src/HOL/Algebra/UnivPoly.thy
 author wenzelm Fri, 16 Apr 2004 13:52:43 +0200 changeset 14590 276ef51cedbf parent 14577 dbb95b825244 child 14651 02b8f3bcf7fe permissions -rw-r--r--
simplified ML code for setsubgoaler;
```
(*
Title:     Univariate Polynomials
Id:        \$Id\$
Author:    Clemens Ballarin, started 9 December 1996
*)

theory UnivPoly = Module:

text {*
Polynomials are formalised as modules with additional operations for
extracting coefficients from polynomials and for obtaining monomials
from coefficients and exponents (record @{text "up_ring"}).
The carrier set is
a set of bounded functions from Nat to the coefficient domain.
Bounded means that these functions return zero above a certain bound
(the degree).  There is a chapter on the formalisation of polynomials
in my PhD thesis (http://www4.in.tum.de/\~{}ballarin/publications/),
which was implemented with axiomatic type classes.  This was later
ported to Locales.
*}

subsection {* The Constructor for Univariate Polynomials *}

(* 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"

lemma mem_upD [dest]:
"f \<in> up R ==> f n \<in> carrier R"

lemma (in cring) bound_upD [dest]:
"f \<in> up R ==> EX n. bound \<zero> n f"

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

"[| 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"
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>"
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>"

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"

"[| 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}"

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"

lemma (in UP_cring) UP_one_closed [simp]:
"\<one>\<^sub>2 \<in> carrier P"

lemma (in UP_cring) UP_zero_closed [intro, simp]:
"\<zero>\<^sub>2 \<in> carrier P"

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"

lemma (in UP) coeff_closed [simp]:
"p \<in> carrier P ==> coeff P p n \<in> carrier R"

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)"

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"

ML_setup {*
simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
*}

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"

ML_setup {*
simpset_ref() := simpset() setsubgoaler asm_simp_tac;
*}

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

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)"

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 (subst l)
done

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_ring:  (* preliminary *)
"ring P"
by (auto intro: ring.intro cring.axioms UP_cring)

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 ring.is_abelian_group [OF UP_ring]])

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 ring.is_abelian_group [OF UP_ring]])
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 ring.is_abelian_group UP_ring)

lemma (in UP_cring) UP_abelian_group:
"abelian_group P"
by (fast intro!: ring.is_abelian_group UP_ring)

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_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] =

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_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 =
ring.r_distr [OF UP_ring]

lemmas (in UP_cring) UP_l_null [simp] =
ring.l_null [OF UP_ring]

lemmas (in UP_cring) UP_r_null [simp] =
ring.r_null [OF UP_ring]

lemmas (in UP_cring) UP_l_minus =
ring.l_minus [OF UP_ring]

lemmas (in UP_cring) UP_r_minus =
ring.r_minus [OF UP_ring]

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 monomials *}

lemma (in UP_cring) monom_zero [simp]:
"monom P \<zero> n = \<zero>\<^sub>2"

ML_setup {*
simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
*}

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)
qed
with R show "coeff P (monom P a 0 \<otimes>\<^sub>2 p) n = coeff P (a \<odot>\<^sub>2 p) n"

ML_setup {*
simpset_ref() := simpset() setsubgoaler asm_simp_tac;
*}

"[| 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 {*
simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
*}

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 -
"!!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"
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 {*
simpset_ref() := simpset() setsubgoaler asm_simp_tac;
*}

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
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))"
also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> m \<otimes>\<^sub>2 monom P \<one> n))"
also from R have "... = a \<odot>\<^sub>2 ((b \<odot>\<^sub>2 monom P \<one> m) \<otimes>\<^sub>2 monom P \<one> n)"
also from R have "... = a \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m))"
also from R have "... = (a \<odot>\<^sub>2 monom P \<one> n) \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m)"
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 *)

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)"
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

ML_setup {*
simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
*}

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
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

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 {*
simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
*}

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

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 {*
simpset_ref() := simpset() setsubgoaler asm_simp_tac;
*}

subsection {* Polynomials over an integral domain form 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>"
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 and Universal Property*}

ML_setup {*
simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
*}

(* alternative congruence rule (possibly 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}"
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}"
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
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
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)}"
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
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])

"[| 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"

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
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
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
also from RS eval_monom1 have "... = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
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)
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)

constdefs
INTEG :: "int ring"
"INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"

lemma cring_INTEG:
"cring INTEG"
by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI

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
```