Theory Polynomials.MPoly_Type_Class

(* Author: Fabian Immler, Alexander Maletzky *)

section ‹Type-Class-Multivariate Polynomials›

theory MPoly_Type_Class
  imports
    Utils
    Power_Products
    More_Modules
begin

text ‹This theory views @{typ "'a 0 'b"} as multivariate polynomials, where type class constraints
  on @{typ 'a} ensure that @{typ 'a} represents something like monomials.›

lemma when_distrib: "f (a when b) = (f a when b)" if "¬ b  f 0 = 0"
  using that by (auto simp: when_def)

definition mapp_2 :: "('a  'b  'c  'd)  ('a 0 'b::zero)  ('a 0 'c::zero)  ('a 0 'd::zero)"
  where "mapp_2 f p q = Abs_poly_mapping (λk. f k (lookup p k) (lookup q k) when k  keys p  keys q)"

lemma lookup_mapp_2:
  "lookup (mapp_2 f p q) k = (f k (lookup p k) (lookup q k) when k  keys p  keys q)"
proof -
  have "lookup (Abs_poly_mapping (λk. f k (lookup p k) (lookup q k) when k  keys p  keys q)) =
        (λk. f k (lookup p k) (lookup q k) when k  keys p  keys q)"
    by (rule Abs_poly_mapping_inverse, simp)
  thus ?thesis by (simp add: mapp_2_def)
qed

lemma lookup_mapp_2_homogenous:
  assumes "f k 0 0 = 0"
  shows "lookup (mapp_2 f p q) k = f k (lookup p k) (lookup q k)"
  by (simp add: lookup_mapp_2 when_def in_keys_iff assms)

lemma mapp_2_cong [fundef_cong]:
  assumes "p = p'" and "q = q'"
  assumes "k. k  keys p'  keys q'  f k (lookup p' k) (lookup q' k) = f' k (lookup p' k) (lookup q' k)"
  shows "mapp_2 f p q = mapp_2 f' p' q'"
  by (rule poly_mapping_eqI, simp add: assms(1, 2) lookup_mapp_2, rule when_cong, fact refl, rule assms(3), blast)

lemma keys_mapp_subset: "keys (mapp_2 f p q)  keys p  keys q"
proof
  fix t
  assume "t  keys (mapp_2 f p q)"
  hence "lookup (mapp_2 f p q) t  0" by (simp add: in_keys_iff) 
  thus "t  keys p  keys q" by (simp add: lookup_mapp_2 when_def split: if_split_asm)
qed

lemma mapp_2_mapp: "mapp_2 (λt a. f t) 0 p = Poly_Mapping.mapp f p"
  by (rule poly_mapping_eqI, simp add: lookup_mapp lookup_mapp_2)

subsection @{const keys}

lemma in_keys_plusI1:
  assumes "t  keys p" and "t  keys q"
  shows "t  keys (p + q)"
  using assms unfolding in_keys_iff lookup_add by simp

lemma in_keys_plusI2:
  assumes "t  keys q" and "t  keys p"
  shows "t  keys (p + q)"
  using assms unfolding in_keys_iff lookup_add by simp

lemma keys_plus_eqI:
  assumes "keys p  keys q = {}"
  shows "keys (p + q) = (keys p  keys q)"
proof
  show "keys (p + q)  keys p  keys q"
    by (simp add: Poly_Mapping.keys_add)
  show "keys p  keys q  keys (p + q)"
    by (simp add: More_MPoly_Type.keys_add assms)
qed
  
lemma keys_uminus: "keys (- p) = keys p"
  by (transfer, auto)

lemma keys_minus: "keys (p - q)  (keys p  keys q)"
  by (transfer, auto)

subsection ‹Monomials›

abbreviation "monomial  (λc t. Poly_Mapping.single t c)"

lemma keys_of_monomial:
  assumes "c  0"
  shows "keys (monomial c t) = {t}"
  using assms by simp

lemma monomial_uminus:
  shows "- monomial c s = monomial (- c) s"
  by (transfer, rule ext, simp add: Poly_Mapping.when_def)

lemma monomial_inj:
  assumes "monomial c s = monomial (d::'b::zero_neq_one) t"
  shows "(c = 0  d = 0)  (c = d  s = t)"
  using assms unfolding poly_mapping_eq_iff
  by (metis (mono_tags, opaque_lifting) lookup_single_eq lookup_single_not_eq)

definition is_monomial :: "('a 0 'b::zero)  bool"
  where "is_monomial p  card (keys p) = 1"

lemma monomial_is_monomial:
  assumes "c  0"
  shows "is_monomial (monomial c t)"
  using keys_single[of t c] assms by (simp add: is_monomial_def)

lemma is_monomial_monomial:
  assumes "is_monomial p"
  obtains c t where "c  0" and "p = monomial c t"
proof -
  from assms have "card (keys p) = 1" unfolding is_monomial_def .
  then obtain t where sp: "keys p = {t}" by (rule card_1_singletonE)
  let ?c = "lookup p t"
  from sp have "?c  0" by fastforce
  show ?thesis
  proof
    show "p = monomial ?c t"
    proof (intro poly_mapping_keys_eqI)
      from sp show "keys p = keys (monomial ?c t)" using ?c  0 by simp
    next
      fix s
      assume "s  keys p"
      with sp have "s = t" by simp
      show "lookup p s = lookup (monomial ?c t) s" by (simp add: s = t)
    qed
  qed fact
qed
  
lemma is_monomial_uminus: "is_monomial (-p)  is_monomial p"
  unfolding is_monomial_def keys_uminus ..

lemma monomial_not_0:
  assumes "is_monomial p"
  shows "p  0"
  using assms unfolding is_monomial_def by auto

lemma keys_subset_singleton_imp_monomial:
  assumes "keys p  {t}"
  shows "monomial (lookup p t) t = p"
proof (rule poly_mapping_eqI, simp add: lookup_single when_def, rule)
  fix s
  assume "t  s"
  hence "s  keys p" using assms by blast
  thus "lookup p s = 0" by (simp add: in_keys_iff) 
qed

lemma monomial_0I:
  assumes "c = 0"
  shows "monomial c t = 0"
  using assms by transfer (auto)

lemma monomial_0D:
  assumes "monomial c t = 0"
  shows "c = 0"
  using assms by transfer (auto simp: fun_eq_iff when_def; meson)

corollary monomial_0_iff: "monomial c t = 0  c = 0"
  by (rule, erule monomial_0D, erule monomial_0I)

lemma lookup_times_monomial_left: "lookup (monomial c t * p) s = (c * lookup p (s - t) when t adds s)"
  for c::"'b::semiring_0" and t::"'a::comm_powerprod"
proof (induct p rule: poly_mapping_except_induct, simp)
  fix p::"'a 0 'b" and w
  assume "p  0" and "w  keys p"
    and IH: "lookup (monomial c t * except p {w}) s =
             (c * lookup (except p {w}) (s - t) when t adds s)" (is "_ = ?x")
  have "monomial c t * p = monomial c t * (monomial (lookup p w) w + except p {w})"
    by (simp only: plus_except[symmetric])
  also have "... = monomial c t * monomial (lookup p w) w + monomial c t * except p {w}"
    by (simp add: algebra_simps)
  also have "... = monomial (c * lookup p w) (t + w) + monomial c t * except p {w}"
    by (simp only: mult_single)
  finally have "lookup (monomial c t * p) s = lookup (monomial (c * lookup p w) (t + w)) s + ?x"
    by (simp only: lookup_add IH)
  also have "... = (lookup (monomial (c * lookup p w) (t + w)) s +
                    c * lookup (except p {w}) (s - t) when t adds s)"
    by (rule when_distrib, auto simp add: lookup_single when_def)
  also from refl have "... = (c * lookup p (s - t) when t adds s)"
  proof (rule when_cong)
    assume "t adds s"
    then obtain u where u: "s = t + u" ..
    show "lookup (monomial (c * lookup p w) (t + w)) s + c * lookup (except p {w}) (s - t) =
          c * lookup p (s - t)"
      by (simp add: u, cases "u = w", simp_all add: lookup_except lookup_single add.commute)
  qed
  finally show "lookup (monomial c t * p) s = (c * lookup p (s - t) when t adds s)" .
qed

lemma lookup_times_monomial_right: "lookup (p * monomial c t) s = (lookup p (s - t) * c when t adds s)"
  for c::"'b::semiring_0" and t::"'a::comm_powerprod"
proof (induct p rule: poly_mapping_except_induct, simp)
  fix p::"'a 0 'b" and w
  assume "p  0" and "w  keys p"
    and IH: "lookup (except p {w} * monomial c t) s =
             ((lookup (except p {w}) (s - t)) * c when t adds s)"
            (is "_ = ?x")
  have "p * monomial c t = (monomial (lookup p w) w + except p {w}) * monomial c t"
    by (simp only: plus_except[symmetric])
  also have "... = monomial (lookup p w) w * monomial c t + except p {w} * monomial c t"
    by (simp add: algebra_simps)
  also have "... = monomial (lookup p w * c) (w + t) + except p {w} * monomial c t"
    by (simp only: mult_single)
  finally have "lookup (p * monomial c t) s = lookup (monomial (lookup p w * c) (w + t)) s + ?x"
    by (simp only: lookup_add IH)
  also have "... = (lookup (monomial (lookup p w * c) (w + t)) s +
                    lookup (except p {w}) (s - t) * c when t adds s)"
    by (rule when_distrib, auto simp add: lookup_single when_def)
  also from refl have "... = (lookup p (s - t) * c when t adds s)"
  proof (rule when_cong)
    assume "t adds s"
    then obtain u where u: "s = t + u" ..
    show "lookup (monomial (lookup p w * c) (w + t)) s + lookup (except p {w}) (s - t) * c =
          lookup p (s - t) * c"
      by (simp add: u, cases "u = w", simp_all add: lookup_except lookup_single add.commute)
  qed
  finally show "lookup (p * monomial c t) s = (lookup p (s - t) * c when t adds s)" .
qed

subsection ‹Vector-Polynomials›

text ‹From now on we consider multivariate vector-polynomials, i.\,e. vectors of scalar polynomials.
  We do this by adding a @{emph ‹component›} to each power-product, yielding
  @{emph ‹terms›}. Vector-polynomials are then again just linear combinations of terms.
  Note that a term is @{emph ‹not›} the same as a vector of power-products!›

text ‹We use define terms in a locale, such that later on we can interpret the
  locale also by ordinary power-products (without components), exploiting the canonical isomorphism
  between @{typ 'a} and @{typ 'a × unit}.›

named_theorems term_simps "simplification rules for terms"

locale term_powerprod =
		fixes pair_of_term::"'t  ('a::comm_powerprod × 'k::linorder)"
		fixes term_of_pair::"('a × 'k)  't"
		assumes term_pair [term_simps]: "term_of_pair (pair_of_term v) = v"
		assumes pair_term [term_simps]: "pair_of_term (term_of_pair p) = p"
begin

lemma pair_of_term_injective:
  assumes "pair_of_term u = pair_of_term v"
  shows "u = v"
proof -
  from assms have "term_of_pair (pair_of_term u) = term_of_pair (pair_of_term v)" by (simp only:)
  thus ?thesis by (simp add: term_simps)
qed

corollary pair_of_term_inj: "inj pair_of_term"
  using pair_of_term_injective by (rule injI)

lemma term_of_pair_injective:
  assumes "term_of_pair p = term_of_pair q"
  shows "p = q"
proof -
  from assms have "pair_of_term (term_of_pair p) = pair_of_term (term_of_pair q)" by (simp only:)
  thus ?thesis by (simp add: term_simps)
qed

corollary term_of_pair_inj: "inj term_of_pair"
  using term_of_pair_injective by (rule injI)

definition pp_of_term :: "'t  'a"
  where "pp_of_term v = fst (pair_of_term v)"

definition component_of_term :: "'t  'k"
  where "component_of_term v = snd (pair_of_term v)"

lemma term_of_pair_pair [term_simps]: "term_of_pair (pp_of_term v, component_of_term v) = v"
  by (simp add: pp_of_term_def component_of_term_def term_pair)

lemma pp_of_term_of_pair [term_simps]: "pp_of_term (term_of_pair (t, k)) = t"
  by (simp add: pp_of_term_def pair_term)

lemma component_of_term_of_pair [term_simps]: "component_of_term (term_of_pair (t, k)) = k"
  by (simp add: component_of_term_def pair_term)

subsubsection ‹Additive Structure of Terms›

definition splus :: "'a  't  't" (infixl "" 75)
  where "splus t v = term_of_pair (t + pp_of_term v, component_of_term v)"

definition sminus :: "'t  'a  't" (infixl "" 75)
  where "sminus v t = term_of_pair (pp_of_term v - t, component_of_term v)"

text ‹Note that the argument order in @{const sminus} is reversed compared to the order in @{const splus}.›

definition adds_pp :: "'a  't  bool" (infix "addsp" 50)
  where "adds_pp t v  t adds pp_of_term v"

definition adds_term :: "'t  't  bool" (infix "addst" 50)
  where "adds_term u v  component_of_term u = component_of_term v  pp_of_term u adds pp_of_term v"

lemma pp_of_term_splus [term_simps]: "pp_of_term (t  v) = t + pp_of_term v"
  by (simp add: splus_def term_simps)

lemma component_of_term_splus [term_simps]: "component_of_term (t  v) = component_of_term v"
  by (simp add: splus_def term_simps)

lemma pp_of_term_sminus [term_simps]: "pp_of_term (v  t) = pp_of_term v - t"
  by (simp add: sminus_def term_simps)

lemma component_of_term_sminus [term_simps]: "component_of_term (v  t) = component_of_term v"
  by (simp add: sminus_def term_simps)

lemma splus_sminus [term_simps]: "(t  v)  t = v"
  by (simp add: sminus_def term_simps)

lemma splus_zero [term_simps]: "0  v = v"
  by (simp add: splus_def term_simps)

lemma sminus_zero [term_simps]: "v  0 = v"
  by (simp add: sminus_def term_simps)

lemma splus_assoc [ac_simps]: "(s + t)  v = s  (t  v)"
  by (simp add: splus_def ac_simps term_simps)

lemma splus_left_commute [ac_simps]: "s  (t  v) = t  (s  v)"
  by (simp add: splus_def ac_simps term_simps)

lemma splus_right_canc [term_simps]: "t  v = s  v  t = s"
  by (metis add_right_cancel pp_of_term_splus)

lemma splus_left_canc [term_simps]: "t  v = t  u  v = u"
  by (metis splus_sminus)

lemma adds_ppI [intro?]:
  assumes "v = t  u"
  shows "t addsp v"
  by (simp add: adds_pp_def assms splus_def term_simps)

lemma adds_ppE [elim?]:
  assumes "t addsp v"
  obtains u where "v = t  u"
proof -
  from assms obtain s where *: "pp_of_term v = t + s" unfolding adds_pp_def ..
  have "v = t  (term_of_pair (s, component_of_term v))"
    by (simp add: splus_def term_simps, metis * add.commute term_of_pair_pair)
  thus ?thesis ..
qed

lemma adds_pp_alt: "t addsp v  (u. v = t  u)"
  by (meson adds_ppE adds_ppI)

lemma adds_pp_refl [term_simps]: "(pp_of_term v) addsp v"
  by (simp add: adds_pp_def)

lemma adds_pp_trans [trans]:
  assumes "s adds t" and "t addsp v"
  shows "s addsp v"
proof -
  note assms(1)
  also from assms(2) have "t adds pp_of_term v" by (simp only: adds_pp_def)
  finally show ?thesis by (simp only: adds_pp_def)
qed

lemma zero_adds_pp [term_simps]: "0 addsp v"
  by (simp add: adds_pp_def)

lemma adds_pp_splus:
  assumes "t addsp v"
  shows "t addsp s  v"
  using assms by (simp add: adds_pp_def term_simps)

lemma adds_pp_triv [term_simps]: "t addsp t  v"
  by (simp add: adds_pp_def term_simps)

lemma plus_adds_pp_mono:
  assumes "s adds t"
    and "u addsp v"
  shows "s + u addsp t  v"
  using assms by (simp add: adds_pp_def term_simps) (rule plus_adds_mono)

lemma plus_adds_pp_left:
  assumes "s + t addsp v"
  shows "s addsp v"
  using assms by (simp add: adds_pp_def plus_adds_left)

lemma plus_adds_pp_right:
  assumes "s + t addsp v"
  shows "t addsp v"
  using assms by (simp add: adds_pp_def plus_adds_right)

lemma adds_pp_sminus:
  assumes "t addsp v"
  shows "t  (v  t) = v"
proof -
  from assms adds_pp_alt[of t v] obtain u where u: "v = t  u" by (auto simp: ac_simps)
  hence "v  t = u" by (simp add: term_simps)
  thus ?thesis using u by simp
qed

lemma adds_pp_canc: "t + s addsp (t  v)  s addsp v"
  by (simp add: adds_pp_def adds_canc_2 term_simps)

lemma adds_pp_canc_2: "s + t addsp (t  v)  s addsp v"
  by (simp add: adds_pp_canc add.commute[of s t])

lemma plus_adds_pp_0:
  assumes "(s + t) addsp v"
  shows "s addsp (v  t)"
  using assms by (simp add: adds_pp_def term_simps) (rule plus_adds_0)

lemma plus_adds_ppI_1:
  assumes "t addsp v" and "s addsp (v  t)"
  shows "(s + t) addsp v"
  using assms by (simp add: adds_pp_def term_simps) (rule plus_adds_2)

lemma plus_adds_ppI_2:
  assumes "t addsp v" and "s addsp (v  t)"
  shows "(t + s) addsp v"
  unfolding add.commute[of t s] using assms by (rule plus_adds_ppI_1)

lemma plus_adds_pp: "(s + t) addsp v  (t addsp v  s addsp (v  t))"
  by (simp add: adds_pp_def plus_adds term_simps)

lemma minus_splus:
  assumes "s adds t"
  shows "(t - s)  v = (t  v)  s"
  by (simp add: assms minus_plus sminus_def splus_def term_simps)

lemma minus_splus_sminus:
  assumes "s adds t" and "u addsp v"
  shows "(t - s)  (v  u) = (t  v)  (s + u)"
  using assms minus_plus_minus term_powerprod.adds_pp_def term_powerprod_axioms sminus_def
    splus_def term_simps by fastforce

lemma minus_splus_sminus_cancel:
  assumes "s adds t" and "t addsp v"
  shows "(t - s)  (v  t) = v  s"
  by (simp add: adds_pp_sminus assms minus_splus)

lemma sminus_plus:
  assumes "s addsp v" and "t addsp (v  s)"
  shows "v  (s + t) = (v  s)  t"
  by (simp add: diff_diff_add sminus_def term_simps)

lemma adds_termI [intro?]:
  assumes "v = t  u"
  shows "u addst v"
  by (simp add: adds_term_def assms splus_def term_simps)

lemma adds_termE [elim?]:
  assumes "u addst v"
  obtains t where "v = t  u"
proof -
  from assms have eq: "component_of_term u = component_of_term v" and "pp_of_term u adds pp_of_term v"
    by (simp_all add: adds_term_def)
  from this(2) obtain s where *: "s + pp_of_term u = pp_of_term v" unfolding adds_term_def
    using adds_minus by blast
  have "v = s  u" by (simp add: splus_def eq * term_simps)
  thus ?thesis ..
qed

lemma adds_term_alt: "u addst v  (t. v = t  u)"
  by (meson adds_termE adds_termI)

lemma adds_term_refl [term_simps]: "v addst v"
  by (simp add: adds_term_def)

lemma adds_term_trans [trans]:
  assumes "u addst v" and "v addst w"
  shows "u addst w"
  using assms unfolding adds_term_def using adds_trans by auto

lemma adds_term_splus:
  assumes "u addst v"
  shows "u addst s  v"
  using assms by (simp add: adds_term_def term_simps)

lemma adds_term_triv [term_simps]: "v addst t  v"
  by (simp add: adds_term_def term_simps)

lemma splus_adds_term_mono:
  assumes "s adds t"
    and "u addst v"
  shows "s  u addst t  v"
  using assms by (auto simp: adds_term_def term_simps intro: plus_adds_mono)

lemma splus_adds_term:
  assumes "t  u addst v"
  shows "u addst v"
  using assms by (auto simp add: adds_term_def term_simps elim: plus_adds_right)

lemma adds_term_adds_pp:
  "u addst v  (component_of_term u = component_of_term v  pp_of_term u addsp v)"
  by (simp add: adds_term_def adds_pp_def)

lemma adds_term_canc: "t  u addst t  v  u addst v"
  by (simp add: adds_term_def adds_canc_2 term_simps)

lemma adds_term_canc_2: "s  v addst t  v  s adds t"
  by (simp add: adds_term_def adds_canc term_simps)

lemma splus_adds_term_0:
  assumes "t  u addst v"
  shows "u addst (v  t)"
  using assms by (simp add: adds_term_def add.commute[of t] term_simps) (auto intro: plus_adds_0)

lemma splus_adds_termI_1:
  assumes "t addsp v" and "u addst (v  t)"
  shows "t  u addst v"
  using assms apply (simp add: adds_term_def term_simps) by (metis add.commute adds_pp_def plus_adds_2)

lemma splus_adds_term_iff: "t  u addst v  (t addsp v  u addst (v  t))"
  by (metis adds_ppI adds_pp_splus adds_termE splus_adds_termI_1 splus_adds_term_0)

lemma adds_minus_splus:
  assumes "pp_of_term u adds t"
  shows "(t - pp_of_term u)  u = term_of_pair (t, component_of_term u)"
  by (simp add: splus_def adds_minus[OF assms])

subsubsection ‹Projections and Conversions›

lift_definition proj_poly :: "'k  ('t 0 'b)  ('a 0 'b::zero)"
  is "λk p t. p (term_of_pair (t, k))"
proof -
  fix k::'k and p::"'t  'b"
  assume fin: "finite {v. p v  0}"
  have "{t. p (term_of_pair (t, k))  0}  pp_of_term ` {v. p v  0}"
  proof (rule, simp)
    fix t
    assume "p (term_of_pair (t, k))  0"
    hence *: "term_of_pair (t, k)  {v. p v  0}" by simp
    have "t = pp_of_term (term_of_pair (t, k))" by (simp add: pp_of_term_def pair_term)
    from this * show "t  pp_of_term ` {v. p v  0}" ..
  qed
  moreover from fin have "finite (pp_of_term ` {v. p v  0})" by (rule finite_imageI)
  ultimately show "finite {t. p (term_of_pair (t, k))  0}" by (rule finite_subset)
qed

definition vectorize_poly :: "('t 0 'b)  ('k 0 ('a 0 'b::zero))"
  where "vectorize_poly p = Abs_poly_mapping (λk. proj_poly k p)"

definition atomize_poly :: "('k 0 ('a 0 'b))  ('t 0 'b::zero)"
  where "atomize_poly p = Abs_poly_mapping (λv. lookup (lookup p (component_of_term v)) (pp_of_term v))"

lemma lookup_proj_poly: "lookup (proj_poly k p) t = lookup p (term_of_pair (t, k))"
  by (transfer, simp)

lemma lookup_vectorize_poly: "lookup (vectorize_poly p) k = proj_poly k p"
proof -
  have "lookup (Abs_poly_mapping (λk. proj_poly k p)) = (λk. proj_poly k p)"
  proof (rule Abs_poly_mapping_inverse, simp)
    have "{k. proj_poly k p  0}  component_of_term ` keys p"
    proof (rule, simp)
      fix k
      assume "proj_poly k p  0"
      hence "keys (proj_poly k p)  {}" using poly_mapping_eq_zeroI by blast
      then obtain t where "lookup (proj_poly k p) t  0" by blast
      hence "term_of_pair (t, k)  keys p" by (simp add: lookup_proj_poly in_keys_iff)
      hence "component_of_term (term_of_pair (t, k))  component_of_term ` keys p" by fastforce
      thus "k  component_of_term ` keys p" by (simp add: term_simps)
    qed
    moreover from finite_keys have "finite (component_of_term ` keys p)" by (rule finite_imageI)
    ultimately show "finite {k. proj_poly k p  0}" by (rule finite_subset)
  qed
  thus ?thesis by (simp add: vectorize_poly_def)
qed

lemma lookup_atomize_poly:
  "lookup (atomize_poly p) v = lookup (lookup p (component_of_term v)) (pp_of_term v)"
proof -
  have "lookup (Abs_poly_mapping (λv. lookup (lookup p (component_of_term v)) (pp_of_term v))) =
        (λv. lookup (lookup p (component_of_term v)) (pp_of_term v))"
  proof (rule Abs_poly_mapping_inverse, simp)
    have "{v. pp_of_term v  keys (lookup p (component_of_term v))} 
          (kkeys p. (λt. term_of_pair (t, k)) ` keys (lookup p k))" (is "_  ?A")
    proof (rule, simp)
      fix v
      assume *: "pp_of_term v  keys (lookup p (component_of_term v))"
      hence "keys (lookup p (component_of_term v))  {}" by blast
      hence "lookup p (component_of_term v)  0" by auto
      hence "component_of_term v  keys p" (is "?k  _") 
        by (simp add: in_keys_iff)
      thus "kkeys p. v  (λt. term_of_pair (t, k)) ` keys (lookup p k)"
      proof
        have "v = term_of_pair (pp_of_term v, component_of_term v)" by (simp add: term_simps)
        from this * show "v  (λt. term_of_pair (t, ?k)) ` keys (lookup p ?k)" ..
      qed
    qed
    moreover have "finite ?A" by (rule, fact finite_keys, rule finite_imageI, rule finite_keys)
    ultimately show "finite {x. lookup (lookup p (component_of_term x)) (pp_of_term x)  0}"
      by (simp add: finite_subset in_keys_iff)
  qed
  thus ?thesis by (simp add: atomize_poly_def)
qed

lemma keys_proj_poly: "keys (proj_poly k p) = pp_of_term ` {xkeys p. component_of_term x = k}"
proof
  show "keys (proj_poly k p)  pp_of_term ` {xkeys p. component_of_term x = k}"
  proof
    fix t
    assume "t  keys (proj_poly k p)"
    hence "lookup (proj_poly k p) t  0" by (simp add: in_keys_iff)
    hence "term_of_pair (t, k)  keys p" by (simp add: in_keys_iff lookup_proj_poly)
    hence "term_of_pair (t, k)  {xkeys p. component_of_term x = k}" by (simp add: term_simps)
    hence "pp_of_term (term_of_pair (t, k))  pp_of_term ` {xkeys p. component_of_term x = k}" by (rule imageI)
    thus "t  pp_of_term ` {xkeys p. component_of_term x = k}" by (simp only: pp_of_term_of_pair)
  qed
next
  show "pp_of_term ` {xkeys p. component_of_term x = k}  keys (proj_poly k p)"
  proof
    fix t
    assume "t  pp_of_term ` {xkeys p. component_of_term x = k}"
    then obtain x where "x  {xkeys p. component_of_term x = k}" and "t = pp_of_term x" ..
    from this(1) have "x  keys p" and "k = component_of_term x" by simp_all
    from this(2) have "x = term_of_pair (t, k)" by (simp add: term_of_pair_pair t = pp_of_term x)
    with x  keys p have "lookup p (term_of_pair (t, k))  0" by (simp add: in_keys_iff)
    hence "lookup (proj_poly k p) t  0" by (simp add: lookup_proj_poly)
    thus "t  keys (proj_poly k p)" by (simp add: in_keys_iff)
  qed
qed

lemma keys_vectorize_poly: "keys (vectorize_poly p) = component_of_term ` keys p"
proof
  show "keys (vectorize_poly p)  component_of_term ` keys p"
  proof
    fix k
    assume "k  keys (vectorize_poly p)"
    hence "lookup (vectorize_poly p) k  0" by (simp add: in_keys_iff)
    hence "proj_poly k p  0" by (simp add: lookup_vectorize_poly)
    then obtain t where "lookup (proj_poly k p) t  0" using aux by blast
    hence "term_of_pair (t, k)  keys p" by (simp add: lookup_proj_poly in_keys_iff)
    hence "component_of_term (term_of_pair (t, k))  component_of_term ` keys p" by (rule imageI)
    thus "k  component_of_term ` keys p" by (simp only: component_of_term_of_pair)
  qed
next
  show "component_of_term ` keys p  keys (vectorize_poly p)"
  proof
    fix k
    assume "k  component_of_term ` keys p"
    then obtain x where "x  keys p" and "k = component_of_term x" ..
    from this(2) have "term_of_pair (pp_of_term x, k) = x" by (simp add: term_of_pair_pair)
    with x  keys p have "lookup p (term_of_pair (pp_of_term x, k))  0" by (simp add: in_keys_iff)
    hence "lookup (proj_poly k p) (pp_of_term x)  0" by (simp add: lookup_proj_poly)
    hence "proj_poly k p  0" by auto
    hence "lookup (vectorize_poly p) k  0" by (simp add: lookup_vectorize_poly)
    thus "k  keys (vectorize_poly p)" by (simp add: in_keys_iff)
  qed
qed

lemma keys_atomize_poly:
  "keys (atomize_poly p) = (kkeys p. (λt. term_of_pair (t, k)) ` keys (lookup p k))" (is "?l = ?r")
proof
  show "?l  ?r"
  proof
    fix v
    assume "v  ?l"
    hence "lookup (atomize_poly p) v  0" by (simp add: in_keys_iff)
    hence *: "pp_of_term v  keys (lookup p (component_of_term v))" by (simp add: in_keys_iff lookup_atomize_poly)
    hence "lookup p (component_of_term v)  0" by fastforce
    hence "component_of_term v  keys p" by (simp add: in_keys_iff)
    thus "v  ?r"
    proof
      from * have "term_of_pair (pp_of_term v, component_of_term v) 
                    (λt. term_of_pair (t, component_of_term v)) ` keys (lookup p (component_of_term v))"
        by (rule imageI)
      thus "v  (λt. term_of_pair (t, component_of_term v)) ` keys (lookup p (component_of_term v))"
        by (simp only: term_of_pair_pair)
    qed
  qed
next
  show "?r  ?l"
  proof
    fix v
    assume "v  ?r"
    then obtain k where "k  keys p" and "v  (λt. term_of_pair (t, k)) ` keys (lookup p k)" ..
    from this(2) obtain t where "t  keys (lookup p k)" and v: "v = term_of_pair (t, k)" ..
    from this(1) have "lookup (atomize_poly p) v  0" by (simp add: v lookup_atomize_poly in_keys_iff term_simps)
    thus "v  ?l" by (simp add: in_keys_iff)
  qed
qed

lemma proj_atomize_poly [term_simps]: "proj_poly k (atomize_poly p) = lookup p k"
  by (rule poly_mapping_eqI, simp add: lookup_proj_poly lookup_atomize_poly term_simps)

lemma vectorize_atomize_poly [term_simps]: "vectorize_poly (atomize_poly p) = p"
  by (rule poly_mapping_eqI, simp add: lookup_vectorize_poly term_simps)

lemma atomize_vectorize_poly [term_simps]: "atomize_poly (vectorize_poly p) = p"
  by (rule poly_mapping_eqI, simp add: lookup_atomize_poly lookup_vectorize_poly lookup_proj_poly term_simps)

lemma proj_zero [term_simps]: "proj_poly k 0 = 0"
  by (rule poly_mapping_eqI, simp add: lookup_proj_poly)

lemma proj_plus: "proj_poly k (p + q) = proj_poly k p + proj_poly k q"
  by (rule poly_mapping_eqI, simp add: lookup_proj_poly lookup_add)

lemma proj_uminus [term_simps]: "proj_poly k (- p) = - proj_poly k p"
  by (rule poly_mapping_eqI, simp add: lookup_proj_poly)

lemma proj_minus: "proj_poly k (p - q) = proj_poly k p - proj_poly k q"
  by (rule poly_mapping_eqI, simp add: lookup_proj_poly lookup_minus)

lemma vectorize_zero [term_simps]: "vectorize_poly 0 = 0"
  by (rule poly_mapping_eqI, simp add: lookup_vectorize_poly term_simps)

lemma vectorize_plus: "vectorize_poly (p + q) = vectorize_poly p + vectorize_poly q"
  by (rule poly_mapping_eqI, simp add: lookup_vectorize_poly lookup_add proj_plus)

lemma vectorize_uminus [term_simps]: "vectorize_poly (- p) = - vectorize_poly p"
  by (rule poly_mapping_eqI, simp add: lookup_vectorize_poly term_simps)

lemma vectorize_minus: "vectorize_poly (p - q) = vectorize_poly p - vectorize_poly q"
  by (rule poly_mapping_eqI, simp add: lookup_vectorize_poly lookup_minus proj_minus)

lemma atomize_zero [term_simps]: "atomize_poly 0 = 0"
  by (rule poly_mapping_eqI, simp add: lookup_atomize_poly)

lemma atomize_plus: "atomize_poly (p + q) = atomize_poly p + atomize_poly q"
  by (rule poly_mapping_eqI, simp add: lookup_atomize_poly lookup_add)

lemma atomize_uminus [term_simps]: "atomize_poly (- p) = - atomize_poly p"
  by (rule poly_mapping_eqI, simp add: lookup_atomize_poly)

lemma atomize_minus: "atomize_poly (p - q) = atomize_poly p - atomize_poly q"
  by (rule poly_mapping_eqI, simp add: lookup_atomize_poly lookup_minus)

lemma proj_monomial:
  "proj_poly k (monomial c v) = (monomial c (pp_of_term v) when component_of_term v = k)"
proof (rule poly_mapping_eqI, simp add: lookup_proj_poly lookup_single when_def term_simps, intro impI)
  fix t
  assume 1: "pp_of_term v = t" and 2: "component_of_term v = k"
  assume "v  term_of_pair (t, k)"
  moreover have "v = term_of_pair (t, k)" by (simp add: 1[symmetric] 2[symmetric] term_simps)
  ultimately show "c = 0" ..
qed

lemma vectorize_monomial:
  "vectorize_poly (monomial c v) = monomial (monomial c (pp_of_term v)) (component_of_term v)"
  by (rule poly_mapping_eqI, simp add: lookup_vectorize_poly proj_monomial lookup_single)

lemma atomize_monomial_monomial:
  "atomize_poly (monomial (monomial c t) k) = monomial c (term_of_pair (t, k))"
proof -
  define v where "v = term_of_pair (t, k)"
  have t: "t = pp_of_term v" and k: "k = component_of_term v" by (simp_all add: v_def term_simps)
  show ?thesis by (simp add: t k vectorize_monomial[symmetric] term_simps)
qed

lemma poly_mapping_eqI_proj:
  assumes "k. proj_poly k p = proj_poly k q"
  shows "p = q"
proof (rule poly_mapping_eqI)
  fix v::'t
  have "proj_poly (component_of_term v) p = proj_poly (component_of_term v) q" by (rule assms)
  hence "lookup (proj_poly (component_of_term v) p) (pp_of_term v) =
          lookup (proj_poly (component_of_term v) q) (pp_of_term v)" by simp
  thus "lookup p v = lookup q v" by (simp add: lookup_proj_poly term_simps)
qed

subsection ‹Scalar Multiplication by Monomials›

definition monom_mult :: "'b::semiring_0  'a::comm_powerprod  ('t 0 'b)  ('t 0 'b)"
  where "monom_mult c t p = Abs_poly_mapping (λv. if t addsp v then c * (lookup p (v  t)) else 0)"

lemma keys_monom_mult_aux:
  "{v. (if t addsp v then c * lookup p (v  t) else 0)  0}  (⊕) t ` keys p" (is "?l  ?r")
  for c::"'b::semiring_0"
proof
  fix v::'t
  assume "v  ?l"
  hence "(if t addsp v then c * lookup p (v  t) else 0)  0" by simp
  hence "t addsp v" and cp_not_zero: "c * lookup p (v  t)  0" by (simp_all split: if_split_asm)
  show "v  ?r"
  proof
    from adds_pp_sminus[OF t addsp v] show "v = t  (v  t)" by simp
  next
    from mult_not_zero[OF cp_not_zero] show "v  t  keys p"
      by (simp add: in_keys_iff)
  qed
qed

lemma lookup_monom_mult:
  "lookup (monom_mult c t p) v = (if t addsp v then c * lookup p (v  t) else 0)"
proof -
  have "lookup (monom_mult c t p) = (λv. if t addsp v then c * lookup p (v  t) else 0)"
    unfolding monom_mult_def
  proof (rule Abs_poly_mapping_inverse)
    from finite_keys have "finite ((⊕) t ` keys p)" ..
    with keys_monom_mult_aux have "finite {v. (if t addsp v then c * lookup p (v  t) else 0)  0}"
      by (rule finite_subset)
    thus "(λv. if t addsp v then c * lookup p (v  t) else 0)  {f. finite {x. f x  0}}" by simp
  qed
  thus ?thesis by simp
qed

lemma lookup_monom_mult_plus:
  "lookup (monom_mult c t p) (t  v) = (c::'b::semiring_0) * lookup p v"
  by (simp add: lookup_monom_mult term_simps)

lemma monom_mult_assoc: "monom_mult c s (monom_mult d t p) = monom_mult (c * d) (s + t) p"
proof (rule poly_mapping_eqI, simp add: lookup_monom_mult sminus_plus ac_simps, intro conjI impI)
  fix v
  assume "s addsp v" and "t addsp v  s"
  hence "s + t addsp v" by (rule plus_adds_ppI_2)
  moreover assume "¬ s + t addsp v"
  ultimately show "c * (d * lookup p (v  s  t)) = 0" by simp
next
  fix v
  assume "s + t addsp v"
  hence "s addsp v" by (rule plus_adds_pp_left)
  moreover assume "¬ s addsp v"
  ultimately show "c * (d * lookup p (v  (s + t))) = 0" by simp
next
  fix v
  assume "s + t addsp v"
  hence "t addsp v  s" by (simp add: add.commute plus_adds_pp_0)
  moreover assume "¬ t addsp v  s"
  ultimately show "c * (d * lookup p (v  (s + t))) = 0" by simp
qed

lemma monom_mult_uminus_left: "monom_mult (- c) t p = - monom_mult (c::'b::ring) t p"
  by (rule poly_mapping_eqI, simp add: lookup_monom_mult)

lemma monom_mult_uminus_right: "monom_mult c t (- p) = - monom_mult (c::'b::ring) t p"
  by (rule poly_mapping_eqI, simp add: lookup_monom_mult)

lemma uminus_monom_mult: "- p = monom_mult (-1::'b::comm_ring_1) 0 p"
  by (rule poly_mapping_eqI, simp add: lookup_monom_mult term_simps)

lemma monom_mult_dist_left: "monom_mult (c + d) t p = (monom_mult c t p) + (monom_mult d t p)"
  by (rule poly_mapping_eqI, simp add: lookup_monom_mult lookup_add algebra_simps)

lemma monom_mult_dist_left_minus:
  "monom_mult (c - d) t p = (monom_mult c t p) - (monom_mult (d::'b::ring) t p)"
  using monom_mult_dist_left[of c "-d" t p] monom_mult_uminus_left[of d t p] by simp

lemma monom_mult_dist_right:
  "monom_mult c t (p + q) = (monom_mult c t p) + (monom_mult c t q)"
  by (rule poly_mapping_eqI, simp add: lookup_monom_mult lookup_add algebra_simps)

lemma monom_mult_dist_right_minus:
  "monom_mult c t (p - q) = (monom_mult c t p) - (monom_mult (c::'b::ring) t q)"
  using monom_mult_dist_right[of c t p "-q"] monom_mult_uminus_right[of c t q] by simp

lemma monom_mult_zero_left [simp]: "monom_mult 0 t p = 0"
  by (rule poly_mapping_eqI, simp add: lookup_monom_mult)

lemma monom_mult_zero_right [simp]: "monom_mult c t 0 = 0"
  by (rule poly_mapping_eqI, simp add: lookup_monom_mult)

lemma monom_mult_one_left [simp]: "(monom_mult (1::'b::semiring_1) 0 p) = p"
  by (rule poly_mapping_eqI, simp add: lookup_monom_mult term_simps)

lemma monom_mult_monomial:
  "monom_mult c s (monomial d v) = monomial (c * (d::'b::semiring_0)) (s  v)"
  by (rule poly_mapping_eqI, auto simp add: lookup_monom_mult lookup_single adds_pp_alt when_def term_simps, metis)

lemma monom_mult_eq_zero_iff: "(monom_mult c t p = 0)  ((c::'b::semiring_no_zero_divisors) = 0  p = 0)"
proof
  assume eq: "monom_mult c t p = 0"
  show "c = 0  p = 0"
  proof (rule ccontr, simp)
    assume "c  0  p  0"
    hence "c  0" and "p  0" by simp_all
    from lookup_zero poly_mapping_eq_iff[of p 0] p  0 obtain v where "lookup p v  0" by fastforce
    from eq lookup_zero have "lookup (monom_mult c t p) (t  v) = 0" by simp
    hence "c * lookup p v = 0" by (simp only: lookup_monom_mult_plus)
    with c  0 lookup p v  0 show False by auto
  qed
next
  assume "c = 0  p = 0"
  with monom_mult_zero_left[of t p] monom_mult_zero_right[of c t] show "monom_mult c t p = 0" by auto
qed

lemma lookup_monom_mult_zero: "lookup (monom_mult c 0 p) t = c * lookup p t"
proof -
  have "lookup (monom_mult c 0 p) t = lookup (monom_mult c 0 p) (0  t)" by (simp add: term_simps)
  also have "... = c * lookup p t" by (rule lookup_monom_mult_plus)
  finally show ?thesis .
qed

lemma monom_mult_inj_1:
  assumes "monom_mult c1 t p = monom_mult c2 t p"
    and "(p::(_ 0 'b::semiring_no_zero_divisors_cancel))  0"
  shows "c1 = c2"
proof -
  from assms(2) have "keys p  {}" using poly_mapping_eq_zeroI by blast
  then obtain v where "v  keys p" by blast
  hence *: "lookup p v  0" by (simp add: in_keys_iff)
  from assms(1) have "lookup (monom_mult c1 t p) (t  v) = lookup (monom_mult c2 t p) (t  v)"
    by simp
  hence "c1 * lookup p v = c2 * lookup p v" by (simp only: lookup_monom_mult_plus)
  with * show ?thesis by auto
qed

text ‹Multiplication by a monomial is injective in the second argument (the power-product) only in
  context @{locale ordered_powerprod}; see lemma monom_mult_inj_2› below.›

lemma monom_mult_inj_3:
  assumes "monom_mult c t p1 = monom_mult c t (p2::(_ 0 'b::semiring_no_zero_divisors_cancel))"
    and "c  0"
  shows "p1 = p2"
proof (rule poly_mapping_eqI)
  fix v
  from assms(1) have "lookup (monom_mult c t p1) (t  v) = lookup (monom_mult c t p2) (t  v)"
    by simp
  hence "c * lookup p1 v = c * lookup p2 v" by (simp only: lookup_monom_mult_plus)
  with assms(2) show "lookup p1 v = lookup p2 v" by simp
qed
    
lemma keys_monom_multI:
  assumes "v  keys p" and "c  (0::'b::semiring_no_zero_divisors)"
  shows "t  v  keys (monom_mult c t p)"
  using assms unfolding in_keys_iff lookup_monom_mult_plus by simp

lemma keys_monom_mult_subset: "keys (monom_mult c t p)  ((⊕) t) ` (keys p)"
proof -
  have "keys (monom_mult c t p)  {v. (if t addsp v then c * lookup p (v  t) else 0)  0}" (is "_  ?A")
  proof
    fix v
    assume "v  keys (monom_mult c t p)"
    hence "lookup (monom_mult c t p) v  0" by (simp add: in_keys_iff)
    thus "v  ?A" unfolding lookup_monom_mult by simp
  qed
  also note keys_monom_mult_aux
  finally show ?thesis .
qed

lemma keys_monom_multE:
  assumes "v  keys (monom_mult c t p)"
  obtains u where "u  keys p" and "v = t  u"
proof -
  note assms
  also have "keys (monom_mult c t p)  ((⊕) t) ` (keys p)" by (fact keys_monom_mult_subset)
  finally have "v  ((⊕) t) ` (keys p)" .
  then obtain u where "u  keys p" and "v = t  u" ..
  thus ?thesis ..
qed

lemma keys_monom_mult:
  assumes "c  (0::'b::semiring_no_zero_divisors)"
  shows "keys (monom_mult c t p) = ((⊕) t) ` (keys p)"
proof (rule, fact keys_monom_mult_subset, rule)
  fix v
  assume "v  (⊕) t ` keys p"
  then obtain u where "u  keys p" and v: "v = t  u" ..
  from u  keys p assms show "v  keys (monom_mult c t p)" unfolding v by (rule keys_monom_multI)
qed

lemma monom_mult_when: "monom_mult c t (p when P) = ((monom_mult c t p) when P)"
  by (cases P, simp_all)

lemma when_monom_mult: "monom_mult (c when P) t p = ((monom_mult c t p) when P)"
  by (cases P, simp_all)

lemma monomial_power: "(monomial c t) ^ n = monomial (c ^ n) (i=0..<n. t)"
  by (induct n, simp_all add: mult_single monom_mult_monomial add.commute)

subsection ‹Component-wise Lifting›

text ‹Component-wise lifting of functions on @{typ "'a 0 'b"} to functions on @{typ "'t 0 'b"}.›

definition lift_poly_fun_2 :: "(('a 0 'b)  ('a 0 'b)  ('a 0 'b))  ('t 0 'b)  ('t 0 'b)  ('t 0 'b::zero)"
  where "lift_poly_fun_2 f p q = atomize_poly (mapp_2 (λ_. f) (vectorize_poly p) (vectorize_poly q))"

definition lift_poly_fun :: "(('a 0 'b)  ('a 0 'b))  ('t 0 'b)  ('t 0 'b::zero)"
  where "lift_poly_fun f p = lift_poly_fun_2 (λ_. f) 0 p"

lemma lookup_lift_poly_fun_2:
  "lookup (lift_poly_fun_2 f p q) v =
    (lookup (f (proj_poly (component_of_term v) p) (proj_poly (component_of_term v) q)) (pp_of_term v)
        when component_of_term v  keys (vectorize_poly p)  keys (vectorize_poly q))"
  by (simp add: lift_poly_fun_2_def lookup_atomize_poly lookup_mapp_2 lookup_vectorize_poly
      when_distrib[of _ "λq. lookup q (pp_of_term v)", OF lookup_zero])

lemma lookup_lift_poly_fun:
  "lookup (lift_poly_fun f p) v =
    (lookup (f (proj_poly (component_of_term v) p)) (pp_of_term v) when component_of_term v  keys (vectorize_poly p))"
  by (simp add: lift_poly_fun_def lookup_lift_poly_fun_2 term_simps)

lemma lookup_lift_poly_fun_2_homogenous:
  assumes "f 0 0 = 0"
  shows "lookup (lift_poly_fun_2 f p q) v =
         lookup (f (proj_poly (component_of_term v) p) (proj_poly (component_of_term v) q)) (pp_of_term v)"
  by (simp add: lookup_lift_poly_fun_2 when_def in_keys_iff lookup_vectorize_poly assms)

lemma proj_lift_poly_fun_2_homogenous:
  assumes "f 0 0 = 0"
  shows "proj_poly k (lift_poly_fun_2 f p q) = f (proj_poly k p) (proj_poly k q)"
  by (rule poly_mapping_eqI,
      simp add: lookup_proj_poly lookup_lift_poly_fun_2_homogenous[of f, OF assms] term_simps)

lemma lookup_lift_poly_fun_homogenous:
  assumes "f 0 = 0"
  shows "lookup (lift_poly_fun f p) v = lookup (f (proj_poly (component_of_term v) p)) (pp_of_term v)"
  by (simp add: lookup_lift_poly_fun when_def in_keys_iff lookup_vectorize_poly assms)

lemma proj_lift_poly_fun_homogenous:
  assumes "f 0 = 0"
  shows "proj_poly k (lift_poly_fun f p) = f (proj_poly k p)"
  by (rule poly_mapping_eqI,
      simp add: lookup_proj_poly lookup_lift_poly_fun_homogenous[of f, OF assms] term_simps)

subsection ‹Component-wise Multiplication›

definition mult_vec :: "('t 0 'b)  ('t 0 'b)  ('t 0 'b::semiring_0)" (infixl "**" 75)
  where "mult_vec = lift_poly_fun_2 (*)"

lemma lookup_mult_vec:
  "lookup (p ** q) v = lookup ((proj_poly (component_of_term v) p) * (proj_poly (component_of_term v) q)) (pp_of_term v)"
  unfolding mult_vec_def by (rule lookup_lift_poly_fun_2_homogenous, simp)

lemma proj_mult_vec [term_simps]: "proj_poly k (p ** q) = (proj_poly k p) * (proj_poly k q)"
  unfolding mult_vec_def by (rule proj_lift_poly_fun_2_homogenous, simp)

lemma mult_vec_zero_left: "0 ** p = 0"
  by (rule poly_mapping_eqI_proj, simp add: term_simps)

lemma mult_vec_zero_right: "p ** 0 = 0"
  by (rule poly_mapping_eqI_proj, simp add: term_simps)

lemma mult_vec_assoc: "(p ** q) ** r = p ** (q ** r)"
  by (rule poly_mapping_eqI_proj, simp add: ac_simps term_simps)

lemma mult_vec_distrib_right: "(p + q) ** r = p ** r + q ** r"
  by (rule poly_mapping_eqI_proj, simp add: algebra_simps proj_plus term_simps)

lemma mult_vec_distrib_left: "r ** (p + q) = r ** p + r ** q"
  by (rule poly_mapping_eqI_proj, simp add: algebra_simps proj_plus term_simps)

lemma mult_vec_minus_mult_left: "(- p) ** q = - (p ** q)"
  by (rule sym, rule minus_unique, simp add: mult_vec_distrib_right[symmetric] mult_vec_zero_left)

lemma mult_vec_minus_mult_right: "p ** (- q) = - (p ** q)"
  by (rule sym, rule minus_unique, simp add: mult_vec_distrib_left [symmetric] mult_vec_zero_right)

lemma minus_mult_vec_minus: "(- p) ** (- q) = p ** q"
  by (simp add: mult_vec_minus_mult_left mult_vec_minus_mult_right)

lemma minus_mult_vec_commute: "(- p) ** q = p ** (- q)"
  by (simp add: mult_vec_minus_mult_left mult_vec_minus_mult_right)

lemma mult_vec_right_diff_distrib: "r ** (p - q) = r ** p - r ** q"
  for r::"_ 0 'b::ring"
  using mult_vec_distrib_left [of r p "- q"] by (simp add: mult_vec_minus_mult_right)

lemma mult_vec_left_diff_distrib: "(p - q) ** r = p ** r - q ** r"
  for p::"_ 0 'b::ring"
  using mult_vec_distrib_right [of p "- q" r] by (simp add: mult_vec_minus_mult_left)

lemma mult_vec_commute: "p ** q = q ** p" for p::"_ 0 'b::comm_semiring_0"
  by (rule poly_mapping_eqI_proj, simp add: term_simps ac_simps)

lemma mult_vec_left_commute: "p ** (q ** r) = q ** (p ** r)"
  for p::"_ 0 'b::comm_semiring_0"
  by (rule poly_mapping_eqI_proj, simp add: term_simps ac_simps)

lemma mult_vec_monomial_monomial:
  "(monomial c u) ** (monomial d v) =
          (monomial (c * d) (term_of_pair (pp_of_term u + pp_of_term v, component_of_term u)) when
            component_of_term u = component_of_term v)"
  by (rule poly_mapping_eqI_proj, simp add: proj_monomial mult_single when_def term_simps)

lemma mult_vec_rec_left: "p ** q = monomial (lookup p v) v ** q + (except p {v}) ** q"
proof -
  from plus_except[of p v] have "p ** q = (monomial (lookup p v) v + except p {v}) ** q" by simp
  also have "... = monomial (lookup p v) v ** q + except p {v} ** q"
    by (simp only: mult_vec_distrib_right)
  finally show ?thesis .
qed

lemma mult_vec_rec_right: "p ** q = p ** monomial (lookup q v) v + p ** except q {v}"
proof -
  have "p ** monomial (lookup q v) v + p ** except q {v} = p ** (monomial (lookup q v) v + except q {v})"
    by (simp only: mult_vec_distrib_left)
  also have "... = p ** q" by (simp only: plus_except[of q v, symmetric])
  finally show ?thesis by simp
qed

lemma in_keys_mult_vecE:
  assumes "w  keys (p ** q)"
  obtains u v where "u  keys p" and "v  keys q" and "component_of_term u = component_of_term v"
    and "w = term_of_pair (pp_of_term u + pp_of_term v, component_of_term u)"
proof -
  from assms have "0  lookup (p ** q) w" by (simp add: in_keys_iff)
  also have "lookup (p ** q) w =
      lookup ((proj_poly (component_of_term w) p) * (proj_poly (component_of_term w) q)) (pp_of_term w)"
    by (fact lookup_mult_vec)
  finally have "pp_of_term w  keys ((proj_poly (component_of_term w) p) * (proj_poly (component_of_term w) q))"
    by (simp add: in_keys_iff)
  from this keys_mult
  have "pp_of_term w  {t + s |t s. t  keys (proj_poly (component_of_term w) p) 
                                   s  keys (proj_poly (component_of_term w) q)}" ..
  then obtain t s where 1: "t  keys (proj_poly (component_of_term w) p)"
    and 2: "s  keys (proj_poly (component_of_term w) q)"
    and eq: "pp_of_term w = t + s" by fastforce
  let ?u = "term_of_pair (t, component_of_term w)"
  let ?v = "term_of_pair (s, component_of_term w)"
  from 1 have "?u  keys p" by (simp only: in_keys_iff lookup_proj_poly not_False_eq_True)
  moreover from 2 have "?v  keys q" by (simp only: in_keys_iff lookup_proj_poly not_False_eq_True)
  moreover have "component_of_term ?u = component_of_term ?v" by (simp add: term_simps)
  moreover have "w = term_of_pair (pp_of_term ?u + pp_of_term ?v, component_of_term ?u)"
    by (simp add: eq[symmetric] term_simps)
  ultimately show ?thesis ..
qed

lemma lookup_mult_vec_monomial_left:
  "lookup (monomial c v ** p) u =
        (c * lookup p (term_of_pair (pp_of_term u - pp_of_term v, component_of_term u)) when v addst u)"
proof -
  have eq1: "lookup ((monomial c (pp_of_term v) when component_of_term v = component_of_term u) * proj_poly (component_of_term u) p)
                (pp_of_term u) =
        (lookup ((monomial c (pp_of_term v)) * proj_poly (component_of_term u) p) (pp_of_term u) when
                component_of_term v = component_of_term u)"
    by (rule when_distrib, simp)
  show ?thesis
    by (simp add: lookup_mult_vec proj_monomial eq1 lookup_times_monomial_left when_when
        adds_term_def lookup_proj_poly conj_commute)
qed

lemma lookup_mult_vec_monomial_right:
  "lookup (p ** monomial c v) u =
        (lookup p (term_of_pair (pp_of_term u - pp_of_term v, component_of_term u)) * c when v addst u)"
proof -
  have eq1: "lookup (proj_poly (component_of_term u) p * (monomial c (pp_of_term v) when component_of_term v = component_of_term u))
                (pp_of_term u) =
        (lookup (proj_poly (component_of_term u) p * (monomial c (pp_of_term v))) (pp_of_term u) when
                component_of_term v = component_of_term u)"
    by (rule when_distrib, simp)
  show ?thesis
    by (simp add: lookup_mult_vec proj_monomial eq1 lookup_times_monomial_right when_when
        adds_term_def lookup_proj_poly conj_commute)
qed

subsection ‹Scalar Multiplication›

definition mult_scalar :: "('a 0 'b)  ('t 0 'b)  ('t 0 'b::semiring_0)" (infixl "" 75)
  where "mult_scalar p = lift_poly_fun ((*) p)"

lemma lookup_mult_scalar:
  "lookup (p  q) v = lookup (p * (proj_poly (component_of_term v) q)) (pp_of_term v)"
  unfolding mult_scalar_def by (rule lookup_lift_poly_fun_homogenous, simp)

lemma lookup_mult_scalar_explicit:
  "lookup (p  q) u = (tkeys p. lookup p t * (vkeys q. lookup q v when u = t  v))"
proof -
  let ?f = "λt s. lookup (proj_poly (component_of_term u) q) s when pp_of_term u = t + s"
  note lookup_mult_scalar
  also have "lookup (p * proj_poly (component_of_term u) q) (pp_of_term u) =
              (t. lookup p t * (Sum_any (?f t)))"
    by (fact lookup_mult)
  also from finite_keys have " = (tkeys p. lookup p t * (Sum_any (?f t)))"
    by (rule Sum_any.expand_superset) (auto simp: in_keys_iff dest: mult_not_zero)
  also from refl have " = (tkeys p. lookup p t * (vkeys q. lookup q v when u = t  v))"
  proof (rule sum.cong)
    fix t
    assume "t  keys p"
    from finite_keys have "Sum_any (?f t) = (skeys (proj_poly (component_of_term u) q). ?f t s)"
      by (rule Sum_any.expand_superset) (auto simp: in_keys_iff)
    also have " = (v{x  keys q. component_of_term x = component_of_term u}. ?f t (pp_of_term v))"
      unfolding keys_proj_poly
    proof (intro sum.reindex[simplified o_def] inj_onI)
      fix v1 v2
      assume "v1  {x  keys q. component_of_term x = component_of_term u}"
        and "v2  {x  keys q. component_of_term x = component_of_term u}"
      hence "component_of_term v1 = component_of_term v2" by simp
      moreover assume "pp_of_term v1 = pp_of_term v2"
      ultimately show "v1 = v2" by (metis term_of_pair_pair)
    qed
    also from finite_keys have " = (vkeys q. lookup q v when u = t  v)"
    proof (intro sum.mono_neutral_cong_left ballI)
      fix v
      assume "v  keys q - {x  keys q. component_of_term x = component_of_term u}"
      hence "u  t  v" by (auto simp: component_of_term_splus)
      thus "(lookup q v when u = t  v) = 0" by simp
    next
      fix v
      assume "v  {x  keys q. component_of_term x = component_of_term u}"
      hence eq[symmetric]: "component_of_term v = component_of_term u" by simp
      have "u = t  v  pp_of_term u = t + pp_of_term v"
      proof
        assume "pp_of_term u = t + pp_of_term v"
        hence "pp_of_term u = pp_of_term (t  v)" by (simp only: pp_of_term_splus)
        moreover have "component_of_term u = component_of_term (t  v)"
          by (simp only: eq component_of_term_splus)
        ultimately show "u = t  v" by (metis term_of_pair_pair)
      qed (simp add: pp_of_term_splus)
      thus "?f t (pp_of_term v) = (lookup q v when u = t  v)"
        by (simp add: lookup_proj_poly eq term_of_pair_pair)
    qed auto
    finally show "lookup p t * (Sum_any (?f t)) = lookup p t * (vkeys q. lookup q v when u = t  v)"
      by (simp only:)
  qed
  finally show ?thesis .
qed

lemma proj_mult_scalar [term_simps]: "proj_poly k (p  q) = p * (proj_poly k q)"
  unfolding mult_scalar_def by (rule proj_lift_poly_fun_homogenous, simp)

lemma mult_scalar_zero_left [simp]: "0  p = 0"
  by (rule poly_mapping_eqI_proj, simp add: term_simps)

lemma mult_scalar_zero_right [simp]: "p  0 = 0"
  by (rule poly_mapping_eqI_proj, simp add: term_simps)

lemma mult_scalar_one [simp]: "(1::_ 0 'b::semiring_1)  p = p"
  by (rule poly_mapping_eqI_proj, simp add: term_simps)

lemma mult_scalar_assoc [ac_simps]: "(p * q)  r = p  (q  r)"
  by (rule poly_mapping_eqI_proj, simp add: ac_simps term_simps)

lemma mult_scalar_distrib_right [algebra_simps]: "(p + q)  r = p  r + q  r"
  by (rule poly_mapping_eqI_proj, simp add: algebra_simps proj_plus term_simps)

lemma mult_scalar_distrib_left [algebra_simps]: "r  (p + q) = r  p + r  q"
  by (rule poly_mapping_eqI_proj, simp add: algebra_simps proj_plus term_simps)

lemma mult_scalar_minus_mult_left [simp]: "(- p)  q = - (p  q)"
  by (rule sym, rule minus_unique, simp add: mult_scalar_distrib_right[symmetric])

lemma mult_scalar_minus_mult_right [simp]: "p  (- q) = - (p  q)"
  by (rule sym, rule minus_unique, simp add: mult_scalar_distrib_left [symmetric])

lemma minus_mult_scalar_minus [simp]: "(- p)  (- q) = p  q"
  by simp

lemma minus_mult_scalar_commute: "(- p)  q = p  (- q)"
  by simp

lemma mult_scalar_right_diff_distrib [algebra_simps]: "r  (p - q) = r  p - r  q"
  for r::"_ 0 'b::ring"
  using mult_scalar_distrib_left [of r p "- q"] by simp

lemma mult_scalar_left_diff_distrib [algebra_simps]: "(p - q)  r = p  r - q  r"
  for p::"_ 0 'b::ring"
  using mult_scalar_distrib_right [of p "- q" r] by simp

lemma sum_mult_scalar_distrib_left: "r  (sum f A) = (aA. r  f a)"
  by (induct A rule: infinite_finite_induct, simp_all add: algebra_simps)

lemma sum_mult_scalar_distrib_right: "(sum f A)  v = (aA. f a  v)"
  by (induct A rule: infinite_finite_induct, simp_all add: algebra_simps)

lemma mult_scalar_monomial_monomial: "(monomial c t)  (monomial d v) = monomial (c * d) (t  v)"
  by (rule poly_mapping_eqI_proj, simp add: proj_monomial mult_single when_def term_simps)

lemma mult_scalar_monomial: "(monomial c t)  p = monom_mult c t p"
  by (rule poly_mapping_eqI_proj, rule poly_mapping_eqI,
      auto simp add: lookup_times_monomial_left lookup_proj_poly lookup_monom_mult when_def
        adds_pp_def sminus_def term_simps)

lemma mult_scalar_rec_left: "p  q = monom_mult (lookup p t) t q + (except p {t})  q"
proof -
  from plus_except[of p t] have "p  q = (monomial (lookup p t) t + except p {t})  q" by simp
  also have "... = monomial (lookup p t) t  q + except p {t}  q" by (simp only: algebra_simps)
  finally show ?thesis by (simp only: mult_scalar_monomial)
qed

lemma mult_scalar_rec_right: "p  q = p  monomial (lookup q v) v + p  except q {v}"
proof -
  have "p  monomial (lookup q v) v + p  except q {v} = p  (monomial (lookup q v) v + except q {v})"
    by (simp only: mult_scalar_distrib_left)
  also have "... = p  q" by (simp only: plus_except[of q v, symmetric])
  finally show ?thesis by simp
qed

lemma in_keys_mult_scalarE:
  assumes "v  keys (p  q)"
  obtains t u where "t  keys p" and "u  keys q" and "v = t  u"
proof -
  from assms have "0  lookup (p  q) v" by (simp add: in_keys_iff)
  also have "lookup (p  q) v = lookup (p * (proj_poly (component_of_term v) q)) (pp_of_term v)"
    by (fact lookup_mult_scalar)
  finally have "pp_of_term v  keys (p * proj_poly (component_of_term v) q)" by (simp add: in_keys_iff)
  from this keys_mult have "pp_of_term v  {t + s |t s. t  keys p  s  keys (proj_poly (component_of_term v) q)}" ..
  then obtain t s where "t  keys p" and *: "s  keys (proj_poly (component_of_term v) q)"
    and eq: "pp_of_term v = t + s" by fastforce
  note this(1)
  moreover from * have "term_of_pair (s, component_of_term v)  keys q"
    by (simp only: in_keys_iff lookup_proj_poly not_False_eq_True)
  moreover have "v = t  term_of_pair (s, component_of_term v)"
    by (simp add: splus_def eq[symmetric] term_simps)
  ultimately show ?thesis ..
qed

lemma lookup_mult_scalar_monomial_right:
  "lookup (p  monomial c v) u = (lookup p (pp_of_term u - pp_of_term v) * c when v addst u)"
proof -
  have eq1: "lookup (p * (monomial c (pp_of_term v) when component_of_term v = component_of_term u)) (pp_of_term u) =
             (lookup (