# Theory UnivPoly

theory UnivPoly
imports Module RingHom
(*  Title:      HOL/Algebra/UnivPoly.thy
Author:     Clemens Ballarin, started 9 December 1996

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 ‹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 ∧ (∃n. bound 𝟬⇘R⇙ n f)}"

definition UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
where "UP R = ⦇
carrier = up R,
mult = (λp∈up R. λq∈up R. λn. ⨁⇘R⇙i ∈ {..n}. p i ⊗⇘R⇙ q (n-i)),
one = (λi. if i=0 then 𝟭⇘R⇙ else 𝟬⇘R⇙),
zero = (λi. 𝟬⇘R⇙),
add = (λp∈up R. λq∈up R. λi. p i ⊕⇘R⇙ q i),
smult = (λa∈carrier R. λp∈up R. λi. a ⊗⇘R⇙ p i),
monom = (λa∈carrier R. λn i. if i=n then a else 𝟬⇘R⇙),
coeff = (λp∈up R. λn. p n)⦈"

text ‹
Properties of the set of polynomials \<^term>‹up›.
›

lemma mem_upI [intro]:
"[| ⋀n. f n ∈ carrier R; ∃n. bound (zero R) n f |] ==> f ∈ up R"

lemma mem_upD [dest]:
"f ∈ up R ==> f n ∈ carrier R"

context ring
begin

lemma bound_upD [dest]: "f ∈ up R ⟹ ∃n. bound 𝟬 n f" by (simp add: up_def)

lemma up_one_closed: "(λn. if n = 0 then 𝟭 else 𝟬) ∈ up R" using up_def by force

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

"[| p ∈ up R; q ∈ up R |] ==> (λi. 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 "∃n. bound 𝟬 n (λi. p i ⊕ q i)"
proof -
from UP obtain n where boundn: "bound 𝟬 n p" by fast
from UP obtain m where boundm: "bound 𝟬 m q" by fast
have "bound 𝟬 (max n m) (λi. p i ⊕ q i)"
proof
fix i
assume "max n m < i"
with boundn and boundm and UP show "p i ⊕ q i = 𝟬" by fastforce
qed
then show ?thesis ..
qed
qed

lemma up_a_inv_closed:
"p ∈ up R ==> (λi. ⊖ (p i)) ∈ up R"
proof
assume R: "p ∈ up R"
then obtain n where "bound 𝟬 n p" by auto
then have "bound 𝟬 n (λi. ⊖ p i)"
then show "∃n. bound 𝟬 n (λi. ⊖ p i)" by auto
qed auto

lemma up_minus_closed:
"[| p ∈ up R; q ∈ up R |] ==> (λi. p i ⊖ q i) ∈ up R"
unfolding a_minus_def
using mem_upD [of p R] mem_upD [of q R] up_add_closed up_a_inv_closed  by auto

lemma up_mult_closed:
"[| p ∈ up R; q ∈ up R |] ==>
(λn. ⨁i ∈ {..n}. p i ⊗ q (n-i)) ∈ up R"
proof
fix n
assume "p ∈ up R" "q ∈ up R"
then show "(⨁i ∈ {..n}. p i ⊗ q (n-i)) ∈ carrier R"
next
assume UP: "p ∈ up R" "q ∈ up R"
show "∃n. bound 𝟬 n (λn. ⨁i ∈ {..n}. p i ⊗ q (n-i))"
proof -
from UP obtain n where boundn: "bound 𝟬 n p" by fast
from UP obtain m where boundm: "bound 𝟬 m q" by fast
have "bound 𝟬 (n + m) (λn. ⨁i ∈ {..n}. p i ⊗ q (n - i))"
proof
fix k assume bound: "n + m < k"
{
fix i
have "p i ⊗ q (k-i) = 𝟬"
proof (cases "n < i")
case True
with boundn have "p i = 𝟬" 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) = 𝟬" by auto
moreover from UP have "p i ∈ carrier R" by auto
ultimately show ?thesis by simp
qed
}
then show "(⨁i ∈ {..k}. p i ⊗ q (k-i)) = 𝟬"
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  (rule P_def)

locale UP_domain = UP + R?: "domain" R

sublocale UP_domain < UP_cring
by intro_locales  (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 𝟬)"
proof -
assume R: "a ∈ carrier R"
then have "(λn. if n = m then a else 𝟬) ∈ up R"
using up_def by force
with R show ?thesis by (simp add: UP_def)
qed

lemma coeff_zero [simp]: "coeff P 𝟬⇘P⇙ n = 𝟬" by (auto simp add: UP_def)

lemma coeff_one [simp]: "coeff P 𝟭⇘P⇙ n = (if n=0 then 𝟭 else 𝟬)"
using up_one_closed by (simp add: UP_def)

lemma coeff_smult [simp]:
"[| a ∈ carrier R; p ∈ carrier P |] ==> coeff P (a ⊙⇘P⇙ p) n = a ⊗ coeff P p n"

"[| p ∈ carrier P; q ∈ carrier P |] ==> coeff P (p ⊕⇘P⇙ q) n = coeff P p n ⊕ coeff P q n"

lemma coeff_mult [simp]:
"[| p ∈ carrier P; q ∈ carrier P |] ==> coeff P (p ⊗⇘P⇙ q) n = (⨁i ∈ {..n}. coeff P p i ⊗ coeff P q (n-i))"

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]:
"𝟭⇘P⇙ ∈ carrier P" by (simp add: UP_def up_one_closed)

lemma UP_zero_closed [intro, simp]:
"𝟬⇘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 ⊙⇘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 "𝟬⇘P⇙ ⊕⇘P⇙ p = p" by (rule up_eqI, simp_all add: R)

lemma UP_l_neg_ex:
assumes R: "p ∈ carrier P"
shows "∃q ∈ carrier P. q ⊕⇘P⇙ p = 𝟬⇘P⇙"
proof -
let ?q = "λi. ⊖ (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 = ⊖ (coeff P p n)"
by (simp add: UP_def P_def up_a_inv_closed)
show ?thesis
proof
show "?q ⊕⇘P⇙ p = 𝟬⇘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 ==>
(⨁j ∈ {..k}. (⨁i ∈ {..j}. a i ⊗ b (j-i)) ⊗ c (n-j)) =
(⨁j ∈ {..k}. a j ⊗ (⨁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"

lemma UP_r_one [simp]:
assumes R: "p ∈ carrier P" shows "p ⊗⇘P⇙ 𝟭⇘P⇙ = p"
proof (rule up_eqI)
fix n
show "coeff P (p ⊗⇘P⇙ 𝟭⇘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⇙ 𝟭⇘P⇙) (Suc nn) = coeff P p (Suc nn)"
proof -
have "coeff P (p ⊗⇘P⇙ 𝟭⇘P⇙) (Suc nn) = (⨁i∈{..Suc nn}. coeff P p i ⊗ (if Suc nn ≤ i then 𝟭 else 𝟬))" using R by simp
also have "… = coeff P p (Suc nn) ⊗ (if Suc nn ≤ Suc nn then 𝟭 else 𝟬) ⊕ (⨁i∈{..nn}. coeff P p i ⊗ (if Suc nn ≤ i then 𝟭 else 𝟬))"
using finsum_Suc [of "(λi::nat. coeff P p i ⊗ (if Suc nn ≤ i then 𝟭 else 𝟬))" "nn"] unfolding Pi_def using R by simp
also have "… = coeff P p (Suc nn) ⊗ (if Suc nn ≤ Suc nn then 𝟭 else 𝟬)"
proof -
have "(⨁i∈{..nn}. coeff P p i ⊗ (if Suc nn ≤ i then 𝟭 else 𝟬)) = (⨁i∈{..nn}. 𝟬)"
using finsum_cong [of "{..nn}" "{..nn}" "(λi::nat. coeff P p i ⊗ (if Suc nn ≤ i then 𝟭 else 𝟬))" "(λi::nat. 𝟬)"] using R
unfolding Pi_def by simp
also have "… = 𝟬" 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

lemma UP_l_one [simp]:
assumes R: "p ∈ carrier P"
shows "𝟭⇘P⇙ ⊗⇘P⇙ p = p"
proof (rule up_eqI)
fix n
show "coeff P (𝟭⇘P⇙ ⊗⇘P⇙ p) n = coeff P p n"
proof (cases n)
case 0 with R show ?thesis by simp
next
case Suc with R show ?thesis
by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
qed

lemma 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)"

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)"

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 ==>
(⨁i ∈ {..k}. a i ⊗ b (n-i)) = (⨁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)

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 ==> ⊖⇘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 (⊖⇘P⇙ p) n = ⊖ (coeff P p n)"
proof -
from R coeff_closed UP_a_inv_closed have
"coeff P (⊖⇘P⇙ p) n = ⊖ coeff P p n ⊕ (coeff P p n ⊕ coeff P (⊖⇘P⇙ p) n)"
by algebra
also from R have "... =  ⊖ (coeff P p n)"
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) ⊙⇘P⇙ p = a ⊙⇘P⇙ p ⊕⇘P⇙ b ⊙⇘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 ⊙⇘P⇙ (p ⊕⇘P⇙ q) = a ⊙⇘P⇙ p ⊕⇘P⇙ a ⊙⇘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) ⊙⇘P⇙ p = a ⊙⇘P⇙ (b ⊙⇘P⇙ p)"
by (rule up_eqI) (simp_all add: R.m_assoc)

lemma UP_smult_zero [simp]:
"p ∈ carrier P ==> 𝟬 ⊙⇘P⇙ p = 𝟬⇘P⇙"
by (rule up_eqI) simp_all

lemma UP_smult_one [simp]:
"p ∈ carrier P ==> 𝟭 ⊙⇘P⇙ p = p"
by (rule up_eqI) simp_all

lemma UP_smult_assoc2:
"[| a ∈ carrier R; p ∈ carrier P; q ∈ carrier P |] ==>
(a ⊙⇘P⇙ p) ⊗⇘P⇙ q = a ⊙⇘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 𝟬 n = 𝟬⇘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 ⊙⇘P⇙ p"
proof (rule up_eqI)
fix n
show "coeff P (monom P a 0 ⊗⇘P⇙ p) n = coeff P (a ⊙⇘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: Pi_def)
qed

lemma monom_one [simp]:
"monom P 𝟭 0 = 𝟭⇘P⇙"
by (rule up_eqI) simp_all

"[| 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 𝟭 (Suc n) = monom P 𝟭 n ⊗⇘P⇙ monom P 𝟭 1"
proof (rule up_eqI)
fix k
show "coeff P (monom P 𝟭 (Suc n)) k = coeff P (monom P 𝟭 n ⊗⇘P⇙ monom P 𝟭 1) k"
proof (cases "k = Suc n")
case True show ?thesis
proof -
fix m
"!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
from True have "coeff P (monom P 𝟭 (Suc n)) k = 𝟭" by simp
also from True
have "... = (⨁i ∈ {..<n} ∪ {n}. coeff P (monom P 𝟭 n) i ⊗
coeff P (monom P 𝟭 1) (k - i))"
by (simp cong: R.finsum_cong add: Pi_def)
also have "... = (⨁i ∈  {..n}. coeff P (monom P 𝟭 n) i ⊗
coeff P (monom P 𝟭 1) (k - i))"
by (simp only: ivl_disj_un_singleton)
also from True
have "... = (⨁i ∈ {..n} ∪ {n<..k}. coeff P (monom P 𝟭 n) i ⊗
coeff P (monom P 𝟭 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 𝟭 n ⊗⇘P⇙ monom P 𝟭 1) k"
finally show ?thesis .
qed
next
case False
note neq = False
let ?s =
"λi. (if n = i then 𝟭 else 𝟬) ⊗ (if Suc 0 = k - i then 𝟭 else 𝟬)"
from neq have "coeff P (monom P 𝟭 (Suc n)) k = 𝟬" by simp
also have "... = (⨁i ∈ {..k}. ?s i)"
proof -
have f1: "(⨁i ∈ {..<n}. ?s i) = 𝟬"
by (simp cong: R.finsum_cong add: Pi_def)
from neq have f2: "(⨁i ∈ {n}. ?s i) = 𝟬"
by (simp cong: R.finsum_cong add: Pi_def) arith
have f3: "n < k ==> (⨁i ∈ {n<..k}. ?s i) = 𝟬"
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 "𝟬 = (⨁i ∈ {..<n} ∪ {n}. ?s i)"
by (simp cong: R.finsum_cong add: Pi_def)
also from True have "... = (⨁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 "𝟬 = (⨁i ∈ {..<n} ∪ {n}. ?s i)"
by (simp add: R.finsum_Un_disjoint f1 f2 Pi_def del: Un_insert_right)
also have "... = (⨁i ∈ {..n}. ?s i)"
by (simp only: ivl_disj_un_singleton)
also from n_less_k neq have "... = (⨁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 "... = (⨁i ∈ {..k}. ?s i)"
by (simp only: ivl_disj_un_one)
finally show ?thesis .
qed
qed
qed
also have "... = coeff P (monom P 𝟭 n ⊗⇘P⇙ monom P 𝟭 1) k" by simp
finally show ?thesis .
qed
qed (simp_all)

lemma monom_one_Suc2:
"monom P 𝟭 (Suc n) = monom P 𝟭 1 ⊗⇘P⇙ monom P 𝟭 n"
proof (induct n)
case 0 show ?case by simp
next
case Suc
{
fix k:: nat
assume hypo: "monom P 𝟭 (Suc k) = monom P 𝟭 1 ⊗⇘P⇙ monom P 𝟭 k"
then show "monom P 𝟭 (Suc (Suc k)) = monom P 𝟭 1 ⊗⇘P⇙ monom P 𝟭 (Suc k)"
proof -
have lhs: "monom P 𝟭 (Suc (Suc k)) = monom P 𝟭 1 ⊗⇘P⇙ monom P 𝟭 k ⊗⇘P⇙ monom P 𝟭 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 𝟭 1 ⊗⇘P⇙ monom P 𝟭 (Suc k) = monom P 𝟭 1 ⊗⇘P⇙ monom P 𝟭 k ⊗⇘P⇙ monom P 𝟭 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 𝟭 k ⊗⇘P⇙ monom P 𝟭 1 = monom P 𝟭 1 ⊗⇘P⇙ monom P 𝟭 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 ⊙⇘P⇙ monom P b n"
by (rule up_eqI) simp_all

lemma monom_one_mult:
"monom P 𝟭 (n + m) = monom P 𝟭 n ⊗⇘P⇙ monom P 𝟭 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 𝟭 n ⊗⇘P⇙ monom P 𝟭 m = monom P 𝟭 m ⊗⇘P⇙ monom P 𝟭 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 𝟬) ⊗ (if m = n + m - i then b else 𝟬))"
"(λi. if n = i then a ⊗ b else 𝟬)"]
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 𝟬) ⊗ (if m = k - i then b else 𝟬))" "(λi. 𝟬)"]
unfolding Pi_def simp_implies_def using a_in_R b_in_R by force
}
qed

lemma monom_a_inv [simp]:
"a ∈ carrier R ==> monom P (⊖ a) n = ⊖⇘P⇙ monom P a n"
by (rule up_eqI) auto

lemma monom_inj:
"inj_on (λa. 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 𝟬⇘R⇙ n (coeff (UP R) p))"

context UP_ring
begin

lemma deg_aboveI:
"[| (!!m. n < m ==> coeff P p m = 𝟬); 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 "(λn. 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 "(λn. 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 = 𝟬"
proof -
from ‹p ∈ carrier P› obtain n where "bound 𝟬 n (coeff P p)"
by (auto simp add: UP_def P_def)
then have "bound 𝟬 (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 ≠ 𝟬"
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 ≠ 𝟬" 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) ≠ 𝟬"
proof -
from R obtain m where "deg R p ≤ m" and m_coeff: "coeff P p m ≠ 𝟬"
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 𝟬 n (coeff P p))"
by (unfold deg_def P_def) simp
then have "¬ bound 𝟬 (deg R p - 1) (coeff P p)" by (rule not_less_Least)
then have "∃m. deg R p - 1 < m ∧ coeff P p m ≠ 𝟬"
by (unfold bound_def) fast
then have "∃m. deg R p ≤ m ∧ coeff P p m ≠ 𝟬" 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 ≠ 𝟬⇘P⇙" and R: "p ∈ carrier P"
shows "coeff P p 0 ≠ 𝟬"
proof -
have "∃m. coeff P p m ≠ 𝟬"
proof (rule classical)
assume "¬ ?thesis"
with R have "p = 𝟬⇘P⇙" by (auto intro: up_eqI)
with nonzero show ?thesis by contradiction
qed
then obtain m where coeff: "coeff P p m ≠ 𝟬" ..
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 ≠ 𝟬⇘P⇙" and R: "p ∈ carrier P"
shows "coeff P p (deg R p) ≠ 𝟬"
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 = 𝟬;
⋀n. n ≠ 0 ⟹ coeff P p n ≠ 𝟬; p ∈ carrier P |] ==> deg R p = n"
by (fast intro: le_antisym deg_aboveI deg_belowI)

text ‹Degree and polynomial operations›

"p ∈ carrier P ⟹ q ∈ carrier P ⟹
deg R (p ⊕⇘P⇙ q) ≤ max (deg R p) (deg R q)"

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 ≠ 𝟬; 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 𝟬⇘P⇙ = 0"
proof (rule le_antisym)
show "deg R 𝟬⇘P⇙ ≤ 0" by (rule deg_aboveI) simp_all
next
show "0 ≤ deg R 𝟬⇘P⇙" by (rule deg_belowI) simp_all
qed

lemma deg_one [simp]:
"deg R 𝟭⇘P⇙ = 0"
proof (rule le_antisym)
show "deg R 𝟭⇘P⇙ ≤ 0" by (rule deg_aboveI) simp_all
next
show "0 ≤ deg R 𝟭⇘P⇙" by (rule deg_belowI) simp_all
qed

lemma deg_uminus [simp]:
assumes R: "p ∈ carrier P" shows "deg R (⊖⇘P⇙ p) = deg R p"
proof (rule le_antisym)
show "deg R (⊖⇘P⇙ p) ≤ deg R p" by (simp add: deg_aboveI deg_aboveD R)
next
show "deg R p ≤ deg R (⊖⇘P⇙ p)"
inj_on_eq_iff [OF R.a_inv_inj, of _ "𝟬", simplified] R)
qed

text‹The following lemma is later \emph{overwritten} by the most
specific one for domains, ‹deg_smult›.›

lemma deg_smult_ring [simp]:
"[| a ∈ carrier R; p ∈ carrier P |] ==>
deg R (a ⊙⇘P⇙ p) ≤ (if a = 𝟬 then 0 else deg R p)"
by (cases "a = 𝟬") (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 ⊙⇘P⇙ p) = (if a = 𝟬 then 0 else deg R p)"
proof (rule le_antisym)
show "deg R (a ⊙⇘P⇙ p) ≤ (if a = 𝟬 then 0 else deg R p)"
using R by (rule deg_smult_ring)
next
show "(if a = 𝟬 then 0 else deg R p) ≤ deg R (a ⊙⇘P⇙ p)"
proof (cases "a = 𝟬")
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) = 𝟬"
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 = 𝟬" by simp

end

context UP_domain
begin

lemma deg_mult [simp]:
"[| p ≠ 𝟬⇘P⇙; q ≠ 𝟬⇘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 = "(λi. 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 ≠ 𝟬⇘P⇙" "q ≠ 𝟬⇘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 "(⨁i ∈ {.. deg R p + deg R q}. ?s i)
= (⨁i ∈ {..< deg R p} ∪ {deg R p .. deg R p + deg R q}. ?s i)"
by (simp only: ivl_disj_un_one)
also have "... = (⨁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
also have "...= (⨁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 "(⨁i ∈ {.. deg R p + deg R q}. ?s i)
= coeff P p (deg R p) ⊗ coeff P q (deg R q)" .
with nz show "(⨁i ∈ {.. deg R p + deg R q}. ?s i) ≠ 𝟬"
by (simp add: integral_iff lcoeff_nonzero 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 = (⨁i ∈ A. coeff P (p i) k)"
using fin by induct (auto simp: Pi_def)

lemma up_repr:
assumes R: "p ∈ carrier P"
shows "(⨁⇘P⇙ i ∈ {..deg R p}. monom P (coeff P p i) i) = p"
proof (rule up_eqI)
let ?s = "(λi. monom P (coeff P p i) i)"
fix k
from R have RR: "!!i. (if i = k then coeff P p i else 𝟬) ∈ carrier R"
by simp
show "coeff P (⨁⇘P⇙ i ∈ {..deg R p}. ?s i) k = coeff P p k"
proof (cases "k ≤ deg R p")
case True
hence "coeff P (⨁⇘P⇙ i ∈ {..deg R p}. ?s i) k =
coeff P (⨁⇘P⇙ i ∈ {..k} ∪ {k<..deg R p}. ?s i) k"
by (simp only: ivl_disj_un_one)
also from True
have "... = coeff P (⨁⇘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 (⨁⇘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 (⨁⇘P⇙ i ∈ {..deg R p}. ?s i) k =
coeff P (⨁⇘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

lemma up_repr_le:
"[| deg R p <= n; p ∈ carrier P |] ==>
(⨁⇘P⇙ i ∈ {..n}. monom P (coeff P p i) i) = p"
proof -
let ?s = "(λi. 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:
"𝟭⇘P⇙ ≠ 𝟬⇘P⇙"
proof
assume "𝟭⇘P⇙ = 𝟬⇘P⇙"
hence "coeff P 𝟭⇘P⇙ 0 = (coeff P 𝟬⇘P⇙ 0)" by simp
hence "𝟭 = 𝟬" by simp
with R.one_not_zero show "False" by contradiction
qed

lemma UP_integral:
"[| p ⊗⇘P⇙ q = 𝟬⇘P⇙; p ∈ carrier P; q ∈ carrier P |] ==> p = 𝟬⇘P⇙ ∨ q = 𝟬⇘P⇙"
proof -
fix p q
assume pq: "p ⊗⇘P⇙ q = 𝟬⇘P⇙" and R: "p ∈ carrier P" "q ∈ carrier P"
show "p = 𝟬⇘P⇙ ∨ q = 𝟬⇘P⇙"
proof (rule classical)
assume c: "¬ (p = 𝟬⇘P⇙ ∨ q = 𝟬⇘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 = (⨁⇘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 = (⨁⇘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 "... = 𝟬" by simp
finally have "coeff P p 0 ⊗ coeff P q 0 = 𝟬" .
with R have "coeff P p 0 = 𝟬 ∨ coeff P q 0 = 𝟬"
with p q show "p = 𝟬⇘P⇙ ∨ q = 𝟬⇘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 𝟬 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 |] ==>
(⨁k ∈ {..n + m}. ⨁i ∈ {..k}. f i ⊗ g (k - i)) =
(⨁k ∈ {..n + m}. ⨁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 ==>
(⨁k ∈ {..j}. ⨁i ∈ {..k}. f i ⊗ g (k - i)) =
(⨁k ∈ {..j}. ⨁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 𝟬 n f" and bg: "bound 𝟬 m g"
and Rf: "f ∈ {..n} → carrier R" and Rg: "g ∈ {..m} → carrier R"
shows "(⨁k ∈ {..n + m}. ⨁i ∈ {..k}. f i ⊗ g (k - i)) =
(⨁i ∈ {..n}. f i) ⊗ (⨁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 "(⨁k ∈ {..n + m}. ⨁i ∈ {..k}. f i ⊗ g (k - i)) =
(⨁k ∈ {..n + m}. ⨁i ∈ {..n + m - k}. f k ⊗ g i)"
also have "... = (⨁k ∈ {..n} ∪ {n<..n + m}. ⨁i ∈ {..n + m - k}. f k ⊗ g i)"
by (simp only: ivl_disj_un_one)
also from f g have "... = (⨁k ∈ {..n}. ⨁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 "... = (⨁k ∈ {..n}. ⨁i ∈ {..m} ∪ {m<..n + m - k}. f k ⊗ g i)"
also from f g have "... = (⨁k ∈ {..n}. ⨁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 "... = (⨁i ∈ {..n}. f i) ⊗ (⨁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:
"(λa. 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).
⨁⇘S⇙i ∈ {..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 = (⨁⇘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

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 ‹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]:
"f ∈ A → carrier R ⟹
h (finsum R f A) = finsum S (h ∘ f) A"
by (induct A rule: infinite_finite_induct, auto simp: Pi_def)

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
"(⨁⇘S ⇙i∈{..deg R (p ⊕⇘P⇙ q)}. h (coeff P (p ⊕⇘P⇙ q) i) ⊗⇘S⇙ s [^]⇘S⇙ i) =
(⨁⇘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
also from R have "... =
(⨁⇘S⇙ i ∈ {..max (deg R p) (deg R q)}.
h (coeff P (p ⊕⇘P⇙ q) i) ⊗⇘S⇙ s [^]⇘S⇙ i)"
also from R S have "... =
(⨁⇘S⇙i∈{..max (deg R p) (deg R q)}. h (coeff P p i) ⊗⇘S⇙ s [^]⇘S⇙ i) ⊕⇘S⇙
(⨁⇘S⇙i∈{..max (deg R p) (deg R q)}. h (coeff P q i) ⊗⇘S⇙ s [^]⇘S⇙ i)"
by (simp cong: S.finsum_cong
also have "... =
(⨁⇘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⇙
(⨁⇘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 max.cobounded1 max.cobounded2)
also from R S have "... =
(⨁⇘S⇙ i ∈ {..deg R p}. h (coeff P p i) ⊗⇘S⇙ s [^]⇘S⇙ i) ⊕⇘S⇙
(⨁⇘S⇙ i ∈ {..deg R q}. h (coeff P q i) ⊗⇘S⇙ s [^]⇘S⇙ i)"
by (simp cong: S.finsum_cong
finally show
"(⨁⇘S⇙i ∈ {..deg R (p ⊕⇘P⇙ q)}. h (coeff P (p ⊕⇘P⇙ q) i) ⊗⇘S⇙ s [^]⇘S⇙ i) =
(⨁⇘S⇙i ∈ {..deg R p}. h (coeff P p i) ⊗⇘S⇙ s [^]⇘S⇙ i) ⊕⇘S⇙
(⨁⇘S⇙i ∈ {..deg R q}. h (coeff P q i) ⊗⇘S⇙ s [^]⇘S⇙ i)" .
qed
next
show "eval R S h s 𝟭⇘P⇙ = 𝟭⇘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
"(⨁⇘S⇙ i ∈ {..deg R (p ⊗⇘P⇙ q)}. h (coeff P (p ⊗⇘P⇙ q) i) ⊗⇘S⇙ s [^]⇘S⇙ i) =
(⨁⇘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
del: coeff_mult)
also from R have "... =
(⨁⇘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 "... =
(⨁⇘S⇙ i ∈ {..deg R p + deg R q}.
⨁⇘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 "... =
(⨁⇘S⇙ i∈{..deg R p}. h (coeff P p i) ⊗⇘S⇙ s [^]⇘S⇙ i) ⊗⇘S⇙
(⨁⇘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
"(⨁⇘S⇙ i ∈ {..deg R (p ⊗⇘P⇙ q)}. h (coeff P (p ⊗⇘P⇙ q) i) ⊗⇘S⇙ s [^]⇘S⇙ i) =
(⨁⇘S⇙ i ∈ {..deg R p}. h (coeff P p i) ⊗⇘S⇙ s [^]⇘S⇙ i) ⊗⇘S⇙
(⨁⇘S⇙ i ∈ {..deg R q}. h (coeff P q i) ⊗⇘S⇙ s [^]⇘S⇙ i)" .
qed
qed

text ‹
The following lemma could be proved in ‹UP_cring› with the additional
assumption that ‹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 𝟭 1) = s"
proof (simp only: eval_on_carrier monom_closed R.one_closed)
from S have
"(⨁⇘S⇙ i∈{..deg R (monom P 𝟭 1)}. h (coeff P (monom P 𝟭 1) i) ⊗⇘S⇙ s [^]⇘S⇙ i) =
(⨁⇘S⇙ i∈{..deg R (monom P 𝟭 1)} ∪ {deg R (monom P 𝟭 1)<..1}.
h (coeff P (monom P 𝟭 1) i) ⊗⇘S⇙ s [^]⇘S⇙ i)"
by (simp cong: S.finsum_cong del: coeff_monom
also have "... =
(⨁⇘S⇙ i ∈ {..1}. h (coeff P (monom P 𝟭 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 = 𝟬⇘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 "(⨁⇘S⇙ i ∈ {..deg R (monom P 𝟭 1)}.
h (coeff P (monom P 𝟭 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
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 𝟭 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 ⊙⇘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 𝟭 (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 𝟭 (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 (⨁⇘P ⇙i ∈ {..deg R p}. monom P (coeff P p i) 0 ⊗⇘P⇙ monom P 𝟭 1 [^]⇘P⇙ i)"
by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
also
have "... =
Psi (⨁⇘P ⇙i∈{..deg R p}. monom P (coeff P p i) 0 ⊗⇘P⇙ monom P 𝟭 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 "∃!Phi. Phi ∈ ring_hom P S ∩ extensional (carrier P) ∧
Phi (monom P 𝟭 1) = s ∧
(∀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 ≠ 𝟬⇘P⇙" shows "lcoeff p ≠ 𝟬"
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 ⊖⇘P⇙ q) n = coeff P p n ⊖ coeff P q n"
by (simp add: a_minus_def p q)

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 ⊙⇘P⇙ f) ≤ deg R f"
using deg_smult_ring [OF a_in_R f_in_P] by (cases "a = 𝟬", 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) = (⨁i∈{..m + n}. (if n = i then c else 𝟬) ⊗ 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 "(⨁i∈{..m + n}. (if n = i then c else 𝟬) ⊗ coeff P p (m + n - i)) =
(⨁i∈{..m + n}. (if n = i then c ⊗ coeff P p (m + n - i) else 𝟬))"
using  R.finsum_cong [of "{..m + n}" "{..m + n}" "(λi::nat. (if n = i then c else 𝟬) ⊗ coeff P p (m + n - i))"
"(λi::nat. (if n = i then c ⊗ coeff P p (m + n - i) else 𝟬))"]
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) = ⊖⇘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 = 𝟬"
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
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 ≠ 𝟬⇘P⇙" by (cases "r = 𝟬⇘P⇙", simp_all)
have "coeff P (p ⊕⇘P⇙ q) (deg R r) = 𝟬⇘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 ≠ 𝟬⇘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 = 𝟬") 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 ≠ 𝟬⇘P⇙"
shows "∃q r (k::nat). (q ∈ carrier P) ∧ (r ∈ carrier P) ∧ (lcoeff g)[^]⇘R⇙k ⊙⇘P⇙ f = g ⊗⇘P⇙ q ⊕⇘P⇙ r ∧ (r = 𝟬⇘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)[^]⇘R⇙k ⊙⇘P⇙ f = g ⊗⇘P⇙ q ⊕⇘P⇙ r ∧ (r = 𝟬⇘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 𝟬⇘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⇙ ⊖⇘P⇙ (?lg ⊙⇘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 𝟬⇘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 ⊙⇘P⇙ f = g ⊗⇘P⇙ ?q ⊕⇘P⇙ ⊖⇘P⇙ ?f1"
OF a_assoc [of "g ⊗⇘P⇙ ?q" "⊖⇘P⇙ (g ⊗⇘P⇙ ?q)" "lcoeff g ⊙⇘P⇙ f"]])
have deg_remainder_l_f: "deg R (⊖⇘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 (⊖⇘P⇙ (?lg ⊙⇘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) = ⊖ coeff P (⊖⇘P⇙ (?lg ⊙⇘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 𝟬))"
"(λi. if deg R g = i then coeff P g i ⊗ ?lf else 𝟬)"]
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
then obtain q' r' k'
where rem_desc: "?lg [^] (k'::nat) ⊙⇘P⇙ (⊖⇘P⇙ ?f1) = g ⊗⇘P⇙ q' ⊕⇘P⇙ r'"
and rem_deg: "(r' = 𝟬⇘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') ⊙⇘P⇙ ?q ⊕⇘P⇙ q')" r' "Suc k'"], intro conjI)
show "(?lg [^] (Suc k')) ⊙⇘P⇙ f = g ⊗⇘P⇙ ((?lg [^] k') ⊙⇘P⇙ ?q ⊕⇘P⇙ q') ⊕⇘P⇙ r'"
proof -
have "(?lg [^] (Suc k')) ⊙⇘P⇙ f = (?lg [^] k') ⊙⇘P⇙ (g ⊗⇘P⇙ ?q ⊕⇘P⇙ ⊖⇘P⇙ ?f1)"
using smult_assoc1 [OF _ _ f_in_P] using exist by simp
also have "… = (?lg [^] k') ⊙⇘P⇙ (g ⊗⇘P⇙ ?q) ⊕⇘P⇙ ((?lg [^] k') ⊙⇘P⇙ ( ⊖⇘P⇙ ?f1))"
using UP_smult_r_distr by simp
also have "… = (?lg [^] k') ⊙⇘P⇙ (g ⊗⇘P⇙ ?q) ⊕⇘P⇙ (g ⊗⇘P⇙ q' ⊕⇘P⇙ r')"
unfolding rem_desc ..
also have "… = (?lg [^] k') ⊙⇘P⇙ (g ⊗⇘P⇙ ?q) ⊕⇘P⇙ g ⊗⇘P⇙ q' ⊕⇘P⇙ r'"
using sym [OF a_assoc [of "?lg [^] k' ⊙⇘P⇙ (g ⊗⇘P⇙ ?q)" "g ⊗⇘P⇙ q'" "r'"]]
using r'_in_carrier q'_in_carrier by simp
also have "… = (?lg [^] k') ⊙⇘P⇙ (?q ⊗⇘P⇙ g) ⊕⇘P⇙ q' ⊗⇘P⇙ g ⊕⇘P⇙ r'"
using q'_in_carrier by (auto simp add: m_comm)
also have "… = (((?lg [^] k') ⊙⇘P⇙ ?q) ⊗⇘P⇙ g) ⊕⇘P⇙ q' ⊗⇘P⇙ g ⊕⇘P⇙ r'"
using smult_assoc2 q'_in_carrier "1.prems" by auto
also have "… = ((?lg [^] k') ⊙⇘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 ≠ {𝟬})"
shows "deg R (monom P 𝟭⇘R⇙ 1 ⊖⇘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 = 𝟬"
using coeff_minus using a by auto algebra
moreover have "deg R ?g ≥ 1"
proof (rule deg_belowI)
show "coeff P ?g 1 ≠ 𝟬"
using a using R.carrier_one_not_zero R_not_trivial by simp algebra
ultimately show ?thesis by simp
qed

lemma lcoeff_monom:
assumes a: "a ∈ carrier R" and R_not_trivial: "(carrier R ≠ {𝟬})"
shows "lcoeff (monom P 𝟭⇘R⇙ 1 ⊖⇘P⇙ monom P a 0) = 𝟭"
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 ≠ 𝟬⇘P⇙"
using deg_zero deg_p_nzero by auto

lemma deg_monom_minus:
assumes a: "a ∈ carrier R"
and R_not_trivial: "carrier R ≠ {𝟬}"
shows "deg R (monom P 𝟭⇘R⇙ 1 ⊖⇘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 = 𝟬"
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
moreover have "1 ≤ deg R ?g"
proof (rule deg_belowI)
show "coeff P ?g 1 ≠ 𝟬"
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
ultimately show ?thesis by simp
qed

lemma eval_monom_expr:
assumes a: "a ∈ carrier R"
shows "eval R R id a (monom P 𝟭⇘R⇙ 1 ⊖⇘P⇙ monom P a 0) = 𝟬"
(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 𝟭⇘R⇙ 1 ∈ carrier P"
and mon0_closed: "monom P a 0 ∈ carrier P"
and min_mon0_closed: "⊖⇘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 𝟭 1) ⊖ eval R R id a (monom P a 0)"
also have "… = a ⊖ 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 "… = 𝟬"
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 ≠ {𝟬}"
shows "∃ q r. (q ∈ carrier P) ∧ (r ∈ carrier P) ∧ f = (monom P 𝟭⇘R⇙ 1 ⊖⇘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 𝟭⇘R⇙ 1 ⊖⇘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 ⊙⇘P⇙ f = ?g ⊗⇘P⇙ q ⊕⇘P⇙ r ∧ (r = 𝟬⇘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 ≠ {𝟬}"
and f_expr: "f = (monom P 𝟭⇘R⇙ 1 ⊖⇘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 standard 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 "… = 𝟬 ⊕ 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 ≠ {𝟬}"
shows "∃ q r. (q ∈ carrier P) ∧ (r ∈ carrier P) ∧
f = (monom P 𝟭⇘R⇙ 1 ⊖⇘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 = (*), one = 1, zero = 0, add = (+)⦈"

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"