(* 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>⇘_{R⇙}i ∈ {..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>⇘_{P⇙}⊕⇘_{P⇙}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>⇘_{P⇙}⊗⇘_{P⇙}p = p"

proof (rule up_eqI)

fix n

show "coeff P (\<one>⇘_{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

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>⇘_{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 = (\<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>⇘_{S⇙}i∈{..max (deg R p) (deg R q)}. h (coeff P p i) ⊗⇘_{S⇙}s (^)⇘_{S⇙}i) ⊕⇘_{S⇙}

(\<Oplus>⇘_{S⇙}i∈{..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>⇘_{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

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)(^)⇘_{R⇙}k \<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)(^)⇘_{R⇙}k \<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