isabelle: comparison src/HOL/Decision_Procs/Reflected_Multivariate

equal deleted inserted replaced

-:85b40f300fab
+:3fe40ff1b921
 *)
 section \<open>Implementation and verification of multivariate polynomials\<close>
 theory Reflected_Multivariate_Polynomial
 imports Complex_Main Rat_Pair Polynomial_List
 begin
 subsection \<open>Datatype of polynomial expressions\<close>
 datatype poly = C Num | Bound nat | Add poly poly | Sub poly poly
 subsection\<open>Boundedness, substitution and all that\<close>
 primrec polysize:: "poly \<Rightarrow> nat"
 where
 "polysize (C c) = 1"
 | "polysize (Bound n) = 1"
 | "polysize (Neg p) = 1 + polysize p"
 | "polysize (Add p q) = 1 + polysize p + polysize q"
 | "polysize (Sub p q) = 1 + polysize p + polysize q"
 | "polysize (Mul p q) = 1 + polysize p + polysize q"
 | "polysize (Pw p n) = 1 + polysize p"
 | "polysize (CN c n p) = 4 + polysize c + polysize p"
 primrec polybound0:: "poly \<Rightarrow> bool" \<comment> \<open>a poly is INDEPENDENT of Bound 0\<close>
 where
 "polybound0 (C c) \<longleftrightarrow> True"
 | "polybound0 (Bound n) \<longleftrightarrow> n > 0"
 | "polybound0 (Neg a) \<longleftrightarrow> polybound0 a"
 | "polybound0 (Add a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
 | "polybound0 (Sub a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
 | "polybound0 (Mul a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
 | "polybound0 (Pw p n) \<longleftrightarrow> polybound0 p"
 | "polybound0 (CN c n p) \<longleftrightarrow> n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p"
 primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" \<comment> \<open>substitute a poly into a poly for Bound 0\<close>
 where
 "polysubst0 t (C c) = C c"
 | "polysubst0 t (Bound n) = (if n = 0 then t else Bound n)"
 | "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
 | "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
 | "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)"
 | "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
 | "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
 | "polysubst0 t (CN c n p) =
 (if n = 0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
 else CN (polysubst0 t c) n (polysubst0 t p))"
 fun decrpoly:: "poly \<Rightarrow> poly"
 where
 "decrpoly (Bound n) = Bound (n - 1)"
 | "decrpoly (Neg a) = Neg (decrpoly a)"
 | "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
 | "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
 | "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
 | "decrpoly (Pw p n) = Pw (decrpoly p) n"
 | "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
 | "decrpoly a = a"
 subsection \<open>Degrees and heads and coefficients\<close>
 fun degree :: "poly \<Rightarrow> nat"
 where
 "degree (CN c 0 p) = 1 + degree p"
 | "degree p = 0"
 fun head :: "poly \<Rightarrow> poly"
 where
 "head (CN c 0 p) = head p"
 | "head p = p"
 text \<open>More general notions of degree and head.\<close>
 fun degreen :: "poly \<Rightarrow> nat \<Rightarrow> nat"
 where
 "degreen (CN c n p) = (\<lambda>m. if n = m then 1 + degreen p n else 0)"
 | "degreen p = (\<lambda>m. 0)"
 fun headn :: "poly \<Rightarrow> nat \<Rightarrow> poly"
 where
 "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
 | "headn p = (\<lambda>m. p)"
 fun coefficients :: "poly \<Rightarrow> poly list"
 where
 "coefficients (CN c 0 p) = c # coefficients p"
 | "coefficients p = [p]"
 fun isconstant :: "poly \<Rightarrow> bool"
 where
 "isconstant (CN c 0 p) = False"
 | "isconstant p = True"
 fun behead :: "poly \<Rightarrow> poly"
 where
 "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
 | "behead p = 0\<^sub>p"
 fun headconst :: "poly \<Rightarrow> Num"
 where
 "headconst (CN c n p) = headconst p"
 | "headconst (C n) = n"
 subsection \<open>Operations for normalization\<close>
 declare if_cong[fundef_cong del]
 declare let_cong[fundef_cong del]
-fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
+fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly"  (infixl "+\<^sub>p" 60)
 where
 "polyadd (C c) (C c') = C (c +\<^sub>N c')"
 | "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
 | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
 | "polyadd (CN c n p) (CN c' n' p') =
 (if n < n' then CN (polyadd c (CN c' n' p')) n p
 else if n' < n then CN (polyadd (CN c n p) c') n' p'
 else
 let
 cc' = polyadd c c';
 pp' = polyadd p p'
 in if pp' = 0\<^sub>p then cc' else CN cc' n pp')"
 | "polyadd a b = Add a b"
 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
 where
 "polyneg (C c) = C (~\<^sub>N c)"
 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
 | "polyneg a = Neg a"
-definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
+definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly"  (infixl "-\<^sub>p" 60)
 where "p -\<^sub>p q = polyadd p (polyneg q)"
-fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
+fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly"  (infixl "*\<^sub>p" 60)
 where
 "polymul (C c) (C c') = C (c *\<^sub>N c')"
 | "polymul (C c) (CN c' n' p') =
 (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
 | "polymul (CN c n p) (C c') =
 (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
 | "polymul (CN c n p) (CN c' n' p') =
 (if n < n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
 else if n' < n then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
 else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"
 | "polymul a b = Mul a b"
 declare if_cong[fundef_cong]
 declare let_cong[fundef_cong]
 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
 where
 "polypow 0 = (\<lambda>p. (1)\<^sub>p)"
 | "polypow n =
 (\<lambda>p.
 let
 q = polypow (n div 2) p;
 d = polymul q q
 in if even n then d else polymul p d)"
-abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
+abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly"  (infixl "^\<^sub>p" 60)
 where "a ^\<^sub>p k \<equiv> polypow k a"
 function polynate :: "poly \<Rightarrow> poly"
 where
 "polynate (Bound n) = CN 0\<^sub>p n (1)\<^sub>p"
 | "polynate (Add p q) = polynate p +\<^sub>p polynate q"
 | "polynate (Sub p q) = polynate p -\<^sub>p polynate q"
 | "polynate (Mul p q) = polynate p *\<^sub>p polynate q"
 | "polynate (Neg p) = ~\<^sub>p (polynate p)"
 | "polynate (Pw p n) = polynate p ^\<^sub>p n"
 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
 | "polynate (C c) = C (normNum c)"
 by pat_completeness auto
 termination by (relation "measure polysize") auto
 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly"
 where
 "poly_cmul y (C x) = C (y *\<^sub>N x)"
 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
 | "poly_cmul y p = C y *\<^sub>p p"
 definition monic :: "poly \<Rightarrow> poly \<times> bool"
-where
+where "monic p =
-"monic p =
 (let h = headconst p
 in if h = 0\<^sub>N then (p, False) else (C (Ninv h) *\<^sub>p p, 0>\<^sub>N h))"
 subsection \<open>Pseudo-division\<close>
 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
 where "polydivide s p = polydivide_aux (head p) (degree p) p 0 s"
 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly"
 where
 "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
 fun poly_deriv :: "poly \<Rightarrow> poly"
 where
 "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
 | "poly_deriv p = 0\<^sub>p"
 subsection \<open>Semantics of the polynomial representation\<close>
 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0,field,power}"
 where
 "Ipoly bs (C c) = INum c"
 | "Ipoly bs (Bound n) = bs!n"
 | "Ipoly bs (Neg a) = - Ipoly bs a"
 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
 | "Ipoly bs (Pw t n) = Ipoly bs t ^ n"
 | "Ipoly bs (CN c n p) = Ipoly bs c + (bs!n) * Ipoly bs p"
 abbreviation Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0,field,power}"  ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
 where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
 lemma Ipoly_CInt: "Ipoly bs (C (i, 1)) = of_int i"
 subsection \<open>Normal form and normalization\<close>
 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
 where
 "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
 | "isnpolyh (CN c n p) = (\<lambda>k. n \<ge> k \<and> isnpolyh c (Suc n) \<and> isnpolyh p n \<and> p \<noteq> 0\<^sub>p)"
 | "isnpolyh p = (\<lambda>k. False)"
 lemma isnpolyh_mono: "n' \<le> n \<Longrightarrow> isnpolyh p n \<Longrightarrow> isnpolyh p n'"
 by (induct p rule: isnpolyh.induct) auto
 definition isnpoly :: "poly \<Rightarrow> bool"
 and np': "isnpolyh p' n'"
 and nc': "isnpolyh c' (Suc n')"
 and nn0: "n \<ge> n0"
 and nn1: "n' \<ge> n1"
 by simp_all
-{
+consider "n < n'" | "n' < n" | "n' = n" by arith
-assume "n < n'"
+then show ?case
+proof cases
+case 1
 with "4.hyps"(4-5)[OF np cnp', of n] and "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
-have ?case by (simp add: min_def)
+show ?thesis by (simp add: min_def)
-} moreover {
+next
-assume "n' < n"
+case 2
 with "4.hyps"(16-17)[OF cnp np', of "n'"] and "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
-have ?case by (cases "Suc n' = n") (simp_all add: min_def)
+show ?thesis by (cases "Suc n' = n") (simp_all add: min_def)
-} moreover {
+next
-assume "n' = n"
+case 3
 with "4.hyps"(16-17)[OF cnp np', of n] and "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
-have ?case
+show ?thesis
-apply (auto intro!: polyadd_normh)
+by (auto intro!: polyadd_normh) (simp_all add: min_def isnpolyh_mono[OF nn0])
-apply (simp_all add: min_def isnpolyh_mono[OF nn0])
+qed
-done
-}
-ultimately show ?case by arith
 next
 fix n0 n1 m
 assume np: "isnpolyh ?cnp n0"
 assume np':"isnpolyh ?cnp' n1"
 assume m: "m \<le> min n0 n1"
 let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
 let ?d1 = "degreen ?cnp m"
 let ?d2 = "degreen ?cnp' m"
 let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
-have "n' < n \<or> n < n' \<or> n' = n" by auto
+consider "n' < n \<or> n < n'" | "n' = n" by linarith
-moreover
+then show ?eq
-{
+proof cases
-assume "n' < n \<or> n < n'"
+case 1
-with "4.hyps"(3,6,18) np np' m have ?eq
+with "4.hyps"(3,6,18) np np' m show ?thesis by auto
-by auto
+next
-}
+case 2
-moreover
+have nn': "n' = n" by fact
-{
-assume nn': "n' = n"
 then have nn: "\<not> n' < n \<and> \<not> n < n'" by arith
 from "4.hyps"(16,18)[of n n' n]
 "4.hyps"(13,14)[of n "Suc n'" n]
 np np' nn'
 have norm:
 "isnpolyh (?cnp *\<^sub>p p') n"
 "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
 "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
 "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
 by (auto simp add: min_def)
-{
+show ?thesis
-assume mn: "m = n"
+proof (cases "m = n")
+case mn: True
 from "4.hyps"(17,18)[OF norm(1,4), of n]
 "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
 have degs:
 "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else ?d1 + degreen c' n)"
 "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n"
 degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
 by simp
 from "4.hyps"(16-18)[OF norm(1,4), of n]
 "4.hyps"(13-15)[OF norm(1,2), of n]
 mn norm m nn' deg
-have ?eq by simp
+show ?thesis by simp
-}
+next
-moreover
+case mn: False
-{
-assume mn: "m \<noteq> n"
 then have mn': "m < n"
 using m np by auto
 from nn' m np have max1: "m \<le> max n n"
 by simp
 then have min1: "m \<le> min n n"
 max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
 using mn nn' np np' by simp
 with "4.hyps"(16-18)[OF norm(1,4) min1]
 "4.hyps"(13-15)[OF norm(1,2) min2]
 degreen_0[OF norm(3) mn']
-have ?eq using nn' mn np np' by clarsimp
+nn' mn np np'
-}
+show ?thesis by clarsimp
-ultimately have ?eq by blast
+qed
-}
+qed
-ultimately show ?eq by blast
 }
 {
 case (2 n0 n1)
 then have np: "isnpolyh ?cnp n0"
 and np': "isnpolyh ?cnp' n1"
 and m: "m \<le> min n0 n1"
 by simp_all
 then have mn: "m \<le> n" by simp
 let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
-{
+have False if C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
-assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
+proof -
-then have nn: "\<not> n' < n \<and> \<not> n < n'"
+from C have nn: "\<not> n' < n \<and> \<not> n < n'"
 by simp
 from "4.hyps"(16-18) [of n n n]
 "4.hyps"(13-15)[of n "Suc n" n]
 np np' C(2) mn
 have norm:
 from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
 by simp
 have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
 using norm by simp
 from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq
-have False by simp
+show ?thesis by simp
-}
+qed
 then show ?case using "4.hyps" by clarsimp
 }
 qed auto
 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = Ipoly bs p * Ipoly bs q"
 by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
 (auto simp: Nsub0[simplified Nsub_def] Let_def)
 text \<open>polypow is a power function and preserves normal forms\<close>
-lemma polypow[simp]:
+lemma polypow[simp]: "Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::{field_char_0,field}) ^ n"
-"Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::{field_char_0,field}) ^ n"
 proof (induct n rule: polypow.induct)
 case 1
-then show ?case
+then show ?case by simp
-by simp
 next
 case (2 n)
 let ?q = "polypow ((Suc n) div 2) p"
 let ?d = "polymul ?q ?q"
-have "odd (Suc n) \<or> even (Suc n)"
+consider "odd (Suc n)" | "even (Suc n)" by auto
-by simp
+then show ?case
-moreover
+proof cases
-{
+case odd: 1
-assume odd: "odd (Suc n)"
+have *: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
-have th: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
 by arith
 from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)"
 by (simp add: Let_def)
 also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2) * (Ipoly bs p)^(Suc n div 2)"
 using "2.hyps" by simp
 also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
 by (simp only: power_add power_one_right) simp
 also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))"
-by (simp only: th)
+by (simp only: *)
-finally have ?case unfolding numeral_2_eq_2 [symmetric]
+finally show ?thesis
-using odd_two_times_div_two_nat [OF odd] by simp
+unfolding numeral_2_eq_2 [symmetric]
-}
+using odd_two_times_div_two_nat [OF odd] by simp
-moreover
+next
-{
+case even: 2
-assume even: "even (Suc n)"
 from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d"
 by (simp add: Let_def)
 also have "\<dots> = (Ipoly bs p) ^ (2 * (Suc n div 2))"
 using "2.hyps" by (simp only: mult_2 power_add) simp
-finally have ?case using even_two_times_div_two [OF even]
+finally show ?thesis
-by simp
+using even_two_times_div_two [OF even] by simp
-}
+qed
-ultimately show ?case by blast
 qed
 lemma polypow_normh:
 assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
 shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
 then show ?case by auto
 next
 case (2 k n)
 let ?q = "polypow (Suc k div 2) p"
 let ?d = "polymul ?q ?q"
-from 2 have th1: "isnpolyh ?q n" and th2: "isnpolyh p n"
+from 2 have *: "isnpolyh ?q n" and **: "isnpolyh p n"
 by blast+
-from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n"
+from polymul_normh[OF * *] have dn: "isnpolyh ?d n"
 by simp
-from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n"
+from polymul_normh[OF ** dn] have on: "isnpolyh (polymul p ?d) n"
 by simp
 from dn on show ?case by (simp, unfold Let_def) auto
 qed
 lemma polypow_norm:
 assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
 shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
 by (simp add: polypow_normh isnpoly_def)
 text \<open>Finally the whole normalization\<close>
-lemma polynate [simp]:
+lemma polynate [simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field})"
-"Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field})"
 by (induct p rule:polynate.induct) auto
 lemma polynate_norm[simp]:
 assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
 shows "isnpoly (polynate p)"
 by (induct p rule: polynate.induct)
 (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
 simp_all add: isnpoly_def)
 text \<open>shift1\<close>
 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
 by (simp add: shift1_def)
 lemma shift1_isnpoly:
 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
 by (induct p) auto
 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
 apply (induct p arbitrary: n0)
 apply auto
 apply atomize
 apply (rename_tac nat a b, erule_tac x = "Suc nat" in allE)
 apply auto
 done
 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
 by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
 primrec maxindex :: "poly \<Rightarrow> nat"
 where
 "maxindex (Bound n) = n + 1"
 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
 | "maxindex (Neg p) = maxindex p"
 | "maxindex (Pw p n) = maxindex p"
 | "maxindex (C x) = 0"
 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
 where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall>c \<in> set (coefficients p). wf_bs bs c"
 proof (induct p rule: coefficients.induct)
 case (1 c p)
 show ?case
 proof
 fix x
-assume xc: "x \<in> set (coefficients (CN c 0 p))"
+assume "x \<in> set (coefficients (CN c 0 p))"
-then have "x = c \<or> x \<in> set (coefficients p)"
+then consider "x = c" | "x \<in> set (coefficients p)"
-by simp
+by auto
-moreover
+then show "wf_bs bs x"
-{
+proof cases
-assume "x = c"
+case prems: 1
-then have "wf_bs bs x"
+then show ?thesis
-using "1.prems" unfolding wf_bs_def by simp
+using "1.prems" by (simp add: wf_bs_def)
-}
+next
-moreover
+case prems: 2
-{
-assume H: "x \<in> set (coefficients p)"
 from "1.prems" have "wf_bs bs p"
-unfolding wf_bs_def by simp
+by (simp add: wf_bs_def)
-with "1.hyps" H have "wf_bs bs x"
+with "1.hyps" prems show ?thesis
 by blast
-}
+qed
-ultimately show "wf_bs bs x"
-by blast
 qed
 qed simp_all
 lemma maxindex_coefficients: "\<forall>c \<in> set (coefficients p). maxindex c \<le> maxindex p"
 by (induct p rule: coefficients.induct) auto
 lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
-unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
+by (induct p) (auto simp add: nth_append wf_bs_def)
 lemma take_maxindex_wf:
 assumes wf: "wf_bs bs p"
 shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
 proof -
 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
 by (induct p) auto
 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
-unfolding wf_bs_def by simp
+by (simp add: wf_bs_def)
 lemma wf_bs_insensitive': "wf_bs (x # bs) p = wf_bs (y # bs) p"
-unfolding wf_bs_def by simp
+by (simp add: wf_bs_def)
 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x # bs) p"
 by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x # bs) p"
 unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists>ys. length (xs @ ys) = n"
-apply (rule exI[where x="replicate (n - length xs) z" for z])
+by (rule exI[where x="replicate (n - length xs) z" for z]) simp
-apply simp
-done
 lemma isnpolyh_Suc_const: "isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
 apply (cases p)
 apply auto
 apply (rename_tac nat a, case_tac "nat")
 apply simp_all
 done
 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
-unfolding wf_bs_def by (induct p q rule: polyadd.induct) (auto simp add: Let_def)
+by (induct p q rule: polyadd.induct) (auto simp add: Let_def wf_bs_def)
 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
-unfolding wf_bs_def
 apply (induct p q arbitrary: bs rule: polymul.induct)
-apply (simp_all add: wf_bs_polyadd)
+apply (simp_all add: wf_bs_polyadd wf_bs_def)
 apply clarsimp
 apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
 apply auto
 done
 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
-unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
+by (induct p rule: polyneg.induct) (auto simp: wf_bs_def)
 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
 unfolding polysub_def split_def fst_conv snd_conv
 using wf_bs_polyadd wf_bs_polyneg by blast
 assumes np: "isnpolyh p n0"
 shows "polypoly (x # bs) p = polypoly' bs p"
 proof -
 let ?cf = "set (coefficients p)"
 from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
-{
+have "polybound0 q" if "q \<in> ?cf" for q
-fix q
+proof -
-assume q: "q \<in> ?cf"
+from that cn_norm have *: "isnpolyh q n0"
-from q cn_norm have th: "isnpolyh q n0"
 by blast
-from coefficients_isconst[OF np] q have "isconstant q"
+from coefficients_isconst[OF np] that have "isconstant q"
 by blast
-with isconstant_polybound0[OF th] have "polybound0 q"
+with isconstant_polybound0[OF *] show ?thesis
 by blast
-}
+qed
 then have "\<forall>q \<in> ?cf. polybound0 q" ..
 then have "\<forall>q \<in> ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)"
 using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
 by auto
 then show ?thesis
 and "polybound0 p"
 shows "polypoly bs p = [Ipoly bs p]"
 using assms
 unfolding polypoly_def
 apply (cases p)
 apply auto
 apply (rename_tac nat a, case_tac nat)
 apply auto
 done
 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
 by (induct p rule: head.induct) auto
 shows "p = 0\<^sub>p"
 using nq eq
 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
 case less
 note np = \<open>isnpolyh p n0\<close> and zp = \<open>\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)\<close>
-{
+show "p = 0\<^sub>p"
-assume nz: "maxindex p = 0"
+proof (cases "maxindex p = 0")
-then obtain c where "p = C c"
+case True
-using np by (cases p) auto
+with np obtain c where "p = C c" by (cases p) auto
-with zp np have "p = 0\<^sub>p"
+with zp np show ?thesis by (simp add: wf_bs_def)
-unfolding wf_bs_def by simp
+next
-}
+case nz: False
-moreover
-{
-assume nz: "maxindex p \<noteq> 0"
 let ?h = "head p"
 let ?hd = "decrpoly ?h"
 let ?ihd = "maxindex ?hd"
 from head_isnpolyh[OF np] head_polybound0[OF np]
 have h: "isnpolyh ?h n0" "polybound0 ?h"
 from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
 have mihn: "maxindex ?h \<le> maxindex p"
 by auto
 with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
 by auto
-{
-fix bs :: "'a list"
+have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" if bs: "wf_bs bs ?hd" for bs :: "'a list"
-assume bs: "wf_bs bs ?hd"
+proof -
 let ?ts = "take ?ihd bs"
 let ?rs = "drop ?ihd bs"
-have ts: "wf_bs ?ts ?hd"
+from bs have ts: "wf_bs ?ts ?hd"
-using bs unfolding wf_bs_def by simp
+by (simp add: wf_bs_def)
 have bs_ts_eq: "?ts @ ?rs = bs"
 by simp
 from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
 by simp
 from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
 by simp
 with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
 by blast
 then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
-unfolding wf_bs_def by simp
+by (simp add: wf_bs_def)
 with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
 by blast
 then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
 by simp
 with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
 then have "poly (polypoly' (?ts @ xs) p) = poly []"
 by auto
 then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
 using poly_zero[where ?'a='a] by (simp add: polypoly'_def)
 with coefficients_head[of p, symmetric]
-have th0: "Ipoly (?ts @ xs) ?hd = 0"
+have *: "Ipoly (?ts @ xs) ?hd = 0"
 by simp
 from bs have wf'': "wf_bs ?ts ?hd"
-unfolding wf_bs_def by simp
+by (simp add: wf_bs_def)
-with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
+with * wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
 by simp
-with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
+with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq show ?thesis
 by simp
-}
+qed
 then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
 by blast
 from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
 by blast
 then have "?h = 0\<^sub>p" by simp
-with head_nz[OF np] have "p = 0\<^sub>p" by simp
+with head_nz[OF np] show ?thesis by simp
-}
+qed
-ultimately show "p = 0\<^sub>p"
-by blast
 qed
 lemma isnpolyh_unique:
 assumes np: "isnpolyh p n0"
 and nq: "isnpolyh q n1"
 shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (\<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup> :: 'a::{field_char_0,field,power})) \<longleftrightarrow> p = q"
 proof auto
-assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
+assume "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
 then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
 by simp
 then have "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
 using wf_bs_polysub[where p=p and q=q] by auto
 with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
 by blast
 qed
-text \<open>consequences of unicity on the algorithms for polynomial normalization\<close>
+text \<open>Consequences of unicity on the algorithms for polynomial normalization.\<close>
 lemma polyadd_commute:
 assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
 and np: "isnpolyh p n0"
 and nq: "isnpolyh q n1"
 "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0,field}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
 using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
 unfolding poly_nate_polypoly' by auto
-subsection \<open>heads, degrees and all that\<close>
+subsection \<open>Heads, degrees and all that\<close>
 lemma degree_eq_degreen0: "degree p = degreen p 0"
 by (induct p rule: degree.induct) simp_all
 lemma degree_polyneg:
 assumes "isnpolyh p n"
 shows "degree (polyneg p) = degree p"
 apply (induct p rule: polyneg.induct)
 using assms
 apply simp_all
 apply (case_tac na)
 apply auto
 done
 lemma degree_polyadd:
 assumes np: "isnpolyh p n0"
 and nq: "isnpolyh q n1"
 from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
 have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"
 by simp
 from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
 by auto
-have "n = n' \<or> n < n' \<or> n > n'"
+consider "n = n'" | "n < n'" | "n > n'"
 by arith
-moreover
+then show ?case
-{
+proof cases
-assume nn': "n = n'"
+case nn': 1
-have "n = 0 \<or> n > 0" by arith
+consider "n = 0" | "n > 0" by arith
-moreover
+then show ?thesis
-{
+proof cases
-assume nz: "n = 0"
+case 1
-then have ?case using 4 nn'
+with 4 nn' show ?thesis
 by (auto simp add: Let_def degcmc')
-}
+next
-moreover
+case 2
-{
+with nn' H(3) have "c = c'" and "p = p'"
-assume nz: "n > 0"
+by (cases n; auto)+
-with nn' H(3) have  cc': "c = c'" and pp': "p = p'"
+with nn' 4 show ?thesis
-by (cases n, auto)+
-then have ?case
 using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
 using polysub_same_0[OF c'nh, simplified polysub_def]
-using nn' 4 by (simp add: Let_def)
+by (simp add: Let_def)
-}
+qed
-ultimately have ?case by blast
+next
-}
+case nn': 2
-moreover
-{
-assume nn': "n < n'"
 then have n'p: "n' > 0"
 by simp
 then have headcnp':"head (CN c' n' p') = CN c' n' p'"
 by (cases n') simp_all
-have degcnp': "degree (CN c' n' p') = 0"
+with 4 nn' have degcnp': "degree (CN c' n' p') = 0"
 and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
-using 4 nn' by (cases n', simp_all)
+by (cases n', simp_all)
 then have "n > 0"
 by (cases n) simp_all
 then have headcnp: "head (CN c n p) = CN c n p"
 by (cases n) auto
-from H(3) headcnp headcnp' nn' have ?case
+from H(3) headcnp headcnp' nn' show ?thesis
 by auto
-}
+next
-moreover
+case nn': 3
-{
-assume nn': "n > n'"
 then have np: "n > 0" by simp
 then have headcnp:"head (CN c n p) = CN c n p"
 by (cases n) simp_all
 from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
 by simp
 by (cases n) simp_all
 with degcnpeq have "n' > 0"
 by (cases n') simp_all
 then have headcnp': "head (CN c' n' p') = CN c' n' p'"
 by (cases n') auto
-from H(3) headcnp headcnp' nn' have ?case by auto
+from H(3) headcnp headcnp' nn' show ?thesis by auto
-}
+qed
-ultimately show ?case by blast
 qed auto
 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
 by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
-lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
+lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
 proof (induct k arbitrary: n0 p)
 case 0
 then show ?case
 by auto
 next
 and nq: "isnpolyh q n1"
 and deg: "degree p \<noteq> degree q"
 shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
 using np nq deg
 apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
 apply simp_all
 apply (case_tac n', simp, simp)
 apply (case_tac n, simp, simp)
 apply (case_tac n, case_tac n', simp add: Let_def)
 apply (auto simp add: polyadd_eq_const_degree)[2]
 apply (metis head_nz)
 apply (metis head_nz)
 apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
 done
 lemma polymul_head_polyeq:
 assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
 next
 case (4 c n p c' n' p' n0 n1)
 then have norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
 "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
 by simp_all
-have "n < n' \<or> n' < n \<or> n = n'" by arith
+consider "n < n'" | "n' < n" | "n' = n" by arith
-moreover
+then show ?case
-{
+proof cases
-assume nn': "n < n'"
+case nn': 1
-then have ?case
+then show ?thesis
 using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
 apply simp
 apply (cases n)
 apply simp
 apply (cases n')
 apply simp_all
 done
-}
+next
-moreover {
+case nn': 2
-assume nn': "n'< n"
+then show ?thesis
-then have ?case
 using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
 apply simp
 apply (cases n')
 apply simp
 apply (cases n)
 apply auto
 done
-}
+next
-moreover
+case nn': 3
-{
-assume nn': "n' = n"
 from nn' polymul_normh[OF norm(5,4)]
-have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
+have ncnpc': "isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
 from nn' polymul_normh[OF norm(5,3)] norm
-have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
+have ncnpp': "isnpolyh (CN c n p *\<^sub>p p') n" by simp
 from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
-have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
+have ncnpp0': "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
 from polyadd_normh[OF ncnpc' ncnpp0']
 have nth: "isnpolyh ((CN c n p *\<^sub>p c') +\<^sub>p (CN 0\<^sub>p n (CN c n p *\<^sub>p p'))) n"
 by (simp add: min_def)
-{
+consider "n > 0" | "n = 0" by auto
-assume np: "n > 0"
+then show ?thesis
+proof cases
+case np: 1
 with nn' head_isnpolyh_Suc'[OF np nth]
 head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
-have ?case by simp
+show ?thesis by simp
-}
+next
-moreover
+case nz: 2
-{
-assume nz: "n = 0"
 from polymul_degreen[OF norm(5,4), where m="0"]
 polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
 norm(5,6) degree_npolyhCN[OF norm(6)]
 have dth: "degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
 by simp
 then have dth': "degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
 by simp
 from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
-have ?case
+show ?thesis
 using norm "4.hyps"(6)[OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] nn' nz
 by simp
-}
+qed
-ultimately have ?case
+qed
-by (cases n) auto
-}
-ultimately show ?case by blast
 qed simp_all
 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
 by (induct p rule: degree.induct) auto
 using funpow_shift1_head[OF np pnz] by simp
 from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
 by (simp add: isnpoly_def)
 from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
 have nakk':"isnpolyh ?akk' 0" by blast
-{
+show ?ths
-assume sz: "s = 0\<^sub>p"
+proof (cases "s = 0\<^sub>p")
-then have ?ths
+case True
-using np polydivide_aux.simps
+with np show ?thesis
-apply clarsimp
+apply (clarsimp simp: polydivide_aux.simps)
 apply (rule exI[where x="0\<^sub>p"])
 apply simp
 done
-}
+next
-moreover
+case sz: False
-{
+show ?thesis
-assume sz: "s \<noteq> 0\<^sub>p"
+proof (cases "degree s < n")
-{
+case True
-assume dn: "degree s < n"
+then show ?thesis
-then have "?ths"
 using ns ndp np polydivide_aux.simps
 apply auto
 apply (rule exI[where x="0\<^sub>p"])
 apply simp
 done
-}
+next
-moreover
+case dn': False
-{
-assume dn': "\<not> degree s < n"
 then have dn: "degree s \<ge> n"
 by arith
 have degsp': "degree s = degree ?p'"
 using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"]
 by simp
-{
+show ?thesis
-assume ba: "?b = a"
+proof (cases "?b = a")
+case ba: True
 then have headsp': "head s = head ?p'"
 using ap headp' by simp
 have nr: "isnpolyh (s -\<^sub>p ?p') 0"
 using polysub_normh[OF ns np'] by simp
 from degree_polysub_samehead[OF ns np' headsp' degsp']
-have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p"
+consider "degree (s -\<^sub>p ?p') < degree s" | "s -\<^sub>p ?p' = 0\<^sub>p" by auto
-by simp
+then show ?thesis
-moreover
+proof cases
-{
+case deglt: 1
-assume deglt:"degree (s -\<^sub>p ?p') < degree s"
 from polydivide_aux.simps sz dn' ba
 have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
 by (simp add: Let_def)
-{
+have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
-assume h1: "polydivide_aux a n p k s = (k', r)"
+if h1: "polydivide_aux a n p k s = (k', r)"
+proof -
 from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
 have kk': "k \<le> k'"
 and nr: "\<exists>nr. isnpolyh r nr"
 and dr: "degree r = 0 \<or> degree r < degree p"
 and q1: "\<exists>q nq. isnpolyh q nq \<and> a ^\<^sub>p k' - k *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
 by simp
 with isnpolyh_unique[OF nakks' nqr']
 have "a ^\<^sub>p (k' - k) *\<^sub>p s =
 p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r"
 by blast
-then have ?qths using nq'
+with nq' have ?qths
 apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
 apply (rule_tac x="0" in exI)
 apply simp
 done
-with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and>
+with kk' nr dr show ?thesis
-(\<exists>nr. isnpolyh r nr) \<and> ?qths"
 by blast
-}
+qed
-then have ?ths by blast
+then show ?thesis by blast
-}
+next
-moreover
+case spz: 2
-{
-assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
 from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{field_char_0,field}"]
 have "\<forall>bs:: 'a::{field_char_0,field} list. Ipoly bs s = Ipoly bs ?p'"
 by simp
-then have "\<forall>bs:: 'a::{field_char_0,field} list. Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
+with np nxdn have "\<forall>bs:: 'a::{field_char_0,field} list. Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
-using np nxdn
+by (simp only: funpow_shift1_1) simp
-apply simp
-apply (simp only: funpow_shift1_1)
-apply simp
-done
 then have sp': "s = ?xdn *\<^sub>p p"
 using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
 by blast
-{
+have ?thesis if h1: "polydivide_aux a n p k s = (k', r)"
-assume h1: "polydivide_aux a n p k s = (k', r)"
+proof -
-from polydivide_aux.simps sz dn' ba
+from sz dn' ba
-have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
+have "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
-by (simp add: Let_def)
+by (simp add: Let_def polydivide_aux.simps)
 also have "\<dots> = (k,0\<^sub>p)"
-using polydivide_aux.simps spz by simp
+using spz by (simp add: polydivide_aux.simps)
 finally have "(k', r) = (k, 0\<^sub>p)"
-using h1 by simp
+by (simp add: h1)
 with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
-polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
+polyadd_0(2)[OF polymul_normh[OF np nxdn]] show ?thesis
 apply auto
 apply (rule exI[where x="?xdn"])
 apply (auto simp add: polymul_commute[of p])
 done
-}
+qed
-}
+then show ?thesis by blast
-ultimately have ?ths by blast
+qed
-}
+next
-moreover
+case ba: False
-{
-assume ba: "?b \<noteq> a"
 from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
 polymul_normh[OF head_isnpolyh[OF ns] np']]
 have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
 by (simp add: min_def)
 have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
 using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
 polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
 funpow_shift1_nz[OF pnz]
 by simp_all
 from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
-polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
+polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz
+funpow_shift1_nz[OF pnz, where n="degree s - n"]
 have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
 using head_head[OF ns] funpow_shift1_head[OF np pnz]
 polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
 by (simp add: ap)
 from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
 funpow_shift1_nz[OF pnz, where n="degree s - n"]
 polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
 ndp dn
 have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
 by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
-{
-assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
+consider "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s" | "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
+using degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
+polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth]
+polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
+head_nz[OF np] pnz sz ap[symmetric]
+by (auto simp add: degree_eq_degreen0[symmetric])
+then show ?thesis
+proof cases
+case dth: 1
 from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
 polymul_normh[OF head_isnpolyh[OF ns]np']] ap
 have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
 by simp
-{
+have ?thesis if h1: "polydivide_aux a n p k s = (k', r)"
-assume h1:"polydivide_aux a n p k s = (k', r)"
+proof -
 from h1 polydivide_aux.simps sz dn' ba
 have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
 by (simp add: Let_def)
 with less(1)[OF dth nasbp', of "Suc k" k' r]
 obtain q nq nr where kk': "Suc k \<le> k'"
 and dr: "degree r = 0 \<or> degree r < degree p"
 and qr: "a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = p *\<^sub>p q +\<^sub>p r"
 by auto
 from kk' have kk'': "Suc (k' - Suc k) = k' - k"
 by arith
-{
+have "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
-fix bs :: "'a::{field_char_0,field} list"
+Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
+for bs :: "'a::{field_char_0,field} list"
+proof -
 from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
 have "Ipoly bs (a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
 by simp
 then have "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
 by (simp add: field_simps)
 then have "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
 by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
-then have "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
+then show ?thesis
-Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
 by (simp add: field_simps)
-}
+qed
-then have ieq:"\<forall>bs :: 'a::{field_char_0,field} list.
+then have ieq: "\<forall>bs :: 'a::{field_char_0,field} list.
 Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
 Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)"
 by auto
 let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
 from polyadd_normh[OF nq polymul_normh[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k"] head_isnpolyh[OF ns], simplified ap] nxdn]]
 have nqw: "isnpolyh ?q 0"
 by simp
 from ieq isnpolyh_unique[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - k"] ns, simplified ap] polyadd_normh[OF polymul_normh[OF np nqw] nr]]
 have asth: "(a ^\<^sub>p (k' - k) *\<^sub>p s) = p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r"
 by blast
-from dr kk' nr h1 asth nqw have ?ths
+from dr kk' nr h1 asth nqw show ?thesis
 apply simp
 apply (rule conjI)
 apply (rule exI[where x="nr"], simp)
 apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
 apply (rule exI[where x="0"], simp)
 done
-}
+qed
-then have ?ths by blast
+then show ?thesis by blast
-}
+next
-moreover
+case spz: 2
-{
+have hth: "\<forall>bs :: 'a::{field_char_0,field} list.
-assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
+Ipoly bs (a *\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
-{
+proof
 fix bs :: "'a::{field_char_0,field} list"
 from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
 have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
 by simp
 then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
 by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
-then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
+then show "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
 by simp
-}
+qed
-then have hth: "\<forall>bs :: 'a::{field_char_0,field} list.
-Ipoly bs (a *\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
 from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
 using isnpolyh_unique[where ?'a = "'a::{field_char_0,field}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
 polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
 simplified ap]
 by simp
-{
+have ?ths if h1: "polydivide_aux a n p k s = (k', r)"
-assume h1: "polydivide_aux a n p k s = (k', r)"
+proof -
 from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
 have "(k', r) = (Suc k, 0\<^sub>p)"
 by (simp add: Let_def)
 with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
 polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
-have ?ths
+show ?thesis
 apply (clarsimp simp add: Let_def)
 apply (rule exI[where x="?b *\<^sub>p ?xdn"])
 apply simp
 apply (rule exI[where x="0"], simp)
 done
-}
+qed
-then have ?ths by blast
+then show ?thesis by blast
-}
+qed
-ultimately have ?ths
+qed
-using degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth] polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
+qed
-head_nz[OF np] pnz sz ap[symmetric]
+qed
-by (auto simp add: degree_eq_degreen0[symmetric])
-}
-ultimately have ?ths by blast
-}
-ultimately have ?ths by blast
-}
-ultimately show ?ths by blast
 qed
 lemma polydivide_properties:
 assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
 and np: "isnpolyh p n0"
 lemma isnonconstant_pnormal_iff:
 assumes "isnonconstant p"
 shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
 proof
 let ?p = "polypoly bs p"
-assume H: "pnormal ?p"
+assume *: "pnormal ?p"
-have csz: "coefficients p \<noteq> []"
+have "coefficients p \<noteq> []"
 using assms by (cases p) auto
-from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] pnormal_last_nonzero[OF H]
+from coefficients_head[of p] last_map[OF this, of "Ipoly bs"] pnormal_last_nonzero[OF *]
 show "Ipoly bs (head p) \<noteq> 0"
 by (simp add: polypoly_def)
 next
-assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
+assume *: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
 let ?p = "polypoly bs p"
 have csz: "coefficients p \<noteq> []"
 using assms by (cases p) auto
 then have pz: "?p \<noteq> []"
 by (simp add: polypoly_def)
 then have lg: "length ?p > 0"
 by simp
-from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
+from * coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
 have lz: "last ?p \<noteq> 0"
 by (simp add: polypoly_def)
 from pnormal_last_length[OF lg lz] show "pnormal ?p" .
 qed
 lemma isnonconstant_nonconstant:
 assumes "isnonconstant p"
 shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
 proof
 let ?p = "polypoly bs p"
-assume nc: "nonconstant ?p"
+assume "nonconstant ?p"
-from isnonconstant_pnormal_iff[OF assms, of bs] nc
+with isnonconstant_pnormal_iff[OF assms, of bs] show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
-show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
 unfolding nonconstant_def by blast
 next
 let ?p = "polypoly bs p"
-assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
+assume "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
-from isnonconstant_pnormal_iff[OF assms, of bs] h
+with isnonconstant_pnormal_iff[OF assms, of bs] have pn: "pnormal ?p"
-have pn: "pnormal ?p"
 by blast
-{
+have False if H: "?p = [x]" for x
-fix x
+proof -
-assume H: "?p = [x]"
 from H have "length (coefficients p) = 1"
-unfolding polypoly_def by auto
+by (auto simp: polypoly_def)
 with isnonconstant_coefficients_length[OF assms]
-have False by arith
+show ?thesis by arith
-}
+qed
 then show "nonconstant ?p"
 using pn unfolding nonconstant_def by blast
 qed
 lemma pnormal_length: "p \<noteq> [] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
 apply (induct p)
 apply (simp_all add: pnormal_def)
 apply (case_tac "p = []")
 apply simp_all
 done
 lemma degree_degree:
 assumes "isnonconstant p"
 shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
+(is "?lhs \<longleftrightarrow> ?rhs")
 proof
 let ?p = "polypoly bs p"
-assume H: "degree p = Polynomial_List.degree ?p"
+{
-from isnonconstant_coefficients_length[OF assms] have pz: "?p \<noteq> []"
+assume ?lhs
-unfolding polypoly_def by auto
+from isnonconstant_coefficients_length[OF assms] have "?p \<noteq> []"
-from H degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
+by (auto simp: polypoly_def)
-have lg: "length (pnormalize ?p) = length ?p"
+from \<open>?lhs\<close> degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
-unfolding Polynomial_List.degree_def polypoly_def by simp
+have "length (pnormalize ?p) = length ?p"
-then have "pnormal ?p"
+by (simp add: Polynomial_List.degree_def polypoly_def)
-using pnormal_length[OF pz] by blast
+with pnormal_length[OF \<open>?p \<noteq> []\<close>] have "pnormal ?p"
-with isnonconstant_pnormal_iff[OF assms] show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
+by blast
-by blast
+with isnonconstant_pnormal_iff[OF assms] show ?rhs
-next
+by blast
-let ?p = "polypoly bs p"
+next
-assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
+assume ?rhs
 with isnonconstant_pnormal_iff[OF assms] have "pnormal ?p"
 by blast
-with degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
+with degree_coefficients[of p] isnonconstant_coefficients_length[OF assms] show ?lhs
-show "degree p = Polynomial_List.degree ?p"
+by (auto simp: polypoly_def pnormal_def Polynomial_List.degree_def)
-unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
+}
 qed
-section \<open>Swaps ; Division by a certain variable\<close>
+section \<open>Swaps -- division by a certain variable\<close>
 primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
 where
 "swap n m (C x) = C x"
 | "swap n m (Bound k) = Bound (if k = n then m else if k = m then n else k)"
 | "swap n m (Neg t) = Neg (swap n m t)"
 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
 | "swap n m (Pw t k) = Pw (swap n m t) k"
 | "swap n m (CN c k p) = CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)"
 lemma swap:
 assumes "n < length bs"
 and "m < length bs"
 shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
 lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
 by (induct p) simp_all
 fun isweaknpoly :: "poly \<Rightarrow> bool"
 where
 "isweaknpoly (C c) = True"
 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
 | "isweaknpoly p = False"
 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
 by (induct p arbitrary: n0) auto
 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"

changeset 67123	3fe40ff1b921
parent 62390	842917225d56
child 68442	477b3f7067c9