keys_cmul)
+  by (metis in_keys_iff lookup_frag_cmul mult_eq_0_iff)
+
+lemma keys_minus [simp]: "Poly_Mapping.keys(-a) = Poly_Mapping.keys a"
+  by (metis (no_types, hide_lams) in_keys_iff lookup_uminus neg_equal_0_iff_equal subsetI subset_antisym)
+
+lemma keys_diff:
+  "Poly_Mapping.keys(a - b) \<subseteq> Poly_Mapping.keys a \<union> Poly_Mapping.keys b"
+  by (auto simp add: in_keys_iff lookup_minus)
+
+lemma keys_eq_empty [simp]: "Poly_Mapping.keys c = {} \<longleftrightarrow> c = 0"
+  by (metis in_keys_iff keys_zero lookup_zero poly_mapping_eqI)
+
+lemma frag_cmul_eq_0_iff [simp]: "frag_cmul k c = 0 \<longleftrightarrow> k=0 \<or> c=0"
+  by auto (metis subsetI subset_antisym keys_cmul_iff keys_eq_empty)
+
+lemma frag_of_eq: "frag_of x = frag_of y \<longleftrightarrow> x = y"
+  by (metis lookup_single_eq lookup_single_not_eq zero_neq_one)
+
+lemma frag_cmul_distrib: "frag_cmul (c+d) a = frag_cmul c a + frag_cmul d a"
+  by (simp add: frag_cmul_def plus_poly_mapping_def int_distrib)
+
+lemma frag_cmul_distrib2: "frag_cmul c (a+b) = frag_cmul c a + frag_cmul c b"
+proof -
+  have "finite {x. poly_mapping.lookup a x + poly_mapping.lookup b x \<noteq> 0}"
+    using keys_add [of a b]
+    by (metis (no_types, lifting) finite_keys finite_subset keys.rep_eq lookup_add mem_Collect_eq subsetI)
+  then show ?thesis
+    by (simp add: frag_cmul_def plus_poly_mapping_def int_distrib)
+qed
+
+lemma frag_cmul_diff_distrib: "frag_cmul (a - b) c = frag_cmul a c - frag_cmul b c"
+  by (auto simp: left_diff_distrib lookup_minus poly_mapping_eqI)
+
+lemma frag_cmul_sum:
+     "frag_cmul a (sum b I) = (\<Sum>i\<in>I. frag_cmul a (b i))"
+proof (induction rule: infinite_finite_induct)
+  case (insert i I)
+  then show ?case
+    by (auto simp: algebra_simps frag_cmul_distrib2)
+qed auto
+
+lemma keys_sum: "Poly_Mapping.keys(sum b I) \<subseteq> (\<Union>i \<in>I. Poly_Mapping.keys(b i))"
+proof (induction I rule: infinite_finite_induct)
+  case (insert i I)
+  then show ?case
+    using keys_add [of "b i" "sum b I"] by auto
+qed auto
+
+
+definition frag_extend :: "('b \<Rightarrow> 'a \<Rightarrow>\<^sub>0 int) \<Rightarrow> ('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 int"
+  where "frag_extend b x \<equiv> (\<Sum>i \<in> Poly_Mapping.keys x. frag_cmul (Poly_Mapping.lookup x i) (b i))"
+
+lemma frag_extend_0 [simp]: "frag_extend b 0 = 0"
+
+lemma frag_extend_of [simp]: "frag_extend f (frag_of a) = f a"
+
+lemma frag_extend_cmul:
+   "frag_extend f (frag_cmul c x) = frag_cmul c (frag_extend f x)"
+  by (auto simp: frag_extend_def frag_cmul_sum intro: sum.mono_neutral_cong_left)
+
+lemma frag_extend_minus:
+   "frag_extend f (- x) = - (frag_extend f x)"
+  using frag_extend_cmul [of f "-1"] by simp
+
+  "frag_extend f (a+b) = (frag_extend f a) + (frag_extend f b)"
+proof -
+  have *: "(\<Sum>i\<in>Poly_Mapping.keys a. frag_cmul (poly_mapping.lookup a i) (f i))
+         = (\<Sum>i\<in>Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup a i) (f i))"
+          "(\<Sum>i\<in>Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup b i) (f i))
+         = (\<Sum>i\<in>Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup b i) (f i))"
+    by (auto simp: in_keys_iff intro: sum.mono_neutral_cong_left)
+  have "frag_extend f (a+b) = (\<Sum>i\<in>Poly_Mapping.keys (a + b).
+       frag_cmul (poly_mapping.lookup a i) (f i) + frag_cmul (poly_mapping.lookup b i) (f i)) "
+    by (auto simp: frag_extend_def Poly_Mapping.lookup_add frag_cmul_distrib)
+  also have "... = (\<Sum>i \<in> Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup a i) (f i)
+                         + frag_cmul (poly_mapping.lookup b i) (f i))"
+    apply (rule sum.mono_neutral_cong_left)
+    using keys_add [of a b]
+     apply (auto simp add: in_keys_iff plus_poly_mapping.rep_eq frag_cmul_distrib [symmetric])
+    done
+  also have "... = (frag_extend f a) + (frag_extend f b)"
+    by (auto simp: * sum.distrib frag_extend_def)
+  finally show ?thesis .
+qed
+
+lemma frag_extend_diff:
+   "frag_extend f (a-b) = (frag_extend f a) - (frag_extend f b)"
+
+lemma frag_extend_sum:
+   "finite I \<Longrightarrow> frag_extend f (\<Sum>i\<in>I. g i) = sum (frag_extend f o g) I"
+
+lemma frag_extend_eq:
+   "(\<And>f. f \<in> Poly_Mapping.keys c \<Longrightarrow> g f = h f) \<Longrightarrow> frag_extend g c = frag_extend h c"
+
+lemma frag_extend_eq_0:
+   "(\<And>x. x \<in> Poly_Mapping.keys c \<Longrightarrow> f x = 0) \<Longrightarrow> frag_extend f c = 0"
+
+lemma keys_frag_extend: "Poly_Mapping.keys(frag_extend f c) \<subseteq> (\<Union>x \<in> Poly_Mapping.keys c. Poly_Mapping.keys(f x))"
+  unfolding frag_extend_def
+  by (smt SUP_mono' keys_cmul keys_sum order_trans)
+
+lemma frag_expansion: "a = frag_extend frag_of a"
+proof -
+  have *: "finite I
+        \<Longrightarrow> Poly_Mapping.lookup (\<Sum>i\<in>I. frag_cmul (Poly_Mapping.lookup a i) (frag_of i)) j =
+            (if j \<in> I then Poly_Mapping.lookup a j else 0)" for I j
+    by (induction I rule: finite_induct) (auto simp: lookup_single lookup_add)
+  show ?thesis
+    unfolding frag_extend_def
+    by (rule poly_mapping_eqI) (fastforce simp add: in_keys_iff *)
+qed
+
+lemma frag_closure_minus_cmul:
+  assumes "P 0" and P: "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> P(x - y)" "P c"
+  shows "P(frag_cmul k c)"
+proof -
+  have "P (frag_cmul (int n) c)" for n
+    apply (induction n)
+     apply (simp_all add: assms frag_cmul_distrib)
+  then show ?thesis
+qed
+
+lemma frag_induction [consumes 1, case_names zero one diff]:
+  assumes supp: "Poly_Mapping.keys c \<subseteq> S"
+    and 0: "P 0" and sing: "\<And>x. x \<in> S \<Longrightarrow> P(frag_of x)"
+    and diff: "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P(a - b)"
+  shows "P c"
+proof -
+  have "P (\<Sum>i\<in>I. frag_cmul (poly_mapping.lookup c i) (frag_of i))"
+    if "I \<subseteq> Poly_Mapping.keys c" for I
+    using finite_subset [OF that finite_keys [of c]] that supp
+  proof (induction I arbitrary: c rule: finite_induct)
+    case empty
+    then show ?case
+      by (auto simp: 0)
+  next
+    case (insert i I c)
+    have ab: "a+b = a - (0 - b)" for a b :: "'a \<Rightarrow>\<^sub>0 int"
+      by simp
+    have Pfrag: "P (frag_cmul (poly_mapping.lookup c i) (frag_of i))"
+      by (metis "0" diff frag_closure_minus_cmul insert.prems insert_subset sing subset_iff)
+    show ?case
+      apply (subst ab)
+      using insert apply (blast intro: assms Pfrag)
+      done
+  qed
+  then show ?thesis
+    by (subst frag_expansion) (auto simp add: frag_extend_def)
+qed
+
+lemma frag_extend_compose:
+  "frag_extend f (frag_extend (frag_of o g) c) = frag_extend (f o g) c"
+  using subset_UNIV
+  by (induction c rule: frag_induction) (auto simp: frag_extend_diff)
+
+lemma frag_split:
+  fixes c :: "'a \<Rightarrow>\<^sub>0 int"
+  assumes "Poly_Mapping.keys c \<subseteq> S \<union> T"
+  obtains d e where "Poly_Mapping.keys d \<subseteq> S" "Poly_Mapping.keys e \<subseteq> T" "d + e = c"
+proof
+  let ?d = "frag_extend (\<lambda>f. if f \<in> S then frag_of f else 0) c"
+  let ?e = "frag_extend (\<lambda>f. if f \<in> S then 0 else frag_of f) c"
+  show "Poly_Mapping.keys ?d \<subseteq> S" "Poly_Mapping.keys ?e \<subseteq> T"
+    using assms by (auto intro!: order_trans [OF keys_frag_extend] split: if_split_asm)
+  show "?d + ?e = c"
+    using assms
+  proof (induction c rule: frag_induction)
+    case (diff a b)
+    then show ?case
\ No newline at end of file`