(* 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 ‹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 𝟬⇘_{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. ⨁⇘_{R⇙}i ∈ {..n}. p i ⊗⇘_{R⇙}q (n-i)), one = (%i. if i=0 then 𝟭⇘_{R⇙}else 𝟬⇘_{R⇙}), zero = (%i. 𝟬⇘_{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 𝟬⇘_{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 𝟬 n f" by (simp add: up_def) lemma up_one_closed: "(%n. if n = 0 then 𝟭 else 𝟬) ∈ 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 𝟬 n (%i. 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) (%i. 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 ==> (%i. ⊖ (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 (%i. ⊖ p i)" by auto then show "EX n. bound 𝟬 n (%i. ⊖ p i)" by auto qed auto lemma up_minus_closed: "[| p ∈ up R; q ∈ up R |] ==> (%i. p i ⊖ 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. ⨁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" by (simp add: mem_upD funcsetI) next assume UP: "p ∈ up R" "q ∈ up R" show "EX n. bound 𝟬 n (%n. ⨁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) (%n. ⨁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" 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 𝟬)" proof - assume R: "a ∈ carrier R" then have "(%n. 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" 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 = (⨁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]: "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 "EX q : carrier P. q ⊕⇘_{P⇙}p = 𝟬⇘_{P⇙}" proof - let ?q = "%i. ⊖ (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" by (simp add: Pi_def) qed (simp_all add: R) 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 qed (simp_all add: R) 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 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 ==> (⨁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) 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 ==> ⊖⇘_{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)" 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) ⊙⇘_{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 qed (simp_all add: R) lemma monom_one [simp]: "monom P 𝟭 0 = 𝟭⇘_{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 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 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 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" by (simp add: ivl_disj_un_one) 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 qed (simp_all add: a_in_R b_in_R) lemma monom_a_inv [simp]: "a ∈ carrier R ==> monom P (⊖ a) n = ⊖⇘_{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 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 "(%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 = 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 "EX m. deg R p - 1 < m & coeff P p m ~= 𝟬" by (unfold bound_def) fast then have "EX 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 "EX 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› 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 ~= 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)" by (simp add: deg_belowI lcoeff_nonzero_deg 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 qed (simp add: R) 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 = "(%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 ~= 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 deg_aboveD less_add_diff R Pi_def) 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 (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 = (⨁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 = "(%i. 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 qed (simp_all add: R Pi_def) 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 = "(%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: "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" by (simp add: R.integral_iff) 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)" by (simp add: diagonal_sum Pi_def) 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)" by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def) 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: "(%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). ⨁⇘_{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 o 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 add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add) also from R have "... = (⨁⇘_{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 "... = (⨁⇘_{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 add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def) 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 add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one 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 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 add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def 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 add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def) 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 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 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 & (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 ≠ 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" 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 ⊙⇘_{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 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 ≠ 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" by (simp add: minus_add r_neg sym [ 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 (simp_all add: deg_f_nzero) 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 qed (simp add: a) 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 qed (simp add: a) 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 qed (simp add: a) 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 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 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)" 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 ⊖ 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 = 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