(* Title: HOL/Algebra/UnivPoly.thy Id: $Id: UnivPoly.thy,v 1.33 2007/06/12 22:01:46 wenzelm Exp $ Author: Clemens Ballarin, started 9 December 1996 Copyright: Clemens Ballarin *) theory UnivPoly imports Module 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" constdefs (structure R) up :: "('a, 'm) ring_scheme => (nat => 'a) set" "up R == {f. f ∈ UNIV -> carrier R & (EX n. bound \<zero> n f)}" UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring" "UP R == (| carrier = up R, mult = (%p:up R. %q:up R. %n. \<Oplus>i ∈ {..n}. p i ⊗ q (n-i)), one = (%i. if i=0 then \<one> else \<zero>), zero = (%i. \<zero>), add = (%p:up R. %q:up R. %i. p i ⊕ q i), smult = (%a:carrier R. %p:up R. %i. a ⊗ p i), monom = (%a:carrier R. %n i. if i=n then a else \<zero>), 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) lemma (in cring) bound_upD [dest]: "f ∈ up R ==> EX n. bound \<zero> n f" by (simp add: up_def) lemma (in cring) up_one_closed: "(%n. if n = 0 then \<one> else \<zero>) ∈ up R" using up_def by force lemma (in cring) up_smult_closed: "[| a ∈ carrier R; p ∈ up R |] ==> (%i. a ⊗ p i) ∈ up R" by force lemma (in cring) 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 fastsimp qed then show ?thesis .. qed qed lemma (in cring) 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 (in cring) 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 subsection {* Effect of Operations on Coefficients *} locale UP = fixes R (structure) and P (structure) defines P_def: "P == UP R" locale UP_cring = UP + cring R locale UP_domain = UP_cring + "domain" R text {* Temporarily declare @{thm [locale=UP] P_def} as simp rule. *} declare (in UP) P_def [simp] lemma (in UP_cring) 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 (in UP_cring) coeff_zero [simp]: "coeff P \<zero>P n = \<zero>" by (auto simp add: UP_def) lemma (in UP_cring) 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 (in UP_cring) 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 (in UP_cring) 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 (in UP_cring) 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) lemma (in UP) 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 subsection {* Polynomials Form a Commutative Ring. *} text {* Operations are closed over @{term P}. *} lemma (in UP_cring) UP_mult_closed [simp]: "[| p ∈ carrier P; q ∈ carrier P |] ==> p ⊗P q ∈ carrier P" by (simp add: UP_def up_mult_closed) lemma (in UP_cring) UP_one_closed [simp]: "\<one>P ∈ carrier P" by (simp add: UP_def up_one_closed) lemma (in UP_cring) UP_zero_closed [intro, simp]: "\<zero>P ∈ carrier P" by (auto simp add: UP_def) lemma (in UP_cring) 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 (in UP_cring) monom_closed [simp]: "a ∈ carrier R ==> monom P a n ∈ carrier P" by (auto simp add: UP_def up_def Pi_def) lemma (in UP_cring) UP_smult_closed [simp]: "[| a ∈ carrier R; p ∈ carrier P |] ==> a \<odot>P p ∈ carrier P" by (simp add: UP_def up_smult_closed) lemma (in UP) coeff_closed [simp]: "p ∈ carrier P ==> coeff P p n ∈ carrier R" by (auto simp add: UP_def) declare (in UP) P_def [simp del] text {* Algebraic ring properties *} lemma (in UP_cring) 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 (in UP_cring) UP_l_zero [simp]: assumes R: "p ∈ carrier P" shows "\<zero>P ⊕P p = p" by (rule up_eqI, simp_all add: R) lemma (in UP_cring) 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 (in UP_cring) 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 (in UP_cring) 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 (in UP_cring) 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 (in UP_cring) 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 (in UP_cring) UP_m_comm: assumes R: "p ∈ carrier P" "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 R show "coeff P (p ⊗P q) n = coeff P (q ⊗P p) n" apply (simp add: Pi_def) apply (subst l) apply (auto simp add: Pi_def) apply (simp add: m_comm) done qed (simp_all add: R) theorem (in UP_cring) UP_cring: "cring P" by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr) lemma (in UP_cring) UP_ring: (* preliminary, we want "UP_ring R P ==> ring P", not "UP_cring R P ==> ring P" *) "ring P" by (auto intro: ring.intro cring.axioms UP_cring) lemma (in UP_cring) 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 (in UP_cring) 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 text {* Interpretation of lemmas from @{term cring}. Saves lifting 43 lemmas manually. *} interpretation UP_cring < cring P by intro_locales (rule cring.axioms ring.axioms abelian_group.axioms comm_monoid.axioms UP_cring)+ subsection {* Polynomials Form an Algebra *} lemma (in UP_cring) 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 (in UP_cring) 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 (in UP_cring) 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 (in UP_cring) UP_smult_one [simp]: "p ∈ carrier P ==> \<one> \<odot>P p = p" by (rule up_eqI) simp_all lemma (in UP_cring) 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) text {* Interpretation of lemmas from @{term algebra}. *} lemma (in cring) cring: "cring R" by (fast intro: cring.intro prems) 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) interpretation UP_cring < algebra R P by intro_locales (rule module.axioms algebra.axioms UP_algebra)+ subsection {* Further Lemmas Involving Monomials *} lemma (in UP_cring) monom_zero [simp]: "monom P \<zero> n = \<zero>P" by (simp add: UP_def P_def) lemma (in UP_cring) 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 have "coeff P (p ⊗P monom P a 0) n = coeff P (a \<odot>P p) n" proof (cases n) case 0 with R show ?thesis by (simp add: R.m_comm) next case Suc with R show ?thesis by (simp cong: R.finsum_cong add: R.r_null Pi_def) (simp add: R.m_comm) qed with R show "coeff P (monom P a 0 ⊗P p) n = coeff P (a \<odot>P p) n" by (simp add: UP_m_comm) qed (simp_all add: R) lemma (in UP_cring) 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 (in UP_cring) 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 - 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: ivl_disj_int_singleton 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 ivl_disj_int_singleton 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 (in UP_cring) 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 (in UP_cring) monom_one [simp]: "monom P \<one> 0 = \<one>P" by (rule up_eqI) simp_all lemma (in UP_cring) 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 by (simp only: add_Suc monom_one_Suc) (simp add: P.m_ac) qed lemma (in UP_cring) monom_mult [simp]: assumes R: "a ∈ carrier R" "b ∈ carrier R" shows "monom P (a ⊗ b) (n + m) = monom P a n ⊗P monom P b m" proof - from R have "monom P (a ⊗ b) (n + m) = monom P (a ⊗ b ⊗ \<one>) (n + m)" by simp also from R have "... = a ⊗ b \<odot>P monom P \<one> (n + m)" by (simp add: monom_mult_smult del: R.r_one) also have "... = a ⊗ b \<odot>P (monom P \<one> n ⊗P monom P \<one> m)" by (simp only: monom_one_mult) also from R have "... = a \<odot>P (b \<odot>P (monom P \<one> n ⊗P monom P \<one> m))" by (simp add: UP_smult_assoc1) also from R have "... = a \<odot>P (b \<odot>P (monom P \<one> m ⊗P monom P \<one> n))" by (simp add: P.m_comm) also from R have "... = a \<odot>P ((b \<odot>P monom P \<one> m) ⊗P monom P \<one> n)" by (simp add: UP_smult_assoc2) also from R have "... = a \<odot>P (monom P \<one> n ⊗P (b \<odot>P monom P \<one> m))" by (simp add: P.m_comm) also from R have "... = (a \<odot>P monom P \<one> n) ⊗P (b \<odot>P monom P \<one> m)" by (simp add: UP_smult_assoc2) also from R have "... = monom P (a ⊗ \<one>) n ⊗P monom P (b ⊗ \<one>) m" by (simp add: monom_mult_smult del: R.r_one) also from R have "... = monom P a n ⊗P monom P b m" by simp finally show ?thesis . qed lemma (in UP_cring) 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 (in UP_cring) 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 subsection {* The Degree Function *} constdefs (structure R) deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat" "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)" lemma (in UP_cring) 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 (in UP_cring) 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 (in UP_cring) 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 (in UP_cring) 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 (* TODO: why does simplification below not work with "1" *) from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))" by (unfold deg_def P_def) arith 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 fastsimp with m_coeff show ?thesis by simp qed lemma (in UP_cring) 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 (in UP_cring) 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 (in UP_cring) 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_anti_sym deg_aboveI deg_belowI) text {* Degree and polynomial operations *} lemma (in UP_cring) deg_add [simp]: assumes R: "p ∈ carrier P" "q ∈ carrier P" shows "deg R (p ⊕P q) <= max (deg R p) (deg R q)" proof (cases "deg R p <= deg R q") case True show ?thesis by (rule deg_aboveI) (simp_all add: True R deg_aboveD) next case False show ?thesis by (rule deg_aboveI) (simp_all add: False R deg_aboveD) qed lemma (in UP_cring) deg_monom_le: "a ∈ carrier R ==> deg R (monom P a n) <= n" by (intro deg_aboveI) simp_all lemma (in UP_cring) deg_monom [simp]: "[| a ~= \<zero>; a ∈ carrier R |] ==> deg R (monom P a n) = n" by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI) lemma (in UP_cring) deg_const [simp]: assumes R: "a ∈ carrier R" shows "deg R (monom P a 0) = 0" proof (rule le_anti_sym) 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 (in UP_cring) deg_zero [simp]: "deg R \<zero>P = 0" proof (rule le_anti_sym) 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 (in UP_cring) deg_one [simp]: "deg R \<one>P = 0" proof (rule le_anti_sym) 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 (in UP_cring) deg_uminus [simp]: assumes R: "p ∈ carrier P" shows "deg R (\<ominus>P p) = deg R p" proof (rule le_anti_sym) 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 lemma (in UP_domain) deg_smult_ring: "[| 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)+ lemma (in UP_domain) 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_anti_sym) 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 lemma (in UP_cring) deg_mult_cring: 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) lemma (in UP_domain) 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_anti_sym) assume "p ∈ carrier P" " q ∈ carrier P" then show "deg R (p ⊗P q) <= deg R p + deg R q" by (rule deg_mult_cring) 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: ivl_disj_int_singleton 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 lemma (in UP_cring) 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 (in UP_cring) 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: ivl_disj_int_singleton 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: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def) finally show ?thesis . qed qed (simp_all add: R Pi_def) lemma (in UP_cring) 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 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 prems del: disjCI) lemma (in UP_domain) 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 one_not_zero show "False" by contradiction qed lemma (in UP_domain) 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 fastsimp qed qed theorem (in UP_domain) UP_domain: "domain P" by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI) text {* Interpretation of theorems from @{term domain}. *} interpretation 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*) theorem (in cring) 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 lemma (in abelian_monoid) boundD_carrier: "[| bound \<zero> n f; n < m |] ==> f m ∈ carrier G" by auto theorem (in cring) 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 lemma (in UP_cring) 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) constdefs (structure S) eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme, 'a => 'b, 'b, nat => 'a] => 'b" "eval R S phi s == λp ∈ carrier (UP R). \<Oplus>i ∈ {..deg R p}. phi (coeff (UP R) p i) ⊗ s (^) i" lemma (in UP) 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 (in UP) eval_extensional: "eval R S phi p ∈ extensional (carrier P)" by (unfold eval_def, fold P_def) simp text {* The universal property of the polynomial ring *} locale UP_pre_univ_prop = ring_hom_cring R S h + UP_cring R P 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" theorem (in UP_pre_univ_prop) 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 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_cring) 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 next fix p q assume R: "p ∈ carrier P" "q ∈ carrier P" then show "eval R S h s (p ⊕P q) = eval R S h s p ⊕S eval R S h s q" proof (simp only: eval_on_carrier P.a_closed) from S R have "(\<Oplus>S i∈{..deg R (p ⊕P q)}. h (coeff P (p ⊕P q) i) ⊗S s (^)S i) = (\<Oplus>S i∈{..deg R (p ⊕P q)} ∪ {deg R (p ⊕P q)<..max (deg R p) (deg R q)}. h (coeff P (p ⊕P q) i) ⊗S s (^)S i)" by (simp cong: S.finsum_cong add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add) also from R have "... = (\<Oplus>S i ∈ {..max (deg R p) (deg R q)}. h (coeff P (p ⊕P q) i) ⊗S s (^)S i)" by (simp add: ivl_disj_un_one) also from R S have "... = (\<Oplus>Si∈{..max (deg R p) (deg R q)}. h (coeff P p i) ⊗S s (^)S i) ⊕S (\<Oplus>Si∈{..max (deg R p) (deg R q)}. h (coeff P q i) ⊗S s (^)S i)" by (simp cong: S.finsum_cong add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def) also have "... = (\<Oplus>S i ∈ {..deg R p} ∪ {deg R p<..max (deg R p) (deg R q)}. h (coeff P p i) ⊗S s (^)S i) ⊕S (\<Oplus>S i ∈ {..deg R q} ∪ {deg R q<..max (deg R p) (deg R q)}. h (coeff P q i) ⊗S s (^)S i)" by (simp only: ivl_disj_un_one le_maxI1 le_maxI2) also from R S have "... = (\<Oplus>S i ∈ {..deg R p}. h (coeff P p i) ⊗S s (^)S i) ⊕S (\<Oplus>S i ∈ {..deg R q}. h (coeff P q i) ⊗S s (^)S i)" by (simp cong: S.finsum_cong add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def) finally show "(\<Oplus>Si ∈ {..deg R (p ⊕P q)}. h (coeff P (p ⊕P q) i) ⊗S s (^)S i) = (\<Oplus>Si ∈ {..deg R p}. h (coeff P p i) ⊗S s (^)S i) ⊕S (\<Oplus>Si ∈ {..deg R q}. h (coeff P q i) ⊗S s (^)S i)" . qed next show "eval R S h s \<one>P = \<one>S" by (simp only: eval_on_carrier UP_one_closed) simp qed text {* Interpretation of ring homomorphism lemmas. *} interpretation UP_univ_prop < ring_hom_cring P S Eval apply (unfold Eval_def) apply intro_locales apply (rule ring_hom_cring.axioms) apply (rule ring_hom_cring.intro) apply unfold_locales apply (rule eval_ring_hom) apply rule done text {* Further properties of the evaluation homomorphism. *} 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 {* 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 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 _] using `UP_pre_univ_prop R S h` 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 prems) lemma (in UP_pre_univ_prop) UP_hom_unique: includes 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" includes 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 - 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 (in UP_pre_univ_prop) ring_homD: assumes Phi: "Phi ∈ ring_hom P S" shows "ring_hom_cring P S Phi" proof (rule ring_hom_cring.intro) show "ring_hom_cring_axioms P S Phi" by (rule ring_hom_cring_axioms.intro) (rule Phi) qed unfold_locales theorem (in UP_pre_univ_prop) 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 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) constdefs INTEG :: "int ring" "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 zadd_zminus_inverse2 zadd_zmult_distrib) 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] apply simp using INTEG_id_eval apply simp done 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
lemma bound_below:
[| bound z m f; f n ≠ z |] ==> n ≤ m
lemma mem_upI:
[| !!n. f n ∈ carrier R; ∃n. bound \<zero>R n f |] ==> f ∈ up R
lemma mem_upD:
f ∈ up R ==> f n ∈ carrier R
lemma bound_upD:
f ∈ up R ==> ∃n. bound \<zero> n f
lemma up_one_closed:
(λn. if n = 0 then \<one> else \<zero>) ∈ up R
lemma up_smult_closed:
[| a ∈ carrier R; p ∈ up R |] ==> (λi. a ⊗ p i) ∈ up R
lemma up_add_closed:
[| p ∈ up R; q ∈ up R |] ==> (λi. p i ⊕ q i) ∈ up R
lemma up_a_inv_closed:
p ∈ up R ==> (λi. \<ominus> p i) ∈ up R
lemma up_mult_closed:
[| p ∈ up R; q ∈ up R |] ==> (λn. \<Oplus>i∈{..n}. p i ⊗ q (n - i)) ∈ up R
lemma coeff_monom:
a ∈ carrier R ==> coeff P (monom P a m) n = (if m = n then a else \<zero>)
lemma coeff_zero:
coeff P \<zero>P n = \<zero>
lemma coeff_one:
coeff P \<one>P n = (if n = 0 then \<one> else \<zero>)
lemma coeff_smult:
[| a ∈ carrier R; p ∈ carrier P |]
==> coeff P (a \<odot>P p) n = a ⊗ coeff P p n
lemma coeff_add:
[| p ∈ carrier P; q ∈ carrier P |]
==> coeff P (p ⊕P q) n = coeff P p n ⊕ coeff P q n
lemma coeff_mult:
[| p ∈ carrier P; q ∈ carrier P |]
==> coeff P (p ⊗P q) n = (\<Oplus>i∈{..n}. coeff P p i ⊗ coeff P q (n - i))
lemma up_eqI:
[| !!n. coeff P p n = coeff P q n; p ∈ carrier P; q ∈ carrier P |] ==> p = q
lemma UP_mult_closed:
[| p ∈ carrier P; q ∈ carrier P |] ==> p ⊗P q ∈ carrier P
lemma UP_one_closed:
\<one>P ∈ carrier P
lemma UP_zero_closed:
\<zero>P ∈ carrier P
lemma UP_a_closed:
[| p ∈ carrier P; q ∈ carrier P |] ==> p ⊕P q ∈ carrier P
lemma monom_closed:
a ∈ carrier R ==> monom P a n ∈ carrier P
lemma UP_smult_closed:
[| a ∈ carrier R; p ∈ carrier P |] ==> a \<odot>P p ∈ carrier P
lemma coeff_closed:
p ∈ carrier P ==> coeff P p n ∈ carrier R
lemma UP_a_assoc:
[| p ∈ carrier P; q ∈ carrier P; r ∈ carrier P |]
==> p ⊕P q ⊕P r = p ⊕P (q ⊕P r)
lemma UP_l_zero:
p ∈ carrier P ==> \<zero>P ⊕P p = p
lemma UP_l_neg_ex:
p ∈ carrier P ==> ∃q∈carrier P. q ⊕P p = \<zero>P
lemma UP_a_comm:
[| p ∈ carrier P; q ∈ carrier P |] ==> p ⊕P q = q ⊕P p
lemma UP_m_assoc:
[| p ∈ carrier P; q ∈ carrier P; r ∈ carrier P |]
==> p ⊗P q ⊗P r = p ⊗P (q ⊗P r)
lemma UP_l_one:
p ∈ carrier P ==> \<one>P ⊗P p = p
lemma UP_l_distr:
[| p ∈ carrier P; q ∈ carrier P; r ∈ carrier P |]
==> (p ⊕P q) ⊗P r = p ⊗P r ⊕P q ⊗P r
lemma UP_m_comm:
[| p ∈ carrier P; q ∈ carrier P |] ==> p ⊗P q = q ⊗P p
theorem UP_cring:
cring P
lemma UP_ring:
Ring.ring P
lemma UP_a_inv_closed:
p ∈ carrier P ==> \<ominus>P p ∈ carrier P
lemma coeff_a_inv:
p ∈ carrier P ==> coeff P (\<ominus>P p) n = \<ominus> coeff P p n
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
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
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)
lemma UP_smult_one:
p ∈ carrier P ==> \<one> \<odot>P p = p
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)
lemma cring:
cring R
lemma UP_algebra:
algebra R P
lemma monom_zero:
monom P \<zero> n = \<zero>P
lemma monom_mult_is_smult:
[| a ∈ carrier R; p ∈ carrier P |] ==> monom P a 0 ⊗P p = a \<odot>P p
lemma monom_add:
[| a ∈ carrier R; b ∈ carrier R |]
==> monom P (a ⊕ b) n = monom P a n ⊕P monom P b n
lemma monom_one_Suc:
monom P \<one> (Suc n) = monom P \<one> n ⊗P monom P \<one> 1
lemma monom_mult_smult:
[| a ∈ carrier R; b ∈ carrier R |]
==> monom P (a ⊗ b) n = a \<odot>P monom P b n
lemma monom_one:
monom P \<one> 0 = \<one>P
lemma monom_one_mult:
monom P \<one> (n + m) = monom P \<one> n ⊗P monom P \<one> m
lemma monom_mult:
[| a ∈ carrier R; b ∈ carrier R |]
==> monom P (a ⊗ b) (n + m) = monom P a n ⊗P monom P b m
lemma monom_a_inv:
a ∈ carrier R ==> monom P (\<ominus> a) n = \<ominus>P monom P a n
lemma monom_inj:
inj_on (λa. monom P a n) (carrier R)
lemma deg_aboveI:
[| !!m. n < m ==> coeff P p m = \<zero>; p ∈ carrier P |] ==> deg R p ≤ n
lemma deg_aboveD:
[| deg R p < m; p ∈ carrier P |] ==> coeff P p m = \<zero>
lemma deg_belowI:
[| n ≠ 0 ==> coeff P p n ≠ \<zero>; p ∈ carrier P |] ==> n ≤ deg R p
lemma lcoeff_nonzero_deg:
[| deg R p ≠ 0; p ∈ carrier P |] ==> coeff P p (deg R p) ≠ \<zero>
lemma lcoeff_nonzero_nonzero:
[| deg R p = 0; p ≠ \<zero>P; p ∈ carrier P |] ==> coeff P p 0 ≠ \<zero>
lemma lcoeff_nonzero:
[| p ≠ \<zero>P; p ∈ carrier P |] ==> coeff P p (deg R p) ≠ \<zero>
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
lemma deg_add:
[| p ∈ carrier P; q ∈ carrier P |] ==> deg R (p ⊕P q) ≤ max (deg R p) (deg R q)
lemma deg_monom_le:
a ∈ carrier R ==> deg R (monom P a n) ≤ n
lemma deg_monom:
[| a ≠ \<zero>; a ∈ carrier R |] ==> deg R (monom P a n) = n
lemma deg_const:
a ∈ carrier R ==> deg R (monom P a 0) = 0
lemma deg_zero:
deg R \<zero>P = 0
lemma deg_one:
deg R \<one>P = 0
lemma deg_uminus:
p ∈ carrier P ==> deg R (\<ominus>P p) = deg R p
lemma deg_smult_ring:
[| a ∈ carrier R; p ∈ carrier P |]
==> deg R (a \<odot>P p) ≤ (if a = \<zero> then 0 else deg R p)
lemma deg_smult:
[| a ∈ carrier R; p ∈ carrier P |]
==> deg R (a \<odot>P p) = (if a = \<zero> then 0 else deg R p)
lemma deg_mult_cring:
[| p ∈ carrier P; q ∈ carrier P |] ==> deg R (p ⊗P q) ≤ deg R p + deg R q
lemma deg_mult:
[| p ≠ \<zero>P; q ≠ \<zero>P; p ∈ carrier P; q ∈ carrier P |]
==> deg R (p ⊗P q) = deg R p + deg R q
lemma coeff_finsum:
[| finite A; p ∈ A -> carrier P |]
==> coeff P (finsum P p A) k = (\<Oplus>i∈A. coeff P (p i) k)
lemma up_repr:
p ∈ carrier P ==> (\<Oplus>Pi∈{..deg R p}. monom P (coeff P p i) i) = p
lemma up_repr_le:
[| deg R p ≤ n; p ∈ carrier P |]
==> (\<Oplus>Pi∈{..n}. monom P (coeff P p i) i) = p
lemma domainI:
[| cring R; \<one>R ≠ \<zero>R;
!!a b. [| a ⊗R b = \<zero>R; a ∈ carrier R; b ∈ carrier R |]
==> a = \<zero>R ∨ b = \<zero>R |]
==> domain R
lemma UP_one_not_zero:
\<one>P ≠ \<zero>P
lemma UP_integral:
[| p ⊗P q = \<zero>P; p ∈ carrier P; q ∈ carrier P |]
==> p = \<zero>P ∨ q = \<zero>P
theorem UP_domain:
domain P
theorem diagonal_sum:
[| f ∈ {..n + m} -> 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)
lemma boundD_carrier:
[| bound \<zero> n f; n < m |] ==> f m ∈ carrier G
theorem cauchy_product:
[| bound \<zero> n f; bound \<zero> m g; f ∈ {..n} -> carrier R;
g ∈ {..m} -> carrier R |]
==> (\<Oplus>k∈{..n + m}. \<Oplus>i∈{..k}. f i ⊗ g (k - i)) =
finsum R f {..n} ⊗ finsum R g {..m}
lemma const_ring_hom:
(λa. monom P a 0) ∈ ring_hom R P
lemma eval_on_carrier:
p ∈ carrier P
==> eval R S phi s p = (\<Oplus>Si∈{..deg R p}. phi (coeff P p i) ⊗S s (^)S i)
lemma eval_extensional:
eval R S phi p ∈ extensional (carrier P)
theorem eval_ring_hom:
s ∈ carrier S ==> eval R S h s ∈ ring_hom P S
lemma eval_const:
[| s ∈ carrier S; r ∈ carrier R |] ==> eval R S h s (monom P r 0) = h r
lemma eval_monom1:
s ∈ carrier S ==> eval R S h s (monom P \<one> 1) = s
lemma monom_pow:
a ∈ carrier R ==> monom P a n (^)P m = monom P (a (^) m) (n * m)
lemma hom_pow:
x ∈ carrier R ==> h (x (^) n) = h x (^)S n
lemma Eval_monom:
r ∈ carrier R ==> Eval (monom P r n) = h r ⊗S s (^)S n
lemma eval_monom:
[| r ∈ carrier R; s ∈ carrier S |]
==> eval R S h s (monom P r n) = h r ⊗S s (^)S n
lemma Eval_smult:
[| r ∈ carrier R; p ∈ carrier P |] ==> Eval (r \<odot>P p) = h r ⊗S Eval p
lemma ring_hom_cringI:
[| cring R; cring S; h ∈ ring_hom R S |] ==> ring_hom_cring R S h
lemma UP_hom_unique:
[| ring_hom_cring P S Phi; Phi (monom P \<one> (Suc 0)) = s;
!!r. r ∈ carrier R ==> Phi (monom P r 0) = h r; ring_hom_cring P S Psi;
Psi (monom P \<one> (Suc 0)) = s;
!!r. r ∈ carrier R ==> Psi (monom P r 0) = h r; p ∈ carrier P;
s ∈ carrier S |]
==> Phi p = Psi p
lemma ring_homD:
Phi ∈ ring_hom P S ==> ring_hom_cring P S Phi
theorem UP_universal_property:
s ∈ carrier S
==> ∃!Phi. Phi ∈ ring_hom P S ∩ extensional (carrier P) ∧
Phi (monom P \<one> 1) = s ∧ (∀r∈carrier R. Phi (monom P r 0) = h r)
lemma UP_pre_univ_propI:
[| cring R; cring S; h ∈ ring_hom R S |] ==> UP_pre_univ_prop R S h
lemma INTEG_cring:
cring INTEG
lemma INTEG_id_eval:
UP_pre_univ_prop INTEG INTEG id
lemma INTEG_closed:
z ∈ carrier INTEG
lemma INTEG_mult:
z ⊗INTEG w = z * w
lemma INTEG_pow:
z (^)INTEG n = z ^ n
lemma
eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500