Theory UnivPoly

theory UnivPoly
imports Module RingHom
(*  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 ==> f m = z"

declare bound.intro [intro!]
and bound.bound [dest]

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"

definition
up :: "('a, 'm) ring_scheme => (nat => 'a) set"
where "up R = {f. f ∈ UNIV -> carrier R & (EX n. bound \<zero>R 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>Ri ∈ {..n}. p i ⊗R q (n-i)),
one = (%i. if i=0 then \<one>R else \<zero>R),
zero = (%i. \<zero>R),
add = (%p:up R. %q:up R. %i. p i ⊕R q i),
smult = (%a:carrier R. %p:up R. %i. a ⊗R 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 ∈ carrier R; EX n. bound (zero R) n f |] ==> f ∈ up R"
by (simp add: up_def Pi_def)

lemma mem_upD [dest]:
"f ∈ up R ==> f n ∈ carrier R"
by (simp add: up_def Pi_def)

context ring
begin

lemma bound_upD [dest]: "f ∈ 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>) ∈ up R" using up_def by force

lemma up_smult_closed: "[| a ∈ carrier R; p ∈ up R |] ==> (%i. a ⊗ p i) ∈ up R" by force

lemma up_add_closed:
"[| p ∈ up R; q ∈ up R |] ==> (%i. p i ⊕ q i) ∈ up R"
proof
fix n
assume "p ∈ up R" and "q ∈ up R"
then show "p n ⊕ q n ∈ carrier R"
by auto
next
assume UP: "p ∈ up R" "q ∈ up R"
show "EX n. bound \<zero> n (%i. p i ⊕ 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 ⊕ q i)"
proof
fix i
assume "max n m < i"
with boundn and boundm and UP show "p i ⊕ q i = \<zero>" by fastforce
qed
then show ?thesis ..
qed
qed

lemma up_a_inv_closed:
"p ∈ up R ==> (%i. \<ominus> (p i)) ∈ up R"
proof
assume R: "p ∈ 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 ∈ up R; q ∈ up R |] ==> (%i. p i \<ominus> q i) ∈ 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 ∈ up R; q ∈ up R |] ==>
(%n. \<Oplus>i ∈ {..n}. p i ⊗ q (n-i)) ∈ up R"

proof
fix n
assume "p ∈ up R" "q ∈ up R"
then show "(\<Oplus>i ∈ {..n}. p i ⊗ q (n-i)) ∈ carrier R"
by (simp add: mem_upD funcsetI)
next
assume UP: "p ∈ up R" "q ∈ up R"
show "EX n. bound \<zero> n (%n. \<Oplus>i ∈ {..n}. p i ⊗ 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 ∈ {..n}. p i ⊗ q (n - i))"
proof
fix k assume bound: "n + m < k"
{
fix i
have "p i ⊗ 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) ∈ 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 ∈ carrier R" by auto
ultimately show ?thesis by simp
qed
}
then show "(\<Oplus>i ∈ {..k}. p i ⊗ 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 ∈ carrier P" "q ∈ 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 ∈ carrier P ==> coeff P p n ∈ 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 ∈ carrier R ==> coeff P (monom P a m) n = (if m=n then a else \<zero>)"
proof -
assume R: "a ∈ carrier R"
then have "(%n. if n = m then a else \<zero>) ∈ up R"
using up_def by force
with R show ?thesis by (simp add: UP_def)
qed

lemma coeff_zero [simp]: "coeff P \<zero>P n = \<zero>" by (auto simp add: UP_def)

lemma coeff_one [simp]: "coeff P \<one>P n = (if n=0 then \<one> else \<zero>)"
using up_one_closed by (simp add: UP_def)

lemma coeff_smult [simp]:
"[| a ∈ carrier R; p ∈ carrier P |] ==> coeff P (a \<odot>P p) n = a ⊗ coeff P p n"
by (simp add: UP_def up_smult_closed)

lemma coeff_add [simp]:
"[| p ∈ carrier P; q ∈ carrier P |] ==> coeff P (p ⊕P q) n = coeff P p n ⊕ coeff P q n"
by (simp add: UP_def up_add_closed)

lemma coeff_mult [simp]:
"[| p ∈ carrier P; q ∈ carrier P |] ==> coeff P (p ⊗P q) n = (\<Oplus>i ∈ {..n}. coeff P p i ⊗ 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 ∈ carrier P; q ∈ carrier P |] ==> p ⊗P q ∈ carrier P" by (simp add: UP_def up_mult_closed)

lemma UP_one_closed [simp]:
"\<one>P ∈ carrier P" by (simp add: UP_def up_one_closed)

lemma UP_zero_closed [intro, simp]:
"\<zero>P ∈ carrier P" by (auto simp add: UP_def)

lemma UP_a_closed [intro, simp]:
"[| p ∈ carrier P; q ∈ carrier P |] ==> p ⊕P q ∈ carrier P" by (simp add: UP_def up_add_closed)

lemma monom_closed [simp]:
"a ∈ carrier R ==> monom P a n ∈ carrier P" by (auto simp add: UP_def up_def Pi_def)

lemma UP_smult_closed [simp]:
"[| a ∈ carrier R; p ∈ carrier P |] ==> a \<odot>P p ∈ 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 ∈ carrier P" "q ∈ carrier P" "r ∈ carrier P"
shows "(p ⊕P q) ⊕P r = p ⊕P (q ⊕P r)" by (rule up_eqI, simp add: a_assoc R, simp_all add: R)

lemma UP_l_zero [simp]:
assumes R: "p ∈ carrier P"
shows "\<zero>PP p = p" by (rule up_eqI, simp_all add: R)

lemma UP_l_neg_ex:
assumes R: "p ∈ carrier P"
shows "EX q : carrier P. q ⊕P p = \<zero>P"
proof -
let ?q = "%i. \<ominus> (p i)"
from R have closed: "?q ∈ 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 ⊕P p = \<zero>P"
by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
qed (rule closed)
qed

lemma UP_a_comm:
assumes R: "p ∈ carrier P" "q ∈ carrier P"
shows "p ⊕P q = q ⊕P p" by (rule up_eqI, simp add: a_comm R, simp_all add: R)

lemma UP_m_assoc:
assumes R: "p ∈ carrier P" "q ∈ carrier P" "r ∈ carrier P"
shows "(p ⊗P q) ⊗P r = p ⊗P (q ⊗P r)"
proof (rule up_eqI)
fix n
{
fix k and a b c :: "nat=>'a"
assume R: "a ∈ UNIV -> carrier R" "b ∈ UNIV -> carrier R"
"c ∈ UNIV -> carrier R"
then have "k <= n ==>
(\<Oplus>j ∈ {..k}. (\<Oplus>i ∈ {..j}. a i ⊗ b (j-i)) ⊗ c (n-j)) =
(\<Oplus>j ∈ {..k}. a j ⊗ (\<Oplus>i ∈ {..k-j}. b i ⊗ c (n-j-i)))"

(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
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 ⊗P q) ⊗P r) n = coeff P (p ⊗P (q ⊗P r)) n"
by (simp add: Pi_def)
qed (simp_all add: R)

lemma UP_r_one [simp]:
assumes R: "p ∈ carrier P" shows "p ⊗P \<one>P = p"
proof (rule up_eqI)
fix n
show "coeff P (p ⊗P \<one>P) 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 ⊗P \<one>P) (Suc nn) = coeff P p (Suc nn)"
proof -
have "coeff P (p ⊗P \<one>P) (Suc nn) = (\<Oplus>i∈{..Suc nn}. coeff P p i ⊗ (if Suc nn ≤ i then \<one> else \<zero>))" using R by simp
also have "… = coeff P p (Suc nn) ⊗ (if Suc nn ≤ Suc nn then \<one> else \<zero>) ⊕ (\<Oplus>i∈{..nn}. coeff P p i ⊗ (if Suc nn ≤ i then \<one> else \<zero>))"
using finsum_Suc [of "(λi::nat. coeff P p i ⊗ (if Suc nn ≤ i then \<one> else \<zero>))" "nn"] unfolding Pi_def using R by simp
also have "… = coeff P p (Suc nn) ⊗ (if Suc nn ≤ Suc nn then \<one> else \<zero>)"
proof -
have "(\<Oplus>i∈{..nn}. coeff P p i ⊗ (if Suc nn ≤ i then \<one> else \<zero>)) = (\<Oplus>i∈{..nn}. \<zero>)"
using finsum_cong [of "{..nn}" "{..nn}" "(λi::nat. coeff P p i ⊗ (if Suc nn ≤ i then \<one> else \<zero>))" "(λi::nat. \<zero>)"] using R
unfolding Pi_def by simp
also have "… = \<zero>" by simp
finally show ?thesis using r_zero R by simp
qed
also have "… = 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 ∈ carrier P"
shows "\<one>PP p = p"
proof (rule up_eqI)
fix n
show "coeff P (\<one>PP 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 ∈ carrier P" "q ∈ carrier P" "r ∈ carrier P"
shows "(p ⊕P q) ⊗P r = (p ⊗P r) ⊕P (q ⊗P r)"
by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)

lemma UP_r_distr:
assumes R: "p ∈ carrier P" "q ∈ carrier P" "r ∈ carrier P"
shows "r ⊗P (p ⊕P q) = (r ⊗P p) ⊕P (r ⊗P 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 ∈ carrier P" and R2: "q ∈ carrier P" shows "p ⊗P q = q ⊗P p"
proof (rule up_eqI)
fix n
{
fix k and a b :: "nat=>'a"
assume R: "a ∈ UNIV -> carrier R" "b ∈ UNIV -> carrier R"
then have "k <= n ==>
(\<Oplus>i ∈ {..k}. a i ⊗ b (n-i)) = (\<Oplus>i ∈ {..k}. a (k-i) ⊗ b (i+n-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 (2) finsum_Suc2) (simp add: Pi_def a_comm)+
qed
}
note l = this
from R1 R2 show "coeff P (p ⊗P q) n = coeff P (q ⊗P p) n"
unfolding coeff_mult [OF R1 R2, of n]
unfolding coeff_mult [OF R2 R1, of n]
using l [of "(λi. coeff P p i)" "(λi. 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 ∈ carrier P ==> \<ominus>P p ∈ 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 ∈ carrier P"
shows "coeff P (\<ominus>P p) n = \<ominus> (coeff P p n)"
proof -
from R coeff_closed UP_a_inv_closed have
"coeff P (\<ominus>P p) n = \<ominus> coeff P p n ⊕ (coeff P p n ⊕ coeff P (\<ominus>P 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 ∈ carrier R; b ∈ carrier R; p ∈ carrier P |] ==>
(a ⊕ b) \<odot>P p = a \<odot>P p ⊕P b \<odot>P p"

by (rule up_eqI) (simp_all add: R.l_distr)

lemma UP_smult_r_distr:
"[| a ∈ carrier R; p ∈ carrier P; q ∈ carrier P |] ==>
a \<odot>P (p ⊕P q) = a \<odot>P p ⊕P a \<odot>P q"

by (rule up_eqI) (simp_all add: R.r_distr)

lemma UP_smult_assoc1:
"[| a ∈ carrier R; b ∈ carrier R; p ∈ carrier P |] ==>
(a ⊗ b) \<odot>P p = a \<odot>P (b \<odot>P p)"

by (rule up_eqI) (simp_all add: R.m_assoc)

lemma UP_smult_zero [simp]:
"p ∈ carrier P ==> \<zero> \<odot>P p = \<zero>P"
by (rule up_eqI) simp_all

lemma UP_smult_one [simp]:
"p ∈ carrier P ==> \<one> \<odot>P p = p"
by (rule up_eqI) simp_all

lemma UP_smult_assoc2:
"[| a ∈ carrier R; p ∈ carrier P; q ∈ carrier P |] ==>
(a \<odot>P p) ⊗P q = a \<odot>P (p ⊗P 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>P" by (simp add: UP_def P_def)

lemma monom_mult_is_smult:
assumes R: "a ∈ carrier R" "p ∈ carrier P"
shows "monom P a 0 ⊗P p = a \<odot>P p"
proof (rule up_eqI)
fix n
show "coeff P (monom P a 0 ⊗P p) n = coeff P (a \<odot>P 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>P"
by (rule up_eqI) simp_all

lemma monom_add [simp]:
"[| a ∈ carrier R; b ∈ carrier R |] ==>
monom P (a ⊕ b) n = monom P a n ⊕P monom P b n"

by (rule up_eqI) simp_all

lemma monom_one_Suc:
"monom P \<one> (Suc n) = monom P \<one> n ⊗P 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 ⊗P 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 ∈ {..<n} ∪ {n}. coeff P (monom P \<one> n) i ⊗
coeff P (monom P \<one> 1) (k - i))"

by (simp cong: R.finsum_cong add: Pi_def)
also have "... = (\<Oplus>i ∈ {..n}. coeff P (monom P \<one> n) i ⊗
coeff P (monom P \<one> 1) (k - i))"

by (simp only: ivl_disj_un_singleton)
also from True
have "... = (\<Oplus>i ∈ {..n} ∪ {n<..k}. coeff P (monom P \<one> n) i ⊗
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 ⊗P monom P \<one> 1) k"
by (simp add: ivl_disj_un_one)
finally show ?thesis .
qed
next
case False
note neq = False
let ?s =
"λi. (if n = i then \<one> else \<zero>) ⊗ (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 ∈ {..k}. ?s i)"
proof -
have f1: "(\<Oplus>i ∈ {..<n}. ?s i) = \<zero>"
by (simp cong: R.finsum_cong add: Pi_def)
from neq have f2: "(\<Oplus>i ∈ {n}. ?s i) = \<zero>"
by (simp cong: R.finsum_cong add: Pi_def) arith
have f3: "n < k ==> (\<Oplus>i ∈ {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 ∈ {..<n} ∪ {n}. ?s i)"
by (simp cong: R.finsum_cong add: Pi_def)
also from True have "... = (\<Oplus>i ∈ {..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 ∈ {..<n} ∪ {n}. ?s i)"
by (simp add: R.finsum_Un_disjoint f1 f2 Pi_def del: Un_insert_right)
also have "... = (\<Oplus>i ∈ {..n}. ?s i)"
by (simp only: ivl_disj_un_singleton)
also from n_less_k neq have "... = (\<Oplus>i ∈ {..n} ∪ {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 ∈ {..k}. ?s i)"
by (simp only: ivl_disj_un_one)
finally show ?thesis .
qed
qed
qed
also have "... = coeff P (monom P \<one> n ⊗P 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 ⊗P 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 ⊗P monom P \<one> k"
then show "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 ⊗P monom P \<one> (Suc k)"
proof -
have lhs: "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 ⊗P monom P \<one> k ⊗P 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 ⊗P monom P \<one> (Suc k) = monom P \<one> 1 ⊗P monom P \<one> k ⊗P 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 ⊗P monom P \<one> 1 = monom P \<one> 1 ⊗P monom P \<one> k"
unfolding monom_one_Suc [symmetric] monom_one_Suc2 [symmetric] ..

lemma monom_mult_smult:
"[| a ∈ carrier R; b ∈ carrier R |] ==> monom P (a ⊗ b) n = a \<odot>P monom P b n"
by (rule up_eqI) simp_all

lemma monom_one_mult:
"monom P \<one> (n + m) = monom P \<one> n ⊗P 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 ⊗P monom P \<one> m = monom P \<one> m ⊗P monom P \<one> n"
unfolding monom_one_mult [symmetric] by (rule up_eqI) simp_all

lemma monom_mult [simp]:
assumes a_in_R: "a ∈ carrier R" and b_in_R: "b ∈ carrier R"
shows "monom P (a ⊗ b) (n + m) = monom P a n ⊗P monom P b m"
proof (rule up_eqI)
fix k
show "coeff P (monom P (a ⊗ b) (n + m)) k = coeff P (monom P a n ⊗P 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}" "(λi. (if n = i then a else \<zero>) ⊗ (if m = n + m - i then b else \<zero>))"
"(λi. if n = i then a ⊗ b else \<zero>)"]
a_in_R b_in_R
unfolding simp_implies_def
using R.finsum_singleton [of n "{.. n + m}" "(λi. a ⊗ 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}" "(λi. (if n = i then a else \<zero>) ⊗ (if m = k - i then b else \<zero>))" "(λi. \<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 ∈ carrier R ==> monom P (\<ominus> a) n = \<ominus>P 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 ∈ carrier R" "y ∈ 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>R n (coeff (UP R) p))"

context UP_ring
begin

lemma deg_aboveI:
"[| (!!m. n < m ==> coeff P p m = \<zero>); p ∈ 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 ∈ carrier P"
shows "coeff P p m = \<zero>"
proof -
from `p ∈ 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 ∈ 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 ∈ 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>P" and R: "p ∈ 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>P" 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>P" and R: "p ∈ 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 ∈ 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 ∈ carrier P ==> q ∈ carrier P ==>
deg R (p ⊕P q) <= max (deg R p) (deg R q)"

by(rule deg_aboveI)(simp_all add: deg_aboveD)

lemma deg_monom_le:
"a ∈ carrier R ==> deg R (monom P a n) <= n"
by (intro deg_aboveI) simp_all

lemma deg_monom [simp]:
"[| a ~= \<zero>; a ∈ 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 ∈ 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>P = 0"
proof (rule le_antisym)
show "deg R \<zero>P <= 0" by (rule deg_aboveI) simp_all
next
show "0 <= deg R \<zero>P" by (rule deg_belowI) simp_all
qed

lemma deg_one [simp]:
"deg R \<one>P = 0"
proof (rule le_antisym)
show "deg R \<one>P <= 0" by (rule deg_aboveI) simp_all
next
show "0 <= deg R \<one>P" by (rule deg_belowI) simp_all
qed

lemma deg_uminus [simp]:
assumes R: "p ∈ carrier P" shows "deg R (\<ominus>P p) = deg R p"
proof (rule le_antisym)
show "deg R (\<ominus>P p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
next
show "deg R p <= deg R (\<ominus>P 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 ∈ carrier R; p ∈ carrier P |] ==>
deg R (a \<odot>P 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 ∈ carrier R" "p ∈ carrier P"
shows "deg R (a \<odot>P p) = (if a = \<zero> then 0 else deg R p)"
proof (rule le_antisym)
show "deg R (a \<odot>P 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>P 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 ∈ carrier P" "q ∈ carrier P"
shows "deg R (p ⊗P 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 ⊗ 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 ⊗P q) m = \<zero>" by simp
qed (simp add: R)

end

context UP_domain
begin

lemma deg_mult [simp]:
"[| p ~= \<zero>P; q ~= \<zero>P; p ∈ carrier P; q ∈ carrier P |] ==>
deg R (p ⊗P q) = deg R p + deg R q"

proof (rule le_antisym)
assume "p ∈ carrier P" " q ∈ carrier P"
then show "deg R (p ⊗P q) <= deg R p + deg R q" by (rule deg_mult_ring)
next
let ?s = "(%i. coeff P p i ⊗ coeff P q (deg R p + deg R q - i))"
assume R: "p ∈ carrier P" "q ∈ carrier P" and nz: "p ~= \<zero>P" "q ~= \<zero>P"
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 ⊗P q)"
proof (rule deg_belowI, simp add: R)
have "(\<Oplus>i ∈ {.. deg R p + deg R q}. ?s i)
= (\<Oplus>i ∈ {..< deg R p} ∪ {deg R p .. deg R p + deg R q}. ?s i)"

by (simp only: ivl_disj_un_one)
also have "... = (\<Oplus>i ∈ {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 ∈ {deg R p} ∪ {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) ⊗ coeff P q (deg R q)"
by (simp cong: R.finsum_cong add: deg_aboveD R Pi_def)
finally have "(\<Oplus>i ∈ {.. deg R p + deg R q}. ?s i)
= coeff P p (deg R p) ⊗ coeff P q (deg R q)"
.
with nz show "(\<Oplus>i ∈ {.. 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 ∈ A -> carrier P ==>
coeff P (finsum P p A) k = (\<Oplus>i ∈ A. coeff P (p i) k)"

using fin by induct (auto simp: Pi_def)

lemma up_repr:
assumes R: "p ∈ carrier P"
shows "(\<Oplus>P i ∈ {..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>) ∈ carrier R"
by simp
show "coeff P (\<Oplus>P i ∈ {..deg R p}. ?s i) k = coeff P p k"
proof (cases "k <= deg R p")
case True
hence "coeff P (\<Oplus>P i ∈ {..deg R p}. ?s i) k =
coeff P (\<Oplus>P i ∈ {..k} ∪ {k<..deg R p}. ?s i) k"

by (simp only: ivl_disj_un_one)
also from True
have "... = coeff P (\<Oplus>P i ∈ {..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>P i ∈ {..<k} ∪ {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>P i ∈ {..deg R p}. ?s i) k =
coeff P (\<Oplus>P i ∈ {..<deg R p} ∪ {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 ∈ carrier P |] ==>
(\<Oplus>P i ∈ {..n}. monom P (coeff P p i) i) = p"

proof -
let ?s = "(%i. monom P (coeff P p i) i)"
assume R: "p ∈ carrier P" and "deg R p <= n"
then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} ∪ {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 ∈ carrier R;
b ∈ 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>P ~= \<zero>P"
proof
assume "\<one>P = \<zero>P"
hence "coeff P \<one>P 0 = (coeff P \<zero>P 0)" by simp
hence "\<one> = \<zero>" by simp
with R.one_not_zero show "False" by contradiction
qed

lemma UP_integral:
"[| p ⊗P q = \<zero>P; p ∈ carrier P; q ∈ carrier P |] ==> p = \<zero>P | q = \<zero>P"
proof -
fix p q
assume pq: "p ⊗P q = \<zero>P" and R: "p ∈ carrier P" "q ∈ carrier P"
show "p = \<zero>P | q = \<zero>P"
proof (rule classical)
assume c: "~ (p = \<zero>P | q = \<zero>P)"
with R have "deg R p + deg R q = deg R (p ⊗P 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>P i ∈ {..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>P i ∈ {..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 ⊗ coeff P q 0 = coeff P (p ⊗P q) 0" by simp
also from pq have "... = \<zero>" by simp
finally have "coeff P p 0 ⊗ 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>P | q = \<zero>P" 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 ∈ A ==> f i ∈ carrier G = True; A = B;
!!i. i ∈ 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 ∈ carrier G"
by auto

context ring
begin

theorem diagonal_sum:
"[| f ∈ {..n + m::nat} -> carrier R; g ∈ {..n + m} -> carrier R |] ==>
(\<Oplus>k ∈ {..n + m}. \<Oplus>i ∈ {..k}. f i ⊗ g (k - i)) =
(\<Oplus>k ∈ {..n + m}. \<Oplus>i ∈ {..n + m - k}. f k ⊗ g i)"

proof -
assume Rf: "f ∈ {..n + m} -> carrier R" and Rg: "g ∈ {..n + m} -> carrier R"
{
fix j
have "j <= n + m ==>
(\<Oplus>k ∈ {..j}. \<Oplus>i ∈ {..k}. f i ⊗ g (k - i)) =
(\<Oplus>k ∈ {..j}. \<Oplus>i ∈ {..j - k}. f k ⊗ 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 ∈ carrier R"
using Suc by (auto intro!: funcset_mem [OF Rg])
have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) ∈ carrier R"
using Suc by (auto intro!: funcset_mem [OF Rg])
have R9: "!!i k. [| k <= Suc j |] ==> f k ∈ carrier R"
using Suc by (auto intro!: funcset_mem [OF Rf])
have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i ∈ carrier R"
using Suc by (auto intro!: funcset_mem [OF Rg])
have R11: "g 0 ∈ 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 ∈ {..n} -> carrier R" and Rg: "g ∈ {..m} -> carrier R"
shows "(\<Oplus>k ∈ {..n + m}. \<Oplus>i ∈ {..k}. f i ⊗ g (k - i)) =
(\<Oplus>i ∈ {..n}. f i) ⊗ (\<Oplus>i ∈ {..m}. g i)"
(* State reverse direction? *)
proof -
have f: "!!x. f x ∈ carrier R"
proof -
fix x
show "f x ∈ carrier R"
using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
qed
have g: "!!x. g x ∈ carrier R"
proof -
fix x
show "g x ∈ carrier R"
using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
qed
from f g have "(\<Oplus>k ∈ {..n + m}. \<Oplus>i ∈ {..k}. f i ⊗ g (k - i)) =
(\<Oplus>k ∈ {..n + m}. \<Oplus>i ∈ {..n + m - k}. f k ⊗ g i)"

by (simp add: diagonal_sum Pi_def)
also have "... = (\<Oplus>k ∈ {..n} ∪ {n<..n + m}. \<Oplus>i ∈ {..n + m - k}. f k ⊗ g i)"
by (simp only: ivl_disj_un_one)
also from f g have "... = (\<Oplus>k ∈ {..n}. \<Oplus>i ∈ {..n + m - k}. f k ⊗ 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 ∈ {..n}. \<Oplus>i ∈ {..m} ∪ {m<..n + m - k}. f k ⊗ g i)"
by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
also from f g have "... = (\<Oplus>k ∈ {..n}. \<Oplus>i ∈ {..m}. f k ⊗ 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 ∈ {..n}. f i) ⊗ (\<Oplus>i ∈ {..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) ∈ 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 = (λp ∈ carrier (UP R).
\<Oplus>Si ∈ {..deg R p}. phi (coeff (UP R) p i) ⊗S s (^)S i)"


context UP
begin

lemma eval_on_carrier:
fixes S (structure)
shows "p ∈ carrier P ==>
eval R S phi s p = (\<Oplus>S i ∈ {..deg R p}. phi (coeff P p i) ⊗S s (^)S i)"

by (unfold eval_def, fold P_def) simp

lemma eval_extensional:
"eval R S phi p ∈ 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 ∈ 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 ∈ 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 ∈ carrier S"
shows "eval R S h s ∈ ring_hom P S"
proof (rule ring_hom_memI)
fix p
assume R: "p ∈ carrier P"
then show "eval R S h s p ∈ carrier S"
by (simp only: eval_on_carrier) (simp add: S Pi_def)
next
fix p q
assume R: "p ∈ carrier P" "q ∈ carrier P"
then show "eval R S h s (p ⊕P q) = eval R S h s p ⊕S eval R S h s q"
proof (simp only: eval_on_carrier P.a_closed)
from S R have
"(\<Oplus>S i∈{..deg R (p ⊕P q)}. h (coeff P (p ⊕P q) i) ⊗S s (^)S i) =
(\<Oplus>S i∈{..deg R (p ⊕P q)} ∪ {deg R (p ⊕P q)<..max (deg R p) (deg R q)}.
h (coeff P (p ⊕P q) i) ⊗S s (^)S 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>S i ∈ {..max (deg R p) (deg R q)}.
h (coeff P (p ⊕P q) i) ⊗S s (^)S i)"

by (simp add: ivl_disj_un_one)
also from R S have "... =
(\<Oplus>Si∈{..max (deg R p) (deg R q)}. h (coeff P p i) ⊗S s (^)S i) ⊕S
(\<Oplus>Si∈{..max (deg R p) (deg R q)}. h (coeff P q i) ⊗S s (^)S i)"

by (simp cong: S.finsum_cong
add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
also have "... =
(\<Oplus>S i ∈ {..deg R p} ∪ {deg R p<..max (deg R p) (deg R q)}.
h (coeff P p i) ⊗S s (^)S i) ⊕S
(\<Oplus>S i ∈ {..deg R q} ∪ {deg R q<..max (deg R p) (deg R q)}.
h (coeff P q i) ⊗S s (^)S i)"

by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
also from R S have "... =
(\<Oplus>S i ∈ {..deg R p}. h (coeff P p i) ⊗S s (^)S i) ⊕S
(\<Oplus>S i ∈ {..deg R q}. h (coeff P q i) ⊗S s (^)S i)"

by (simp cong: S.finsum_cong
add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
finally show
"(\<Oplus>Si ∈ {..deg R (p ⊕P q)}. h (coeff P (p ⊕P q) i) ⊗S s (^)S i) =
(\<Oplus>Si ∈ {..deg R p}. h (coeff P p i) ⊗S s (^)S i) ⊕S
(\<Oplus>Si ∈ {..deg R q}. h (coeff P q i) ⊗S s (^)S i)"
.
qed
next
show "eval R S h s \<one>P = \<one>S"
by (simp only: eval_on_carrier UP_one_closed) simp
next
fix p q
assume R: "p ∈ carrier P" "q ∈ carrier P"
then show "eval R S h s (p ⊗P q) = eval R S h s p ⊗S eval R S h s q"
proof (simp only: eval_on_carrier UP_mult_closed)
from R S have
"(\<Oplus>S i ∈ {..deg R (p ⊗P q)}. h (coeff P (p ⊗P q) i) ⊗S s (^)S i) =
(\<Oplus>S i ∈ {..deg R (p ⊗P q)} ∪ {deg R (p ⊗P q)<..deg R p + deg R q}.
h (coeff P (p ⊗P q) i) ⊗S s (^)S 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>S i ∈ {..deg R p + deg R q}. h (coeff P (p ⊗P q) i) ⊗S s (^)S i)"

by (simp only: ivl_disj_un_one deg_mult_ring)
also from R S have "... =
(\<Oplus>S i ∈ {..deg R p + deg R q}.
\<Oplus>S k ∈ {..i}.
h (coeff P p k) ⊗S h (coeff P q (i - k)) ⊗S
(s (^)S k ⊗S s (^)S (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>S i∈{..deg R p}. h (coeff P p i) ⊗S s (^)S i) ⊗S
(\<Oplus>S i∈{..deg R q}. h (coeff P q i) ⊗S s (^)S i)"

by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
Pi_def)
finally show
"(\<Oplus>S i ∈ {..deg R (p ⊗P q)}. h (coeff P (p ⊗P q) i) ⊗S s (^)S i) =
(\<Oplus>S i ∈ {..deg R p}. h (coeff P p i) ⊗S s (^)S i) ⊗S
(\<Oplus>S i ∈ {..deg R q}. h (coeff P q i) ⊗S s (^)S 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 ∈ carrier S; r ∈ 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 ∈ 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>S i∈{..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) ⊗S s (^)S i) =
(\<Oplus>S i∈{..deg R (monom P \<one> 1)} ∪ {deg R (monom P \<one> 1)<..1}.
h (coeff P (monom P \<one> 1) i) ⊗S s (^)S 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>S i ∈ {..1}. h (coeff P (monom P \<one> 1) i) ⊗S s (^)S i)"

by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
also have "... = s"
proof (cases "s = \<zero>S")
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>S i ∈ {..deg R (monom P \<one> 1)}.
h (coeff P (monom P \<one> 1) i) ⊗S s (^)S 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 ∈ carrier R"
shows "(monom P a n) (^)P m = monom P (a (^) m) (n * m)"
proof (induct m)
case 0 from R show ?case by simp
next
case Suc with R show ?case
by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
qed

lemma (in ring_hom_cring) hom_pow [simp]:
"x ∈ carrier R ==> h (x (^) n) = h x (^)S (n::nat)"
by (induct n) simp_all

lemma (in UP_univ_prop) Eval_monom:
"r ∈ carrier R ==> Eval (monom P r n) = h r ⊗S s (^)S n"
proof -
assume R: "r ∈ carrier R"
from R have "Eval (monom P r n) = Eval (monom P r 0 ⊗P (monom P \<one> 1) (^)P n)"
by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)
also
from R eval_monom1 [where s = s, folded Eval_def]
have "... = h r ⊗S s (^)S n"
by (simp add: eval_const [where s = s, folded Eval_def])
finally show ?thesis .
qed

lemma (in UP_pre_univ_prop) eval_monom:
assumes R: "r ∈ carrier R" and S: "s ∈ carrier S"
shows "eval R S h s (monom P r n) = h r ⊗S s (^)S n"
proof -
interpret UP_univ_prop R S h P s "eval R S h s"
using UP_pre_univ_prop_axioms P_def R S
by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)
from R
show ?thesis by (rule Eval_monom)
qed

lemma (in UP_univ_prop) Eval_smult:
"[| r ∈ carrier R; p ∈ carrier P |] ==> Eval (r \<odot>P p) = h r ⊗S Eval p"
proof -
assume R: "r ∈ carrier R" and P: "p ∈ carrier P"
then show ?thesis
by (simp add: monom_mult_is_smult [THEN sym]
eval_const [where s = s, folded Eval_def])
qed

lemma ring_hom_cringI:
assumes "cring R"
and "cring S"
and "h ∈ 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 assms)

context UP_pre_univ_prop
begin

lemma UP_hom_unique:
assumes "ring_hom_cring P S Phi"
assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"
"!!r. r ∈ carrier R ==> Phi (monom P r 0) = h r"
assumes "ring_hom_cring P S Psi"
assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"
"!!r. r ∈ carrier R ==> Psi (monom P r 0) = h r"
and P: "p ∈ carrier P" and S: "s ∈ carrier S"
shows "Phi p = Psi p"
proof -
interpret ring_hom_cring P S Phi by fact
interpret ring_hom_cring P S Psi by fact
have "Phi p =
Phi (\<Oplus>P i ∈ {..deg R p}. monom P (coeff P p i) 0 ⊗P monom P \<one> 1 (^)P i)"

by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
also
have "... =
Psi (\<Oplus>P i∈{..deg R p}. monom P (coeff P p i) 0 ⊗P monom P \<one> 1 (^)P i)"

by (simp add: Phi Psi P Pi_def comp_def)
also have "... = Psi p"
by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
finally show ?thesis .
qed

lemma ring_homD:
assumes Phi: "Phi ∈ ring_hom P S"
shows "ring_hom_cring P S Phi"
by unfold_locales (rule Phi)

theorem UP_universal_property:
assumes S: "s ∈ carrier S"
shows "EX! Phi. Phi ∈ ring_hom P S ∩ extensional (carrier P) &
Phi (monom P \<one> 1) = s &
(ALL r : carrier R. Phi (monom P r 0) = h r)"

using S eval_monom1
apply (auto intro: eval_ring_hom eval_const eval_extensional)
apply (rule extensionalityI)
apply (auto intro: UP_hom_unique ring_homD)
done

end

text{*JE: The following lemma was added by me; it might be even lifted to a simpler locale*}

context monoid
begin

lemma nat_pow_eone[simp]: assumes x_in_G: "x ∈ carrier G" shows "x (^) (1::nat) = x"
using nat_pow_Suc [of x 0] unfolding nat_pow_0 [of x] unfolding l_one [OF x_in_G] by simp

end

context UP_ring
begin

abbreviation lcoeff :: "(nat =>'a) => 'a" where "lcoeff p == coeff P p (deg R p)"

lemma lcoeff_nonzero2: assumes p_in_R: "p ∈ carrier P" and p_not_zero: "p ≠ \<zero>P" shows "lcoeff p ≠ \<zero>"
using lcoeff_nonzero [OF p_not_zero p_in_R] .


subsection{*The long division algorithm: some previous facts.*}

lemma coeff_minus [simp]:
assumes p: "p ∈ carrier P" and q: "q ∈ carrier P" shows "coeff P (p \<ominus>P q) n = coeff P p n \<ominus> coeff P q n"
unfolding a_minus_def [OF p q] unfolding coeff_add [OF p a_inv_closed [OF q]] unfolding coeff_a_inv [OF q]
using coeff_closed [OF p, of n] using coeff_closed [OF q, of n] by algebra

lemma lcoeff_closed [simp]: assumes p: "p ∈ carrier P" shows "lcoeff p ∈ carrier R"
using coeff_closed [OF p, of "deg R p"] by simp

lemma deg_smult_decr: assumes a_in_R: "a ∈ carrier R" and f_in_P: "f ∈ carrier P" shows "deg R (a \<odot>P f) ≤ deg R f"
using deg_smult_ring [OF a_in_R f_in_P] by (cases "a = \<zero>", auto)

lemma coeff_monom_mult: assumes R: "c ∈ carrier R" and P: "p ∈ carrier P"
shows "coeff P (monom P c n ⊗P p) (m + n) = c ⊗ (coeff P p m)"
proof -
have "coeff P (monom P c n ⊗P p) (m + n) = (\<Oplus>i∈{..m + n}. (if n = i then c else \<zero>) ⊗ coeff P p (m + n - i))"
unfolding coeff_mult [OF monom_closed [OF R, of n] P, of "m + n"] unfolding coeff_monom [OF R, of n] by simp
also have "(\<Oplus>i∈{..m + n}. (if n = i then c else \<zero>) ⊗ coeff P p (m + n - i)) =
(\<Oplus>i∈{..m + n}. (if n = i then c ⊗ coeff P p (m + n - i) else \<zero>))"

using R.finsum_cong [of "{..m + n}" "{..m + n}" "(λi::nat. (if n = i then c else \<zero>) ⊗ coeff P p (m + n - i))"
"(λi::nat. (if n = i then c ⊗ coeff P p (m + n - i) else \<zero>))"]
using coeff_closed [OF P] unfolding Pi_def simp_implies_def using R by auto
also have "… = c ⊗ coeff P p m" using R.finsum_singleton [of n "{..m + n}" "(λi. c ⊗ coeff P p (m + n - i))"]
unfolding Pi_def using coeff_closed [OF P] using P R by auto
finally show ?thesis by simp
qed

lemma deg_lcoeff_cancel:
assumes p_in_P: "p ∈ carrier P" and q_in_P: "q ∈ carrier P" and r_in_P: "r ∈ carrier P"
and deg_r_nonzero: "deg R r ≠ 0"
and deg_R_p: "deg R p ≤ deg R r" and deg_R_q: "deg R q ≤ deg R r"
and coeff_R_p_eq_q: "coeff P p (deg R r) = \<ominus>R (coeff P q (deg R r))"
shows "deg R (p ⊕P q) < deg R r"
proof -
have deg_le: "deg R (p ⊕P q) ≤ deg R r"
proof (rule deg_aboveI)
fix m
assume deg_r_le: "deg R r < m"
show "coeff P (p ⊕P q) m = \<zero>"
proof -
have slp: "deg R p < m" and "deg R q < m" using deg_R_p deg_R_q using deg_r_le by auto
then have max_sl: "max (deg R p) (deg R q) < m" by simp
then have "deg R (p ⊕P q) < m" using deg_add [OF p_in_P q_in_P] by arith
with deg_R_p deg_R_q show ?thesis using coeff_add [OF p_in_P q_in_P, of m]
using deg_aboveD [of "p ⊕P q" m] using p_in_P q_in_P by simp
qed
qed (simp add: p_in_P q_in_P)
moreover have deg_ne: "deg R (p ⊕P q) ≠ deg R r"
proof (rule ccontr)
assume nz: "¬ deg R (p ⊕P q) ≠ deg R r" then have deg_eq: "deg R (p ⊕P q) = deg R r" by simp
from deg_r_nonzero have r_nonzero: "r ≠ \<zero>P" by (cases "r = \<zero>P", simp_all)
have "coeff P (p ⊕P q) (deg R r) = \<zero>R" using coeff_add [OF p_in_P q_in_P, of "deg R r"] using coeff_R_p_eq_q
using coeff_closed [OF p_in_P, of "deg R r"] coeff_closed [OF q_in_P, of "deg R r"] by algebra
with lcoeff_nonzero [OF r_nonzero r_in_P] and deg_eq show False using lcoeff_nonzero [of "p ⊕P q"] using p_in_P q_in_P
using deg_r_nonzero by (cases "p ⊕P q ≠ \<zero>P", auto)
qed
ultimately show ?thesis by simp
qed

lemma monom_deg_mult:
assumes f_in_P: "f ∈ carrier P" and g_in_P: "g ∈ carrier P" and deg_le: "deg R g ≤ deg R f"
and a_in_R: "a ∈ carrier R"
shows "deg R (g ⊗P monom P a (deg R f - deg R g)) ≤ deg R f"
using deg_mult_ring [OF g_in_P monom_closed [OF a_in_R, of "deg R f - deg R g"]]
apply (cases "a = \<zero>") using g_in_P apply simp
using deg_monom [OF _ a_in_R, of "deg R f - deg R g"] using deg_le by simp

lemma deg_zero_impl_monom:
assumes f_in_P: "f ∈ carrier P" and deg_f: "deg R f = 0"
shows "f = monom P (coeff P f 0) 0"
apply (rule up_eqI) using coeff_monom [OF coeff_closed [OF f_in_P], of 0 0]
using f_in_P deg_f using deg_aboveD [of f _] by auto

end


subsection {* The long division proof for commutative rings *}

context UP_cring
begin

lemma exI3: assumes exist: "Pred x y z"
shows "∃ x y z. Pred x y z"
using exist by blast

text {* Jacobson's Theorem 2.14 *}

lemma long_div_theorem:
assumes g_in_P [simp]: "g ∈ carrier P" and f_in_P [simp]: "f ∈ carrier P"
and g_not_zero: "g ≠ \<zero>P"
shows "∃ q r (k::nat). (q ∈ carrier P) ∧ (r ∈ carrier P) ∧ (lcoeff g)(^)Rk \<odot>P f = g ⊗P q ⊕P r ∧ (r = \<zero>P | deg R r < deg R g)"
using f_in_P
proof (induct "deg R f" arbitrary: "f" rule: nat_less_induct)
case (1 f)
note f_in_P [simp] = "1.prems"
let ?pred = "(λ q r (k::nat).
(q ∈ carrier P) ∧ (r ∈ carrier P)
∧ (lcoeff g)(^)Rk \<odot>P f = g ⊗P q ⊕P r ∧ (r = \<zero>P | deg R r < deg R g))"

let ?lg = "lcoeff g" and ?lf = "lcoeff f"
show ?case
proof (cases "deg R f < deg R g")
case True
have "?pred \<zero>P f 0" using True by force
then show ?thesis by blast
next
case False then have deg_g_le_deg_f: "deg R g ≤ deg R f" by simp
{
let ?k = "1::nat"
let ?f1 = "(g ⊗P (monom P (?lf) (deg R f - deg R g))) ⊕P \<ominus>P (?lg \<odot>P f)"
let ?q = "monom P (?lf) (deg R f - deg R g)"
have f1_in_carrier: "?f1 ∈ carrier P" and q_in_carrier: "?q ∈ carrier P" by simp_all
show ?thesis
proof (cases "deg R f = 0")
case True
{
have deg_g: "deg R g = 0" using True using deg_g_le_deg_f by simp
have "?pred f \<zero>P 1"
using deg_zero_impl_monom [OF g_in_P deg_g]
using sym [OF monom_mult_is_smult [OF coeff_closed [OF g_in_P, of 0] f_in_P]]
using deg_g by simp
then show ?thesis by blast
}
next
case False note deg_f_nzero = False
{
have exist: "lcoeff g (^) ?k \<odot>P f = g ⊗P ?q ⊕P \<ominus>P ?f1"
by (simp add: minus_add r_neg sym [
OF a_assoc [of "g ⊗P ?q" "\<ominus>P (g ⊗P ?q)" "lcoeff g \<odot>P f"]])
have deg_remainder_l_f: "deg R (\<ominus>P ?f1) < deg R f"
proof (unfold deg_uminus [OF f1_in_carrier])
show "deg R ?f1 < deg R f"
proof (rule deg_lcoeff_cancel)
show "deg R (\<ominus>P (?lg \<odot>P f)) ≤ deg R f"
using deg_smult_ring [of ?lg f]
using lcoeff_nonzero2 [OF g_in_P g_not_zero] by simp
show "deg R (g ⊗P ?q) ≤ deg R f"
by (simp add: monom_deg_mult [OF f_in_P g_in_P deg_g_le_deg_f, of ?lf])
show "coeff P (g ⊗P ?q) (deg R f) = \<ominus> coeff P (\<ominus>P (?lg \<odot>P f)) (deg R f)"
unfolding coeff_mult [OF g_in_P monom_closed
[OF lcoeff_closed [OF f_in_P],
of "deg R f - deg R g"], of "deg R f"]
unfolding coeff_monom [OF lcoeff_closed
[OF f_in_P], of "(deg R f - deg R g)"]
using R.finsum_cong' [of "{..deg R f}" "{..deg R f}"
"(λi. coeff P g i ⊗ (if deg R f - deg R g = deg R f - i then ?lf else \<zero>))"
"(λi. if deg R g = i then coeff P g i ⊗ ?lf else \<zero>)"]
using R.finsum_singleton [of "deg R g" "{.. deg R f}" "(λi. coeff P g i ⊗ ?lf)"]
unfolding Pi_def using deg_g_le_deg_f by force
qed (simp_all add: deg_f_nzero)
qed
then obtain q' r' k'
where rem_desc: "?lg (^) (k'::nat) \<odot>P (\<ominus>P ?f1) = g ⊗P q' ⊕P r'"
and rem_deg: "(r' = \<zero>P ∨ deg R r' < deg R g)"
and q'_in_carrier: "q' ∈ carrier P" and r'_in_carrier: "r' ∈ carrier P"
using "1.hyps" using f1_in_carrier by blast
show ?thesis
proof (rule exI3 [of _ "((?lg (^) k') \<odot>P ?q ⊕P q')" r' "Suc k'"], intro conjI)
show "(?lg (^) (Suc k')) \<odot>P f = g ⊗P ((?lg (^) k') \<odot>P ?q ⊕P q') ⊕P r'"
proof -
have "(?lg (^) (Suc k')) \<odot>P f = (?lg (^) k') \<odot>P (g ⊗P ?q ⊕P \<ominus>P ?f1)"
using smult_assoc1 [OF _ _ f_in_P] using exist by simp
also have "… = (?lg (^) k') \<odot>P (g ⊗P ?q) ⊕P ((?lg (^) k') \<odot>P ( \<ominus>P ?f1))"
using UP_smult_r_distr by simp
also have "… = (?lg (^) k') \<odot>P (g ⊗P ?q) ⊕P (g ⊗P q' ⊕P r')"
unfolding rem_desc ..
also have "… = (?lg (^) k') \<odot>P (g ⊗P ?q) ⊕P g ⊗P q' ⊕P r'"
using sym [OF a_assoc [of "?lg (^) k' \<odot>P (g ⊗P ?q)" "g ⊗P q'" "r'"]]
using r'_in_carrier q'_in_carrier by simp
also have "… = (?lg (^) k') \<odot>P (?q ⊗P g) ⊕P q' ⊗P g ⊕P r'"
using q'_in_carrier by (auto simp add: m_comm)
also have "… = (((?lg (^) k') \<odot>P ?q) ⊗P g) ⊕P q' ⊗P g ⊕P r'"
using smult_assoc2 q'_in_carrier "1.prems" by auto
also have "… = ((?lg (^) k') \<odot>P ?q ⊕P q') ⊗P g ⊕P r'"
using sym [OF l_distr] and q'_in_carrier by auto
finally show ?thesis using m_comm q'_in_carrier by auto
qed
qed (simp_all add: rem_deg q'_in_carrier r'_in_carrier)
}
qed
}
qed
qed

end


text {*The remainder theorem as corollary of the long division theorem.*}

context UP_cring
begin

lemma deg_minus_monom:
assumes a: "a ∈ carrier R"
and R_not_trivial: "(carrier R ≠ {\<zero>})"
shows "deg R (monom P \<one>R 1 \<ominus>P monom P a 0) = 1"
(is "deg R ?g = 1")
proof -
have "deg R ?g ≤ 1"
proof (rule deg_aboveI)
fix m
assume "(1::nat) < m"
then show "coeff P ?g m = \<zero>"
using coeff_minus using a by auto algebra
qed (simp add: a)
moreover have "deg R ?g ≥ 1"
proof (rule deg_belowI)
show "coeff P ?g 1 ≠ \<zero>"
using a using R.carrier_one_not_zero R_not_trivial by simp algebra
qed (simp add: a)
ultimately show ?thesis by simp
qed

lemma lcoeff_monom:
assumes a: "a ∈ carrier R" and R_not_trivial: "(carrier R ≠ {\<zero>})"
shows "lcoeff (monom P \<one>R 1 \<ominus>P monom P a 0) = \<one>"
using deg_minus_monom [OF a R_not_trivial]
using coeff_minus a by auto algebra

lemma deg_nzero_nzero:
assumes deg_p_nzero: "deg R p ≠ 0"
shows "p ≠ \<zero>P"
using deg_zero deg_p_nzero by auto

lemma deg_monom_minus:
assumes a: "a ∈ carrier R"
and R_not_trivial: "carrier R ≠ {\<zero>}"
shows "deg R (monom P \<one>R 1 \<ominus>P monom P a 0) = 1"
(is "deg R ?g = 1")
proof -
have "deg R ?g ≤ 1"
proof (rule deg_aboveI)
fix m::nat assume "1 < m" then show "coeff P ?g m = \<zero>"
using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of m]
using coeff_monom [OF R.one_closed, of 1 m] using coeff_monom [OF a, of 0 m] by auto algebra
qed (simp add: a)
moreover have "1 ≤ deg R ?g"
proof (rule deg_belowI)
show "coeff P ?g 1 ≠ \<zero>"
using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of 1]
using coeff_monom [OF R.one_closed, of 1 1] using coeff_monom [OF a, of 0 1]
using R_not_trivial using R.carrier_one_not_zero
by auto algebra
qed (simp add: a)
ultimately show ?thesis by simp
qed

lemma eval_monom_expr:
assumes a: "a ∈ carrier R"
shows "eval R R id a (monom P \<one>R 1 \<ominus>P monom P a 0) = \<zero>"
(is "eval R R id a ?g = _")
proof -
interpret UP_pre_univ_prop R R id by unfold_locales simp
have eval_ring_hom: "eval R R id a ∈ ring_hom P R" using eval_ring_hom [OF a] by simp
interpret ring_hom_cring P R "eval R R id a" by unfold_locales (rule eval_ring_hom)
have mon1_closed: "monom P \<one>R 1 ∈ carrier P"
and mon0_closed: "monom P a 0 ∈ carrier P"
and min_mon0_closed: "\<ominus>P monom P a 0 ∈ carrier P"
using a R.a_inv_closed by auto
have "eval R R id a ?g = eval R R id a (monom P \<one> 1) \<ominus> eval R R id a (monom P a 0)"
unfolding P.minus_eq [OF mon1_closed mon0_closed]
unfolding hom_add [OF mon1_closed min_mon0_closed]
unfolding hom_a_inv [OF mon0_closed]
using R.minus_eq [symmetric] mon1_closed mon0_closed by auto
also have "… = a \<ominus> a"
using eval_monom [OF R.one_closed a, of 1] using eval_monom [OF a a, of 0] using a by simp
also have "… = \<zero>"
using a by algebra
finally show ?thesis by simp
qed

lemma remainder_theorem_exist:
assumes f: "f ∈ carrier P" and a: "a ∈ carrier R"
and R_not_trivial: "carrier R ≠ {\<zero>}"
shows "∃ q r. (q ∈ carrier P) ∧ (r ∈ carrier P) ∧ f = (monom P \<one>R 1 \<ominus>P monom P a 0) ⊗P q ⊕P r ∧ (deg R r = 0)"
(is "∃ q r. (q ∈ carrier P) ∧ (r ∈ carrier P) ∧ f = ?g ⊗P q ⊕P r ∧ (deg R r = 0)")
proof -
let ?g = "monom P \<one>R 1 \<ominus>P monom P a 0"
from deg_minus_monom [OF a R_not_trivial]
have deg_g_nzero: "deg R ?g ≠ 0" by simp
have "∃q r (k::nat). q ∈ carrier P ∧ r ∈ carrier P ∧
lcoeff ?g (^) k \<odot>P f = ?g ⊗P q ⊕P r ∧ (r = \<zero>P ∨ deg R r < deg R ?g)"

using long_div_theorem [OF _ f deg_nzero_nzero [OF deg_g_nzero]] a
by auto
then show ?thesis
unfolding lcoeff_monom [OF a R_not_trivial]
unfolding deg_monom_minus [OF a R_not_trivial]
using smult_one [OF f] using deg_zero by force
qed

lemma remainder_theorem_expression:
assumes f [simp]: "f ∈ carrier P" and a [simp]: "a ∈ carrier R"
and q [simp]: "q ∈ carrier P" and r [simp]: "r ∈ carrier P"
and R_not_trivial: "carrier R ≠ {\<zero>}"
and f_expr: "f = (monom P \<one>R 1 \<ominus>P monom P a 0) ⊗P q ⊕P r"
(is "f = ?g ⊗P q ⊕P r" is "f = ?gq ⊕P r")
and deg_r_0: "deg R r = 0"
shows "r = monom P (eval R R id a f) 0"
proof -
interpret UP_pre_univ_prop R R id P by default simp
have eval_ring_hom: "eval R R id a ∈ ring_hom P R"
using eval_ring_hom [OF a] by simp
have "eval R R id a f = eval R R id a ?gq ⊕R eval R R id a r"
unfolding f_expr using ring_hom_add [OF eval_ring_hom] by auto
also have "… = ((eval R R id a ?g) ⊗ (eval R R id a q)) ⊕R eval R R id a r"
using ring_hom_mult [OF eval_ring_hom] by auto
also have "… = \<zero> ⊕ eval R R id a r"
unfolding eval_monom_expr [OF a] using eval_ring_hom
unfolding ring_hom_def using q unfolding Pi_def by simp
also have "… = eval R R id a r"
using eval_ring_hom unfolding ring_hom_def using r unfolding Pi_def by simp
finally have eval_eq: "eval R R id a f = eval R R id a r" by simp
from deg_zero_impl_monom [OF r deg_r_0]
have "r = monom P (coeff P r 0) 0" by simp
with eval_const [OF a, of "coeff P r 0"] eval_eq
show ?thesis by auto
qed

corollary remainder_theorem:
assumes f [simp]: "f ∈ carrier P" and a [simp]: "a ∈ carrier R"
and R_not_trivial: "carrier R ≠ {\<zero>}"
shows "∃ q r. (q ∈ carrier P) ∧ (r ∈ carrier P) ∧
f = (monom P \<one>R 1 \<ominus>P monom P a 0) ⊗P q ⊕P monom P (eval R R id a f) 0"

(is "∃ q r. (q ∈ carrier P) ∧ (r ∈ carrier P) ∧ f = ?g ⊗P q ⊕P monom P (eval R R id a f) 0")
proof -
from remainder_theorem_exist [OF f a R_not_trivial]
obtain q r
where q_r: "q ∈ carrier P ∧ r ∈ carrier P ∧ f = ?g ⊗P q ⊕P r"
and deg_r: "deg R r = 0" by force
with remainder_theorem_expression [OF f a _ _ R_not_trivial, of q r]
show ?thesis by auto
qed

end


subsection {* Sample Application of Evaluation Homomorphism *}

lemma UP_pre_univ_propI:
assumes "cring R"
and "cring S"
and "h ∈ ring_hom R S"
shows "UP_pre_univ_prop R S h"
using assms
by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro
ring_hom_cring_axioms.intro UP_cring.intro)

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

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

lemma INTEG_id_eval:
"UP_pre_univ_prop INTEG INTEG id"
by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)

text {*
Interpretation now enables to import all theorems and lemmas
valid in the context of homomorphisms between @{term INTEG} and @{term
"UP INTEG"} globally.
*}


interpretation INTEG: UP_pre_univ_prop INTEG INTEG id "UP INTEG"
using INTEG_id_eval by simp_all

lemma INTEG_closed [intro, simp]:
"z ∈ 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: INTEG.eval_monom)

end