# HG changeset patch # User chaieb # Date 1256457456 -3600 # Node ID 92080294beb804eed89cdc2ba00a6384e9cafa55 # Parent 241cfaed158f52a50a0f31b2f6fd86935bb7a658 A theory of polynomials based on lists diff -r 241cfaed158f -r 92080294beb8 src/HOL/Decision_Procs/Polynomial_List.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Decision_Procs/Polynomial_List.thy Sun Oct 25 08:57:36 2009 +0100 @@ -0,0 +1,783 @@ +(* Title: HOL/Decision_Procs/Polynomial_List.thy + Author: Amine Chaieb +*) + +header{*Univariate Polynomials as Lists *} + +theory Polynomial_List +imports Main +begin + +text{* Application of polynomial as a real function. *} + +consts poly :: "'a list => 'a => ('a::{comm_ring})" +primrec + poly_Nil: "poly [] x = 0" + poly_Cons: "poly (h#t) x = h + x * poly t x" + + +subsection{*Arithmetic Operations on Polynomials*} + +text{*addition*} +consts padd :: "['a list, 'a list] => ('a::comm_ring_1) list" (infixl "+++" 65) +primrec + padd_Nil: "[] +++ l2 = l2" + padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t + else (h + hd l2)#(t +++ tl l2))" + +text{*Multiplication by a constant*} +consts cmult :: "['a :: comm_ring_1, 'a list] => 'a list" (infixl "%*" 70) +primrec + cmult_Nil: "c %* [] = []" + cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)" + +text{*Multiplication by a polynomial*} +consts pmult :: "['a list, 'a list] => ('a::comm_ring_1) list" (infixl "***" 70) +primrec + pmult_Nil: "[] *** l2 = []" + pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2 + else (h %* l2) +++ ((0) # (t *** l2)))" + +text{*Repeated multiplication by a polynomial*} +consts mulexp :: "[nat, 'a list, 'a list] => ('a ::comm_ring_1) list" +primrec + mulexp_zero: "mulexp 0 p q = q" + mulexp_Suc: "mulexp (Suc n) p q = p *** mulexp n p q" + +text{*Exponential*} +consts pexp :: "['a list, nat] => ('a::comm_ring_1) list" (infixl "%^" 80) +primrec + pexp_0: "p %^ 0 = [1]" + pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)" + +text{*Quotient related value of dividing a polynomial by x + a*} +(* Useful for divisor properties in inductive proofs *) +consts "pquot" :: "['a list, 'a::field] => 'a list" +primrec + pquot_Nil: "pquot [] a= []" + pquot_Cons: "pquot (h#t) a = (if t = [] then [h] + else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))" + + +text{*normalization of polynomials (remove extra 0 coeff)*} +consts pnormalize :: "('a::comm_ring_1) list => 'a list" +primrec + pnormalize_Nil: "pnormalize [] = []" + pnormalize_Cons: "pnormalize (h#p) = (if ( (pnormalize p) = []) + then (if (h = 0) then [] else [h]) + else (h#(pnormalize p)))" + +definition "pnormal p = ((pnormalize p = p) \ p \ [])" +definition "nonconstant p = (pnormal p \ (\x. p \ [x]))" +text{*Other definitions*} + +definition + poly_minus :: "'a list => ('a :: comm_ring_1) list" ("-- _" [80] 80) where + "-- p = (- 1) %* p" + +definition + divides :: "[('a::comm_ring_1) list, 'a list] => bool" (infixl "divides" 70) where + "p1 divides p2 = (\q. poly p2 = poly(p1 *** q))" + +definition + order :: "('a::comm_ring_1) => 'a list => nat" where + --{*order of a polynomial*} + "order a p = (SOME n. ([-a, 1] %^ n) divides p & + ~ (([-a, 1] %^ (Suc n)) divides p))" + +definition + degree :: "('a::comm_ring_1) list => nat" where + --{*degree of a polynomial*} + "degree p = length (pnormalize p) - 1" + +definition + rsquarefree :: "('a::comm_ring_1) list => bool" where + --{*squarefree polynomials --- NB with respect to real roots only.*} + "rsquarefree p = (poly p \ poly [] & + (\a. (order a p = 0) | (order a p = 1)))" + +lemma padd_Nil2: "p +++ [] = p" +by (induct p) auto +declare padd_Nil2 [simp] + +lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)" +by auto + +lemma pminus_Nil: "-- [] = []" +by (simp add: poly_minus_def) +declare pminus_Nil [simp] + +lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" +by simp + +lemma poly_ident_mult: "1 %* t = t" +by (induct "t", auto) +declare poly_ident_mult [simp] + +lemma poly_simple_add_Cons: "[a] +++ ((0)#t) = (a#t)" +by simp +declare poly_simple_add_Cons [simp] + +text{*Handy general properties*} + +lemma padd_commut: "b +++ a = a +++ b" +apply (subgoal_tac "\a. b +++ a = a +++ b") +apply (induct_tac [2] "b", auto) +apply (rule padd_Cons [THEN ssubst]) +apply (case_tac "aa", auto) +done + +lemma padd_assoc [rule_format]: "\b c. (a +++ b) +++ c = a +++ (b +++ c)" +apply (induct "a", simp, clarify) +apply (case_tac b, simp_all) +done + +lemma poly_cmult_distr [rule_format]: + "\q. a %* ( p +++ q) = (a %* p +++ a %* q)" +apply (induct "p", simp, clarify) +apply (case_tac "q") +apply (simp_all add: right_distrib) +done + +lemma pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)" +apply (induct "t", simp) +by (auto simp add: mult_zero_left poly_ident_mult padd_commut) + + +text{*properties of evaluation of polynomials.*} + +lemma poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x" +apply (subgoal_tac "\p2. poly (p1 +++ p2) x = poly (p1) x + poly (p2) x") +apply (induct_tac [2] "p1", auto) +apply (case_tac "p2") +apply (auto simp add: right_distrib) +done + +lemma poly_cmult: "poly (c %* p) x = c * poly p x" +apply (induct "p") +apply (case_tac [2] "x=0") +apply (auto simp add: right_distrib mult_ac) +done + +lemma poly_minus: "poly (-- p) x = - (poly p x)" +apply (simp add: poly_minus_def) +apply (auto simp add: poly_cmult) +done + +lemma poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x" +apply (subgoal_tac "\p2. poly (p1 *** p2) x = poly p1 x * poly p2 x") +apply (simp (no_asm_simp)) +apply (induct "p1") +apply (auto simp add: poly_cmult) +apply (case_tac p1) +apply (auto simp add: poly_cmult poly_add left_distrib right_distrib mult_ac) +done + +lemma poly_exp: "poly (p %^ n) (x::'a::comm_ring_1) = (poly p x) ^ n" +apply (induct "n") +apply (auto simp add: poly_cmult poly_mult power_Suc) +done + +text{*More Polynomial Evaluation Lemmas*} + +lemma poly_add_rzero: "poly (a +++ []) x = poly a x" +by simp +declare poly_add_rzero [simp] + +lemma poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x" + by (simp add: poly_mult mult_assoc) + +lemma poly_mult_Nil2: "poly (p *** []) x = 0" +by (induct "p", auto) +declare poly_mult_Nil2 [simp] + +lemma poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x" +apply (induct "n") +apply (auto simp add: poly_mult mult_assoc) +done + +subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides + @{term "p(x)"} *} + +lemma lemma_poly_linear_rem: "\h. \q r. h#t = [r] +++ [-a, 1] *** q" +apply (induct "t", safe) +apply (rule_tac x = "[]" in exI) +apply (rule_tac x = h in exI, simp) +apply (drule_tac x = aa in spec, safe) +apply (rule_tac x = "r#q" in exI) +apply (rule_tac x = "a*r + h" in exI) +apply (case_tac "q", auto) +done + +lemma poly_linear_rem: "\q r. h#t = [r] +++ [-a, 1] *** q" +by (cut_tac t = t and a = a in lemma_poly_linear_rem, auto) + + +lemma poly_linear_divides: "(poly p a = 0) = ((p = []) | (\q. p = [-a, 1] *** q))" +apply (auto simp add: poly_add poly_cmult right_distrib) +apply (case_tac "p", simp) +apply (cut_tac h = aa and t = list and a = a in poly_linear_rem, safe) +apply (case_tac "q", auto) +apply (drule_tac x = "[]" in spec, simp) +apply (auto simp add: poly_add poly_cmult add_assoc) +apply (drule_tac x = "aa#lista" in spec, auto) +done + +lemma lemma_poly_length_mult: "\h k a. length (k %* p +++ (h # (a %* p))) = Suc (length p)" +by (induct "p", auto) +declare lemma_poly_length_mult [simp] + +lemma lemma_poly_length_mult2: "\h k. length (k %* p +++ (h # p)) = Suc (length p)" +by (induct "p", auto) +declare lemma_poly_length_mult2 [simp] + +lemma poly_length_mult: "length([-a,1] *** q) = Suc (length q)" +by auto +declare poly_length_mult [simp] + + +subsection{*Polynomial length*} + +lemma poly_cmult_length: "length (a %* p) = length p" +by (induct "p", auto) +declare poly_cmult_length [simp] + +lemma poly_add_length [rule_format]: + "\p2. length (p1 +++ p2) = + (if (length p1 < length p2) then length p2 else length p1)" +apply (induct "p1", simp_all) +apply arith +done + +lemma poly_root_mult_length: "length([a,b] *** p) = Suc (length p)" +by (simp add: poly_cmult_length poly_add_length) +declare poly_root_mult_length [simp] + +lemma poly_mult_not_eq_poly_Nil: "(poly (p *** q) x \ poly [] x) = + (poly p x \ poly [] x & poly q x \ poly [] (x::'a::idom))" +apply (auto simp add: poly_mult) +done +declare poly_mult_not_eq_poly_Nil [simp] + +lemma poly_mult_eq_zero_disj: "(poly (p *** q) (x::'a::idom) = 0) = (poly p x = 0 | poly q x = 0)" +by (auto simp add: poly_mult) + +text{*Normalisation Properties*} + +lemma poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)" +by (induct "p", auto) + +text{*A nontrivial polynomial of degree n has no more than n roots*} + +lemma poly_roots_index_lemma0 [rule_format]: + "\p x. poly p x \ poly [] x & length p = n + --> (\i. \x. (poly p x = (0::'a::idom)) --> (\m. (m \ n & x = i m)))" +apply (induct "n", safe) +apply (rule ccontr) +apply (subgoal_tac "\a. poly p a = 0", safe) +apply (drule poly_linear_divides [THEN iffD1], safe) +apply (drule_tac x = q in spec) +apply (drule_tac x = x in spec) +apply (simp del: poly_Nil pmult_Cons) +apply (erule exE) +apply (drule_tac x = "%m. if m = Suc n then a else i m" in spec, safe) +apply (drule poly_mult_eq_zero_disj [THEN iffD1], safe) +apply (drule_tac x = "Suc (length q)" in spec) +apply (auto simp add: ring_simps) +apply (drule_tac x = xa in spec) +apply (clarsimp simp add: ring_simps) +apply (drule_tac x = m in spec) +apply (auto simp add:ring_simps) +done +lemmas poly_roots_index_lemma1 = conjI [THEN poly_roots_index_lemma0, standard] + +lemma poly_roots_index_length0: "poly p (x::'a::idom) \ poly [] x ==> + \i. \x. (poly p x = 0) --> (\n. n \ length p & x = i n)" +by (blast intro: poly_roots_index_lemma1) + +lemma poly_roots_finite_lemma: "poly p (x::'a::idom) \ poly [] x ==> + \N i. \x. (poly p x = 0) --> (\n. (n::nat) < N & x = i n)" +apply (drule poly_roots_index_length0, safe) +apply (rule_tac x = "Suc (length p)" in exI) +apply (rule_tac x = i in exI) +apply (simp add: less_Suc_eq_le) +done + + +lemma real_finite_lemma: + assumes P: "\x. P x --> (\n. n < length j & x = j!n)" + shows "finite {(x::'a::idom). P x}" +proof- + let ?M = "{x. P x}" + let ?N = "set j" + have "?M \ ?N" using P by auto + thus ?thesis using finite_subset by auto +qed + +lemma poly_roots_index_lemma [rule_format]: + "\p x. poly p x \ poly [] x & length p = n + --> (\i. \x. (poly p x = (0::'a::{idom})) --> x \ set i)" +apply (induct "n", safe) +apply (rule ccontr) +apply (subgoal_tac "\a. poly p a = 0", safe) +apply (drule poly_linear_divides [THEN iffD1], safe) +apply (drule_tac x = q in spec) +apply (drule_tac x = x in spec) +apply (auto simp del: poly_Nil pmult_Cons) +apply (drule_tac x = "a#i" in spec) +apply (auto simp only: poly_mult List.list.size) +apply (drule_tac x = xa in spec) +apply (clarsimp simp add: ring_simps) +done + +lemmas poly_roots_index_lemma2 = conjI [THEN poly_roots_index_lemma, standard] + +lemma poly_roots_index_length: "poly p (x::'a::idom) \ poly [] x ==> + \i. \x. (poly p x = 0) --> x \ set i" +by (blast intro: poly_roots_index_lemma2) + +lemma poly_roots_finite_lemma': "poly p (x::'a::idom) \ poly [] x ==> + \i. \x. (poly p x = 0) --> x \ set i" +by (drule poly_roots_index_length, safe) + +lemma UNIV_nat_infinite: "\ finite (UNIV :: nat set)" + unfolding finite_conv_nat_seg_image +proof(auto simp add: expand_set_eq image_iff) + fix n::nat and f:: "nat \ nat" + let ?N = "{i. i < n}" + let ?fN = "f ` ?N" + let ?y = "Max ?fN + 1" + from nat_seg_image_imp_finite[of "?fN" "f" n] + have thfN: "finite ?fN" by simp + {assume "n =0" hence "\x. \xa f xa" by auto} + moreover + {assume nz: "n \ 0" + hence thne: "?fN \ {}" by (auto simp add: neq0_conv) + have "\x\ ?fN. Max ?fN \ x" using nz Max_ge_iff[OF thfN thne] by auto + hence "\x\ ?fN. ?y > x" by (auto simp add: less_Suc_eq_le) + hence "?y \ ?fN" by auto + hence "\x. \xa f xa" by auto } + ultimately show "\x. \xa f xa" by blast +qed + +lemma UNIV_ring_char_0_infinte: "\ finite (UNIV:: ('a::ring_char_0) set)" +proof + assume F: "finite (UNIV :: 'a set)" + have th0: "of_nat ` UNIV \ (UNIV:: 'a set)" by simp + from finite_subset[OF th0 F] have th: "finite (of_nat ` UNIV :: 'a set)" . + have th': "inj_on (of_nat::nat \ 'a) (UNIV)" + unfolding inj_on_def by auto + from finite_imageD[OF th th'] UNIV_nat_infinite + show False by blast +qed + +lemma poly_roots_finite: "(poly p \ poly []) = + finite {x. poly p x = (0::'a::{idom, ring_char_0})}" +proof + assume H: "poly p \ poly []" + show "finite {x. poly p x = (0::'a)}" + using H + apply - + apply (erule contrapos_np, rule ext) + apply (rule ccontr) + apply (clarify dest!: poly_roots_finite_lemma') + using finite_subset + proof- + fix x i + assume F: "\ finite {x. poly p x = (0\'a)}" + and P: "\x. poly p x = (0\'a) \ x \ set i" + let ?M= "{x. poly p x = (0\'a)}" + from P have "?M \ set i" by auto + with finite_subset F show False by auto + qed +next + assume F: "finite {x. poly p x = (0\'a)}" + show "poly p \ poly []" using F UNIV_ring_char_0_infinte by auto +qed + +text{*Entirety and Cancellation for polynomials*} + +lemma poly_entire_lemma: "[| poly (p:: ('a::{idom,ring_char_0}) list) \ poly [] ; poly q \ poly [] |] + ==> poly (p *** q) \ poly []" +by (auto simp add: poly_roots_finite poly_mult Collect_disj_eq) + +lemma poly_entire: "(poly (p *** q) = poly ([]::('a::{idom,ring_char_0}) list)) = ((poly p = poly []) | (poly q = poly []))" +apply (auto intro: ext dest: fun_cong simp add: poly_entire_lemma poly_mult) +apply (blast intro: ccontr dest: poly_entire_lemma poly_mult [THEN subst]) +done + +lemma poly_entire_neg: "(poly (p *** q) \ poly ([]::('a::{idom,ring_char_0}) list)) = ((poly p \ poly []) & (poly q \ poly []))" +by (simp add: poly_entire) + +lemma fun_eq: " (f = g) = (\x. f x = g x)" +by (auto intro!: ext) + +lemma poly_add_minus_zero_iff: "(poly (p +++ -- q) = poly []) = (poly p = poly q)" +by (auto simp add: ring_simps poly_add poly_minus_def fun_eq poly_cmult) + +lemma poly_add_minus_mult_eq: "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))" +by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult right_distrib) + +lemma poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly ([]::('a::{idom, ring_char_0}) list) | poly q = poly r)" +apply (rule_tac p1 = "p *** q" in poly_add_minus_zero_iff [THEN subst]) +apply (auto intro: ext simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff) +done + +lemma poly_exp_eq_zero: + "(poly (p %^ n) = poly ([]::('a::idom) list)) = (poly p = poly [] & n \ 0)" +apply (simp only: fun_eq add: all_simps [symmetric]) +apply (rule arg_cong [where f = All]) +apply (rule ext) +apply (induct_tac "n") +apply (auto simp add: poly_mult) +done +declare poly_exp_eq_zero [simp] + +lemma poly_prime_eq_zero: "poly [a,(1::'a::comm_ring_1)] \ poly []" +apply (simp add: fun_eq) +apply (rule_tac x = "1 - a" in exI, simp) +done +declare poly_prime_eq_zero [simp] + +lemma poly_exp_prime_eq_zero: "(poly ([a, (1::'a::idom)] %^ n) \ poly [])" +by auto +declare poly_exp_prime_eq_zero [simp] + +text{*A more constructive notion of polynomials being trivial*} + +lemma poly_zero_lemma': "poly (h # t) = poly [] ==> h = (0::'a::{idom,ring_char_0}) & poly t = poly []" +apply(simp add: fun_eq) +apply (case_tac "h = 0") +apply (drule_tac [2] x = 0 in spec, auto) +apply (case_tac "poly t = poly []", simp) +proof- + fix x + assume H: "\x. x = (0\'a) \ poly t x = (0\'a)" and pnz: "poly t \ poly []" + let ?S = "{x. poly t x = 0}" + from H have "\x. x \0 \ poly t x = 0" by blast + hence th: "?S \ UNIV - {0}" by auto + from poly_roots_finite pnz have th': "finite ?S" by blast + from finite_subset[OF th th'] UNIV_ring_char_0_infinte[where ?'a = 'a] + show "poly t x = (0\'a)" by simp + qed + +lemma poly_zero: "(poly p = poly []) = list_all (%c. c = (0::'a::{idom,ring_char_0})) p" +apply (induct "p", simp) +apply (rule iffI) +apply (drule poly_zero_lemma', auto) +done + + + +text{*Basics of divisibility.*} + +lemma poly_primes: "([a, (1::'a::idom)] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)" +apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult left_distrib [symmetric]) +apply (drule_tac x = "-a" in spec) +apply (auto simp add: poly_linear_divides poly_add poly_cmult left_distrib [symmetric]) +apply (rule_tac x = "qa *** q" in exI) +apply (rule_tac [2] x = "p *** qa" in exI) +apply (auto simp add: poly_add poly_mult poly_cmult mult_ac) +done + +lemma poly_divides_refl: "p divides p" +apply (simp add: divides_def) +apply (rule_tac x = "[1]" in exI) +apply (auto simp add: poly_mult fun_eq) +done +declare poly_divides_refl [simp] + +lemma poly_divides_trans: "[| p divides q; q divides r |] ==> p divides r" +apply (simp add: divides_def, safe) +apply (rule_tac x = "qa *** qaa" in exI) +apply (auto simp add: poly_mult fun_eq mult_assoc) +done + +lemma poly_divides_exp: "m \ n ==> (p %^ m) divides (p %^ n)" +apply (auto simp add: le_iff_add) +apply (induct_tac k) +apply (rule_tac [2] poly_divides_trans) +apply (auto simp add: divides_def) +apply (rule_tac x = p in exI) +apply (auto simp add: poly_mult fun_eq mult_ac) +done + +lemma poly_exp_divides: "[| (p %^ n) divides q; m\n |] ==> (p %^ m) divides q" +by (blast intro: poly_divides_exp poly_divides_trans) + +lemma poly_divides_add: + "[| p divides q; p divides r |] ==> p divides (q +++ r)" +apply (simp add: divides_def, auto) +apply (rule_tac x = "qa +++ qaa" in exI) +apply (auto simp add: poly_add fun_eq poly_mult right_distrib) +done + +lemma poly_divides_diff: + "[| p divides q; p divides (q +++ r) |] ==> p divides r" +apply (simp add: divides_def, auto) +apply (rule_tac x = "qaa +++ -- qa" in exI) +apply (auto simp add: poly_add fun_eq poly_mult poly_minus right_diff_distrib algebra_simps) +done + +lemma poly_divides_diff2: "[| p divides r; p divides (q +++ r) |] ==> p divides q" +apply (erule poly_divides_diff) +apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac) +done + +lemma poly_divides_zero: "poly p = poly [] ==> q divides p" +apply (simp add: divides_def) +apply (rule exI[where x="[]"]) +apply (auto simp add: fun_eq poly_mult) +done + +lemma poly_divides_zero2: "q divides []" +apply (simp add: divides_def) +apply (rule_tac x = "[]" in exI) +apply (auto simp add: fun_eq) +done +declare poly_divides_zero2 [simp] + +text{*At last, we can consider the order of a root.*} + + +lemma poly_order_exists_lemma [rule_format]: + "\p. length p = d --> poly p \ poly [] + --> (\n q. p = mulexp n [-a, (1::'a::{idom,ring_char_0})] q & poly q a \ 0)" +apply (induct "d") +apply (simp add: fun_eq, safe) +apply (case_tac "poly p a = 0") +apply (drule_tac poly_linear_divides [THEN iffD1], safe) +apply (drule_tac x = q in spec) +apply (drule_tac poly_entire_neg [THEN iffD1], safe, force) +apply (rule_tac x = "Suc n" in exI) +apply (rule_tac x = qa in exI) +apply (simp del: pmult_Cons) +apply (rule_tac x = 0 in exI, force) +done + +(* FIXME: Tidy up *) +lemma poly_order_exists: + "[| length p = d; poly p \ poly [] |] + ==> \n. ([-a, 1] %^ n) divides p & + ~(([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p)" +apply (drule poly_order_exists_lemma [where a=a], assumption, clarify) +apply (rule_tac x = n in exI, safe) +apply (unfold divides_def) +apply (rule_tac x = q in exI) +apply (induct_tac "n", simp) +apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult right_distrib mult_ac) +apply safe +apply (subgoal_tac "poly (mulexp n [- a, 1] q) \ poly ([- a, 1] %^ Suc n *** qa)") +apply simp +apply (induct_tac "n") +apply (simp del: pmult_Cons pexp_Suc) +apply (erule_tac Q = "poly q a = 0" in contrapos_np) +apply (simp add: poly_add poly_cmult) +apply (rule pexp_Suc [THEN ssubst]) +apply (rule ccontr) +apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc) +done + +lemma poly_one_divides: "[1] divides p" +by (simp add: divides_def, auto) +declare poly_one_divides [simp] + +lemma poly_order: "poly p \ poly [] + ==> EX! n. ([-a, (1::'a::{idom,ring_char_0})] %^ n) divides p & + ~(([-a, 1] %^ (Suc n)) divides p)" +apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc) +apply (cut_tac x = y and y = n in less_linear) +apply (drule_tac m = n in poly_exp_divides) +apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides] + simp del: pmult_Cons pexp_Suc) +done + +text{*Order*} + +lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n" +by (blast intro: someI2) + +lemma order: + "(([-a, (1::'a::{idom,ring_char_0})] %^ n) divides p & + ~(([-a, 1] %^ (Suc n)) divides p)) = + ((n = order a p) & ~(poly p = poly []))" +apply (unfold order_def) +apply (rule iffI) +apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order) +apply (blast intro!: poly_order [THEN [2] some1_equalityD]) +done + +lemma order2: "[| poly p \ poly [] |] + ==> ([-a, (1::'a::{idom,ring_char_0})] %^ (order a p)) divides p & + ~(([-a, 1] %^ (Suc(order a p))) divides p)" +by (simp add: order del: pexp_Suc) + +lemma order_unique: "[| poly p \ poly []; ([-a, 1] %^ n) divides p; + ~(([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p) + |] ==> (n = order a p)" +by (insert order [of a n p], auto) + +lemma order_unique_lemma: "(poly p \ poly [] & ([-a, 1] %^ n) divides p & + ~(([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p)) + ==> (n = order a p)" +by (blast intro: order_unique) + +lemma order_poly: "poly p = poly q ==> order a p = order a q" +by (auto simp add: fun_eq divides_def poly_mult order_def) + +lemma pexp_one: "p %^ (Suc 0) = p" +apply (induct "p") +apply (auto simp add: numeral_1_eq_1) +done +declare pexp_one [simp] + +lemma lemma_order_root [rule_format]: + "\p a. 0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p + --> poly p a = 0" +apply (induct "n", blast) +apply (auto simp add: divides_def poly_mult simp del: pmult_Cons) +done + +lemma order_root: "(poly p a = (0::'a::{idom,ring_char_0})) = ((poly p = poly []) | order a p \ 0)" +apply (case_tac "poly p = poly []", auto) +apply (simp add: poly_linear_divides del: pmult_Cons, safe) +apply (drule_tac [!] a = a in order2) +apply (rule ccontr) +apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast) +using neq0_conv +apply (blast intro: lemma_order_root) +done + +lemma order_divides: "(([-a, 1::'a::{idom,ring_char_0}] %^ n) divides p) = ((poly p = poly []) | n \ order a p)" +apply (case_tac "poly p = poly []", auto) +apply (simp add: divides_def fun_eq poly_mult) +apply (rule_tac x = "[]" in exI) +apply (auto dest!: order2 [where a=a] + intro: poly_exp_divides simp del: pexp_Suc) +done + +lemma order_decomp: + "poly p \ poly [] + ==> \q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) & + ~([-a, 1::'a::{idom,ring_char_0}] divides q)" +apply (unfold divides_def) +apply (drule order2 [where a = a]) +apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe) +apply (rule_tac x = q in exI, safe) +apply (drule_tac x = qa in spec) +apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons) +done + +text{*Important composition properties of orders.*} + +lemma order_mult: "poly (p *** q) \ poly [] + ==> order a (p *** q) = order a p + order (a::'a::{idom,ring_char_0}) q" +apply (cut_tac a = a and p = "p***q" and n = "order a p + order a q" in order) +apply (auto simp add: poly_entire simp del: pmult_Cons) +apply (drule_tac a = a in order2)+ +apply safe +apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe) +apply (rule_tac x = "qa *** qaa" in exI) +apply (simp add: poly_mult mult_ac del: pmult_Cons) +apply (drule_tac a = a in order_decomp)+ +apply safe +apply (subgoal_tac "[-a,1] divides (qa *** qaa) ") +apply (simp add: poly_primes del: pmult_Cons) +apply (auto simp add: divides_def simp del: pmult_Cons) +apply (rule_tac x = qb in exI) +apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))") +apply (drule poly_mult_left_cancel [THEN iffD1], force) +apply (subgoal_tac "poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** (qa *** qaa))) = poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))) ") +apply (drule poly_mult_left_cancel [THEN iffD1], force) +apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons) +done + + + +lemma order_root2: "poly p \ poly [] ==> (poly p a = 0) = (order (a::'a::{idom,ring_char_0}) p \ 0)" +by (rule order_root [THEN ssubst], auto) + + +lemma pmult_one: "[1] *** p = p" +by auto +declare pmult_one [simp] + +lemma poly_Nil_zero: "poly [] = poly [0]" +by (simp add: fun_eq) + +lemma rsquarefree_decomp: + "[| rsquarefree p; poly p a = (0::'a::{idom,ring_char_0}) |] + ==> \q. (poly p = poly ([-a, 1] *** q)) & poly q a \ 0" +apply (simp add: rsquarefree_def, safe) +apply (frule_tac a = a in order_decomp) +apply (drule_tac x = a in spec) +apply (drule_tac a = a in order_root2 [symmetric]) +apply (auto simp del: pmult_Cons) +apply (rule_tac x = q in exI, safe) +apply (simp add: poly_mult fun_eq) +apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1]) +apply (simp add: divides_def del: pmult_Cons, safe) +apply (drule_tac x = "[]" in spec) +apply (auto simp add: fun_eq) +done + + +text{*Normalization of a polynomial.*} + +lemma poly_normalize: "poly (pnormalize p) = poly p" +apply (induct "p") +apply (auto simp add: fun_eq) +done +declare poly_normalize [simp] + + +text{*The degree of a polynomial.*} + +lemma lemma_degree_zero: + "list_all (%c. c = 0) p \ pnormalize p = []" +by (induct "p", auto) + +lemma degree_zero: "(poly p = poly ([]:: (('a::{idom,ring_char_0}) list))) \ (degree p = 0)" +apply (simp add: degree_def) +apply (case_tac "pnormalize p = []") +apply (auto simp add: poly_zero lemma_degree_zero ) +done + +lemma pnormalize_sing: "(pnormalize [x] = [x]) \ x \ 0" by simp +lemma pnormalize_pair: "y \ 0 \ (pnormalize [x, y] = [x, y])" by simp +lemma pnormal_cons: "pnormal p \ pnormal (c#p)" + unfolding pnormal_def by simp +lemma pnormal_tail: "p\[] \ pnormal (c#p) \ pnormal p" + unfolding pnormal_def + apply (cases "pnormalize p = []", auto) + by (cases "c = 0", auto) +lemma pnormal_last_nonzero: "pnormal p ==> last p \ 0" + apply (induct p, auto simp add: pnormal_def) + apply (case_tac "pnormalize p = []", auto) + by (case_tac "a=0", auto) +lemma pnormal_length: "pnormal p \ 0 < length p" + unfolding pnormal_def length_greater_0_conv by blast +lemma pnormal_last_length: "\0 < length p ; last p \ 0\ \ pnormal p" + apply (induct p, auto) + apply (case_tac "p = []", auto) + apply (simp add: pnormal_def) + by (rule pnormal_cons, auto) +lemma pnormal_id: "pnormal p \ (0 < length p \ last p \ 0)" + using pnormal_last_length pnormal_length pnormal_last_nonzero by blast + +text{*Tidier versions of finiteness of roots.*} + +lemma poly_roots_finite_set: "poly p \ poly [] ==> finite {x::'a::{idom,ring_char_0}. poly p x = 0}" +unfolding poly_roots_finite . + +text{*bound for polynomial.*} + +lemma poly_mono: "abs(x) \ k ==> abs(poly p (x::'a::{ordered_idom})) \ poly (map abs p) k" +apply (induct "p", auto) +apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans) +apply (rule abs_triangle_ineq) +apply (auto intro!: mult_mono simp add: abs_mult) +done + +lemma poly_Sing: "poly [c] x = c" by simp +end