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src/HOL/Algebra/UnivPoly.thy

author | ballarin |

Mon, 26 Jul 2004 15:48:50 +0200 | |

changeset 15076 | 4b3d280ef06a |

parent 15045 | d59f7e2e18d3 |

child 15095 | 63f5f4c265dd |

permissions | -rw-r--r-- |

New prover for transitive and reflexive-transitive closure of relations.
- Code in Provers/trancl.ML
- HOL: Simplifier set up to use it as solver

(* Title: HOL/Algebra/UnivPoly.thy Id: $Id$ Author: Clemens Ballarin, started 9 December 1996 Copyright: Clemens Ballarin *) header {* Univariate Polynomials *} theory UnivPoly = Module: 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 *} locale bound = fixes z :: 'a and n :: nat and f :: "nat => 'a" assumes bound: "!!m. n < m \<Longrightarrow> f m = z" declare bound.intro [intro!] and bound.bound [dest] lemma bound_below: assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> 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 :: "_ => (nat => 'a) set" "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero> n f)}" UP :: "_ => ('a, nat => 'a) up_ring" "UP R == (| carrier = up R, mult = (%p:up R. %q:up R. %n. \<Oplus>i \<in> {..n}. p i \<otimes> 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 \<oplus> q i), smult = (%a:carrier R. %p:up R. %i. a \<otimes> 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 \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R" by (simp add: up_def Pi_def) lemma mem_upD [dest]: "f \<in> up R ==> f n \<in> carrier R" by (simp add: up_def Pi_def) lemma (in cring) bound_upD [dest]: "f \<in> 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>) \<in> up R" using up_def by force lemma (in cring) up_smult_closed: "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R" by force lemma (in cring) up_add_closed: "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R" proof fix n assume "p \<in> up R" and "q \<in> up R" then show "p n \<oplus> q n \<in> carrier R" by auto next assume UP: "p \<in> up R" "q \<in> up R" show "EX n. bound \<zero> n (%i. p i \<oplus> 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 \<oplus> q i)" proof fix i assume "max n m < i" with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp qed then show ?thesis .. qed qed lemma (in cring) up_a_inv_closed: "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R" proof assume R: "p \<in> 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 \<in> up R; q \<in> up R |] ==> (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R" proof fix n assume "p \<in> up R" "q \<in> up R" then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R" by (simp add: mem_upD funcsetI) next assume UP: "p \<in> up R" "q \<in> up R" show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> 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 \<in> {..n}. p i \<otimes> q (n - i))" proof fix k assume bound: "n + m < k" { fix i have "p i \<otimes> 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) \<in> 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 \<in> carrier R" by auto ultimately show ?thesis by simp qed } then show "(\<Oplus>i \<in> {..k}. p i \<otimes> 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 = struct R + struct P + defines P_def: "P == UP R" locale UP_cring = UP + cring R locale UP_domain = UP_cring + "domain" R text {* Temporarily declare @{text UP.P_def} as simp rule. *} (* TODO: use antiquotation once text (in locale) is supported. *) declare (in UP) P_def [simp] lemma (in UP_cring) coeff_monom [simp]: "a \<in> carrier R ==> coeff P (monom P a m) n = (if m=n then a else \<zero>)" proof - assume R: "a \<in> carrier R" then have "(%n. if n = m then a else \<zero>) \<in> 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>\<^sub>2 n = \<zero>" by (auto simp add: UP_def) lemma (in UP_cring) coeff_one [simp]: "coeff P \<one>\<^sub>2 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 \<in> carrier R; p \<in> carrier P |] ==> coeff P (a \<odot>\<^sub>2 p) n = a \<otimes> coeff P p n" by (simp add: UP_def up_smult_closed) lemma (in UP_cring) coeff_add [simp]: "[| p \<in> carrier P; q \<in> carrier P |] ==> coeff P (p \<oplus>\<^sub>2 q) n = coeff P p n \<oplus> coeff P q n" by (simp add: UP_def up_add_closed) lemma (in UP_cring) coeff_mult [simp]: "[| p \<in> carrier P; q \<in> carrier P |] ==> coeff P (p \<otimes>\<^sub>2 q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> 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 \<in> carrier P" "q \<in> 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 \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^sub>2 q \<in> carrier P" by (simp add: UP_def up_mult_closed) lemma (in UP_cring) UP_one_closed [simp]: "\<one>\<^sub>2 \<in> carrier P" by (simp add: UP_def up_one_closed) lemma (in UP_cring) UP_zero_closed [intro, simp]: "\<zero>\<^sub>2 \<in> carrier P" by (auto simp add: UP_def) lemma (in UP_cring) UP_a_closed [intro, simp]: "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^sub>2 q \<in> carrier P" by (simp add: UP_def up_add_closed) lemma (in UP_cring) monom_closed [simp]: "a \<in> carrier R ==> monom P a n \<in> carrier P" by (auto simp add: UP_def up_def Pi_def) lemma (in UP_cring) UP_smult_closed [simp]: "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^sub>2 p \<in> carrier P" by (simp add: UP_def up_smult_closed) lemma (in UP) coeff_closed [simp]: "p \<in> carrier P ==> coeff P p n \<in> 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 \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P" shows "(p \<oplus>\<^sub>2 q) \<oplus>\<^sub>2 r = p \<oplus>\<^sub>2 (q \<oplus>\<^sub>2 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 \<in> carrier P" shows "\<zero>\<^sub>2 \<oplus>\<^sub>2 p = p" by (rule up_eqI, simp_all add: R) lemma (in UP_cring) UP_l_neg_ex: assumes R: "p \<in> carrier P" shows "EX q : carrier P. q \<oplus>\<^sub>2 p = \<zero>\<^sub>2" proof - let ?q = "%i. \<ominus> (p i)" from R have closed: "?q \<in> 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 \<oplus>\<^sub>2 p = \<zero>\<^sub>2" 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 \<in> carrier P" "q \<in> carrier P" shows "p \<oplus>\<^sub>2 q = q \<oplus>\<^sub>2 p" by (rule up_eqI, simp add: a_comm R, simp_all add: R) ML_setup {* simpset_ref() := simpset() setsubgoaler asm_full_simp_tac; *} lemma (in UP_cring) UP_m_assoc: assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P" shows "(p \<otimes>\<^sub>2 q) \<otimes>\<^sub>2 r = p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 r)" proof (rule up_eqI) fix n { fix k and a b c :: "nat=>'a" assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R" "c \<in> UNIV -> carrier R" then have "k <= n ==> (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) = (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))" (concl 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 then 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 \<otimes>\<^sub>2 q) \<otimes>\<^sub>2 r) n = coeff P (p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 r)) n" by (simp add: Pi_def) qed (simp_all add: R) ML_setup {* simpset_ref() := simpset() setsubgoaler asm_simp_tac; *} lemma (in UP_cring) UP_l_one [simp]: assumes R: "p \<in> carrier P" shows "\<one>\<^sub>2 \<otimes>\<^sub>2 p = p" proof (rule up_eqI) fix n show "coeff P (\<one>\<^sub>2 \<otimes>\<^sub>2 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 \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P" shows "(p \<oplus>\<^sub>2 q) \<otimes>\<^sub>2 r = (p \<otimes>\<^sub>2 r) \<oplus>\<^sub>2 (q \<otimes>\<^sub>2 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 \<in> carrier P" "q \<in> carrier P" shows "p \<otimes>\<^sub>2 q = q \<otimes>\<^sub>2 p" proof (rule up_eqI) fix n { fix k and a b :: "nat=>'a" assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R" then have "k <= n ==> (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) = (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))" (concl 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 finsum_Suc2) (simp add: Pi_def a_comm)+ qed } note l = this from R show "coeff P (p \<otimes>\<^sub>2 q) n = coeff P (q \<otimes>\<^sub>2 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 *) "ring P" by (auto intro: ring.intro cring.axioms UP_cring) lemma (in UP_cring) UP_a_inv_closed [intro, simp]: "p \<in> carrier P ==> \<ominus>\<^sub>2 p \<in> 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 \<in> carrier P" shows "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> (coeff P p n)" proof - from R coeff_closed UP_a_inv_closed have "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^sub>2 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 {* Instantiation of lemmas from @{term cring}. *} lemma (in UP_cring) UP_monoid: "monoid P" by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro UP_cring) (* TODO: provide cring.is_monoid *) lemma (in UP_cring) UP_comm_monoid: "comm_monoid P" by (fast intro!: cring.is_comm_monoid UP_cring) lemma (in UP_cring) UP_abelian_monoid: "abelian_monoid P" by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring) lemma (in UP_cring) UP_abelian_group: "abelian_group P" by (fast intro!: ring.is_abelian_group UP_ring) lemmas (in UP_cring) UP_r_one [simp] = monoid.r_one [OF UP_monoid] lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] = monoid.nat_pow_closed [OF UP_monoid] lemmas (in UP_cring) UP_nat_pow_0 [simp] = monoid.nat_pow_0 [OF UP_monoid] lemmas (in UP_cring) UP_nat_pow_Suc [simp] = monoid.nat_pow_Suc [OF UP_monoid] lemmas (in UP_cring) UP_nat_pow_one [simp] = monoid.nat_pow_one [OF UP_monoid] lemmas (in UP_cring) UP_nat_pow_mult = monoid.nat_pow_mult [OF UP_monoid] lemmas (in UP_cring) UP_nat_pow_pow = monoid.nat_pow_pow [OF UP_monoid] lemmas (in UP_cring) UP_m_lcomm = comm_monoid.m_lcomm [OF UP_comm_monoid] lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm lemmas (in UP_cring) UP_nat_pow_distr = comm_monoid.nat_pow_distr [OF UP_comm_monoid] lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid] lemmas (in UP_cring) UP_r_zero [simp] = abelian_monoid.r_zero [OF UP_abelian_monoid] lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm lemmas (in UP_cring) UP_finsum_empty [simp] = abelian_monoid.finsum_empty [OF UP_abelian_monoid] lemmas (in UP_cring) UP_finsum_insert [simp] = abelian_monoid.finsum_insert [OF UP_abelian_monoid] lemmas (in UP_cring) UP_finsum_zero [simp] = abelian_monoid.finsum_zero [OF UP_abelian_monoid] lemmas (in UP_cring) UP_finsum_closed [simp] = abelian_monoid.finsum_closed [OF UP_abelian_monoid] lemmas (in UP_cring) UP_finsum_Un_Int = abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid] lemmas (in UP_cring) UP_finsum_Un_disjoint = abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid] lemmas (in UP_cring) UP_finsum_addf = abelian_monoid.finsum_addf [OF UP_abelian_monoid] lemmas (in UP_cring) UP_finsum_cong' = abelian_monoid.finsum_cong' [OF UP_abelian_monoid] lemmas (in UP_cring) UP_finsum_0 [simp] = abelian_monoid.finsum_0 [OF UP_abelian_monoid] lemmas (in UP_cring) UP_finsum_Suc [simp] = abelian_monoid.finsum_Suc [OF UP_abelian_monoid] lemmas (in UP_cring) UP_finsum_Suc2 = abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid] lemmas (in UP_cring) UP_finsum_add [simp] = abelian_monoid.finsum_add [OF UP_abelian_monoid] lemmas (in UP_cring) UP_finsum_cong = abelian_monoid.finsum_cong [OF UP_abelian_monoid] lemmas (in UP_cring) UP_minus_closed [intro, simp] = abelian_group.minus_closed [OF UP_abelian_group] lemmas (in UP_cring) UP_a_l_cancel [simp] = abelian_group.a_l_cancel [OF UP_abelian_group] lemmas (in UP_cring) UP_a_r_cancel [simp] = abelian_group.a_r_cancel [OF UP_abelian_group] lemmas (in UP_cring) UP_l_neg = abelian_group.l_neg [OF UP_abelian_group] lemmas (in UP_cring) UP_r_neg = abelian_group.r_neg [OF UP_abelian_group] lemmas (in UP_cring) UP_minus_zero [simp] = abelian_group.minus_zero [OF UP_abelian_group] lemmas (in UP_cring) UP_minus_minus [simp] = abelian_group.minus_minus [OF UP_abelian_group] lemmas (in UP_cring) UP_minus_add = abelian_group.minus_add [OF UP_abelian_group] lemmas (in UP_cring) UP_r_neg2 = abelian_group.r_neg2 [OF UP_abelian_group] lemmas (in UP_cring) UP_r_neg1 = abelian_group.r_neg1 [OF UP_abelian_group] lemmas (in UP_cring) UP_r_distr = ring.r_distr [OF UP_ring] lemmas (in UP_cring) UP_l_null [simp] = ring.l_null [OF UP_ring] lemmas (in UP_cring) UP_r_null [simp] = ring.r_null [OF UP_ring] lemmas (in UP_cring) UP_l_minus = ring.l_minus [OF UP_ring] lemmas (in UP_cring) UP_r_minus = ring.r_minus [OF UP_ring] lemmas (in UP_cring) UP_finsum_ldistr = cring.finsum_ldistr [OF UP_cring] lemmas (in UP_cring) UP_finsum_rdistr = cring.finsum_rdistr [OF UP_cring] subsection {* Polynomials form an Algebra *} lemma (in UP_cring) UP_smult_l_distr: "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==> (a \<oplus> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 b \<odot>\<^sub>2 p" by (rule up_eqI) (simp_all add: R.l_distr) lemma (in UP_cring) UP_smult_r_distr: "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==> a \<odot>\<^sub>2 (p \<oplus>\<^sub>2 q) = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 a \<odot>\<^sub>2 q" by (rule up_eqI) (simp_all add: R.r_distr) lemma (in UP_cring) UP_smult_assoc1: "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==> (a \<otimes> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 p)" by (rule up_eqI) (simp_all add: R.m_assoc) lemma (in UP_cring) UP_smult_one [simp]: "p \<in> carrier P ==> \<one> \<odot>\<^sub>2 p = p" by (rule up_eqI) simp_all lemma (in UP_cring) UP_smult_assoc2: "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==> (a \<odot>\<^sub>2 p) \<otimes>\<^sub>2 q = a \<odot>\<^sub>2 (p \<otimes>\<^sub>2 q)" by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def) text {* Instantiation of lemmas from @{term algebra}. *} (* TODO: move to CRing.thy, really a fact missing from the locales package *) 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 cring UP_cring UP_smult_l_distr UP_smult_r_distr UP_smult_assoc1 UP_smult_assoc2) lemmas (in UP_cring) UP_smult_l_null [simp] = algebra.smult_l_null [OF UP_algebra] lemmas (in UP_cring) UP_smult_r_null [simp] = algebra.smult_r_null [OF UP_algebra] lemmas (in UP_cring) UP_smult_l_minus = algebra.smult_l_minus [OF UP_algebra] lemmas (in UP_cring) UP_smult_r_minus = algebra.smult_r_minus [OF UP_algebra] subsection {* Further lemmas involving monomials *} lemma (in UP_cring) monom_zero [simp]: "monom P \<zero> n = \<zero>\<^sub>2" by (simp add: UP_def P_def) ML_setup {* simpset_ref() := simpset() setsubgoaler asm_full_simp_tac; *} lemma (in UP_cring) monom_mult_is_smult: assumes R: "a \<in> carrier R" "p \<in> carrier P" shows "monom P a 0 \<otimes>\<^sub>2 p = a \<odot>\<^sub>2 p" proof (rule up_eqI) fix n have "coeff P (p \<otimes>\<^sub>2 monom P a 0) n = coeff P (a \<odot>\<^sub>2 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: finsum_cong add: R.r_null Pi_def) (simp add: m_comm) qed with R show "coeff P (monom P a 0 \<otimes>\<^sub>2 p) n = coeff P (a \<odot>\<^sub>2 p) n" by (simp add: UP_m_comm) qed (simp_all add: R) ML_setup {* simpset_ref() := simpset() setsubgoaler asm_simp_tac; *} lemma (in UP_cring) monom_add [simp]: "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<oplus> b) n = monom P a n \<oplus>\<^sub>2 monom P b n" by (rule up_eqI) simp_all ML_setup {* simpset_ref() := simpset() setsubgoaler asm_full_simp_tac; *} lemma (in UP_cring) monom_one_Suc: "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^sub>2 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 \<otimes>\<^sub>2 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 \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes> coeff P (monom P \<one> 1) (k - i))" by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def) also have "... = (\<Oplus>i \<in> {..n}. coeff P (monom P \<one> n) i \<otimes> coeff P (monom P \<one> 1) (k - i))" by (simp only: ivl_disj_un_singleton) also from True have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes> coeff P (monom P \<one> 1) (k - i))" by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one order_less_imp_not_eq Pi_def) also from True have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k" by (simp add: ivl_disj_un_one) finally show ?thesis . qed next case False note neq = False let ?s = "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (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 \<in> {..k}. ?s i)" proof - have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>" by (simp cong: finsum_cong add: Pi_def) from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>" by (simp cong: finsum_cong add: Pi_def) arith have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>" by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def) show ?thesis proof (cases "k < n") case True then show ?thesis by (simp cong: 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 \<in> {..<n} \<union> {n}. ?s i)" by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_singleton Pi_def) also from True have "... = (\<Oplus>i \<in> {..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 \<in> {..<n} \<union> {n}. ?s i)" by (simp add: finsum_Un_disjoint f1 f2 ivl_disj_int_singleton Pi_def del: Un_insert_right) also have "... = (\<Oplus>i \<in> {..n}. ?s i)" by (simp only: ivl_disj_un_singleton) also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)" by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def) also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)" by (simp only: ivl_disj_un_one) finally show ?thesis . qed qed qed also have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k" by simp finally show ?thesis . qed qed (simp_all) ML_setup {* simpset_ref() := simpset() setsubgoaler asm_simp_tac; *} lemma (in UP_cring) monom_mult_smult: "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^sub>2 monom P b n" by (rule up_eqI) simp_all lemma (in UP_cring) monom_one [simp]: "monom P \<one> 0 = \<one>\<^sub>2" by (rule up_eqI) simp_all lemma (in UP_cring) monom_one_mult: "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^sub>2 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: UP_m_ac) qed lemma (in UP_cring) monom_mult [simp]: assumes R: "a \<in> carrier R" "b \<in> carrier R" shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^sub>2 monom P b m" proof - from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp also from R have "... = a \<otimes> b \<odot>\<^sub>2 monom P \<one> (n + m)" by (simp add: monom_mult_smult del: r_one) also have "... = a \<otimes> b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m)" by (simp only: monom_one_mult) also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m))" by (simp add: UP_smult_assoc1) also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> m \<otimes>\<^sub>2 monom P \<one> n))" by (simp add: UP_m_comm) also from R have "... = a \<odot>\<^sub>2 ((b \<odot>\<^sub>2 monom P \<one> m) \<otimes>\<^sub>2 monom P \<one> n)" by (simp add: UP_smult_assoc2) also from R have "... = a \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m))" by (simp add: UP_m_comm) also from R have "... = (a \<odot>\<^sub>2 monom P \<one> n) \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m)" by (simp add: UP_smult_assoc2) also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^sub>2 monom P (b \<otimes> \<one>) m" by (simp add: monom_mult_smult del: r_one) also from R have "... = monom P a n \<otimes>\<^sub>2 monom P b m" by simp finally show ?thesis . qed lemma (in UP_cring) monom_a_inv [simp]: "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^sub>2 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 \<in> carrier R" "y \<in> 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 :: "[_, 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 \<in> 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: "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>" proof - assume R: "p \<in> carrier P" and "deg R p < m" from R 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) then show ?thesis .. qed lemma (in UP_cring) deg_belowI: assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>" and R: "p \<in> 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 \<in> 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 proof 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 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>\<^sub>2" and R: "p \<in> 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>\<^sub>2" by (auto intro: up_eqI) with nonzero show ?thesis by contradiction qed then obtain m where coeff: "coeff P p m ~= \<zero>" .. then 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>\<^sub>2" and R: "p \<in> 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 \<in> carrier P |] ==> deg R p = n" by (fast intro: le_anti_sym deg_aboveI deg_belowI) (* Degree and polynomial operations *) lemma (in UP_cring) deg_add [simp]: assumes R: "p \<in> carrier P" "q \<in> carrier P" shows "deg R (p \<oplus>\<^sub>2 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 \<in> 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 \<in> 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 \<in> 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>\<^sub>2 = 0" proof (rule le_anti_sym) show "deg R \<zero>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all next show "0 <= deg R \<zero>\<^sub>2" by (rule deg_belowI) simp_all qed lemma (in UP_cring) deg_one [simp]: "deg R \<one>\<^sub>2 = 0" proof (rule le_anti_sym) show "deg R \<one>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all next show "0 <= deg R \<one>\<^sub>2" by (rule deg_belowI) simp_all qed lemma (in UP_cring) deg_uminus [simp]: assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^sub>2 p) = deg R p" proof (rule le_anti_sym) show "deg R (\<ominus>\<^sub>2 p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R) next show "deg R p <= deg R (\<ominus>\<^sub>2 p)" by (simp add: deg_belowI lcoeff_nonzero_deg inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R) qed lemma (in UP_domain) deg_smult_ring: "[| a \<in> carrier R; p \<in> carrier P |] ==> deg R (a \<odot>\<^sub>2 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 \<in> carrier R" "p \<in> carrier P" shows "deg R (a \<odot>\<^sub>2 p) = (if a = \<zero> then 0 else deg R p)" proof (rule le_anti_sym) show "deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)" by (rule deg_smult_ring) next show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^sub>2 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 \<in> carrier P" "q \<in> carrier P" shows "deg R (p \<otimes>\<^sub>2 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 \<otimes> 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 \<otimes>\<^sub>2 q) m = \<zero>" by simp qed (simp add: R) ML_setup {* simpset_ref() := simpset() setsubgoaler asm_full_simp_tac; *} lemma (in UP_domain) deg_mult [simp]: "[| p ~= \<zero>\<^sub>2; q ~= \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==> deg R (p \<otimes>\<^sub>2 q) = deg R p + deg R q" proof (rule le_anti_sym) assume "p \<in> carrier P" " q \<in> carrier P" show "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q" by (rule deg_mult_cring) next let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))" assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^sub>2" "q ~= \<zero>\<^sub>2" 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 \<otimes>\<^sub>2 q)" proof (rule deg_belowI, simp add: R) have "finsum R ?s {.. deg R p + deg R q} = finsum R ?s ({..< deg R p} Un {deg R p .. deg R p + deg R q})" by (simp only: ivl_disj_un_one) also have "... = finsum R ?s {deg R p .. deg R p + deg R q}" by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one deg_aboveD less_add_diff R Pi_def) also have "...= finsum R ?s ({deg R p} Un {deg R p <.. deg R p + deg R q})" by (simp only: ivl_disj_un_singleton) also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_singleton deg_aboveD R Pi_def) finally have "finsum R ?s {.. deg R p + deg R q} = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" . with nz show "finsum R ?s {.. deg R p + deg R q} ~= \<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 \<in> A -> carrier P ==> coeff P (finsum P p A) k == finsum R (%i. coeff P (p i) k) A" using fin by induct (auto simp: Pi_def) ML_setup {* simpset_ref() := simpset() setsubgoaler asm_full_simp_tac; *} lemma (in UP_cring) up_repr: assumes R: "p \<in> carrier P" shows "(\<Oplus>\<^sub>2 i \<in> {..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>) \<in> carrier R" by simp show "coeff P (finsum P ?s {..deg R p}) k = coeff P p k" proof (cases "k <= deg R p") case True hence "coeff P (finsum P ?s {..deg R p}) k = coeff P (finsum P ?s ({..k} Un {k<..deg R p})) k" by (simp only: ivl_disj_un_one) also from True have "... = coeff P (finsum P ?s {..k}) k" by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def) also have "... = coeff P (finsum P ?s ({..<k} Un {k})) k" by (simp only: ivl_disj_un_singleton) also have "... = coeff P p k" by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def) finally show ?thesis . next case False hence "coeff P (finsum P ?s {..deg R p}) k = coeff P (finsum P ?s ({..<deg R p} Un {deg R p})) k" by (simp only: ivl_disj_un_singleton) also from False have "... = coeff P p k" by (simp cong: finsum_cong add: setsum_Un_disjoint 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 \<in> carrier P |] ==> finsum P (%i. monom P (coeff P p i) i) {..n} = p" proof - let ?s = "(%i. monom P (coeff P p i) i)" assume R: "p \<in> carrier P" and "deg R p <= n" then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} Un {deg R p<..n})" by (simp only: ivl_disj_un_one) also have "... = finsum P ?s {..deg R p}" by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one deg_aboveD R Pi_def) also have "... = p" by (rule up_repr) finally show ?thesis . qed ML_setup {* simpset_ref() := simpset() setsubgoaler asm_simp_tac; *} subsection {* Polynomials over an integral domain form an integral domain *} 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 \<in> carrier R; b \<in> 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>\<^sub>2 ~= \<zero>\<^sub>2" proof assume "\<one>\<^sub>2 = \<zero>\<^sub>2" hence "coeff P \<one>\<^sub>2 0 = (coeff P \<zero>\<^sub>2 0)" by simp hence "\<one> = \<zero>" by simp with one_not_zero show "False" by contradiction qed lemma (in UP_domain) UP_integral: "[| p \<otimes>\<^sub>2 q = \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2" proof - fix p q assume pq: "p \<otimes>\<^sub>2 q = \<zero>\<^sub>2" and R: "p \<in> carrier P" "q \<in> carrier P" show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2" proof (rule classical) assume c: "~ (p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2)" with R have "deg R p + deg R q = deg R (p \<otimes>\<^sub>2 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>\<^sub>2 i \<in> {..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>\<^sub>2 i \<in> {..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 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^sub>2 q) 0" by simp also from pq have "... = \<zero>" by simp finally have "coeff P p 0 \<otimes> 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>\<^sub>2 | q = \<zero>\<^sub>2" 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 {* Instantiation of results from @{term domain}. *} lemmas (in UP_domain) UP_zero_not_one [simp] = domain.zero_not_one [OF UP_domain] lemmas (in UP_domain) UP_integral_iff = domain.integral_iff [OF UP_domain] lemmas (in UP_domain) UP_m_lcancel = domain.m_lcancel [OF UP_domain] lemmas (in UP_domain) UP_m_rcancel = domain.m_rcancel [OF UP_domain] lemma (in UP_domain) smult_integral: "[| a \<odot>\<^sub>2 p = \<zero>\<^sub>2; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^sub>2" by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff inj_on_iff [OF monom_inj, of _ "\<zero>", simplified]) subsection {* Evaluation Homomorphism and Universal Property*} (* alternative congruence rule (possibly more efficient) lemma (in abelian_monoid) finsum_cong2: "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B; !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B" sorry*) ML_setup {* simpset_ref() := simpset() setsubgoaler asm_full_simp_tac; *} theorem (in cring) diagonal_sum: "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==> (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) = (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)" proof - assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R" { fix j have "j <= n + m ==> (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) = (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)" proof (induct j) case 0 from Rf Rg show ?case by (simp add: Pi_def) next case (Suc j) (* The following could be simplified if there was a reasoner for total orders integrated with simp. *) have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R" using Suc by (auto intro!: funcset_mem [OF Rg]) arith have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R" using Suc by (auto intro!: funcset_mem [OF Rg]) arith have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R" using Suc by (auto intro!: funcset_mem [OF Rf]) have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R" using Suc by (auto intro!: funcset_mem [OF Rg]) arith have R11: "g 0 \<in> 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 \<in> carrier G" by auto theorem (in cring) cauchy_product: assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g" and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R" shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)" (* State revese direction? *) proof - have f: "!!x. f x \<in> carrier R" proof - fix x show "f x \<in> carrier R" using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def) qed have g: "!!x. g x \<in> carrier R" proof - fix x show "g x \<in> carrier R" using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def) qed from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) = (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)" by (simp add: diagonal_sum Pi_def) also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)" by (simp only: ivl_disj_un_one) also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> 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 \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)" by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def) also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> 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 \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..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) \<in> ring_hom R P" by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult) constdefs (structure S) eval :: "[_, _, 'a => 'b, 'b, nat => 'a] => 'b" "eval R S phi s == \<lambda>p \<in> carrier (UP R). \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> pow S s i" (* "eval R S phi s p == if p \<in> carrier (UP R) then finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p} else arbitrary" *) locale ring_hom_UP_cring = ring_hom_cring R S + UP_cring R lemma (in ring_hom_UP_cring) eval_on_carrier: "p \<in> carrier P ==> eval R S phi s p = (\<Oplus>\<^sub>2 i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^sub>2 pow S s i)" by (unfold eval_def, fold P_def) simp lemma (in ring_hom_UP_cring) eval_extensional: "eval R S phi s \<in> extensional (carrier P)" by (unfold eval_def, fold P_def) simp theorem (in ring_hom_UP_cring) eval_ring_hom: "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S" proof (rule ring_hom_memI) fix p assume RS: "p \<in> carrier P" "s \<in> carrier S" then show "eval R S h s p \<in> carrier S" by (simp only: eval_on_carrier) (simp add: Pi_def) next fix p q assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S" then show "eval R S h s (p \<otimes>\<^sub>3 q) = eval R S h s p \<otimes>\<^sub>2 eval R S h s q" proof (simp only: eval_on_carrier UP_mult_closed) from RS have "(\<Oplus>\<^sub>2 i \<in> {..deg R (p \<otimes>\<^sub>3 q)}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) = (\<Oplus>\<^sub>2 i \<in> {..deg R (p \<otimes>\<^sub>3 q)} \<union> {deg R (p \<otimes>\<^sub>3 q)<..deg R p + deg R q}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" by (simp cong: finsum_cong add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_mult) also from RS have "... = (\<Oplus>\<^sub>2 i \<in> {..deg R p + deg R q}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" by (simp only: ivl_disj_un_one deg_mult_cring) also from RS have "... = (\<Oplus>\<^sub>2 i \<in> {..deg R p + deg R q}. \<Oplus>\<^sub>2 k \<in> {..i}. h (coeff P p k) \<otimes>\<^sub>2 h (coeff P q (i - k)) \<otimes>\<^sub>2 (s (^)\<^sub>2 k \<otimes>\<^sub>2 s (^)\<^sub>2 (i - k)))" by (simp cong: finsum_cong add: nat_pow_mult Pi_def S.m_ac S.finsum_rdistr) also from RS have "... = (\<Oplus>\<^sub>2i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<otimes>\<^sub>2 (\<Oplus>\<^sub>2i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac Pi_def) finally show "(\<Oplus>\<^sub>2 i \<in> {..deg R (p \<otimes>\<^sub>3 q)}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) = (\<Oplus>\<^sub>2 i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<otimes>\<^sub>2 (\<Oplus>\<^sub>2 i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" . qed next fix p q assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S" then show "eval R S h s (p \<oplus>\<^sub>3 q) = eval R S h s p \<oplus>\<^sub>2 eval R S h s q" proof (simp only: eval_on_carrier UP_a_closed) from RS have "(\<Oplus>\<^sub>2i \<in> {..deg R (p \<oplus>\<^sub>3 q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) = (\<Oplus>\<^sub>2i \<in> {..deg R (p \<oplus>\<^sub>3 q)} \<union> {deg R (p \<oplus>\<^sub>3 q)<..max (deg R p) (deg R q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" by (simp cong: finsum_cong add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add) also from RS have "... = (\<Oplus>\<^sub>2 i \<in> {..max (deg R p) (deg R q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" by (simp add: ivl_disj_un_one) also from RS have "... = (\<Oplus>\<^sub>2i\<in>{..max (deg R p) (deg R q)}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2 (\<Oplus>\<^sub>2i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" by (simp cong: finsum_cong add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def) also have "... = (\<Oplus>\<^sub>2 i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2 (\<Oplus>\<^sub>2 i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" by (simp only: ivl_disj_un_one le_maxI1 le_maxI2) also from RS have "... = (\<Oplus>\<^sub>2 i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2 (\<Oplus>\<^sub>2 i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" by (simp cong: finsum_cong add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def) finally show "(\<Oplus>\<^sub>2i \<in> {..deg R (p \<oplus>\<^sub>3 q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) = (\<Oplus>\<^sub>2i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2 (\<Oplus>\<^sub>2i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" . qed next assume S: "s \<in> carrier S" then show "eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2" by (simp only: eval_on_carrier UP_one_closed) simp qed text {* Instantiation of ring homomorphism lemmas. *} lemma (in ring_hom_UP_cring) ring_hom_cring_P_S: "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)" by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems intro: ring_hom_cring_axioms.intro eval_ring_hom) lemma (in ring_hom_UP_cring) UP_hom_closed [intro, simp]: "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S" by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S]) lemma (in ring_hom_UP_cring) UP_hom_mult [simp]: "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==> eval R S h s (p \<otimes>\<^sub>3 q) = eval R S h s p \<otimes>\<^sub>2 eval R S h s q" by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S]) lemma (in ring_hom_UP_cring) UP_hom_add [simp]: "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==> eval R S h s (p \<oplus>\<^sub>3 q) = eval R S h s p \<oplus>\<^sub>2 eval R S h s q" by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S]) lemma (in ring_hom_UP_cring) UP_hom_one [simp]: "s \<in> carrier S ==> eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2" by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S]) lemma (in ring_hom_UP_cring) UP_hom_zero [simp]: "s \<in> carrier S ==> eval R S h s \<zero>\<^sub>3 = \<zero>\<^sub>2" by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S]) lemma (in ring_hom_UP_cring) UP_hom_a_inv [simp]: "[| s \<in> carrier S; p \<in> carrier P |] ==> (eval R S h s) (\<ominus>\<^sub>3 p) = \<ominus>\<^sub>2 (eval R S h s) p" by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S]) lemma (in ring_hom_UP_cring) UP_hom_finsum [simp]: "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==> (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A" by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S]) lemma (in ring_hom_UP_cring) UP_hom_finprod [simp]: "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==> (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A" by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S]) text {* Further properties of the evaluation homomorphism. *} (* The following lemma could be proved in UP\_cring with the additional assumption that h is closed. *) lemma (in ring_hom_UP_cring) eval_const: "[| s \<in> carrier S; r \<in> 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 ring_hom_UP_cring) eval_monom1: "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s" proof (simp only: eval_on_carrier monom_closed R.one_closed) assume S: "s \<in> carrier S" then have "(\<Oplus>\<^sub>2 i \<in> {..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) = (\<Oplus>\<^sub>2i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" by (simp cong: finsum_cong del: coeff_monom add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def) also have "... = (\<Oplus>\<^sub>2 i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" by (simp only: ivl_disj_un_one deg_monom_le R.one_closed) also have "... = s" proof (cases "s = \<zero>\<^sub>2") case True then show ?thesis by (simp add: Pi_def) next case False with S show ?thesis by (simp add: Pi_def) qed finally show "(\<Oplus>\<^sub>2 i \<in> {..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) = s" . qed lemma (in UP_cring) monom_pow: assumes R: "a \<in> carrier R" shows "(monom P a n) (^)\<^sub>2 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 \<in> carrier R ==> h (x (^) n) = h x (^)\<^sub>2 (n::nat)" by (induct n) simp_all lemma (in ring_hom_UP_cring) UP_hom_pow [simp]: "[| s \<in> carrier S; p \<in> carrier P |] ==> (eval R S h s) (p (^)\<^sub>3 n) = eval R S h s p (^)\<^sub>2 (n::nat)" by (rule ring_hom_cring.hom_pow [OF ring_hom_cring_P_S]) lemma (in ring_hom_UP_cring) eval_monom: "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r n) = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n" proof - assume RS: "s \<in> carrier S" "r \<in> carrier R" then have "eval R S h s (monom P r n) = eval R S h s (monom P r 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 n)" by (simp del: monom_mult UP_hom_mult UP_hom_pow add: monom_mult [THEN sym] monom_pow) also from RS eval_monom1 have "... = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n" by (simp add: eval_const) finally show ?thesis . qed lemma (in ring_hom_UP_cring) eval_smult: "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==> eval R S h s (r \<odot>\<^sub>3 p) = h r \<otimes>\<^sub>2 eval R S h s p" by (simp add: monom_mult_is_smult [THEN sym] eval_const) lemma ring_hom_cringI: assumes "cring R" and "cring S" and "h \<in> 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 ring_hom_UP_cring) UP_hom_unique: assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s" "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r" and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s" "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r" and RS: "s \<in> carrier S" "p \<in> carrier P" shows "Phi p = Psi p" proof - have Phi_hom: "ring_hom_cring P S Phi" by (auto intro: ring_hom_cringI UP_cring S.cring Phi) have Psi_hom: "ring_hom_cring P S Psi" by (auto intro: ring_hom_cringI UP_cring S.cring Psi) have "Phi p = Phi (\<Oplus>\<^sub>3i \<in> {..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^sub>3 monom P \<one> 1 (^)\<^sub>3 i)" by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult) also have "... = Psi (\<Oplus>\<^sub>3i\<in>{..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^sub>3 monom P \<one> 1 (^)\<^sub>3 i)" by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom] ring_hom_cring.hom_mult [OF Phi_hom] ring_hom_cring.hom_pow [OF Phi_hom] Phi ring_hom_cring.hom_finsum [OF Psi_hom] ring_hom_cring.hom_mult [OF Psi_hom] ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def) also have "... = Psi p" by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult) finally show ?thesis . qed theorem (in ring_hom_UP_cring) UP_universal_property: "s \<in> carrier S ==> EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) & Phi (monom P \<one> 1) = s & (ALL r : carrier R. Phi (monom P r 0) = h r)" using eval_monom1 apply (auto intro: eval_ring_hom eval_const eval_extensional) apply (rule extensionalityI) apply (auto intro: UP_hom_unique) done subsection {* Sample application of evaluation homomorphism *} lemma ring_hom_UP_cringI: assumes "cring R" and "cring S" and "h \<in> ring_hom R S" shows "ring_hom_UP_cring R S h" by (fast intro: ring_hom_UP_cring.intro ring_hom_cring_axioms.intro cring.axioms prems) constdefs INTEG :: "int ring" "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)" lemma cring_INTEG: "cring INTEG" by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI zadd_zminus_inverse2 zadd_zmult_distrib) lemma INTEG_id: "ring_hom_UP_cring INTEG INTEG id" by (fast intro: ring_hom_UP_cringI cring_INTEG id_ring_hom) text {* An instantiation mechanism would now import all theorems and lemmas valid in the context of homomorphisms between @{term INTEG} and @{term "UP INTEG"}. *} lemma INTEG_closed [intro, simp]: "z \<in> 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: ring_hom_UP_cring.eval_monom [OF INTEG_id]) end