(* Title: HOL/Algebra/UnivPoly.thy
Author: Clemens Ballarin, started 9 December 1996
Copyright: Clemens Ballarin
Contributions, in particular on long division, by Jesus Aransay.
*)
theory UnivPoly
imports Module RingHom
begin
section {* Univariate Polynomials *}
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 the PhD thesis \cite{Ballarin:1999},
which was implemented with axiomatic type classes. This was later
ported to Locales.
*}
subsection {* The Constructor for Univariate Polynomials *}
text {*
Functions with finite support.
*}
locale bound =
fixes z :: 'a
and n :: nat
and f :: "nat => 'a"
assumes bound: "!!m. n < m \<Longrightarrow> f m = z"
declare bound.intro [intro!]
and bound.bound [dest]
lemma bound_below:
assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> 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"
definition
up :: "('a, 'm) ring_scheme => (nat => 'a) set"
where "up R = {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero>\<^bsub>R\<^esub> n f)}"
definition UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
where "UP R = (|
carrier = up R,
mult = (%p:up R. %q:up R. %n. \<Oplus>\<^bsub>R\<^esub>i \<in> {..n}. p i \<otimes>\<^bsub>R\<^esub> q (n-i)),
one = (%i. if i=0 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>),
zero = (%i. \<zero>\<^bsub>R\<^esub>),
add = (%p:up R. %q:up R. %i. p i \<oplus>\<^bsub>R\<^esub> q i),
smult = (%a:carrier R. %p:up R. %i. a \<otimes>\<^bsub>R\<^esub> p i),
monom = (%a:carrier R. %n i. if i=n then a else \<zero>\<^bsub>R\<^esub>),
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)
context ring
begin
lemma bound_upD [dest]: "f \<in> up R ==> EX n. bound \<zero> n f" by (simp add: up_def)
lemma up_one_closed: "(%n. if n = 0 then \<one> else \<zero>) \<in> up R" using up_def by force
lemma up_smult_closed: "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R" by force
lemma 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 fastforce
qed
then show ?thesis ..
qed
qed
lemma 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 up_minus_closed:
"[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<ominus> q i) \<in> up R"
using mem_upD [of p R] mem_upD [of q R] up_add_closed up_a_inv_closed a_minus_def [of _ R]
by auto
lemma up_mult_closed:
"[| p \<in> up R; q \<in> up R |] ==>
(%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"
proof
fix n
assume "p \<in> up R" "q \<in> up R"
then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<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. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-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> (n + m) (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"
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 "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"
by (simp add: Pi_def)
qed
then show ?thesis by fast
qed
qed
end
subsection {* Effect of Operations on Coefficients *}
locale UP =
fixes R (structure) and P (structure)
defines P_def: "P == UP R"
locale UP_ring = UP + R: ring R
locale UP_cring = UP + R: cring R
sublocale UP_cring < UP_ring
by intro_locales [1] (rule P_def)
locale UP_domain = UP + R: "domain" R
sublocale UP_domain < UP_cring
by intro_locales [1] (rule P_def)
context UP
begin
text {*Temporarily declare @{thm P_def} as simp rule.*}
declare P_def [simp]
lemma 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
lemma coeff_closed [simp]:
"p \<in> carrier P ==> coeff P p n \<in> carrier R" by (auto simp add: UP_def)
end
context UP_ring
begin
(* Theorems generalised from commutative rings to rings by Jesus Aransay. *)
lemma 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 coeff_zero [simp]: "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>" by (auto simp add: UP_def)
lemma coeff_one [simp]: "coeff P \<one>\<^bsub>P\<^esub> n = (if n=0 then \<one> else \<zero>)"
using up_one_closed by (simp add: UP_def)
lemma coeff_smult [simp]:
"[| a \<in> carrier R; p \<in> carrier P |] ==> coeff P (a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> coeff P p n"
by (simp add: UP_def up_smult_closed)
lemma coeff_add [simp]:
"[| p \<in> carrier P; q \<in> carrier P |] ==> coeff P (p \<oplus>\<^bsub>P\<^esub> q) n = coeff P p n \<oplus> coeff P q n"
by (simp add: UP_def up_add_closed)
lemma coeff_mult [simp]:
"[| p \<in> carrier P; q \<in> carrier P |] ==> coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> coeff P q (n-i))"
by (simp add: UP_def up_mult_closed)
end
subsection {* Polynomials Form a Ring. *}
context UP_ring
begin
text {* Operations are closed over @{term P}. *}
lemma UP_mult_closed [simp]:
"[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^bsub>P\<^esub> q \<in> carrier P" by (simp add: UP_def up_mult_closed)
lemma UP_one_closed [simp]:
"\<one>\<^bsub>P\<^esub> \<in> carrier P" by (simp add: UP_def up_one_closed)
lemma UP_zero_closed [intro, simp]:
"\<zero>\<^bsub>P\<^esub> \<in> carrier P" by (auto simp add: UP_def)
lemma UP_a_closed [intro, simp]:
"[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^bsub>P\<^esub> q \<in> carrier P" by (simp add: UP_def up_add_closed)
lemma 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 UP_smult_closed [simp]:
"[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^bsub>P\<^esub> p \<in> carrier P" by (simp add: UP_def up_smult_closed)
end
declare (in UP) P_def [simp del]
text {* Algebraic ring properties *}
context UP_ring
begin
lemma UP_a_assoc:
assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
shows "(p \<oplus>\<^bsub>P\<^esub> q) \<oplus>\<^bsub>P\<^esub> r = p \<oplus>\<^bsub>P\<^esub> (q \<oplus>\<^bsub>P\<^esub> r)" by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
lemma UP_l_zero [simp]:
assumes R: "p \<in> carrier P"
shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p" by (rule up_eqI, simp_all add: R)
lemma UP_l_neg_ex:
assumes R: "p \<in> carrier P"
shows "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
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>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
qed (rule closed)
qed
lemma UP_a_comm:
assumes R: "p \<in> carrier P" "q \<in> carrier P"
shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p" by (rule up_eqI, simp add: a_comm R, simp_all add: R)
lemma UP_m_assoc:
assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> 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 ==>
(\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
(\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
(is "_ \<Longrightarrow> ?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
from this R 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>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r) n = coeff P (p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)) n"
by (simp add: Pi_def)
qed (simp_all add: R)
lemma UP_r_one [simp]:
assumes R: "p \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub> = p"
proof (rule up_eqI)
fix n
show "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) n = coeff P p n"
proof (cases n)
case 0
{
with R show ?thesis by simp
}
next
case Suc
{
(*JE: in the locale UP_cring the proof was solved only with "by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)", but I did not get it to work here*)
fix nn assume Succ: "n = Suc nn"
have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = coeff P p (Suc nn)"
proof -
have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = (\<Oplus>i\<in>{..Suc nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" using R by simp
also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>) \<oplus> (\<Oplus>i\<in>{..nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))"
using finsum_Suc [of "(\<lambda>i::nat. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" "nn"] unfolding Pi_def using R by simp
also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>)"
proof -
have "(\<Oplus>i\<in>{..nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>)) = (\<Oplus>i\<in>{..nn}. \<zero>)"
using finsum_cong [of "{..nn}" "{..nn}" "(\<lambda>i::nat. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" "(\<lambda>i::nat. \<zero>)"] using R
unfolding Pi_def by simp
also have "\<dots> = \<zero>" by simp
finally show ?thesis using r_zero R by simp
qed
also have "\<dots> = coeff P p (Suc nn)" using R by simp
finally show ?thesis by simp
qed
then show ?thesis using Succ by simp
}
qed
qed (simp_all add: R)
lemma UP_l_one [simp]:
assumes R: "p \<in> carrier P"
shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
proof (rule up_eqI)
fix n
show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> 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 UP_l_distr:
assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
shows "(p \<oplus>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = (p \<otimes>\<^bsub>P\<^esub> r) \<oplus>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
lemma UP_r_distr:
assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
shows "r \<otimes>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = (r \<otimes>\<^bsub>P\<^esub> p) \<oplus>\<^bsub>P\<^esub> (r \<otimes>\<^bsub>P\<^esub> q)"
by (rule up_eqI) (simp add: r_distr R Pi_def, simp_all add: R)
theorem UP_ring: "ring P"
by (auto intro!: ringI abelian_groupI monoidI UP_a_assoc)
(auto intro: UP_a_comm UP_l_neg_ex UP_m_assoc UP_l_distr UP_r_distr)
end
subsection {* Polynomials Form a Commutative Ring. *}
context UP_cring
begin
lemma UP_m_comm:
assumes R1: "p \<in> carrier P" and R2: "q \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> 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 ==>
(\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) = (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
(is "_ \<Longrightarrow> ?eq k")
proof (induct k)
case 0 then show ?case by (simp add: Pi_def)
next
case (Suc k) then show ?case
by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
qed
}
note l = this
from R1 R2 show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
unfolding coeff_mult [OF R1 R2, of n]
unfolding coeff_mult [OF R2 R1, of n]
using l [of "(\<lambda>i. coeff P p i)" "(\<lambda>i. coeff P q i)" "n"] by (simp add: Pi_def m_comm)
qed (simp_all add: R1 R2)
subsection {*Polynomials over a commutative ring for a commutative ring*}
theorem UP_cring:
"cring P" using UP_ring unfolding cring_def by (auto intro!: comm_monoidI UP_m_assoc UP_m_comm)
end
context UP_ring
begin
lemma UP_a_inv_closed [intro, simp]:
"p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
by (rule abelian_group.a_inv_closed [OF ring.is_abelian_group [OF UP_ring]])
lemma coeff_a_inv [simp]:
assumes R: "p \<in> carrier P"
shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
proof -
from R coeff_closed UP_a_inv_closed have
"coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^bsub>P\<^esub> p) n)"
by algebra
also from R have "... = \<ominus> (coeff P p n)"
by (simp del: coeff_add add: coeff_add [THEN sym]
abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
finally show ?thesis .
qed
end
sublocale UP_ring < P: ring P using UP_ring .
sublocale UP_cring < P: cring P using UP_cring .
subsection {* Polynomials Form an Algebra *}
context UP_ring
begin
lemma UP_smult_l_distr:
"[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
(a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
by (rule up_eqI) (simp_all add: R.l_distr)
lemma UP_smult_r_distr:
"[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
a \<odot>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> a \<odot>\<^bsub>P\<^esub> q"
by (rule up_eqI) (simp_all add: R.r_distr)
lemma UP_smult_assoc1:
"[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
(a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
by (rule up_eqI) (simp_all add: R.m_assoc)
lemma UP_smult_zero [simp]:
"p \<in> carrier P ==> \<zero> \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
by (rule up_eqI) simp_all
lemma UP_smult_one [simp]:
"p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
by (rule up_eqI) simp_all
lemma UP_smult_assoc2:
"[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
(a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
end
text {*
Interpretation of lemmas from @{term algebra}.
*}
lemma (in cring) cring:
"cring R" ..
lemma (in UP_cring) UP_algebra:
"algebra R P" by (auto intro!: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr
UP_smult_assoc1 UP_smult_assoc2)
sublocale UP_cring < algebra R P using UP_algebra .
subsection {* Further Lemmas Involving Monomials *}
context UP_ring
begin
lemma monom_zero [simp]:
"monom P \<zero> n = \<zero>\<^bsub>P\<^esub>" by (simp add: UP_def P_def)
lemma monom_mult_is_smult:
assumes R: "a \<in> carrier R" "p \<in> carrier P"
shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
proof (rule up_eqI)
fix n
show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
proof (cases n)
case 0 with R show ?thesis by simp
next
case Suc with R show ?thesis
using R.finsum_Suc2 by (simp del: R.finsum_Suc add: R.r_null Pi_def)
qed
qed (simp_all add: R)
lemma monom_one [simp]:
"monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
by (rule up_eqI) simp_all
lemma monom_add [simp]:
"[| a \<in> carrier R; b \<in> carrier R |] ==>
monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
by (rule up_eqI) simp_all
lemma monom_one_Suc:
"monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> 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>\<^bsub>P\<^esub> monom P \<one> 1) k"
proof (cases "k = Suc n")
case True show ?thesis
proof -
fix m
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 "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
coeff P (monom P \<one> 1) (k - i))"
by (simp cong: R.finsum_cong add: Pi_def)
also have "... = (\<Oplus>i \<in> {..n}. coeff P (monom P \<one> n) i \<otimes>
coeff P (monom P \<one> 1) (k - i))"
by (simp only: ivl_disj_un_singleton)
also from True
have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
coeff P (monom P \<one> 1) (k - i))"
by (simp cong: R.finsum_cong add: R.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>\<^bsub>P\<^esub> 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 "... = (\<Oplus>i \<in> {..k}. ?s i)"
proof -
have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
by (simp cong: R.finsum_cong add: Pi_def)
from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
by (simp cong: R.finsum_cong add: Pi_def) arith
have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)
show ?thesis
proof (cases "k < n")
case True then show ?thesis by (simp cong: R.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> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
by (simp cong: R.finsum_cong add: Pi_def)
also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
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> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
by (simp add: R.finsum_Un_disjoint f1 f2 Pi_def del: Un_insert_right)
also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
by (simp only: ivl_disj_un_singleton)
also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
by (simp only: ivl_disj_un_one)
finally show ?thesis .
qed
qed
qed
also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
finally show ?thesis .
qed
qed (simp_all)
lemma monom_one_Suc2:
"monom P \<one> (Suc n) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> n"
proof (induct n)
case 0 show ?case by simp
next
case Suc
{
fix k:: nat
assume hypo: "monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"
then show "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k)"
proof -
have lhs: "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
unfolding monom_one_Suc [of "Suc k"] unfolding hypo ..
note cl = monom_closed [OF R.one_closed, of 1]
note clk = monom_closed [OF R.one_closed, of k]
have rhs: "monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
unfolding monom_one_Suc [of k] unfolding sym [OF m_assoc [OF cl clk cl]] ..
from lhs rhs show ?thesis by simp
qed
}
qed
text{*The following corollary follows from lemmas @{thm "monom_one_Suc"}
and @{thm "monom_one_Suc2"}, and is trivial in @{term UP_cring}*}
corollary monom_one_comm: shows "monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"
unfolding monom_one_Suc [symmetric] monom_one_Suc2 [symmetric] ..
lemma monom_mult_smult:
"[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
by (rule up_eqI) simp_all
lemma monom_one_mult:
"monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
proof (induct n)
case 0 show ?case by simp
next
case Suc then show ?case
unfolding add_Suc unfolding monom_one_Suc unfolding Suc.hyps
using m_assoc monom_one_comm [of m] by simp
qed
lemma monom_one_mult_comm: "monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m = monom P \<one> m \<otimes>\<^bsub>P\<^esub> monom P \<one> n"
unfolding monom_one_mult [symmetric] by (rule up_eqI) simp_all
lemma monom_mult [simp]:
assumes a_in_R: "a \<in> carrier R" and b_in_R: "b \<in> carrier R"
shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
proof (rule up_eqI)
fix k
show "coeff P (monom P (a \<otimes> b) (n + m)) k = coeff P (monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m) k"
proof (cases "n + m = k")
case True
{
show ?thesis
unfolding True [symmetric]
coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of "n + m"]
coeff_monom [OF a_in_R, of n] coeff_monom [OF b_in_R, of m]
using R.finsum_cong [of "{.. n + m}" "{.. n + m}" "(\<lambda>i. (if n = i then a else \<zero>) \<otimes> (if m = n + m - i then b else \<zero>))"
"(\<lambda>i. if n = i then a \<otimes> b else \<zero>)"]
a_in_R b_in_R
unfolding simp_implies_def
using R.finsum_singleton [of n "{.. n + m}" "(\<lambda>i. a \<otimes> b)"]
unfolding Pi_def by auto
}
next
case False
{
show ?thesis
unfolding coeff_monom [OF R.m_closed [OF a_in_R b_in_R], of "n + m" k] apply (simp add: False)
unfolding coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of k]
unfolding coeff_monom [OF a_in_R, of n] unfolding coeff_monom [OF b_in_R, of m] using False
using R.finsum_cong [of "{..k}" "{..k}" "(\<lambda>i. (if n = i then a else \<zero>) \<otimes> (if m = k - i then b else \<zero>))" "(\<lambda>i. \<zero>)"]
unfolding Pi_def simp_implies_def using a_in_R b_in_R by force
}
qed
qed (simp_all add: a_in_R b_in_R)
lemma monom_a_inv [simp]:
"a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
by (rule up_eqI) simp_all
lemma 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
end
subsection {* The Degree Function *}
definition
deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
where "deg R p = (LEAST n. bound \<zero>\<^bsub>R\<^esub> n (coeff (UP R) p))"
context UP_ring
begin
lemma 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 deg_aboveD:
assumes "deg R p < m" and "p \<in> carrier P"
shows "coeff P p m = \<zero>"
proof -
from `p \<in> carrier P` 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)
from this and `deg R p < m` show ?thesis ..
qed
lemma 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 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
from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
by (unfold deg_def P_def) simp
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 intro: that)
qed
with deg_belowI R have "deg R p = m" by fastforce
with m_coeff show ?thesis by simp
qed
lemma lcoeff_nonzero_nonzero:
assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" 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>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
with nonzero show ?thesis by contradiction
qed
then obtain m where coeff: "coeff P p m ~= \<zero>" ..
from this and R 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 lcoeff_nonzero:
assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" 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 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_antisym deg_aboveI deg_belowI)
text {* Degree and polynomial operations *}
lemma deg_add [simp]:
"p \<in> carrier P \<Longrightarrow> q \<in> carrier P \<Longrightarrow>
deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
by(rule deg_aboveI)(simp_all add: deg_aboveD)
lemma deg_monom_le:
"a \<in> carrier R ==> deg R (monom P a n) <= n"
by (intro deg_aboveI) simp_all
lemma deg_monom [simp]:
"[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
by (fastforce intro: le_antisym deg_aboveI deg_belowI)
lemma deg_const [simp]:
assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
proof (rule le_antisym)
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 deg_zero [simp]:
"deg R \<zero>\<^bsub>P\<^esub> = 0"
proof (rule le_antisym)
show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
next
show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
qed
lemma deg_one [simp]:
"deg R \<one>\<^bsub>P\<^esub> = 0"
proof (rule le_antisym)
show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
next
show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
qed
lemma deg_uminus [simp]:
assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
proof (rule le_antisym)
show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
next
show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
by (simp add: deg_belowI lcoeff_nonzero_deg
inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
qed
text{*The following lemma is later \emph{overwritten} by the most
specific one for domains, @{text deg_smult}.*}
lemma deg_smult_ring [simp]:
"[| a \<in> carrier R; p \<in> carrier P |] ==>
deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
end
context UP_domain
begin
lemma deg_smult [simp]:
assumes R: "a \<in> carrier R" "p \<in> carrier P"
shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
proof (rule le_antisym)
show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
using R by (rule deg_smult_ring)
next
show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
proof (cases "a = \<zero>")
qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
qed
end
context UP_ring
begin
lemma deg_mult_ring:
assumes R: "p \<in> carrier P" "q \<in> carrier P"
shows "deg R (p \<otimes>\<^bsub>P\<^esub> 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>\<^bsub>P\<^esub> q) m = \<zero>" by simp
qed (simp add: R)
end
context UP_domain
begin
lemma deg_mult [simp]:
"[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
proof (rule le_antisym)
assume "p \<in> carrier P" " q \<in> carrier P"
then show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_ring)
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>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
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>\<^bsub>P\<^esub> q)"
proof (rule deg_belowI, simp add: R)
have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
= (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
by (simp only: ivl_disj_un_one)
also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
deg_aboveD less_add_diff R Pi_def)
also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
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: R.finsum_cong add: deg_aboveD R Pi_def)
finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
= coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
by (simp add: integral_iff lcoeff_nonzero R)
qed (simp add: R)
qed
end
text{*The following lemmas also can be lifted to @{term UP_ring}.*}
context UP_ring
begin
lemma coeff_finsum:
assumes fin: "finite A"
shows "p \<in> A -> carrier P ==>
coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
using fin by induct (auto simp: Pi_def)
lemma up_repr:
assumes R: "p \<in> carrier P"
shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = 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 (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
proof (cases "k <= deg R p")
case True
hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
by (simp only: ivl_disj_un_one)
also from True
have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
also
have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
by (simp only: ivl_disj_un_singleton)
also have "... = coeff P p k"
by (simp cong: R.finsum_cong add: coeff_finsum deg_aboveD R RR Pi_def)
finally show ?thesis .
next
case False
hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
by (simp only: ivl_disj_un_singleton)
also from False have "... = coeff P p k"
by (simp cong: R.finsum_cong add: coeff_finsum deg_aboveD R Pi_def)
finally show ?thesis .
qed
qed (simp_all add: R Pi_def)
lemma up_repr_le:
"[| deg R p <= n; p \<in> carrier P |] ==>
(\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = 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} \<union> {deg R p<..n})"
by (simp only: ivl_disj_un_one)
also have "... = finsum P ?s {..deg R p}"
by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one
deg_aboveD R Pi_def)
also have "... = p" using R by (rule up_repr)
finally show ?thesis .
qed
end
subsection {* Polynomials over Integral Domains *}
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 assms
del: disjCI)
context UP_domain
begin
lemma UP_one_not_zero:
"\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
proof
assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
hence "\<one> = \<zero>" by simp
with R.one_not_zero show "False" by contradiction
qed
lemma UP_integral:
"[| p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
proof -
fix p q
assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
proof (rule classical)
assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> 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 = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
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 = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
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>\<^bsub>P\<^esub> 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>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastforce
qed
qed
theorem UP_domain:
"domain P"
by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
end
text {*
Interpretation of theorems from @{term domain}.
*}
sublocale UP_domain < "domain" P
by intro_locales (rule domain.axioms UP_domain)+
subsection {* The Evaluation Homomorphism and Universal Property*}
(* 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*)
lemma (in abelian_monoid) boundD_carrier:
"[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
by auto
context ring
begin
theorem diagonal_sum:
"[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
proof -
assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
{
fix j
have "j <= n + m ==>
(\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
(\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
proof (induct j)
case 0 from Rf Rg show ?case by (simp add: Pi_def)
next
case (Suc j)
have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
using Suc by (auto intro!: funcset_mem [OF Rg])
have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
using Suc by (auto intro!: funcset_mem [OF Rg])
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])
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
theorem 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 "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
(\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)" (* State reverse 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 "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
by (simp add: diagonal_sum Pi_def)
also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
by (simp only: ivl_disj_un_one)
also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
by (simp cong: finsum_cong
add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
also from f g
have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
by (simp cong: finsum_cong
add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
by (simp add: finsum_ldistr diagonal_sum Pi_def,
simp cong: finsum_cong add: finsum_rdistr Pi_def)
finally show ?thesis .
qed
end
lemma (in UP_ring) 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)
definition
eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
'a => 'b, 'b, nat => 'a] => 'b"
where "eval R S phi s = (\<lambda>p \<in> carrier (UP R).
\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
context UP
begin
lemma eval_on_carrier:
fixes S (structure)
shows "p \<in> carrier P ==>
eval R S phi s p = (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
by (unfold eval_def, fold P_def) simp
lemma eval_extensional:
"eval R S phi p \<in> extensional (carrier P)"
by (unfold eval_def, fold P_def) simp
end
text {* The universal property of the polynomial ring *}
locale UP_pre_univ_prop = ring_hom_cring + UP_cring
(* FIXME print_locale ring_hom_cring fails *)
locale UP_univ_prop = UP_pre_univ_prop +
fixes s and Eval
assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
defines Eval_def: "Eval == eval R S h s"
text{*JE: I have moved the following lemma from Ring.thy and lifted then to the locale @{term ring_hom_ring} from @{term ring_hom_cring}.*}
text{*JE: I was considering using it in @{text eval_ring_hom}, but that property does not hold for non commutative rings, so
maybe it is not that necessary.*}
lemma (in ring_hom_ring) hom_finsum [simp]:
"[| finite A; f \<in> A -> carrier R |] ==>
h (finsum R f A) = finsum S (h o f) A"
proof (induct set: finite)
case empty then show ?case by simp
next
case insert then show ?case by (simp add: Pi_def)
qed
context UP_pre_univ_prop
begin
theorem eval_ring_hom:
assumes S: "s \<in> carrier S"
shows "eval R S h s \<in> ring_hom P S"
proof (rule ring_hom_memI)
fix p
assume R: "p \<in> carrier P"
then show "eval R S h s p \<in> carrier S"
by (simp only: eval_on_carrier) (simp add: S Pi_def)
next
fix p q
assume R: "p \<in> carrier P" "q \<in> carrier P"
then show "eval R S h s (p \<oplus>\<^bsub>P\<^esub> q) = eval R S h s p \<oplus>\<^bsub>S\<^esub> eval R S h s q"
proof (simp only: eval_on_carrier P.a_closed)
from S R have
"(\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
(\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<oplus>\<^bsub>P\<^esub> q)<..max (deg R p) (deg R q)}.
h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
by (simp cong: S.finsum_cong
add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add)
also from R have "... =
(\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
by (simp add: ivl_disj_un_one)
also from R S have "... =
(\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
(\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
by (simp cong: S.finsum_cong
add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
also have "... =
(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
also from R S have "... =
(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
by (simp cong: S.finsum_cong
add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
finally show
"(\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
(\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
(\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
qed
next
show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
by (simp only: eval_on_carrier UP_one_closed) simp
next
fix p q
assume R: "p \<in> carrier P" "q \<in> carrier P"
then show "eval R S h s (p \<otimes>\<^bsub>P\<^esub> q) = eval R S h s p \<otimes>\<^bsub>S\<^esub> eval R S h s q"
proof (simp only: eval_on_carrier UP_mult_closed)
from R S have
"(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<otimes>\<^bsub>P\<^esub> q)<..deg R p + deg R q}.
h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
by (simp cong: S.finsum_cong
add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
del: coeff_mult)
also from R have "... =
(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
by (simp only: ivl_disj_un_one deg_mult_ring)
also from R S have "... =
(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
\<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
(s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def
S.m_ac S.finsum_rdistr)
also from R S have "... =
(\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
(\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
Pi_def)
finally show
"(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
qed
qed
text {*
The following lemma could be proved in @{text UP_cring} with the additional
assumption that @{text h} is closed. *}
lemma (in UP_pre_univ_prop) 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 {* Further properties of the evaluation homomorphism. *}
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 UP_pre_univ_prop) eval_monom1:
assumes S: "s \<in> carrier S"
shows "eval R S h s (monom P \<one> 1) = s"
proof (simp only: eval_on_carrier monom_closed R.one_closed)
from S have
"(\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
(\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
by (simp cong: S.finsum_cong del: coeff_monom
add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
also have "... =
(\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
also have "... = s"
proof (cases "s = \<zero>\<^bsub>S\<^esub>")
case True then show ?thesis by (simp add: Pi_def)
next
case False then show ?thesis by (simp add: S Pi_def)
qed
finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
qed
end
text {* Interpretation of ring homomorphism lemmas. *}
sublocale UP_univ_prop < ring_hom_cring P S Eval
unfolding Eval_def
by unfold_locales (fast intro: eval_ring_hom)
lemma (in UP_cring) monom_pow:
assumes R: "a \<in> carrier R"
shows "(monom P a n) (^)\<^bsub>P\<^esub> 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)