author ballarin Wed, 30 Apr 2003 18:32:06 +0200 changeset 13940 c67798653056 parent 13939 b3ef90abbd02 child 13941 2ae108fcd068
HOL-Algebra: New polynomial development added.
 src/HOL/Algebra/Coset.thy file | annotate | diff | comparison | revisions src/HOL/Algebra/Group.thy file | annotate | diff | comparison | revisions src/HOL/Algebra/Module.thy file | annotate | diff | comparison | revisions src/HOL/Algebra/ROOT.ML file | annotate | diff | comparison | revisions src/HOL/Algebra/UnivPoly.thy file | annotate | diff | comparison | revisions
```--- a/src/HOL/Algebra/Coset.thy	Wed Apr 30 18:31:38 2003 +0200
+++ b/src/HOL/Algebra/Coset.thy	Wed Apr 30 18:32:06 2003 +0200
@@ -468,7 +468,7 @@
"[|H <| G; M \<in> rcosets G H|] ==> set_inv G M <#> M = H"
by (auto simp add: setrcos_eq rcos_inv rcos_sum normal_imp_subgroup
subgroup.subset)
-
+(*
lemma (in group) factorgroup_is_magma:
"H <| G ==> magma (G Mod H)"
by rule (simp add: FactGroup_def coset.setmult_closed [OF is_coset])
@@ -477,7 +477,7 @@
"H <| G ==> semigroup_axioms (G Mod H)"
by rule (simp add: FactGroup_def coset.setrcos_assoc [OF is_coset]
coset.setmult_closed [OF is_coset])
-
+*)
theorem (in group) factorgroup_is_group:
"H <| G ==> group (G Mod H)"
apply (insert is_coset) ```
```--- a/src/HOL/Algebra/Group.thy	Wed Apr 30 18:31:38 2003 +0200
+++ b/src/HOL/Algebra/Group.thy	Wed Apr 30 18:32:06 2003 +0200
@@ -93,6 +93,10 @@
finally show ?thesis .
qed

+lemma (in monoid) Units_one_closed [intro, simp]:
+  "\<one> \<in> Units G"
+  by (unfold Units_def) auto
+
lemma (in monoid) Units_inv_closed [intro, simp]:
"x \<in> Units G ==> inv x \<in> carrier G"
apply (unfold Units_def m_inv_def)
@@ -162,6 +166,15 @@
with G show "x = y" by simp
qed

+lemma (in monoid) Units_inv_comm:
+  assumes inv: "x \<otimes> y = \<one>"
+    and G: "x \<in> Units G" "y \<in> Units G"
+  shows "y \<otimes> x = \<one>"
+proof -
+  from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed)
+  with G show ?thesis by (simp del: r_one add: m_assoc Units_closed)
+qed
+
text {* Power *}

lemma (in monoid) nat_pow_closed [intro, simp]:
@@ -287,16 +300,7 @@
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
(x \<otimes> y = x \<otimes> z) = (y = z)"
using Units_l_inv by simp
-(*
-lemma (in group) r_one [simp]:
-  "x \<in> carrier G ==> x \<otimes> \<one> = x"
-proof -
-  assume x: "x \<in> carrier G"
-  then have "inv x \<otimes> (x \<otimes> \<one>) = inv x \<otimes> x"
-    by (simp add: m_assoc [symmetric] l_inv)
-  with x show ?thesis by simp
-qed
-*)
+
lemma (in group) r_inv:
"x \<in> carrier G ==> x \<otimes> inv x = \<one>"
proof -
@@ -346,6 +350,10 @@
with G show ?thesis by simp
qed

+lemma (in group) inv_comm:
+  "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
+  by (rule Units_inv_comm) auto
+
text {* Power *}

lemma (in group) int_pow_def2:```
```--- a/src/HOL/Algebra/Module.thy	Wed Apr 30 18:31:38 2003 +0200
+++ b/src/HOL/Algebra/Module.thy	Wed Apr 30 18:32:06 2003 +0200
@@ -139,7 +139,7 @@
finally show ?thesis .
qed

-subsection {* Every Abelian Group is a \$\mathbb{Z}\$-module *}
+subsection {* Every Abelian Group is a Z-module *}

text {* Not finished. *}
```
```--- a/src/HOL/Algebra/ROOT.ML	Wed Apr 30 18:31:38 2003 +0200
+++ b/src/HOL/Algebra/ROOT.ML	Wed Apr 30 18:32:06 2003 +0200
@@ -6,8 +6,9 @@

(* New development, based on explicit structures *)

+no_document use_thy "FuncSet";
use_thy "Sylow";		(* Groups *)
-(* use_thy "UnivPoly"; *)	(* Rings and polynomials *)
+use_thy "UnivPoly";		(* Rings and polynomials *)

(* Old development, based on axiomatic type classes.
Will be withdrawn in future. *)```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Algebra/UnivPoly.thy	Wed Apr 30 18:32:06 2003 +0200
@@ -0,0 +1,1546 @@
+(*
+  Title:     Univariate Polynomials
+  Id:        \$Id\$
+  Author:    Clemens Ballarin, started 9 December 1996
+  Copyright: Clemens Ballarin
+*)
+
+theory UnivPoly = Module:
+
+section {* Univariate Polynomials *}
+
+subsection
+  {* Definition of the Constructor for Univariate Polynomials @{term UP} *}
+
+(* Could alternatively use locale ...
+locale bound = cring + var bound +
+  defines ...
+*)
+
+constdefs
+  bound  :: "['a, nat, nat => 'a] => bool"
+  "bound z n f == (ALL i. n < i --> f i = z)"
+
+lemma boundI [intro!]:
+  "[| !! m. n < m ==> f m = z |] ==> bound z n f"
+  by (unfold bound_def) fast
+
+lemma boundE [elim?]:
+  "[| bound z n f; (!! m. n < m ==> f m = z) ==> P |] ==> P"
+  by (unfold bound_def) fast
+
+lemma boundD [dest]:
+  "[| bound z n f; n < m |] ==> f m = z"
+  by (unfold bound_def) fast
+
+lemma bound_below:
+  assumes bound: "bound z m f" and nonzero: "f n ~= z" shows "n <= m"
+proof (rule classical)
+  assume "~ ?thesis"
+  then have "m < n" by arith
+  with bound have "f n = z" ..
+  with nonzero show ?thesis by contradiction
+qed
+
+record ('a, 'p) up_ring = "('a, 'p) module" +
+  monom :: "['a, nat] => 'p"
+  coeff :: "['p, nat] => 'a"
+
+constdefs
+  up :: "('a, 'm) ring_scheme => (nat => 'a) set"
+  "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound (zero R) n f)}"
+  UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
+  "UP R == (|
+    carrier = up R,
+    mult = (%p:up R. %q:up R. %n. finsum R (%i. mult R (p i) (q (n-i))) {..n}),
+    one = (%i. if i=0 then one R else zero R),
+    zero = (%i. zero R),
+    add = (%p:up R. %q:up R. %i. add R (p i) (q i)),
+    smult = (%a:carrier R. %p:up R. %i. mult R a (p i)),
+    monom = (%a:carrier R. %n i. if i=n then a else zero R),
+    coeff = (%p:up R. %n. p n) |)"
+
+text {*
+  Properties of the set of polynomials @{term up}.
+*}
+
+lemma mem_upI [intro]:
+  "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"
+  by (simp add: up_def Pi_def)
+
+lemma mem_upD [dest]:
+  "f \<in> up R ==> f n \<in> carrier R"
+  by (simp add: up_def Pi_def)
+
+lemma (in cring) bound_upD [dest]:
+  "f \<in> up R ==> EX n. bound \<zero> n f"
+  by (simp add: up_def)
+
+lemma (in cring) up_one_closed:
+   "(%n. if n = 0 then \<one> else \<zero>) \<in> up R"
+  using up_def by force
+
+lemma (in cring) up_smult_closed:
+  "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R"
+  by force
+
+lemma (in cring) up_add_closed:
+  "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R"
+proof
+  fix n
+  assume "p \<in> up R" and "q \<in> up R"
+  then show "p n \<oplus> q n \<in> carrier R"
+    by auto
+next
+  assume UP: "p \<in> up R" "q \<in> up R"
+  show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"
+  proof -
+    from UP obtain n where boundn: "bound \<zero> n p" by fast
+    from UP obtain m where boundm: "bound \<zero> m q" by fast
+    have "bound \<zero> (max n m) (%i. p i \<oplus> q i)"
+    proof
+      fix i
+      assume "max n m < i"
+      with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp
+    qed
+    then show ?thesis ..
+  qed
+qed
+
+lemma (in cring) up_a_inv_closed:
+  "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"
+proof
+  assume R: "p \<in> up R"
+  then obtain n where "bound \<zero> n p" by auto
+  then have "bound \<zero> n (%i. \<ominus> p i)" by auto
+  then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto
+qed auto
+
+lemma (in cring) up_mult_closed:
+  "[| p \<in> up R; q \<in> up R |] ==>
+  (%n. finsum R (%i. p i \<otimes> q (n-i)) {..n}) \<in> up R"
+proof
+  fix n
+  assume "p \<in> up R" "q \<in> up R"
+  then show "finsum R (%i. p i \<otimes> q (n-i)) {..n} \<in> carrier R"
+    by (simp add: mem_upD  funcsetI)
+next
+  assume UP: "p \<in> up R" "q \<in> up R"
+  show "EX n. bound \<zero> n (%n. finsum R (%i. p i \<otimes> q (n - i)) {..n})"
+  proof -
+    from UP obtain n where boundn: "bound \<zero> n p" by fast
+    from UP obtain m where boundm: "bound \<zero> m q" by fast
+    have "bound \<zero> (n + m) (%n. finsum R (%i. p i \<otimes> q (n - i)) {..n})"
+    proof
+      fix k
+      assume bound: "n + m < k"
+      {
+	fix i
+	have "p i \<otimes> q (k-i) = \<zero>"
+	proof (cases "n < i")
+	  case True
+	  with boundn have "p i = \<zero>" by auto
+          moreover from UP have "q (k-i) \<in> carrier R" by auto
+	  ultimately show ?thesis by simp
+	next
+	  case False
+	  with bound have "m < k-i" by arith
+	  with boundm have "q (k-i) = \<zero>" by auto
+	  moreover from UP have "p i \<in> carrier R" by auto
+	  ultimately show ?thesis by simp
+	qed
+      }
+      then show "finsum R (%i. p i \<otimes> q (k-i)) {..k} = \<zero>"
+	by (simp add: Pi_def)
+    qed
+    then show ?thesis by fast
+  qed
+qed
+
+subsection {* Effect of operations on coefficients *}
+
+locale UP = struct R + struct P +
+  defines P_def: "P == UP R"
+
+locale UP_cring = UP + cring R
+
+locale UP_domain = UP_cring + "domain" R
+
+text {*
+  Temporarily declare UP.P\_def as simp rule.
+*}
+(* TODO: use antiquotation once text (in locale) is supported. *)
+
+declare (in UP) P_def [simp]
+
+lemma (in UP_cring) coeff_monom [simp]:
+  "a \<in> carrier R ==>
+  coeff P (monom P a m) n = (if m=n then a else \<zero>)"
+proof -
+  assume R: "a \<in> carrier R"
+  then have "(%n. if n = m then a else \<zero>) \<in> up R"
+    using up_def by force
+  with R show ?thesis by (simp add: UP_def)
+qed
+
+lemma (in UP_cring) coeff_zero [simp]:
+  "coeff P \<zero>\<^sub>2 n = \<zero>"
+  by (auto simp add: UP_def)
+
+lemma (in UP_cring) coeff_one [simp]:
+  "coeff P \<one>\<^sub>2 n = (if n=0 then \<one> else \<zero>)"
+  using up_one_closed by (simp add: UP_def)
+
+lemma (in UP_cring) coeff_smult [simp]:
+  "[| a \<in> carrier R; p \<in> carrier P |] ==>
+  coeff P (a \<odot>\<^sub>2 p) n = a \<otimes> coeff P p n"
+  by (simp add: UP_def up_smult_closed)
+
+lemma (in UP_cring) coeff_add [simp]:
+  "[| p \<in> carrier P; q \<in> carrier P |] ==>
+  coeff P (p \<oplus>\<^sub>2 q) n = coeff P p n \<oplus> coeff P q n"
+
+lemma (in UP_cring) coeff_mult [simp]:
+  "[| p \<in> carrier P; q \<in> carrier P |] ==>
+  coeff P (p \<otimes>\<^sub>2 q) n = finsum R (%i. coeff P p i \<otimes> coeff P q (n-i)) {..n}"
+  by (simp add: UP_def up_mult_closed)
+
+lemma (in UP) up_eqI:
+  assumes prem: "!!n. coeff P p n = coeff P q n"
+    and R: "p \<in> carrier P" "q \<in> carrier P"
+  shows "p = q"
+proof
+  fix x
+  from prem and R show "p x = q x" by (simp add: UP_def)
+qed
+
+subsection {* Polynomials form a commutative ring. *}
+
+text {* Operations are closed over @{term "P"}. *}
+
+lemma (in UP_cring) UP_mult_closed [simp]:
+  "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^sub>2 q \<in> carrier P"
+  by (simp add: UP_def up_mult_closed)
+
+lemma (in UP_cring) UP_one_closed [simp]:
+  "\<one>\<^sub>2 \<in> carrier P"
+  by (simp add: UP_def up_one_closed)
+
+lemma (in UP_cring) UP_zero_closed [intro, simp]:
+  "\<zero>\<^sub>2 \<in> carrier P"
+  by (auto simp add: UP_def)
+
+lemma (in UP_cring) UP_a_closed [intro, simp]:
+  "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^sub>2 q \<in> carrier P"
+
+lemma (in UP_cring) monom_closed [simp]:
+  "a \<in> carrier R ==> monom P a n \<in> carrier P"
+  by (auto simp add: UP_def up_def Pi_def)
+
+lemma (in UP_cring) UP_smult_closed [simp]:
+  "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^sub>2 p \<in> carrier P"
+  by (simp add: UP_def up_smult_closed)
+
+lemma (in UP) coeff_closed [simp]:
+  "p \<in> carrier P ==> coeff P p n \<in> carrier R"
+  by (auto simp add: UP_def)
+
+declare (in UP) P_def [simp del]
+
+text {* Algebraic ring properties *}
+
+lemma (in UP_cring) UP_a_assoc:
+  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
+  shows "(p \<oplus>\<^sub>2 q) \<oplus>\<^sub>2 r = p \<oplus>\<^sub>2 (q \<oplus>\<^sub>2 r)"
+  by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
+
+lemma (in UP_cring) UP_l_zero [simp]:
+  assumes R: "p \<in> carrier P"
+  shows "\<zero>\<^sub>2 \<oplus>\<^sub>2 p = p"
+  by (rule up_eqI, simp_all add: R)
+
+lemma (in UP_cring) UP_l_neg_ex:
+  assumes R: "p \<in> carrier P"
+  shows "EX q : carrier P. q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
+proof -
+  let ?q = "%i. \<ominus> (p i)"
+  from R have closed: "?q \<in> carrier P"
+    by (simp add: UP_def P_def up_a_inv_closed)
+  from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
+    by (simp add: UP_def P_def up_a_inv_closed)
+  show ?thesis
+  proof
+    show "?q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
+      by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
+  qed (rule closed)
+qed
+
+lemma (in UP_cring) UP_a_comm:
+  assumes R: "p \<in> carrier P" "q \<in> carrier P"
+  shows "p \<oplus>\<^sub>2 q = q \<oplus>\<^sub>2 p"
+  by (rule up_eqI, simp add: a_comm R, simp_all add: R)
+
+ML_setup {*
+Context.>> (fn thy => (simpset_ref_of thy :=
+  simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
+
+lemma (in UP_cring) UP_m_assoc:
+  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
+  shows "(p \<otimes>\<^sub>2 q) \<otimes>\<^sub>2 r = p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 r)"
+proof (rule up_eqI)
+  fix n
+  {
+    fix k and a b c :: "nat=>'a"
+    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
+      "c \<in> UNIV -> carrier R"
+    then have "k <= n ==>
+      finsum R (%j. finsum R (%i. a i \<otimes> b (j-i)) {..j} \<otimes> c (n-j)) {..k} =
+      finsum R (%j. a j \<otimes> finsum R (%i. b i \<otimes> c (n-j-i)) {..k-j}) {..k}"
+      (is "_ ==> ?eq k")
+    proof (induct k)
+      case 0 then show ?case by (simp add: Pi_def m_assoc)
+    next
+      case (Suc k)
+      then have "k <= n" by arith
+      then have "?eq k" by (rule Suc)
+      with R show ?case
+	by (simp cong: finsum_cong
+             add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
+          (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
+    qed
+  }
+  with R show "coeff P ((p \<otimes>\<^sub>2 q) \<otimes>\<^sub>2 r) n = coeff P (p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 r)) n"
+    by (simp add: Pi_def)
+qed (simp_all add: R)
+
+ML_setup {*
+Context.>> (fn thy => (simpset_ref_of thy :=
+  simpset_of thy setsubgoaler asm_simp_tac; thy)) *}
+
+lemma (in UP_cring) UP_l_one [simp]:
+  assumes R: "p \<in> carrier P"
+  shows "\<one>\<^sub>2 \<otimes>\<^sub>2 p = p"
+proof (rule up_eqI)
+  fix n
+  show "coeff P (\<one>\<^sub>2 \<otimes>\<^sub>2 p) n = coeff P p n"
+  proof (cases n)
+    case 0 with R show ?thesis by simp
+  next
+    case Suc with R show ?thesis
+      by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
+  qed
+qed (simp_all add: R)
+
+lemma (in UP_cring) UP_l_distr:
+  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
+  shows "(p \<oplus>\<^sub>2 q) \<otimes>\<^sub>2 r = (p \<otimes>\<^sub>2 r) \<oplus>\<^sub>2 (q \<otimes>\<^sub>2 r)"
+  by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
+
+lemma (in UP_cring) UP_m_comm:
+  assumes R: "p \<in> carrier P" "q \<in> carrier P"
+  shows "p \<otimes>\<^sub>2 q = q \<otimes>\<^sub>2 p"
+proof (rule up_eqI)
+  fix n
+  {
+    fix k and a b :: "nat=>'a"
+    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
+    then have "k <= n ==>
+      finsum R (%i. a i \<otimes> b (n-i)) {..k} =
+      finsum R (%i. a (k-i) \<otimes> b (i+n-k)) {..k}"
+      (is "_ ==> ?eq k")
+    proof (induct k)
+      case 0 then show ?case by (simp add: Pi_def)
+    next
+      case (Suc k) then show ?case
+	by (subst finsum_Suc2) (simp add: Pi_def a_comm)+
+    qed
+  }
+  note l = this
+  from R show "coeff P (p \<otimes>\<^sub>2 q) n =  coeff P (q \<otimes>\<^sub>2 p) n"
+    apply (simp add: Pi_def)
+    apply (subst l)
+    apply (auto simp add: Pi_def)
+    apply (simp add: m_comm)
+    done
+qed (simp_all add: R)
+
+theorem (in UP_cring) UP_cring:
+  "cring P"
+  by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
+    UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
+
+lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
+  "p \<in> carrier P ==> \<ominus>\<^sub>2 p \<in> carrier P"
+  by (rule abelian_group.a_inv_closed
+    [OF cring.is_abelian_group [OF UP_cring]])
+
+lemma (in UP_cring) coeff_a_inv [simp]:
+  assumes R: "p \<in> carrier P"
+  shows "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> (coeff P p n)"
+proof -
+  from R coeff_closed UP_a_inv_closed have
+    "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^sub>2 p) n)"
+    by algebra
+  also from R have "... =  \<ominus> (coeff P p n)"
+      abelian_group.r_neg [OF cring.is_abelian_group [OF UP_cring]])
+  finally show ?thesis .
+qed
+
+text {*
+  Instantiation of lemmas from @{term cring}.
+*}
+
+lemma (in UP_cring) UP_monoid:
+  "monoid P"
+  by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
+    UP_cring)
+(* TODO: provide cring.is_monoid *)
+
+lemma (in UP_cring) UP_comm_semigroup:
+  "comm_semigroup P"
+  by (fast intro!: cring.is_comm_monoid comm_monoid.axioms comm_semigroup.intro
+    UP_cring)
+
+lemma (in UP_cring) UP_comm_monoid:
+  "comm_monoid P"
+  by (fast intro!: cring.is_comm_monoid UP_cring)
+
+lemma (in UP_cring) UP_abelian_monoid:
+  "abelian_monoid P"
+  by (fast intro!: abelian_group.axioms cring.is_abelian_group UP_cring)
+
+lemma (in UP_cring) UP_abelian_group:
+  "abelian_group P"
+  by (fast intro!: cring.is_abelian_group UP_cring)
+
+lemmas (in UP_cring) UP_r_one [simp] =
+  monoid.r_one [OF UP_monoid]
+
+lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
+  monoid.nat_pow_closed [OF UP_monoid]
+
+lemmas (in UP_cring) UP_nat_pow_0 [simp] =
+  monoid.nat_pow_0 [OF UP_monoid]
+
+lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
+  monoid.nat_pow_Suc [OF UP_monoid]
+
+lemmas (in UP_cring) UP_nat_pow_one [simp] =
+  monoid.nat_pow_one [OF UP_monoid]
+
+lemmas (in UP_cring) UP_nat_pow_mult =
+  monoid.nat_pow_mult [OF UP_monoid]
+
+lemmas (in UP_cring) UP_nat_pow_pow =
+  monoid.nat_pow_pow [OF UP_monoid]
+
+lemmas (in UP_cring) UP_m_lcomm =
+  comm_semigroup.m_lcomm [OF UP_comm_semigroup]
+
+lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
+
+lemmas (in UP_cring) UP_nat_pow_distr =
+  comm_monoid.nat_pow_distr [OF UP_comm_monoid]
+
+lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_r_zero [simp] =
+  abelian_monoid.r_zero [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
+
+lemmas (in UP_cring) UP_finsum_empty [simp] =
+  abelian_monoid.finsum_empty [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_finsum_insert [simp] =
+  abelian_monoid.finsum_insert [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_finsum_zero [simp] =
+  abelian_monoid.finsum_zero [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_finsum_closed [simp] =
+  abelian_monoid.finsum_closed [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_finsum_Un_Int =
+  abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_finsum_Un_disjoint =
+  abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_finsum_addf =
+  abelian_monoid.finsum_addf [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_finsum_cong' =
+  abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_finsum_0 [simp] =
+  abelian_monoid.finsum_0 [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_finsum_Suc [simp] =
+  abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_finsum_Suc2 =
+  abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_finsum_add [simp] =
+  abelian_monoid.finsum_add [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_finsum_cong =
+  abelian_monoid.finsum_cong [OF UP_abelian_monoid]
+
+lemmas (in UP_cring) UP_minus_closed [intro, simp] =
+  abelian_group.minus_closed [OF UP_abelian_group]
+
+lemmas (in UP_cring) UP_a_l_cancel [simp] =
+  abelian_group.a_l_cancel [OF UP_abelian_group]
+
+lemmas (in UP_cring) UP_a_r_cancel [simp] =
+  abelian_group.a_r_cancel [OF UP_abelian_group]
+
+lemmas (in UP_cring) UP_l_neg =
+  abelian_group.l_neg [OF UP_abelian_group]
+
+lemmas (in UP_cring) UP_r_neg =
+  abelian_group.r_neg [OF UP_abelian_group]
+
+lemmas (in UP_cring) UP_minus_zero [simp] =
+  abelian_group.minus_zero [OF UP_abelian_group]
+
+lemmas (in UP_cring) UP_minus_minus [simp] =
+  abelian_group.minus_minus [OF UP_abelian_group]
+
+lemmas (in UP_cring) UP_minus_add =
+  abelian_group.minus_add [OF UP_abelian_group]
+
+lemmas (in UP_cring) UP_r_neg2 =
+  abelian_group.r_neg2 [OF UP_abelian_group]
+
+lemmas (in UP_cring) UP_r_neg1 =
+  abelian_group.r_neg1 [OF UP_abelian_group]
+
+lemmas (in UP_cring) UP_r_distr =
+  cring.r_distr [OF UP_cring]
+
+lemmas (in UP_cring) UP_l_null [simp] =
+  cring.l_null [OF UP_cring]
+
+lemmas (in UP_cring) UP_r_null [simp] =
+  cring.r_null [OF UP_cring]
+
+lemmas (in UP_cring) UP_l_minus =
+  cring.l_minus [OF UP_cring]
+
+lemmas (in UP_cring) UP_r_minus =
+  cring.r_minus [OF UP_cring]
+
+lemmas (in UP_cring) UP_finsum_ldistr =
+  cring.finsum_ldistr [OF UP_cring]
+
+lemmas (in UP_cring) UP_finsum_rdistr =
+  cring.finsum_rdistr [OF UP_cring]
+
+subsection {* Polynomials form an Algebra *}
+
+lemma (in UP_cring) UP_smult_l_distr:
+  "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
+  (a \<oplus> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 b \<odot>\<^sub>2 p"
+  by (rule up_eqI) (simp_all add: R.l_distr)
+
+lemma (in UP_cring) UP_smult_r_distr:
+  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
+  a \<odot>\<^sub>2 (p \<oplus>\<^sub>2 q) = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 a \<odot>\<^sub>2 q"
+  by (rule up_eqI) (simp_all add: R.r_distr)
+
+lemma (in UP_cring) UP_smult_assoc1:
+      "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
+      (a \<otimes> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 p)"
+  by (rule up_eqI) (simp_all add: R.m_assoc)
+
+lemma (in UP_cring) UP_smult_one [simp]:
+      "p \<in> carrier P ==> \<one> \<odot>\<^sub>2 p = p"
+  by (rule up_eqI) simp_all
+
+lemma (in UP_cring) UP_smult_assoc2:
+  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
+  (a \<odot>\<^sub>2 p) \<otimes>\<^sub>2 q = a \<odot>\<^sub>2 (p \<otimes>\<^sub>2 q)"
+  by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
+
+text {*
+  Instantiation of lemmas from @{term algebra}.
+*}
+
+(* TODO: move to CRing.thy, really a fact missing from the locales package *)
+
+lemma (in cring) cring:
+  "cring R"
+  by (fast intro: cring.intro prems)
+
+lemma (in UP_cring) UP_algebra:
+  "algebra R P"
+  by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
+    UP_smult_assoc1 UP_smult_assoc2)
+
+lemmas (in UP_cring) UP_smult_l_null [simp] =
+  algebra.smult_l_null [OF UP_algebra]
+
+lemmas (in UP_cring) UP_smult_r_null [simp] =
+  algebra.smult_r_null [OF UP_algebra]
+
+lemmas (in UP_cring) UP_smult_l_minus =
+  algebra.smult_l_minus [OF UP_algebra]
+
+lemmas (in UP_cring) UP_smult_r_minus =
+  algebra.smult_r_minus [OF UP_algebra]
+
+subsection {* Further Lemmas Involving @{term monom} *}
+
+lemma (in UP_cring) monom_zero [simp]:
+  "monom P \<zero> n = \<zero>\<^sub>2"
+  by (simp add: UP_def P_def)
+
+ML_setup {*
+Context.>> (fn thy => (simpset_ref_of thy :=
+  simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
+
+lemma (in UP_cring) monom_mult_is_smult:
+  assumes R: "a \<in> carrier R" "p \<in> carrier P"
+  shows "monom P a 0 \<otimes>\<^sub>2 p = a \<odot>\<^sub>2 p"
+proof (rule up_eqI)
+  fix n
+  have "coeff P (p \<otimes>\<^sub>2 monom P a 0) n = coeff P (a \<odot>\<^sub>2 p) n"
+  proof (cases n)
+    case 0 with R show ?thesis by (simp add: R.m_comm)
+  next
+    case Suc with R show ?thesis
+      by (simp cong: finsum_cong add: R.r_null Pi_def)
+        (simp add: m_comm)
+  qed
+  with R show "coeff P (monom P a 0 \<otimes>\<^sub>2 p) n = coeff P (a \<odot>\<^sub>2 p) n"
+    by (simp add: UP_m_comm)
+qed (simp_all add: R)
+
+ML_setup {*
+Context.>> (fn thy => (simpset_ref_of thy :=
+  simpset_of thy setsubgoaler asm_simp_tac; thy)) *}
+
+lemma (in UP_cring) monom_add [simp]:
+  "[| a \<in> carrier R; b \<in> carrier R |] ==>
+  monom P (a \<oplus> b) n = monom P a n \<oplus>\<^sub>2 monom P b n"
+  by (rule up_eqI) simp_all
+
+ML_setup {*
+Context.>> (fn thy => (simpset_ref_of thy :=
+  simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
+
+lemma (in UP_cring) monom_one_Suc:
+  "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1"
+proof (rule up_eqI)
+  fix k
+  show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
+  proof (cases "k = Suc n")
+    case True show ?thesis
+    proof -
+      from True have less_add_diff:
+	"!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
+      from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
+      also from True
+      have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
+	coeff P (monom P \<one> 1) (k - i)) ({..n(} Un {n})"
+	by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
+      also have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
+	coeff P (monom P \<one> 1) (k - i)) {..n}"
+	by (simp only: ivl_disj_un_singleton)
+      also from True have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
+	coeff P (monom P \<one> 1) (k - i)) ({..n} Un {)n..k})"
+	by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
+	  order_less_imp_not_eq Pi_def)
+      also from True have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
+	by (simp add: ivl_disj_un_one)
+      finally show ?thesis .
+    qed
+  next
+    case False
+    note neq = False
+    let ?s =
+      "(\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>))"
+    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
+    also have "... = finsum R ?s {..k}"
+    proof -
+      have f1: "finsum R ?s {..n(} = \<zero>" by (simp cong: finsum_cong add: Pi_def)
+      from neq have f2: "finsum R ?s {n} = \<zero>"
+	by (simp cong: finsum_cong add: Pi_def) arith
+      have f3: "n < k ==> finsum R ?s {)n..k} = \<zero>"
+	by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
+      show ?thesis
+      proof (cases "k < n")
+	case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
+      next
+	case False then have n_le_k: "n <= k" by arith
+	show ?thesis
+	proof (cases "n = k")
+	  case True
+	  then have "\<zero> = finsum R ?s ({..n(} \<union> {n})"
+	    by (simp cong: finsum_cong add: finsum_Un_disjoint
+	      ivl_disj_int_singleton Pi_def)
+	  also from True have "... = finsum R ?s {..k}"
+	    by (simp only: ivl_disj_un_singleton)
+	  finally show ?thesis .
+	next
+	  case False with n_le_k have n_less_k: "n < k" by arith
+	  with neq have "\<zero> = finsum R ?s ({..n(} \<union> {n})"
+	    by (simp add: finsum_Un_disjoint f1 f2
+	      ivl_disj_int_singleton Pi_def del: Un_insert_right)
+	  also have "... = finsum R ?s {..n}"
+	    by (simp only: ivl_disj_un_singleton)
+	  also from n_less_k neq have "... = finsum R ?s ({..n} \<union> {)n..k})"
+	    by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
+	  also from n_less_k have "... = finsum R ?s {..k}"
+	    by (simp only: ivl_disj_un_one)
+	  finally show ?thesis .
+	qed
+      qed
+    qed
+    also have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k" by simp
+    finally show ?thesis .
+  qed
+qed (simp_all)
+
+ML_setup {*
+Context.>> (fn thy => (simpset_ref_of thy :=
+  simpset_of thy setsubgoaler asm_simp_tac; thy)) *}
+
+lemma (in UP_cring) monom_mult_smult:
+  "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^sub>2 monom P b n"
+  by (rule up_eqI) simp_all
+
+lemma (in UP_cring) monom_one [simp]:
+  "monom P \<one> 0 = \<one>\<^sub>2"
+  by (rule up_eqI) simp_all
+
+lemma (in UP_cring) monom_one_mult:
+  "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m"
+proof (induct n)
+  case 0 show ?case by simp
+next
+  case Suc then show ?case
+    by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
+qed
+
+lemma (in UP_cring) monom_mult [simp]:
+  assumes R: "a \<in> carrier R" "b \<in> carrier R"
+  shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^sub>2 monom P b m"
+proof -
+  from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
+  also from R have "... = a \<otimes> b \<odot>\<^sub>2 monom P \<one> (n + m)"
+    by (simp add: monom_mult_smult del: r_one)
+  also have "... = a \<otimes> b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m)"
+    by (simp only: monom_one_mult)
+  also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m))"
+    by (simp add: UP_smult_assoc1)
+  also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> m \<otimes>\<^sub>2 monom P \<one> n))"
+    by (simp add: UP_m_comm)
+  also from R have "... = a \<odot>\<^sub>2 ((b \<odot>\<^sub>2 monom P \<one> m) \<otimes>\<^sub>2 monom P \<one> n)"
+    by (simp add: UP_smult_assoc2)
+  also from R have "... = a \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m))"
+    by (simp add: UP_m_comm)
+  also from R have "... = (a \<odot>\<^sub>2 monom P \<one> n) \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m)"
+    by (simp add: UP_smult_assoc2)
+  also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^sub>2 monom P (b \<otimes> \<one>) m"
+    by (simp add: monom_mult_smult del: r_one)
+  also from R have "... = monom P a n \<otimes>\<^sub>2 monom P b m" by simp
+  finally show ?thesis .
+qed
+
+lemma (in UP_cring) monom_a_inv [simp]:
+  "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^sub>2 monom P a n"
+  by (rule up_eqI) simp_all
+
+lemma (in UP_cring) monom_inj:
+  "inj_on (%a. monom P a n) (carrier R)"
+proof (rule inj_onI)
+  fix x y
+  assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
+  then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
+  with R show "x = y" by simp
+qed
+
+subsection {* The Degree Function *}
+
+constdefs
+  deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
+  "deg R p == LEAST n. bound (zero R) n (coeff (UP R) p)"
+
+lemma (in UP_cring) deg_aboveI:
+  "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
+  by (unfold deg_def P_def) (fast intro: Least_le)
+(*
+lemma coeff_bound_ex: "EX n. bound n (coeff p)"
+proof -
+  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
+  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
+  then show ?thesis ..
+qed
+
+lemma bound_coeff_obtain:
+  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
+proof -
+  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
+  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
+  with prem show P .
+qed
+*)
+lemma (in UP_cring) deg_aboveD:
+  "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
+proof -
+  assume R: "p \<in> carrier P" and "deg R p < m"
+  from R obtain n where "bound \<zero> n (coeff P p)"
+    by (auto simp add: UP_def P_def)
+  then have "bound \<zero> (deg R p) (coeff P p)"
+    by (auto simp: deg_def P_def dest: LeastI)
+  then show ?thesis by (rule boundD)
+qed
+
+lemma (in UP_cring) deg_belowI:
+  assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
+    and R: "p \<in> carrier P"
+  shows "n <= deg R p"
+-- {* Logically, this is a slightly stronger version of
+  @{thm [source] deg_aboveD} *}
+proof (cases "n=0")
+  case True then show ?thesis by simp
+next
+  case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
+  then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
+  then show ?thesis by arith
+qed
+
+lemma (in UP_cring) lcoeff_nonzero_deg:
+  assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
+  shows "coeff P p (deg R p) ~= \<zero>"
+proof -
+  from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
+  proof -
+    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
+      by arith
+(* TODO: why does proof not work with "1" *)
+    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
+      by (unfold deg_def P_def) arith
+    then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
+    then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
+      by (unfold bound_def) fast
+    then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
+    then show ?thesis by auto
+  qed
+  with deg_belowI R have "deg R p = m" by fastsimp
+  with m_coeff show ?thesis by simp
+qed
+
+lemma (in UP_cring) lcoeff_nonzero_nonzero:
+  assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
+  shows "coeff P p 0 ~= \<zero>"
+proof -
+  have "EX m. coeff P p m ~= \<zero>"
+  proof (rule classical)
+    assume "~ ?thesis"
+    with R have "p = \<zero>\<^sub>2" by (auto intro: up_eqI)
+    with nonzero show ?thesis by contradiction
+  qed
+  then obtain m where coeff: "coeff P p m ~= \<zero>" ..
+  then have "m <= deg R p" by (rule deg_belowI)
+  then have "m = 0" by (simp add: deg)
+  with coeff show ?thesis by simp
+qed
+
+lemma (in UP_cring) lcoeff_nonzero:
+  assumes neq: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
+  shows "coeff P p (deg R p) ~= \<zero>"
+proof (cases "deg R p = 0")
+  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
+next
+  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
+qed
+
+lemma (in UP_cring) deg_eqI:
+  "[| !!m. n < m ==> coeff P p m = \<zero>;
+      !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
+by (fast intro: le_anti_sym deg_aboveI deg_belowI)
+
+(* Degree and polynomial operations *)
+
+lemma (in UP_cring) deg_add [simp]:
+  assumes R: "p \<in> carrier P" "q \<in> carrier P"
+  shows "deg R (p \<oplus>\<^sub>2 q) <= max (deg R p) (deg R q)"
+proof (cases "deg R p <= deg R q")
+  case True show ?thesis
+    by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
+next
+  case False show ?thesis
+    by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
+qed
+
+lemma (in UP_cring) deg_monom_le:
+  "a \<in> carrier R ==> deg R (monom P a n) <= n"
+  by (intro deg_aboveI) simp_all
+
+lemma (in UP_cring) deg_monom [simp]:
+  "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
+  by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
+
+lemma (in UP_cring) deg_const [simp]:
+  assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
+proof (rule le_anti_sym)
+  show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
+next
+  show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
+qed
+
+lemma (in UP_cring) deg_zero [simp]:
+  "deg R \<zero>\<^sub>2 = 0"
+proof (rule le_anti_sym)
+  show "deg R \<zero>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
+next
+  show "0 <= deg R \<zero>\<^sub>2" by (rule deg_belowI) simp_all
+qed
+
+lemma (in UP_cring) deg_one [simp]:
+  "deg R \<one>\<^sub>2 = 0"
+proof (rule le_anti_sym)
+  show "deg R \<one>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
+next
+  show "0 <= deg R \<one>\<^sub>2" by (rule deg_belowI) simp_all
+qed
+
+lemma (in UP_cring) deg_uminus [simp]:
+  assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^sub>2 p) = deg R p"
+proof (rule le_anti_sym)
+  show "deg R (\<ominus>\<^sub>2 p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
+next
+  show "deg R p <= deg R (\<ominus>\<^sub>2 p)"
+    by (simp add: deg_belowI lcoeff_nonzero_deg
+      inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
+qed
+
+lemma (in UP_domain) deg_smult_ring:
+  "[| a \<in> carrier R; p \<in> carrier P |] ==>
+  deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
+  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
+
+lemma (in UP_domain) deg_smult [simp]:
+  assumes R: "a \<in> carrier R" "p \<in> carrier P"
+  shows "deg R (a \<odot>\<^sub>2 p) = (if a = \<zero> then 0 else deg R p)"
+proof (rule le_anti_sym)
+  show "deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
+    by (rule deg_smult_ring)
+next
+  show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^sub>2 p)"
+  proof (cases "a = \<zero>")
+  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
+qed
+
+lemma (in UP_cring) deg_mult_cring:
+  assumes R: "p \<in> carrier P" "q \<in> carrier P"
+  shows "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q"
+proof (rule deg_aboveI)
+  fix m
+  assume boundm: "deg R p + deg R q < m"
+  {
+    fix k i
+    assume boundk: "deg R p + deg R q < k"
+    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
+    proof (cases "deg R p < i")
+      case True then show ?thesis by (simp add: deg_aboveD R)
+    next
+      case False with boundk have "deg R q < k - i" by arith
+      then show ?thesis by (simp add: deg_aboveD R)
+    qed
+  }
+  with boundm R show "coeff P (p \<otimes>\<^sub>2 q) m = \<zero>" by simp
+qed (simp add: R)
+
+ML_setup {*
+Context.>> (fn thy => (simpset_ref_of thy :=
+  simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
+
+lemma (in UP_domain) deg_mult [simp]:
+  "[| p ~= \<zero>\<^sub>2; q ~= \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==>
+  deg R (p \<otimes>\<^sub>2 q) = deg R p + deg R q"
+proof (rule le_anti_sym)
+  assume "p \<in> carrier P" " q \<in> carrier P"
+  show "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q" by (rule deg_mult_cring)
+next
+  let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
+  assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^sub>2" "q ~= \<zero>\<^sub>2"
+  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
+  show "deg R p + deg R q <= deg R (p \<otimes>\<^sub>2 q)"
+  proof (rule deg_belowI, simp add: R)
+    have "finsum R ?s {.. deg R p + deg R q}
+      = finsum R ?s ({.. deg R p(} Un {deg R p .. deg R p + deg R q})"
+      by (simp only: ivl_disj_un_one)
+    also have "... = finsum R ?s {deg R p .. deg R p + deg R q}"
+      by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
+        deg_aboveD less_add_diff R Pi_def)
+    also have "...= finsum R ?s ({deg R p} Un {)deg R p .. deg R p + deg R q})"
+      by (simp only: ivl_disj_un_singleton)
+    also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
+      by (simp cong: finsum_cong add: finsum_Un_disjoint
+	ivl_disj_int_singleton deg_aboveD R Pi_def)
+    finally have "finsum R ?s {.. deg R p + deg R q}
+      = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
+    with nz show "finsum R ?s {.. deg R p + deg R q} ~= \<zero>"
+      by (simp add: integral_iff lcoeff_nonzero R)
+    qed (simp add: R)
+  qed
+
+lemma (in UP_cring) coeff_finsum:
+  assumes fin: "finite A"
+  shows "p \<in> A -> carrier P ==>
+    coeff P (finsum P p A) k == finsum R (%i. coeff P (p i) k) A"
+  using fin by induct (auto simp: Pi_def)
+
+ML_setup {*
+Context.>> (fn thy => (simpset_ref_of thy :=
+  simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
+
+lemma (in UP_cring) up_repr:
+  assumes R: "p \<in> carrier P"
+  shows "finsum P (%i. monom P (coeff P p i) i) {..deg R p} = p"
+proof (rule up_eqI)
+  let ?s = "(%i. monom P (coeff P p i) i)"
+  fix k
+  from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
+    by simp
+  show "coeff P (finsum P ?s {..deg R p}) k = coeff P p k"
+  proof (cases "k <= deg R p")
+    case True
+    hence "coeff P (finsum P ?s {..deg R p}) k =
+          coeff P (finsum P ?s ({..k} Un {)k..deg R p})) k"
+      by (simp only: ivl_disj_un_one)
+    also from True
+    have "... = coeff P (finsum P ?s {..k}) k"
+      by (simp cong: finsum_cong add: finsum_Un_disjoint
+	ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
+    also
+    have "... = coeff P (finsum P ?s ({..k(} Un {k})) k"
+      by (simp only: ivl_disj_un_singleton)
+    also have "... = coeff P p k"
+      by (simp cong: finsum_cong add: setsum_Un_disjoint
+	ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
+    finally show ?thesis .
+  next
+    case False
+    hence "coeff P (finsum P ?s {..deg R p}) k =
+          coeff P (finsum P ?s ({..deg R p(} Un {deg R p})) k"
+      by (simp only: ivl_disj_un_singleton)
+    also from False have "... = coeff P p k"
+      by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
+        coeff_finsum deg_aboveD R Pi_def)
+    finally show ?thesis .
+  qed
+qed (simp_all add: R Pi_def)
+
+lemma (in UP_cring) up_repr_le:
+  "[| deg R p <= n; p \<in> carrier P |] ==>
+  finsum P (%i. monom P (coeff P p i) i) {..n} = p"
+proof -
+  let ?s = "(%i. monom P (coeff P p i) i)"
+  assume R: "p \<in> carrier P" and "deg R p <= n"
+  then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} Un {)deg R p..n})"
+    by (simp only: ivl_disj_un_one)
+  also have "... = finsum P ?s {..deg R p}"
+    by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
+      deg_aboveD R Pi_def)
+  also have "... = p" by (rule up_repr)
+  finally show ?thesis .
+qed
+
+ML_setup {*
+Context.>> (fn thy => (simpset_ref_of thy :=
+  simpset_of thy setsubgoaler asm_simp_tac; thy)) *}
+
+subsection {* Polynomial over an Integral Domain are an Integral Domain *}
+
+lemma domainI:
+  assumes cring: "cring R"
+    and one_not_zero: "one R ~= zero R"
+    and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
+      b \<in> carrier R |] ==> a = zero R | b = zero R"
+  shows "domain R"
+  by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
+    del: disjCI)
+
+lemma (in UP_domain) UP_one_not_zero:
+  "\<one>\<^sub>2 ~= \<zero>\<^sub>2"
+proof
+  assume "\<one>\<^sub>2 = \<zero>\<^sub>2"
+  hence "coeff P \<one>\<^sub>2 0 = (coeff P \<zero>\<^sub>2 0)" by simp
+  hence "\<one> = \<zero>" by simp
+  with one_not_zero show "False" by contradiction
+qed
+
+lemma (in UP_domain) UP_integral:
+  "[| p \<otimes>\<^sub>2 q = \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
+proof -
+  fix p q
+  assume pq: "p \<otimes>\<^sub>2 q = \<zero>\<^sub>2" and R: "p \<in> carrier P" "q \<in> carrier P"
+  show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
+  proof (rule classical)
+    assume c: "~ (p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2)"
+    with R have "deg R p + deg R q = deg R (p \<otimes>\<^sub>2 q)" by simp
+    also from pq have "... = 0" by simp
+    finally have "deg R p + deg R q = 0" .
+    then have f1: "deg R p = 0 & deg R q = 0" by simp
+    from f1 R have "p = finsum P (%i. (monom P (coeff P p i) i)) {..0}"
+      by (simp only: up_repr_le)
+    also from R have "... = monom P (coeff P p 0) 0" by simp
+    finally have p: "p = monom P (coeff P p 0) 0" .
+    from f1 R have "q = finsum P (%i. (monom P (coeff P q i) i)) {..0}"
+      by (simp only: up_repr_le)
+    also from R have "... = monom P (coeff P q 0) 0" by simp
+    finally have q: "q = monom P (coeff P q 0) 0" .
+    from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^sub>2 q) 0" by simp
+    also from pq have "... = \<zero>" by simp
+    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
+    with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
+      by (simp add: R.integral_iff)
+    with p q show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2" by fastsimp
+  qed
+qed
+
+theorem (in UP_domain) UP_domain:
+  "domain P"
+  by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
+
+text {*
+  Instantiation of results from @{term domain}.
+*}
+
+lemmas (in UP_domain) UP_zero_not_one [simp] =
+  domain.zero_not_one [OF UP_domain]
+
+lemmas (in UP_domain) UP_integral_iff =
+  domain.integral_iff [OF UP_domain]
+
+lemmas (in UP_domain) UP_m_lcancel =
+  domain.m_lcancel [OF UP_domain]
+
+lemmas (in UP_domain) UP_m_rcancel =
+  domain.m_rcancel [OF UP_domain]
+
+lemma (in UP_domain) smult_integral:
+  "[| a \<odot>\<^sub>2 p = \<zero>\<^sub>2; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^sub>2"
+  by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
+    inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
+
+subsection {* Evaluation Homomorphism *}
+
+ML_setup {*
+Context.>> (fn thy => (simpset_ref_of thy :=
+  simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
+
+(* alternative congruence rule (more efficient)
+lemma (in abelian_monoid) finsum_cong2:
+  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
+  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
+  sorry
+*)
+
+theorem (in cring) diagonal_sum:
+  "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
+  finsum R (%k. finsum R (%i. f i \<otimes> g (k - i)) {..k}) {..n + m} =
+  finsum R (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n + m}"
+proof -
+  assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
+  {
+    fix j
+    have "j <= n + m ==>
+      finsum R (%k. finsum R (%i. f i \<otimes> g (k - i)) {..k}) {..j} =
+      finsum R (%k. finsum R (%i. f k \<otimes> g i) {..j - k}) {..j}"
+    proof (induct j)
+      case 0 from Rf Rg show ?case by (simp add: Pi_def)
+    next
+      case (Suc j)
+      (* The following could be simplified if there was a reasoner for
+	total orders integrated with simip. *)
+      have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
+	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
+      have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
+	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
+      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
+	using Suc by (auto intro!: funcset_mem [OF Rf])
+      have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
+	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
+      have R11: "g 0 \<in> carrier R"
+	using Suc by (auto intro!: funcset_mem [OF Rg])
+      from Suc show ?case
+	by (simp cong: finsum_cong add: Suc_diff_le a_ac
+	  Pi_def R6 R8 R9 R10 R11)
+    qed
+  }
+  then show ?thesis by fast
+qed
+
+lemma (in abelian_monoid) boundD_carrier:
+  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
+  by auto
+
+theorem (in cring) cauchy_product:
+  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
+    and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
+  shows "finsum R (%k. finsum R (%i. f i \<otimes> g (k-i)) {..k}) {..n + m} =
+    finsum R f {..n} \<otimes> finsum R g {..m}"
+(* State revese direction? *)
+proof -
+  have f: "!!x. f x \<in> carrier R"
+  proof -
+    fix x
+    show "f x \<in> carrier R"
+      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
+  qed
+  have g: "!!x. g x \<in> carrier R"
+  proof -
+    fix x
+    show "g x \<in> carrier R"
+      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
+  qed
+  from f g have "finsum R (%k. finsum R (%i. f i \<otimes> g (k-i)) {..k}) {..n + m} =
+    finsum R (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n + m}"
+    by (simp add: diagonal_sum Pi_def)
+  also have "... = finsum R
+      (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) ({..n} Un {)n..n + m})"
+    by (simp only: ivl_disj_un_one)
+  also from f g have "... = finsum R
+      (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n}"
+    by (simp cong: finsum_cong
+      add: boundD [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
+  also from f g have "... = finsum R
+      (%k. finsum R (%i. f k \<otimes> g i) ({..m} Un {)m..n + m - k})) {..n}"
+    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
+  also from f g have "... = finsum R
+      (%k. finsum R (%i. f k \<otimes> g i) {..m}) {..n}"
+    by (simp cong: finsum_cong
+      add: boundD [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
+  also from f g have "... = finsum R f {..n} \<otimes> finsum R g {..m}"
+    by (simp add: finsum_ldistr diagonal_sum Pi_def,
+      simp cong: finsum_cong add: finsum_rdistr Pi_def)
+  finally show ?thesis .
+qed
+
+lemma (in UP_cring) const_ring_hom:
+  "(%a. monom P a 0) \<in> ring_hom R P"
+  by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
+
+constdefs
+  eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
+          'a => 'b, 'b, nat => 'a] => 'b"
+  "eval R S phi s == (\<lambda>p \<in> carrier (UP R).
+    finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p})"
+(*
+  "eval R S phi s p == if p \<in> carrier (UP R)
+  then finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p}
+  else arbitrary"
+*)
+
+locale ring_hom_UP_cring = ring_hom_cring R S + UP_cring R
+
+lemma (in ring_hom_UP_cring) eval_on_carrier:
+  "p \<in> carrier P ==>
+    eval R S phi s p =
+    finsum S (%i. mult S (phi (coeff P p i)) (pow S s i)) {..deg R p}"
+  by (unfold eval_def, fold P_def) simp
+
+lemma (in ring_hom_UP_cring) eval_extensional:
+  "eval R S phi s \<in> extensional (carrier P)"
+  by (unfold eval_def, fold P_def) simp
+
+theorem (in ring_hom_UP_cring) eval_ring_hom:
+  "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
+proof (rule ring_hom_memI)
+  fix p
+  assume RS: "p \<in> carrier P" "s \<in> carrier S"
+  then show "eval R S h s p \<in> carrier S"
+    by (simp only: eval_on_carrier) (simp add: Pi_def)
+next
+  fix p q
+  assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
+  then show "eval R S h s (p \<otimes>\<^sub>3 q) = eval R S h s p \<otimes>\<^sub>2 eval R S h s q"
+  proof (simp only: eval_on_carrier UP_mult_closed)
+    from RS have
+      "finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<otimes>\<^sub>3 q)} =
+      finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
+        ({..deg R (p \<otimes>\<^sub>3 q)} Un {)deg R (p \<otimes>\<^sub>3 q)..deg R p + deg R q})"
+      by (simp cong: finsum_cong
+	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
+	del: coeff_mult)
+    also from RS have "... =
+      finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p + deg R q}"
+      by (simp only: ivl_disj_un_one deg_mult_cring)
+    also from RS have "... =
+      finsum S (%i.
+        finsum S (%k.
+        (h (coeff P p k) \<otimes>\<^sub>2 h (coeff P q (i-k))) \<otimes>\<^sub>2 (s (^)\<^sub>2 k \<otimes>\<^sub>2 s (^)\<^sub>2 (i-k)))
+      {..i}) {..deg R p + deg R q}"
+      by (simp cong: finsum_cong add: nat_pow_mult Pi_def
+	S.m_ac S.finsum_rdistr)
+    also from RS have "... =
+      finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<otimes>\<^sub>2
+      finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
+      by (simp add: S.cauchy_product [THEN sym] boundI deg_aboveD S.m_ac
+	Pi_def)
+    finally show
+      "finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<otimes>\<^sub>3 q)} =
+      finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<otimes>\<^sub>2
+      finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}" .
+  qed
+next
+  fix p q
+  assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
+  then show "eval R S h s (p \<oplus>\<^sub>3 q) = eval R S h s p \<oplus>\<^sub>2 eval R S h s q"
+  proof (simp only: eval_on_carrier UP_a_closed)
+    from RS have
+      "finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<oplus>\<^sub>3 q)} =
+      finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
+        ({..deg R (p \<oplus>\<^sub>3 q)} Un {)deg R (p \<oplus>\<^sub>3 q)..max (deg R p) (deg R q)})"
+      by (simp cong: finsum_cong
+	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
+    also from RS have "... =
+      finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
+        {..max (deg R p) (deg R q)}"
+      by (simp add: ivl_disj_un_one)
+    also from RS have "... =
+      finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..max (deg R p) (deg R q)} \<oplus>\<^sub>2
+      finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..max (deg R p) (deg R q)}"
+      by (simp cong: finsum_cong
+	add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
+    also have "... =
+      finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
+        ({..deg R p} Un {)deg R p..max (deg R p) (deg R q)}) \<oplus>\<^sub>2
+      finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
+        ({..deg R q} Un {)deg R q..max (deg R p) (deg R q)})"
+      by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
+    also from RS have "... =
+      finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<oplus>\<^sub>2
+      finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
+      by (simp cong: finsum_cong
+	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
+    finally show
+      "finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<oplus>\<^sub>3 q)} =
+      finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<oplus>\<^sub>2
+      finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
+      .
+  qed
+next
+  assume S: "s \<in> carrier S"
+  then show "eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
+    by (simp only: eval_on_carrier UP_one_closed) simp
+qed
+
+text {* Instantiation of ring homomorphism lemmas. *}
+
+lemma (in ring_hom_UP_cring) ring_hom_cring_P_S:
+  "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
+  by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
+  intro: ring_hom_cring_axioms.intro eval_ring_hom)
+
+lemma (in ring_hom_UP_cring) UP_hom_closed [intro, simp]:
+  "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
+  by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
+
+lemma (in ring_hom_UP_cring) UP_hom_mult [simp]:
+  "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
+  eval R S h s (p \<otimes>\<^sub>3 q) = eval R S h s p \<otimes>\<^sub>2 eval R S h s q"
+  by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
+
+lemma (in ring_hom_UP_cring) UP_hom_add [simp]:
+  "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
+  eval R S h s (p \<oplus>\<^sub>3 q) = eval R S h s p \<oplus>\<^sub>2 eval R S h s q"
+  by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
+
+lemma (in ring_hom_UP_cring) UP_hom_one [simp]:
+  "s \<in> carrier S ==> eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
+  by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
+
+lemma (in ring_hom_UP_cring) UP_hom_zero [simp]:
+  "s \<in> carrier S ==> eval R S h s \<zero>\<^sub>3 = \<zero>\<^sub>2"
+  by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
+
+lemma (in ring_hom_UP_cring) UP_hom_a_inv [simp]:
+  "[| s \<in> carrier S; p \<in> carrier P |] ==>
+  (eval R S h s) (\<ominus>\<^sub>3 p) = \<ominus>\<^sub>2 (eval R S h s) p"
+  by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
+
+lemma (in ring_hom_UP_cring) UP_hom_finsum [simp]:
+  "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
+  (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
+  by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
+
+lemma (in ring_hom_UP_cring) UP_hom_finprod [simp]:
+  "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
+  (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
+  by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
+
+text {* Further properties of the evaluation homomorphism. *}
+
+(* The following lemma could be proved in UP\_cring with the additional
+   assumption that h is closed. *)
+
+lemma (in ring_hom_UP_cring) eval_const:
+  "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
+  by (simp only: eval_on_carrier monom_closed) simp
+
+text {* The following proof is complicated by the fact that in arbitrary
+  rings one might have @{term "one R = zero R"}. *}
+
+(* TODO: simplify by cases "one R = zero R" *)
+
+lemma (in ring_hom_UP_cring) eval_monom1:
+  "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
+proof (simp only: eval_on_carrier monom_closed R.one_closed)
+  assume S: "s \<in> carrier S"
+  then have "finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
+      {..deg R (monom P \<one> 1)} =
+    finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
+      ({..deg R (monom P \<one> 1)} Un {)deg R (monom P \<one> 1)..1})"
+    by (simp cong: finsum_cong del: coeff_monom
+      add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
+  also have "... =
+    finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..1}"
+    by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
+  also have "... = s"
+  proof (cases "s = \<zero>\<^sub>2")
+    case True then show ?thesis by (simp add: Pi_def)
+  next
+    case False with S show ?thesis by (simp add: Pi_def)
+  qed
+  finally show "finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
+      {..deg R (monom P \<one> 1)} = s" .
+qed
+
+lemma (in UP_cring) monom_pow:
+  assumes R: "a \<in> carrier R"
+  shows "(monom P a n) (^)\<^sub>2 m = monom P (a (^) m) (n * m)"
+proof (induct m)
+  case 0 from R show ?case by simp
+next
+  case Suc with R show ?case
+    by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
+qed
+
+lemma (in ring_hom_cring) hom_pow [simp]:
+  "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^sub>2 (n::nat)"
+  by (induct n) simp_all
+
+lemma (in ring_hom_UP_cring) UP_hom_pow [simp]:
+  "[| s \<in> carrier S; p \<in> carrier P |] ==>
+  (eval R S h s) (p (^)\<^sub>3 n) = eval R S h s p (^)\<^sub>2 (n::nat)"
+  by (rule ring_hom_cring.hom_pow [OF ring_hom_cring_P_S])
+
+lemma (in ring_hom_UP_cring) eval_monom:
+  "[| s \<in> carrier S; r \<in> carrier R |] ==>
+  eval R S h s (monom P r n) = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
+proof -
+  assume RS: "s \<in> carrier S" "r \<in> carrier R"
+  then have "eval R S h s (monom P r n) =
+    eval R S h s (monom P r 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 n)"
+    by (simp del: monom_mult UP_hom_mult UP_hom_pow
+      add: monom_mult [THEN sym] monom_pow)
+  also from RS eval_monom1 have "... = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
+    by (simp add: eval_const)
+  finally show ?thesis .
+qed
+
+lemma (in ring_hom_UP_cring) eval_smult:
+  "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
+  eval R S h s (r \<odot>\<^sub>3 p) = h r \<otimes>\<^sub>2 eval R S h s p"
+  by (simp add: monom_mult_is_smult [THEN sym] eval_const)
+
+lemma ring_hom_cringI:
+  assumes "cring R"
+    and "cring S"
+    and "h \<in> ring_hom R S"
+  shows "ring_hom_cring R S h"
+  by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
+    cring.axioms prems)
+
+lemma (in ring_hom_UP_cring) UP_hom_unique:
+  assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
+      "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
+    and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
+      "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
+    and RS: "s \<in> carrier S" "p \<in> carrier P"
+  shows "Phi p = Psi p"
+proof -
+  have Phi_hom: "ring_hom_cring P S Phi"
+    by (auto intro: ring_hom_cringI UP_cring S.cring Phi)
+  have Psi_hom: "ring_hom_cring P S Psi"
+    by (auto intro: ring_hom_cringI UP_cring S.cring Psi)
+thm monom_mult
+  have "Phi p = Phi (finsum P
+    (%i. monom P (coeff P p i) 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 i) {..deg R p})"
+    by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
+  also have "... = Psi (finsum P
+    (%i. monom P (coeff P p i) 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 i) {..deg R p})"
+    by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
+      ring_hom_cring.hom_mult [OF Phi_hom]
+      ring_hom_cring.hom_pow [OF Phi_hom] Phi
+      ring_hom_cring.hom_finsum [OF Psi_hom]
+      ring_hom_cring.hom_mult [OF Psi_hom]
+      ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
+  also have "... = Psi p"
+    by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
+  finally show ?thesis .
+qed
+
+
+theorem (in ring_hom_UP_cring) UP_universal_property:
+  "s \<in> carrier S ==>
+  EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
+    Phi (monom P \<one> 1) = s &
+    (ALL r : carrier R. Phi (monom P r 0) = h r)"
+  using eval_monom1
+  apply (auto intro: eval_ring_hom eval_const eval_extensional)
+  apply (rule extensionalityI)
+  apply (auto intro: UP_hom_unique)
+  done
+
+subsection {* Sample application of evaluation homomorphism *}
+
+lemma ring_hom_UP_cringI:
+  assumes "cring R"
+    and "cring S"
+    and "h \<in> ring_hom R S"
+  shows "ring_hom_UP_cring R S h"
+  by (fast intro: ring_hom_UP_cring.intro ring_hom_cring_axioms.intro
+    cring.axioms prems)
+
+lemma INTEG_id:
+  "ring_hom_UP_cring INTEG INTEG id"
+  by (fast intro: ring_hom_UP_cringI cring_INTEG id_ring_hom)
+
+text {*
+  An instantiation mechanism would now import all theorems and lemmas
+  valid in the context of homomorphisms between @{term INTEG} and @{term
+  "UP INTEG"}.  *}
+
+lemma INTEG_closed [intro, simp]:
+  "z \<in> carrier INTEG"
+  by (unfold INTEG_def) simp
+
+lemma INTEG_mult [simp]:
+  "mult INTEG z w = z * w"
+  by (unfold INTEG_def) simp
+
+lemma INTEG_pow [simp]:
+  "pow INTEG z n = z ^ n"
+  by (induct n) (simp_all add: INTEG_def nat_pow_def)
+
+lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
+  by (simp add: ring_hom_UP_cring.eval_monom [OF INTEG_id])
+
+-- {* Calculates @{term "x = 500"} *}
+
+
+end
\ No newline at end of file```