author | berghofe |
Mon, 30 Sep 2002 16:14:02 +0200 | |
changeset 13601 | fd3e3d6b37b2 |
parent 12481 | ea5d6da573c5 |
child 14266 | 08b34c902618 |
permissions | -rw-r--r-- |
12224 | 1 |
(* Title: Poly.ML |
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Author: Jacques D. Fleuriot |
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Copyright: 2000 University of Edinburgh |
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Description: Properties of real polynomials following |
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John Harrison's HOL-Light implementation. |
|
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Some early theorems by Lucas Dixon |
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*) |
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||
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Goal "p +++ [] = p"; |
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by (induct_tac "p" 1); |
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by Auto_tac; |
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qed "padd_Nil2"; |
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Addsimps [padd_Nil2]; |
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||
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Goal "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"; |
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by Auto_tac; |
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qed "padd_Cons_Cons"; |
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||
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Goal "-- [] = []"; |
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by (rewtac poly_minus_def); |
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by (Auto_tac); |
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qed "pminus_Nil"; |
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Addsimps [pminus_Nil]; |
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||
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Goal "[h1] *** p1 = h1 %* p1"; |
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by (Simp_tac 1); |
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qed "pmult_singleton"; |
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||
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Goal "1 %* t = t"; |
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by (induct_tac "t" 1); |
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by Auto_tac; |
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qed "poly_ident_mult"; |
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Addsimps [poly_ident_mult]; |
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||
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Goal "[a] +++ ((0)#t) = (a#t)"; |
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by (Simp_tac 1); |
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qed "poly_simple_add_Cons"; |
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Addsimps [poly_simple_add_Cons]; |
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||
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(*-------------------------------------------------------------------------*) |
|
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(* Handy general properties *) |
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(*-------------------------------------------------------------------------*) |
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||
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Goal "b +++ a = a +++ b"; |
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by (subgoal_tac "ALL a. b +++ a = a +++ b" 1); |
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by (induct_tac "b" 2); |
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by Auto_tac; |
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by (rtac (padd_Cons RS ssubst) 1); |
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by (case_tac "aa" 1); |
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by Auto_tac; |
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qed "padd_commut"; |
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||
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Goal "(a +++ b) +++ c = a +++ (b +++ c)"; |
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by (subgoal_tac "ALL b c. (a +++ b) +++ c = a +++ (b +++ c)" 1); |
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by (Asm_simp_tac 1); |
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by (induct_tac "a" 1); |
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by (Step_tac 2); |
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by (case_tac "b" 2); |
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by (Asm_simp_tac 2); |
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by (Asm_simp_tac 2); |
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by (Asm_simp_tac 1); |
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qed "padd_assoc"; |
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||
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Goal "a %* ( p +++ q ) = (a %* p +++ a %* q)"; |
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by (subgoal_tac "ALL q. a %* ( p +++ q ) = (a %* p +++ a %* q) " 1); |
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by (induct_tac "p" 2); |
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by (Simp_tac 2); |
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by (rtac allI 2 ); |
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by (case_tac "q" 2); |
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by (Asm_simp_tac 2); |
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by (asm_simp_tac (simpset() addsimps [real_add_mult_distrib2] ) 2); |
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by (Asm_simp_tac 1); |
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qed "poly_cmult_distr"; |
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||
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Goal "[0, 1] *** t = ((0)#t)"; |
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by (induct_tac "t" 1); |
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by (Simp_tac 1); |
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by (simp_tac (simpset() addsimps [poly_ident_mult, padd_commut]) 1); |
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by (case_tac "list" 1); |
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by (Asm_simp_tac 1); |
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by (asm_full_simp_tac (simpset() addsimps [poly_ident_mult, padd_commut]) 1); |
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qed "pmult_by_x"; |
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Addsimps [pmult_by_x]; |
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||
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||
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(*-------------------------------------------------------------------------*) |
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(* properties of evaluation of polynomials. *) |
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(*-------------------------------------------------------------------------*) |
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||
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Goal "poly (p1 +++ p2) x = poly p1 x + poly p2 x"; |
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by (subgoal_tac "ALL p2. poly (p1 +++ p2) x = poly( p1 ) x + poly( p2 ) x" 1); |
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by (induct_tac "p1" 2); |
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by Auto_tac; |
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by (case_tac "p2" 1); |
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by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib2])); |
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qed "poly_add"; |
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||
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Goal "poly (c %* p) x = c * poly p x"; |
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by (induct_tac "p" 1); |
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by (case_tac "x=0" 2); |
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by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib2] |
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@ real_mult_ac)); |
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qed "poly_cmult"; |
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||
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Goalw [poly_minus_def] "poly (-- p) x = - (poly p x)"; |
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by (auto_tac (claset(),simpset() addsimps [poly_cmult])); |
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qed "poly_minus"; |
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||
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Goal "poly (p1 *** p2) x = poly p1 x * poly p2 x"; |
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by (subgoal_tac "ALL p2. poly (p1 *** p2) x = poly p1 x * poly p2 x" 1); |
|
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by (Asm_simp_tac 1); |
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by (induct_tac "p1" 1); |
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by (auto_tac (claset(),simpset() addsimps [poly_cmult])); |
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by (case_tac "list" 1); |
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by (auto_tac (claset(),simpset() addsimps [poly_cmult,poly_add, |
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real_add_mult_distrib,real_add_mult_distrib2] @ real_mult_ac)); |
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qed "poly_mult"; |
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||
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Goal "poly (p %^ n) x = (poly p x) ^ n"; |
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by (induct_tac "n" 1); |
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by (auto_tac (claset(),simpset() addsimps [poly_cmult, poly_mult])); |
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qed "poly_exp"; |
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||
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(*-------------------------------------------------------------------------*) |
|
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(* More Polynomial Evaluation Lemmas *) |
|
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(*-------------------------------------------------------------------------*) |
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||
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Goal "poly (a +++ []) x = poly a x"; |
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by (Simp_tac 1); |
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qed "poly_add_rzero"; |
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Addsimps [poly_add_rzero]; |
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||
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Goal "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"; |
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by (simp_tac (simpset() addsimps [poly_mult,real_mult_assoc]) 1); |
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qed "poly_mult_assoc"; |
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||
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Goal "poly (p *** []) x = 0"; |
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by (induct_tac "p" 1); |
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by Auto_tac; |
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qed "poly_mult_Nil2"; |
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Addsimps [poly_mult_Nil2]; |
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||
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Goal "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d ) x"; |
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by (induct_tac "n" 1); |
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by (auto_tac (claset(), simpset() addsimps [poly_mult,real_mult_assoc])); |
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qed "poly_exp_add"; |
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||
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(*-------------------------------------------------------------------------*) |
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(* The derivative *) |
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(*-------------------------------------------------------------------------*) |
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||
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Goalw [pderiv_def] "pderiv [] = []"; |
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by (Simp_tac 1); |
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qed "pderiv_Nil"; |
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Addsimps [pderiv_Nil]; |
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||
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Goalw [pderiv_def] "pderiv [c] = []"; |
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by (Simp_tac 1); |
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qed "pderiv_singleton"; |
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Addsimps [pderiv_singleton]; |
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||
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Goalw [pderiv_def] "pderiv (h#t) = pderiv_aux 1 t"; |
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by (Simp_tac 1); |
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qed "pderiv_Cons"; |
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||
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Goal "DERIV f x :> D ==> DERIV (%x. (f x) * c) x :> D * c"; |
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by (auto_tac (claset() addIs [DERIV_cmult,real_mult_commute RS subst], |
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simpset() addsimps [real_mult_commute])); |
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qed "DERIV_cmult2"; |
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Goal "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"; |
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by (rtac lemma_DERIV_subst 1 THEN rtac DERIV_pow 1); |
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by (Simp_tac 1); |
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qed "DERIV_pow2"; |
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Addsimps [DERIV_pow2,DERIV_pow]; |
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Goal "ALL n. DERIV (%x. (x ^ (Suc n) * poly p x)) x :> \ |
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\ x ^ n * poly (pderiv_aux (Suc n) p) x "; |
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by (induct_tac "p" 1); |
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by (auto_tac (claset() addSIs [DERIV_add,DERIV_cmult2],simpset() addsimps |
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[pderiv_def,real_add_mult_distrib2,real_mult_assoc RS sym] delsimps |
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[realpow_Suc])); |
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by (rtac (real_mult_commute RS subst) 1); |
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by (simp_tac (simpset() delsimps [realpow_Suc]) 1); |
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by (asm_full_simp_tac (simpset() addsimps [real_mult_commute,realpow_Suc RS sym] |
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delsimps [realpow_Suc]) 1); |
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qed "lemma_DERIV_poly1"; |
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Goal "DERIV (%x. (x ^ (Suc n) * poly p x)) x :> \ |
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\ x ^ n * poly (pderiv_aux (Suc n) p) x "; |
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by (simp_tac (simpset() addsimps [lemma_DERIV_poly1] delsimps [realpow_Suc]) 1); |
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qed "lemma_DERIV_poly"; |
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Goal "DERIV f x :> D ==> DERIV (%x. a + f x) x :> D"; |
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by (rtac lemma_DERIV_subst 1 THEN rtac DERIV_add 1); |
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by Auto_tac; |
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qed "DERIV_add_const"; |
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Goal "DERIV (%x. poly p x) x :> poly (pderiv p) x"; |
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by (induct_tac "p" 1); |
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by (auto_tac (claset(),simpset() addsimps [pderiv_Cons])); |
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by (rtac DERIV_add_const 1); |
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by (rtac lemma_DERIV_subst 1); |
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by (rtac (full_simplify (simpset()) |
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(read_instantiate [("n","0")] lemma_DERIV_poly)) 1); |
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by (simp_tac (simpset() addsimps [CLAIM "1 = 1"]) 1); |
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qed "poly_DERIV"; |
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Addsimps [poly_DERIV]; |
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||
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||
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(*-------------------------------------------------------------------------*) |
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(* Consequences of the derivative theorem above *) |
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(*-------------------------------------------------------------------------*) |
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||
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Goalw [differentiable_def] "(%x. poly p x) differentiable x"; |
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by (blast_tac (claset() addIs [poly_DERIV]) 1); |
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qed "poly_differentiable"; |
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Addsimps [poly_differentiable]; |
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||
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Goal "isCont (%x. poly p x) x"; |
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by (rtac (poly_DERIV RS DERIV_isCont) 1); |
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qed "poly_isCont"; |
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Addsimps [poly_isCont]; |
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||
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Goal "[| a < b; poly p a < 0; 0 < poly p b |] \ |
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\ ==> EX x. a < x & x < b & (poly p x = 0)"; |
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by (cut_inst_tac [("f","%x. poly p x"),("a","a"),("b","b"),("y","0")] |
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IVT_objl 1); |
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by (auto_tac (claset(),simpset() addsimps [real_le_less])); |
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qed "poly_IVT_pos"; |
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Goal "[| a < b; 0 < poly p a; poly p b < 0 |] \ |
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\ ==> EX x. a < x & x < b & (poly p x = 0)"; |
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by (blast_tac (claset() addIs [full_simplify (simpset() |
|
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addsimps [poly_minus, rename_numerals real_minus_zero_less_iff2]) |
|
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(read_instantiate [("p","-- p")] poly_IVT_pos)]) 1); |
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qed "poly_IVT_neg"; |
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238 |
||
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Goal "a < b ==> \ |
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\ EX x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"; |
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by (dres_inst_tac [("f","poly p")] MVT 1); |
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by Auto_tac; |
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by (res_inst_tac [("x","z")] exI 1); |
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by (auto_tac (claset() addDs [ARITH_PROVE |
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"[| a < z; z < b |] ==> (b - (a::real)) ~= 0"],simpset() |
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addsimps [real_mult_left_cancel,poly_DERIV RS DERIV_unique])); |
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qed "poly_MVT"; |
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248 |
||
249 |
||
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(*-------------------------------------------------------------------------*) |
|
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(* Lemmas for Derivatives *) |
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(*-------------------------------------------------------------------------*) |
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253 |
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Goal "ALL p2 n. poly (pderiv_aux n (p1 +++ p2)) x = \ |
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\ poly (pderiv_aux n p1 +++ pderiv_aux n p2) x"; |
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by (induct_tac "p1" 1); |
|
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by (Step_tac 2); |
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by (case_tac "p2" 2); |
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by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib2])); |
|
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qed "lemma_poly_pderiv_aux_add"; |
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261 |
||
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Goal "poly (pderiv_aux n (p1 +++ p2)) x = \ |
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\ poly (pderiv_aux n p1 +++ pderiv_aux n p2) x"; |
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by (simp_tac (simpset() addsimps [lemma_poly_pderiv_aux_add]) 1); |
|
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qed "poly_pderiv_aux_add"; |
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266 |
||
267 |
Goal "ALL n. poly (pderiv_aux n (c %* p) ) x = poly (c %* pderiv_aux n p) x"; |
|
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by (induct_tac "p" 1); |
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by (auto_tac (claset(),simpset() addsimps [poly_cmult] @ real_mult_ac)); |
|
270 |
qed "lemma_poly_pderiv_aux_cmult"; |
|
271 |
||
272 |
Goal "poly (pderiv_aux n (c %* p) ) x = poly (c %* pderiv_aux n p) x"; |
|
273 |
by (simp_tac (simpset() addsimps [lemma_poly_pderiv_aux_cmult]) 1); |
|
274 |
qed "poly_pderiv_aux_cmult"; |
|
275 |
||
276 |
Goalw [poly_minus_def] |
|
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"poly (pderiv_aux n (-- p)) x = poly (-- pderiv_aux n p) x"; |
|
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by (simp_tac (simpset() addsimps [poly_pderiv_aux_cmult]) 1); |
|
279 |
qed "poly_pderiv_aux_minus"; |
|
280 |
||
281 |
Goal "ALL n. poly (pderiv_aux (Suc n) p) x = poly ((pderiv_aux n p) +++ p) x"; |
|
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by (induct_tac "p" 1); |
|
283 |
by (auto_tac (claset(),simpset() addsimps [real_of_nat_Suc, |
|
284 |
real_add_mult_distrib])); |
|
285 |
qed "lemma_poly_pderiv_aux_mult1"; |
|
286 |
||
287 |
Goal "poly (pderiv_aux (Suc n) p) x = poly ((pderiv_aux n p) +++ p) x"; |
|
288 |
by (simp_tac (simpset() addsimps [lemma_poly_pderiv_aux_mult1]) 1); |
|
289 |
qed "lemma_poly_pderiv_aux_mult"; |
|
290 |
||
291 |
Goal "ALL q. poly (pderiv (p +++ q)) x = poly (pderiv p +++ pderiv q) x"; |
|
292 |
by (induct_tac "p" 1); |
|
293 |
by (Step_tac 2); |
|
294 |
by (case_tac "q" 2); |
|
295 |
by (auto_tac (claset(),simpset() addsimps [poly_pderiv_aux_add,poly_add, |
|
296 |
pderiv_def])); |
|
297 |
qed "lemma_poly_pderiv_add"; |
|
298 |
||
299 |
Goal "poly (pderiv (p +++ q)) x = poly (pderiv p +++ pderiv q) x"; |
|
300 |
by (simp_tac (simpset() addsimps [lemma_poly_pderiv_add]) 1); |
|
301 |
qed "poly_pderiv_add"; |
|
302 |
||
303 |
Goal "poly (pderiv (c %* p)) x = poly (c %* (pderiv p)) x"; |
|
304 |
by (induct_tac "p" 1); |
|
305 |
by (auto_tac (claset(),simpset() addsimps [poly_pderiv_aux_cmult,poly_cmult, |
|
306 |
pderiv_def])); |
|
307 |
qed "poly_pderiv_cmult"; |
|
308 |
||
309 |
Goalw [poly_minus_def] "poly (pderiv (--p)) x = poly (--(pderiv p)) x"; |
|
310 |
by (simp_tac (simpset() addsimps [poly_pderiv_cmult]) 1); |
|
311 |
qed "poly_pderiv_minus"; |
|
312 |
||
313 |
Goalw [pderiv_def] |
|
314 |
"poly (pderiv (h#t)) x = poly ((0 # (pderiv t)) +++ t) x"; |
|
315 |
by (induct_tac "t" 1); |
|
316 |
by (auto_tac (claset(),simpset() addsimps [poly_add, |
|
317 |
lemma_poly_pderiv_aux_mult])); |
|
318 |
qed "lemma_poly_mult_pderiv"; |
|
319 |
||
320 |
Goal "ALL q. poly (pderiv (p *** q)) x = \ |
|
321 |
\ poly (p *** (pderiv q) +++ q *** (pderiv p)) x"; |
|
322 |
by (induct_tac "p" 1); |
|
323 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_cmult, |
|
324 |
poly_pderiv_cmult,poly_pderiv_add,poly_mult])); |
|
325 |
by (rtac (lemma_poly_mult_pderiv RS ssubst) 1); |
|
326 |
by (rtac (lemma_poly_mult_pderiv RS ssubst) 1); |
|
327 |
by (rtac (poly_add RS ssubst) 1); |
|
328 |
by (rtac (poly_add RS ssubst) 1); |
|
329 |
by (asm_simp_tac (simpset() addsimps [poly_mult,real_add_mult_distrib2] |
|
330 |
@ real_add_ac @ real_mult_ac) 1); |
|
331 |
qed "poly_pderiv_mult"; |
|
332 |
||
333 |
Goal "poly (pderiv (p %^ (Suc n))) x = \ |
|
334 |
\ poly ((real (Suc n)) %* (p %^ n) *** pderiv p ) x"; |
|
335 |
by (induct_tac "n" 1); |
|
336 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_pderiv_cmult, |
|
337 |
poly_cmult,poly_pderiv_mult,real_of_nat_zero,poly_mult, |
|
338 |
real_of_nat_Suc,real_add_mult_distrib2,real_add_mult_distrib] |
|
339 |
@ real_mult_ac)); |
|
340 |
qed "poly_pderiv_exp"; |
|
341 |
||
342 |
Goal "poly (pderiv ([-a, 1] %^ (Suc n))) x = \ |
|
343 |
\ poly (real (Suc n) %* ([-a, 1] %^ n)) x"; |
|
344 |
by (simp_tac (simpset() addsimps [poly_pderiv_exp,poly_mult] |
|
345 |
delsimps [pexp_Suc]) 1); |
|
346 |
by (simp_tac (simpset() addsimps [poly_cmult,pderiv_def]) 1); |
|
347 |
qed "poly_pderiv_exp_prime"; |
|
348 |
||
349 |
(* ----------------------------------------------------------------------- *) |
|
350 |
(* Key property that f(a) = 0 ==> (x - a) divides p(x). *) |
|
351 |
(* ----------------------------------------------------------------------- *) |
|
352 |
||
353 |
Goal "ALL h. EX q r. h#t = [r] +++ [-a, 1] *** q"; |
|
354 |
by (induct_tac "t" 1); |
|
355 |
by (Step_tac 1); |
|
356 |
by (res_inst_tac [("x","[]")] exI 1); |
|
357 |
by (res_inst_tac [("x","h")] exI 1); |
|
358 |
by (Simp_tac 1); |
|
359 |
by (dres_inst_tac [("x","aa")] spec 1); |
|
360 |
by (Step_tac 1); |
|
361 |
by (res_inst_tac [("x","r#q")] exI 1); |
|
362 |
by (res_inst_tac [("x","a*r + h")] exI 1); |
|
363 |
by (case_tac "q" 1); |
|
12481
ea5d6da573c5
mods due to reorienting and renaming of real_minus_mult_eq1/2
nipkow
parents:
12224
diff
changeset
|
364 |
by (Auto_tac); |
12224 | 365 |
qed "lemma_poly_linear_rem"; |
366 |
||
367 |
Goal "EX q r. h#t = [r] +++ [-a, 1] *** q"; |
|
368 |
by (cut_inst_tac [("t","t"),("a","a")] lemma_poly_linear_rem 1); |
|
369 |
by Auto_tac; |
|
370 |
qed "poly_linear_rem"; |
|
371 |
||
372 |
||
373 |
Goal "(poly p a = 0) = ((p = []) | (EX q. p = [-a, 1] *** q))"; |
|
374 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_cmult, |
|
375 |
real_add_mult_distrib2])); |
|
376 |
by (case_tac "p" 1); |
|
377 |
by (cut_inst_tac [("h","aa"),("t","list"),("a","a")] poly_linear_rem 2); |
|
378 |
by (Step_tac 2); |
|
379 |
by (case_tac "q" 1); |
|
380 |
by Auto_tac; |
|
381 |
by (dres_inst_tac [("x","[]")] spec 1); |
|
382 |
by (Asm_full_simp_tac 1); |
|
383 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_cmult, |
|
384 |
real_add_assoc])); |
|
385 |
by (dres_inst_tac [("x","aa#lista")] spec 1); |
|
386 |
by Auto_tac; |
|
387 |
qed "poly_linear_divides"; |
|
388 |
||
389 |
Goal "ALL h k a. length (k %* p +++ (h # (a %* p))) = Suc (length p)"; |
|
390 |
by (induct_tac "p" 1); |
|
391 |
by Auto_tac; |
|
392 |
qed "lemma_poly_length_mult"; |
|
393 |
Addsimps [lemma_poly_length_mult]; |
|
394 |
||
395 |
Goal "ALL h k. length (k %* p +++ (h # p)) = Suc (length p)"; |
|
396 |
by (induct_tac "p" 1); |
|
397 |
by Auto_tac; |
|
398 |
qed "lemma_poly_length_mult2"; |
|
399 |
Addsimps [lemma_poly_length_mult2]; |
|
400 |
||
401 |
Goal "length([-a ,1] *** q) = Suc (length q)"; |
|
402 |
by Auto_tac; |
|
403 |
qed "poly_length_mult"; |
|
404 |
Addsimps [poly_length_mult]; |
|
405 |
||
406 |
||
407 |
(*-------------------------------------------------------------------------*) |
|
408 |
(* Polynomial length *) |
|
409 |
(*-------------------------------------------------------------------------*) |
|
410 |
||
411 |
Goal "length (a %* p) = length p"; |
|
412 |
by (induct_tac "p" 1); |
|
413 |
by Auto_tac; |
|
414 |
qed "poly_cmult_length"; |
|
415 |
Addsimps [poly_cmult_length]; |
|
416 |
||
417 |
Goal "length (p1 +++ p2) = (if (length( p1 ) < length( p2 )) \ |
|
418 |
\ then (length( p2 )) else (length( p1) ))"; |
|
419 |
by (subgoal_tac "ALL p2. length (p1 +++ p2) = (if (length( p1 ) < \ |
|
420 |
\ length( p2 )) then (length( p2 )) else (length( p1) ))" 1); |
|
421 |
by (induct_tac "p1" 2); |
|
422 |
by (Simp_tac 2); |
|
423 |
by (Simp_tac 2); |
|
424 |
by (Step_tac 2); |
|
425 |
by (Asm_full_simp_tac 2); |
|
426 |
by (arith_tac 2); |
|
427 |
by (Asm_full_simp_tac 2); |
|
428 |
by (arith_tac 2); |
|
429 |
by (induct_tac "p2" 1); |
|
430 |
by (Asm_full_simp_tac 1); |
|
431 |
by (Asm_full_simp_tac 1); |
|
432 |
qed "poly_add_length"; |
|
433 |
||
434 |
Goal "length([a,b] *** p) = Suc (length p)"; |
|
435 |
by (asm_full_simp_tac (simpset() addsimps [poly_cmult_length, |
|
436 |
poly_add_length]) 1); |
|
437 |
qed "poly_root_mult_length"; |
|
438 |
Addsimps [poly_root_mult_length]; |
|
439 |
||
440 |
Goal "(poly (p *** q) x ~= poly [] x) = \ |
|
441 |
\ (poly p x ~= poly [] x & poly q x ~= poly [] x)"; |
|
442 |
by (auto_tac (claset(),simpset() addsimps [poly_mult,rename_numerals |
|
443 |
real_mult_not_zero])); |
|
444 |
qed "poly_mult_not_eq_poly_Nil"; |
|
445 |
Addsimps [poly_mult_not_eq_poly_Nil]; |
|
446 |
||
447 |
Goal "(poly (p *** q) x = 0) = (poly p x = 0 | poly q x = 0)"; |
|
448 |
by (auto_tac (claset() addDs [CLAIM "x * y = 0 ==> x = 0 | y = (0::real)"], |
|
449 |
simpset() addsimps [poly_mult])); |
|
450 |
qed "poly_mult_eq_zero_disj"; |
|
451 |
||
452 |
(*-------------------------------------------------------------------------*) |
|
453 |
(* Normalisation Properties *) |
|
454 |
(*-------------------------------------------------------------------------*) |
|
455 |
||
456 |
Goal "(pnormalize p = []) --> (poly p x = 0)"; |
|
457 |
by (induct_tac "p" 1); |
|
458 |
by Auto_tac; |
|
459 |
qed "poly_normalized_nil"; |
|
460 |
||
461 |
(*-------------------------------------------------------------------------*) |
|
462 |
(* A nontrivial polynomial of degree n has no more than n roots *) |
|
463 |
(*-------------------------------------------------------------------------*) |
|
464 |
||
465 |
Goal |
|
466 |
"ALL p x. (poly p x ~= poly [] x & length p = n \ |
|
467 |
\ --> (EX i. ALL x. (poly p x = (0::real)) --> (EX m. (m <= n & x = i m))))"; |
|
468 |
by (induct_tac "n" 1); |
|
469 |
by (Step_tac 1); |
|
470 |
by (rtac ccontr 1); |
|
471 |
by (subgoal_tac "EX a. poly p a = 0" 1 THEN Step_tac 1); |
|
472 |
by (dtac (poly_linear_divides RS iffD1) 1); |
|
473 |
by (Step_tac 1); |
|
474 |
by (dres_inst_tac [("x","q")] spec 1); |
|
475 |
by (dres_inst_tac [("x","x")] spec 1); |
|
476 |
by (asm_full_simp_tac (simpset() delsimps [poly_Nil,pmult_Cons]) 1); |
|
477 |
by (etac exE 1); |
|
478 |
by (dres_inst_tac [("x","%m. if m = Suc n then a else i m")] spec 1); |
|
479 |
by (Step_tac 1); |
|
480 |
by (dtac (poly_mult_eq_zero_disj RS iffD1) 1); |
|
481 |
by (Step_tac 1); |
|
482 |
by (dres_inst_tac [("x","Suc(length q)")] spec 1); |
|
483 |
by (Asm_full_simp_tac 1); |
|
484 |
by (dres_inst_tac [("x","xa")] spec 1 THEN Step_tac 1); |
|
485 |
by (dres_inst_tac [("x","m")] spec 1); |
|
486 |
by (Asm_full_simp_tac 1); |
|
487 |
by (Blast_tac 1); |
|
488 |
qed_spec_mp "poly_roots_index_lemma"; |
|
489 |
bind_thm ("poly_roots_index_lemma2",conjI RS poly_roots_index_lemma); |
|
490 |
||
491 |
Goal "poly p x ~= poly [] x ==> \ |
|
492 |
\ EX i. ALL x. (poly p x = 0) --> (EX n. n <= length p & x = i n)"; |
|
493 |
by (blast_tac (claset() addIs [poly_roots_index_lemma2]) 1); |
|
494 |
qed "poly_roots_index_length"; |
|
495 |
||
496 |
Goal "poly p x ~= poly [] x ==> \ |
|
497 |
\ EX N i. ALL x. (poly p x = 0) --> (EX n. (n::nat) < N & x = i n)"; |
|
498 |
by (dtac poly_roots_index_length 1 THEN Step_tac 1); |
|
499 |
by (res_inst_tac [("x","Suc (length p)")] exI 1); |
|
500 |
by (res_inst_tac [("x","i")] exI 1); |
|
501 |
by (auto_tac (claset(),simpset() addsimps |
|
502 |
[ARITH_PROVE "(m < Suc n) = (m <= n)"])); |
|
503 |
qed "poly_roots_finite_lemma"; |
|
504 |
||
505 |
(* annoying proof *) |
|
506 |
Goal "ALL P. (ALL x. P x --> (EX n. (n::nat) < N & x = (j::nat=>real) n)) \ |
|
507 |
\ --> (EX a. ALL x. P x --> x < a)"; |
|
508 |
by (induct_tac "N" 1); |
|
509 |
by (Asm_full_simp_tac 1); |
|
510 |
by (Step_tac 1); |
|
511 |
by (dres_inst_tac [("x","%z. P z & (z ~= (j::nat=>real) n)")] spec 1); |
|
512 |
by Auto_tac; |
|
513 |
by (dres_inst_tac [("x","x")] spec 1); |
|
514 |
by (Step_tac 1); |
|
515 |
by (res_inst_tac [("x","na")] exI 1); |
|
516 |
by (auto_tac (claset() addDs [ARITH_PROVE "na < Suc n ==> na = n | na < n"], |
|
517 |
simpset())); |
|
518 |
by (res_inst_tac [("x","abs a + abs(j n) + 1")] exI 1); |
|
519 |
by (Step_tac 1); |
|
520 |
by (dres_inst_tac [("x","x")] spec 1); |
|
521 |
by (Step_tac 1); |
|
522 |
by (dres_inst_tac [("x","j na")] spec 1); |
|
523 |
by (Step_tac 1); |
|
524 |
by (ALLGOALS(arith_tac)); |
|
525 |
qed_spec_mp "real_finite_lemma"; |
|
526 |
||
527 |
Goal "(poly p ~= poly []) = \ |
|
528 |
\ (EX N j. ALL x. poly p x = 0 --> (EX n. (n::nat) < N & x = j n))"; |
|
529 |
by (Step_tac 1); |
|
530 |
by (etac swap 1 THEN rtac ext 1); |
|
531 |
by (rtac ccontr 1); |
|
532 |
by (clarify_tac (claset() addSDs [poly_roots_finite_lemma]) 1); |
|
533 |
by (clarify_tac (claset() addSDs [real_finite_lemma]) 1); |
|
534 |
by (dres_inst_tac [("x","a")] fun_cong 1); |
|
535 |
by Auto_tac; |
|
536 |
qed "poly_roots_finite"; |
|
537 |
||
538 |
(*-------------------------------------------------------------------------*) |
|
539 |
(* Entirety and Cancellation for polynomials *) |
|
540 |
(*-------------------------------------------------------------------------*) |
|
541 |
||
542 |
Goal "[| poly p ~= poly [] ; poly q ~= poly [] |] \ |
|
543 |
\ ==> poly (p *** q) ~= poly []"; |
|
544 |
by (auto_tac (claset(),simpset() addsimps [poly_roots_finite])); |
|
545 |
by (res_inst_tac [("x","N + Na")] exI 1); |
|
546 |
by (res_inst_tac [("x","%n. if n < N then j n else ja (n - N)")] exI 1); |
|
547 |
by (auto_tac (claset(),simpset() addsimps [poly_mult_eq_zero_disj])); |
|
548 |
by (flexflex_tac THEN rotate_tac 1 1); |
|
549 |
by (dtac spec 1 THEN Auto_tac); |
|
550 |
qed "poly_entire_lemma"; |
|
551 |
||
552 |
Goal "(poly (p *** q) = poly []) = ((poly p = poly []) | (poly q = poly []))"; |
|
553 |
by (auto_tac (claset() addIs [ext] addDs [fun_cong],simpset() |
|
554 |
addsimps [poly_entire_lemma,poly_mult])); |
|
555 |
by (blast_tac (claset() addIs [ccontr] addDs [poly_entire_lemma, |
|
556 |
poly_mult RS subst]) 1); |
|
557 |
qed "poly_entire"; |
|
558 |
||
559 |
Goal "(poly (p *** q) ~= poly []) = ((poly p ~= poly []) & (poly q ~= poly []))"; |
|
560 |
by (asm_full_simp_tac (simpset() addsimps [poly_entire]) 1); |
|
561 |
qed "poly_entire_neg"; |
|
562 |
||
563 |
Goal " (f = g) = (ALL x. f x = g x)"; |
|
564 |
by (auto_tac (claset() addSIs [ext],simpset())); |
|
565 |
qed "fun_eq"; |
|
566 |
||
567 |
Goal "(poly (p +++ -- q) = poly []) = (poly p = poly q)"; |
|
568 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_minus_def, |
|
569 |
fun_eq,poly_cmult,ARITH_PROVE "(p + -q = 0) = (p = (q::real))"])); |
|
570 |
qed "poly_add_minus_zero_iff"; |
|
571 |
||
572 |
Goal "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"; |
|
573 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_minus_def, |
|
574 |
fun_eq,poly_mult,poly_cmult,real_add_mult_distrib2])); |
|
575 |
qed "poly_add_minus_mult_eq"; |
|
576 |
||
577 |
Goal "(poly (p *** q) = poly (p *** r)) = (poly p = poly [] | poly q = poly r)"; |
|
578 |
by (res_inst_tac [("p1","p *** q")] (poly_add_minus_zero_iff RS subst) 1); |
|
579 |
by (auto_tac (claset() addIs [ext], simpset() addsimps [poly_add_minus_mult_eq, |
|
580 |
poly_entire,poly_add_minus_zero_iff])); |
|
581 |
qed "poly_mult_left_cancel"; |
|
582 |
||
583 |
Goal "(x * y = 0) = (x = (0::real) | y = 0)"; |
|
584 |
by (auto_tac (claset() addDs [CLAIM "x * y = 0 ==> x = 0 | y = (0::real)"], |
|
585 |
simpset())); |
|
586 |
qed "real_mult_zero_disj_iff"; |
|
587 |
||
588 |
Goal "(poly (p %^ n) = poly []) = (poly p = poly [] & n ~= 0)"; |
|
589 |
by (simp_tac (simpset() addsimps [fun_eq]) 1); |
|
590 |
by (rtac (CLAIM "((ALL x. P x) & Q) = (ALL x. P x & Q)" RS ssubst) 1); |
|
591 |
by (rtac (CLAIM "f = g ==> (ALL x. f x) = (ALL x. g x)") 1); |
|
592 |
by (rtac ext 1); |
|
593 |
by (induct_tac "n" 1); |
|
594 |
by (auto_tac (claset(),simpset() addsimps [poly_mult, |
|
595 |
real_mult_zero_disj_iff])); |
|
596 |
qed "poly_exp_eq_zero"; |
|
597 |
Addsimps [poly_exp_eq_zero]; |
|
598 |
||
599 |
Goal "poly [a,1] ~= poly []"; |
|
600 |
by (simp_tac (simpset() addsimps [fun_eq]) 1); |
|
601 |
by (res_inst_tac [("x","1 - a")] exI 1); |
|
602 |
by (Simp_tac 1); |
|
603 |
qed "poly_prime_eq_zero"; |
|
604 |
Addsimps [poly_prime_eq_zero]; |
|
605 |
||
606 |
Goal "(poly ([a, 1] %^ n) ~= poly [])"; |
|
607 |
by Auto_tac; |
|
608 |
qed "poly_exp_prime_eq_zero"; |
|
609 |
Addsimps [poly_exp_prime_eq_zero]; |
|
610 |
||
611 |
(*-------------------------------------------------------------------------*) |
|
612 |
(* A more constructive notion of polynomials being trivial *) |
|
613 |
(*-------------------------------------------------------------------------*) |
|
614 |
||
615 |
Goal "poly (h # t) = poly [] ==> h = 0 & poly t = poly []"; |
|
616 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq]) 1); |
|
617 |
by (case_tac "h = 0" 1); |
|
618 |
by (dres_inst_tac [("x","0")] spec 2); |
|
619 |
by (rtac conjI 1); |
|
620 |
by (rtac ((simplify (simpset()) (read_instantiate [("g","poly []")] fun_eq)) |
|
621 |
RS iffD1) 2 THEN rtac ccontr 2); |
|
622 |
by (auto_tac (claset(),simpset() addsimps [poly_roots_finite, |
|
623 |
real_mult_zero_disj_iff])); |
|
624 |
by (dtac real_finite_lemma 1 THEN Step_tac 1); |
|
625 |
by (REPEAT(dres_inst_tac [("x","abs a + 1")] spec 1)); |
|
626 |
by (arith_tac 1); |
|
627 |
qed "poly_zero_lemma"; |
|
628 |
||
629 |
Goal "(poly p = poly []) = list_all (%c. c = 0) p"; |
|
630 |
by (induct_tac "p" 1); |
|
631 |
by (Asm_full_simp_tac 1); |
|
632 |
by (rtac iffI 1); |
|
633 |
by (dtac poly_zero_lemma 1); |
|
634 |
by Auto_tac; |
|
635 |
qed "poly_zero"; |
|
636 |
||
637 |
Addsimps [real_mult_zero_disj_iff]; |
|
638 |
Goal "ALL n. (list_all (%c. c = 0) (pderiv_aux (Suc n) p) = \ |
|
639 |
\ list_all (%c. c = 0) p)"; |
|
640 |
by (induct_tac "p" 1); |
|
641 |
by Auto_tac; |
|
642 |
qed_spec_mp "pderiv_aux_iszero"; |
|
643 |
Addsimps [pderiv_aux_iszero]; |
|
644 |
||
645 |
Goal "(number_of n :: nat) ~= 0 \ |
|
646 |
\ ==> (list_all (%c. c = 0) (pderiv_aux (number_of n) p) = \ |
|
647 |
\ list_all (%c. c = 0) p)"; |
|
648 |
by (res_inst_tac [("n1","number_of n"),("m1","0")] (less_imp_Suc_add RS exE) 1); |
|
649 |
by (Force_tac 1); |
|
650 |
by (res_inst_tac [("n1","0 + x")] (pderiv_aux_iszero RS subst) 1); |
|
651 |
by (asm_simp_tac (simpset() delsimps [pderiv_aux_iszero]) 1); |
|
652 |
qed "pderiv_aux_iszero_num"; |
|
653 |
||
654 |
Goal "poly (pderiv p) = poly [] --> (EX h. poly p = poly [h])"; |
|
655 |
by (asm_full_simp_tac (simpset() addsimps [poly_zero]) 1); |
|
656 |
by (induct_tac "p" 1); |
|
657 |
by (Force_tac 1); |
|
658 |
by (asm_full_simp_tac (simpset() addsimps [pderiv_Cons, |
|
659 |
pderiv_aux_iszero_num] delsimps [poly_Cons]) 1); |
|
660 |
by (auto_tac (claset(),simpset() addsimps [poly_zero RS sym])); |
|
661 |
qed_spec_mp "pderiv_iszero"; |
|
662 |
||
663 |
Goal "poly p = poly [] --> (poly (pderiv p) = poly [])"; |
|
664 |
by (asm_full_simp_tac (simpset() addsimps [poly_zero]) 1); |
|
665 |
by (induct_tac "p" 1); |
|
666 |
by (Force_tac 1); |
|
667 |
by (asm_full_simp_tac (simpset() addsimps [pderiv_Cons, |
|
668 |
pderiv_aux_iszero_num] delsimps [poly_Cons]) 1); |
|
669 |
qed "pderiv_zero_obj"; |
|
670 |
||
671 |
Goal "poly p = poly [] ==> (poly (pderiv p) = poly [])"; |
|
672 |
by (blast_tac (claset() addEs [pderiv_zero_obj RS impE]) 1); |
|
673 |
qed "pderiv_zero"; |
|
674 |
Addsimps [pderiv_zero]; |
|
675 |
||
676 |
Goal "poly p = poly q ==> (poly (pderiv p) = poly (pderiv q))"; |
|
677 |
by (cut_inst_tac [("p","p +++ --q")] pderiv_zero_obj 1); |
|
678 |
by (auto_tac (claset() addIs [ ARITH_PROVE "x + - y = 0 ==> x = (y::real)"], |
|
679 |
simpset() addsimps [fun_eq,poly_add,poly_minus,poly_pderiv_add, |
|
680 |
poly_pderiv_minus] delsimps [pderiv_zero])); |
|
681 |
qed "poly_pderiv_welldef"; |
|
682 |
||
683 |
(* ------------------------------------------------------------------------- *) |
|
684 |
(* Basics of divisibility. *) |
|
685 |
(* ------------------------------------------------------------------------- *) |
|
686 |
||
687 |
Goal "([a, 1] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)"; |
|
688 |
by (auto_tac (claset(),simpset() addsimps [divides_def,fun_eq,poly_mult, |
|
689 |
poly_add,poly_cmult,real_add_mult_distrib RS sym])); |
|
690 |
by (dres_inst_tac [("x","-a")] spec 1); |
|
691 |
by (auto_tac (claset(),simpset() addsimps [poly_linear_divides,poly_add, |
|
692 |
poly_cmult,real_add_mult_distrib RS sym])); |
|
693 |
by (res_inst_tac [("x","qa *** q")] exI 1); |
|
694 |
by (res_inst_tac [("x","p *** qa")] exI 2); |
|
695 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_mult, |
|
696 |
poly_cmult] @ real_mult_ac)); |
|
697 |
qed "poly_primes"; |
|
698 |
||
699 |
Goalw [divides_def] "p divides p"; |
|
700 |
by (res_inst_tac [("x","[1]")] exI 1); |
|
701 |
by (auto_tac (claset(),simpset() addsimps [poly_mult,fun_eq])); |
|
702 |
qed "poly_divides_refl"; |
|
703 |
Addsimps [poly_divides_refl]; |
|
704 |
||
705 |
Goalw [divides_def] "[| p divides q; q divides r |] ==> p divides r"; |
|
706 |
by (Step_tac 1); |
|
707 |
by (res_inst_tac [("x","qa *** qaa")] exI 1); |
|
708 |
by (auto_tac (claset(),simpset() addsimps [poly_mult,fun_eq, |
|
709 |
real_mult_assoc])); |
|
710 |
qed "poly_divides_trans"; |
|
711 |
||
712 |
Goal "(m::nat) <= n = (EX d. n = m + d)"; |
|
713 |
by (auto_tac (claset(),simpset() addsimps [le_eq_less_or_eq, |
|
714 |
less_iff_Suc_add])); |
|
715 |
qed "le_iff_add"; |
|
716 |
||
717 |
Goal "m <= n ==> (p %^ m) divides (p %^ n)"; |
|
718 |
by (auto_tac (claset(),simpset() addsimps [le_iff_add])); |
|
719 |
by (induct_tac "d" 1); |
|
720 |
by (rtac poly_divides_trans 2); |
|
721 |
by (auto_tac (claset(),simpset() addsimps [divides_def])); |
|
722 |
by (res_inst_tac [("x","p")] exI 1); |
|
723 |
by (auto_tac (claset(),simpset() addsimps [poly_mult,fun_eq] |
|
724 |
@ real_mult_ac)); |
|
725 |
qed "poly_divides_exp"; |
|
726 |
||
727 |
Goal "[| (p %^ n) divides q; m <= n |] ==> (p %^ m) divides q"; |
|
728 |
by (blast_tac (claset() addIs [poly_divides_exp,poly_divides_trans]) 1); |
|
729 |
qed "poly_exp_divides"; |
|
730 |
||
731 |
Goalw [divides_def] |
|
732 |
"[| p divides q; p divides r |] ==> p divides (q +++ r)"; |
|
733 |
by Auto_tac; |
|
734 |
by (res_inst_tac [("x","qa +++ qaa")] exI 1); |
|
735 |
by (auto_tac (claset(),simpset() addsimps [poly_add,fun_eq,poly_mult, |
|
736 |
real_add_mult_distrib2])); |
|
737 |
qed "poly_divides_add"; |
|
738 |
||
739 |
Goalw [divides_def] |
|
740 |
"[| p divides q; p divides (q +++ r) |] ==> p divides r"; |
|
741 |
by Auto_tac; |
|
742 |
by (res_inst_tac [("x","qaa +++ -- qa")] exI 1); |
|
743 |
by (auto_tac (claset(),simpset() addsimps [poly_add,fun_eq,poly_mult, |
|
12481
ea5d6da573c5
mods due to reorienting and renaming of real_minus_mult_eq1/2
nipkow
parents:
12224
diff
changeset
|
744 |
poly_minus,real_add_mult_distrib2, |
12224 | 745 |
ARITH_PROVE "(x = y + -z) = (z + x = (y::real))"])); |
746 |
qed "poly_divides_diff"; |
|
747 |
||
748 |
Goal "[| p divides r; p divides (q +++ r) |] ==> p divides q"; |
|
749 |
by (etac poly_divides_diff 1); |
|
750 |
by (auto_tac (claset(),simpset() addsimps [poly_add,fun_eq,poly_mult, |
|
751 |
divides_def] @ real_add_ac)); |
|
752 |
qed "poly_divides_diff2"; |
|
753 |
||
754 |
Goalw [divides_def] "poly p = poly [] ==> q divides p"; |
|
755 |
by (auto_tac (claset(),simpset() addsimps [fun_eq,poly_mult])); |
|
756 |
qed "poly_divides_zero"; |
|
757 |
||
758 |
Goalw [divides_def] "q divides []"; |
|
759 |
by (res_inst_tac [("x","[]")] exI 1); |
|
760 |
by (auto_tac (claset(),simpset() addsimps [fun_eq])); |
|
761 |
qed "poly_divides_zero2"; |
|
762 |
Addsimps [poly_divides_zero2]; |
|
763 |
||
764 |
(* ------------------------------------------------------------------------- *) |
|
765 |
(* At last, we can consider the order of a root. *) |
|
766 |
(* ------------------------------------------------------------------------- *) |
|
767 |
||
768 |
(* FIXME: Tidy up *) |
|
769 |
Goal "[| length p = d; poly p ~= poly [] |] \ |
|
770 |
\ ==> EX n. ([-a, 1] %^ n) divides p & \ |
|
771 |
\ ~(([-a, 1] %^ (Suc n)) divides p)"; |
|
772 |
by (subgoal_tac "ALL p. length p = d & poly p ~= poly [] \ |
|
773 |
\ --> (EX n q. p = mulexp n [-a, 1] q & poly q a ~= 0)" 1); |
|
774 |
by (induct_tac "d" 2); |
|
775 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq]) 2); |
|
776 |
by (Step_tac 2); |
|
777 |
by (case_tac "poly pa a = 0" 2); |
|
778 |
by (dtac (poly_linear_divides RS iffD1) 2); |
|
779 |
by (Step_tac 2); |
|
780 |
by (dres_inst_tac [("x","q")] spec 2); |
|
781 |
by (dtac (poly_entire_neg RS iffD1) 2); |
|
782 |
by (Step_tac 2); |
|
783 |
by (Force_tac 2 THEN Blast_tac 2); |
|
784 |
by (res_inst_tac [("x","Suc na")] exI 2); |
|
785 |
by (res_inst_tac [("x","qa")] exI 2); |
|
786 |
by (asm_full_simp_tac (simpset() delsimps [pmult_Cons]) 2); |
|
787 |
by (res_inst_tac [("x","0")] exI 2); |
|
788 |
by (Force_tac 2); |
|
789 |
by (dres_inst_tac [("x","p")] spec 1 THEN Step_tac 1); |
|
790 |
by (res_inst_tac [("x","n")] exI 1 THEN Step_tac 1); |
|
791 |
by (rewtac divides_def); |
|
792 |
by (res_inst_tac [("x","q")] exI 1); |
|
793 |
by (induct_tac "n" 1); |
|
794 |
by (Simp_tac 1); |
|
795 |
by (asm_simp_tac (simpset() addsimps [poly_add,poly_cmult,poly_mult, |
|
796 |
real_add_mult_distrib2] @ real_mult_ac) 1); |
|
797 |
by (Step_tac 1); |
|
798 |
by (rotate_tac 2 1); |
|
799 |
by (rtac swap 1 THEN assume_tac 2); |
|
800 |
by (induct_tac "n" 1); |
|
801 |
by (asm_full_simp_tac (simpset() delsimps [pmult_Cons,pexp_Suc]) 1); |
|
802 |
by (eres_inst_tac [("Pa","poly q a = 0")] swap 1); |
|
12481
ea5d6da573c5
mods due to reorienting and renaming of real_minus_mult_eq1/2
nipkow
parents:
12224
diff
changeset
|
803 |
by (asm_full_simp_tac (simpset() addsimps [poly_add,poly_cmult]) 1); |
12224 | 804 |
by (rtac (pexp_Suc RS ssubst) 1); |
805 |
by (rtac ccontr 1); |
|
806 |
by (asm_full_simp_tac (simpset() addsimps [poly_mult_left_cancel, |
|
807 |
poly_mult_assoc] delsimps [pmult_Cons,pexp_Suc]) 1); |
|
808 |
qed "poly_order_exists"; |
|
809 |
||
810 |
Goalw [divides_def] "[1] divides p"; |
|
811 |
by Auto_tac; |
|
812 |
qed "poly_one_divides"; |
|
813 |
Addsimps [poly_one_divides]; |
|
814 |
||
815 |
Goal "poly p ~= poly [] \ |
|
816 |
\ ==> EX! n. ([-a, 1] %^ n) divides p & \ |
|
817 |
\ ~(([-a, 1] %^ (Suc n)) divides p)"; |
|
818 |
by (auto_tac (claset() addIs [poly_order_exists], |
|
819 |
simpset() addsimps [less_linear] delsimps [pmult_Cons,pexp_Suc])); |
|
820 |
by (cut_inst_tac [("m","y"),("n","n")] less_linear 1); |
|
821 |
by (dres_inst_tac [("m","n")] poly_exp_divides 1); |
|
822 |
by (auto_tac (claset() addDs [ARITH_PROVE "n < m ==> Suc n <= m" RSN |
|
823 |
(2,poly_exp_divides)],simpset() delsimps [pmult_Cons,pexp_Suc])); |
|
824 |
qed "poly_order"; |
|
825 |
||
826 |
(* ------------------------------------------------------------------------- *) |
|
827 |
(* Order *) |
|
828 |
(* ------------------------------------------------------------------------- *) |
|
829 |
||
830 |
Goal "[| n = (@n. P n); EX! n. P n |] ==> P n"; |
|
831 |
by (blast_tac (claset() addIs [someI2]) 1); |
|
832 |
qed "some1_equalityD"; |
|
833 |
||
834 |
Goalw [order_def] |
|
835 |
"(([-a, 1] %^ n) divides p & \ |
|
836 |
\ ~(([-a, 1] %^ (Suc n)) divides p)) = \ |
|
837 |
\ ((n = order a p) & ~(poly p = poly []))"; |
|
838 |
by (rtac iffI 1); |
|
839 |
by (blast_tac (claset() addDs [poly_divides_zero] |
|
840 |
addSIs [some1_equality RS sym, poly_order]) 1); |
|
841 |
by (blast_tac (claset() addSIs [poly_order RSN (2,some1_equalityD)]) 1); |
|
842 |
qed "order"; |
|
843 |
||
844 |
Goal "[| poly p ~= poly [] |] \ |
|
845 |
\ ==> ([-a, 1] %^ (order a p)) divides p & \ |
|
846 |
\ ~(([-a, 1] %^ (Suc(order a p))) divides p)"; |
|
847 |
by (asm_full_simp_tac (simpset() addsimps [order] delsimps [pexp_Suc]) 1); |
|
848 |
qed "order2"; |
|
849 |
||
850 |
Goal "[| poly p ~= poly []; ([-a, 1] %^ n) divides p; \ |
|
851 |
\ ~(([-a, 1] %^ (Suc n)) divides p) \ |
|
852 |
\ |] ==> (n = order a p)"; |
|
853 |
by (cut_inst_tac [("p","p"),("a","a"),("n","n")] order 1); |
|
854 |
by Auto_tac; |
|
855 |
qed "order_unique"; |
|
856 |
||
857 |
Goal "(poly p ~= poly [] & ([-a, 1] %^ n) divides p & \ |
|
858 |
\ ~(([-a, 1] %^ (Suc n)) divides p)) \ |
|
859 |
\ ==> (n = order a p)"; |
|
860 |
by (blast_tac (claset() addIs [order_unique]) 1); |
|
861 |
qed "order_unique_lemma"; |
|
862 |
||
863 |
Goal "poly p = poly q ==> order a p = order a q"; |
|
864 |
by (auto_tac (claset(),simpset() addsimps [fun_eq,divides_def,poly_mult, |
|
865 |
order_def])); |
|
866 |
qed "order_poly"; |
|
867 |
||
868 |
Goal "p %^ (Suc 0) = p"; |
|
869 |
by (induct_tac "p" 1); |
|
870 |
by (auto_tac (claset(),simpset() addsimps [numeral_1_eq_1])); |
|
871 |
qed "pexp_one"; |
|
872 |
Addsimps [pexp_one]; |
|
873 |
||
874 |
Goal "ALL p a. 0 < n & [- a, 1] %^ n divides p & \ |
|
875 |
\ ~ [- a, 1] %^ (Suc n) divides p \ |
|
876 |
\ --> poly p a = 0"; |
|
877 |
by (induct_tac "n" 1); |
|
878 |
by (Blast_tac 1); |
|
879 |
by (auto_tac (claset(),simpset() addsimps [divides_def,poly_mult] |
|
880 |
delsimps [pmult_Cons])); |
|
881 |
qed_spec_mp "lemma_order_root"; |
|
882 |
||
883 |
Goal "(poly p a = 0) = ((poly p = poly []) | order a p ~= 0)"; |
|
884 |
by (case_tac "poly p = poly []" 1); |
|
885 |
by Auto_tac; |
|
886 |
by (asm_full_simp_tac (simpset() addsimps [poly_linear_divides] |
|
887 |
delsimps [pmult_Cons]) 1); |
|
888 |
by (Step_tac 1); |
|
889 |
by (ALLGOALS(dres_inst_tac [("a","a")] order2)); |
|
890 |
by (rtac ccontr 1); |
|
891 |
by (asm_full_simp_tac (simpset() addsimps [divides_def,poly_mult,fun_eq] |
|
892 |
delsimps [pmult_Cons]) 1); |
|
893 |
by (Blast_tac 1); |
|
894 |
by (blast_tac (claset() addIs [lemma_order_root]) 1); |
|
895 |
qed "order_root"; |
|
896 |
||
897 |
Goal "(([-a, 1] %^ n) divides p) = ((poly p = poly []) | n <= order a p)"; |
|
898 |
by (case_tac "poly p = poly []" 1); |
|
899 |
by Auto_tac; |
|
900 |
by (asm_full_simp_tac (simpset() addsimps [divides_def,fun_eq,poly_mult]) 1); |
|
901 |
by (res_inst_tac [("x","[]")] exI 1); |
|
902 |
by (TRYALL(dres_inst_tac [("a","a")] order2)); |
|
903 |
by (auto_tac (claset() addIs [poly_exp_divides],simpset() |
|
904 |
delsimps [pexp_Suc])); |
|
905 |
qed "order_divides"; |
|
906 |
||
907 |
Goalw [divides_def] |
|
908 |
"poly p ~= poly [] \ |
|
909 |
\ ==> EX q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) & \ |
|
910 |
\ ~([-a, 1] divides q)"; |
|
911 |
by (dres_inst_tac [("a","a")] order2 1); |
|
912 |
by (asm_full_simp_tac (simpset() addsimps [divides_def] |
|
913 |
delsimps [pexp_Suc,pmult_Cons]) 1); |
|
914 |
by (Step_tac 1); |
|
915 |
by (res_inst_tac [("x","q")] exI 1); |
|
916 |
by (Step_tac 1); |
|
917 |
by (dres_inst_tac [("x","qa")] spec 1); |
|
918 |
by (auto_tac (claset(),simpset() addsimps [poly_mult,fun_eq,poly_exp] |
|
919 |
@ real_mult_ac delsimps [pmult_Cons])); |
|
920 |
qed "order_decomp"; |
|
921 |
||
922 |
(* ------------------------------------------------------------------------- *) |
|
923 |
(* Important composition properties of orders. *) |
|
924 |
(* ------------------------------------------------------------------------- *) |
|
925 |
||
926 |
Goal "poly (p *** q) ~= poly [] \ |
|
927 |
\ ==> order a (p *** q) = order a p + order a q"; |
|
928 |
by (cut_inst_tac [("a","a"),("p","p***q"),("n","order a p + order a q")] |
|
929 |
order 1); |
|
930 |
by (auto_tac (claset(),simpset() addsimps [poly_entire] delsimps [pmult_Cons])); |
|
931 |
by (REPEAT(dres_inst_tac [("a","a")] order2 1)); |
|
932 |
by (Step_tac 1); |
|
933 |
by (asm_full_simp_tac (simpset() addsimps [divides_def,fun_eq,poly_exp_add, |
|
934 |
poly_mult] delsimps [pmult_Cons]) 1); |
|
935 |
by (Step_tac 1); |
|
936 |
by (res_inst_tac [("x","qa *** qaa")] exI 1); |
|
937 |
by (asm_full_simp_tac (simpset() addsimps [poly_mult] @ real_mult_ac |
|
938 |
delsimps [pmult_Cons]) 1); |
|
939 |
by (REPEAT(dres_inst_tac [("a","a")] order_decomp 1)); |
|
940 |
by (Step_tac 1); |
|
941 |
by (subgoal_tac "[-a,1] divides (qa *** qaa)" 1); |
|
942 |
by (asm_full_simp_tac (simpset() addsimps [poly_primes] |
|
943 |
delsimps [pmult_Cons]) 1); |
|
944 |
by (auto_tac (claset(),simpset() addsimps [divides_def] |
|
945 |
delsimps [pmult_Cons])); |
|
946 |
by (res_inst_tac [("x","qb")] exI 1); |
|
947 |
by (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = \ |
|
948 |
\ poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))" 1); |
|
949 |
by (dtac (poly_mult_left_cancel RS iffD1) 1); |
|
950 |
by (Force_tac 1); |
|
951 |
by (subgoal_tac "poly ([-a, 1] %^ (order a q) *** \ |
|
952 |
\ ([-a, 1] %^ (order a p) *** (qa *** qaa))) = \ |
|
953 |
\ poly ([-a, 1] %^ (order a q) *** \ |
|
954 |
\ ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb)))" 1); |
|
955 |
by (dtac (poly_mult_left_cancel RS iffD1) 1); |
|
956 |
by (Force_tac 1); |
|
957 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq,poly_exp_add,poly_mult] |
|
958 |
@ real_mult_ac delsimps [pmult_Cons]) 1); |
|
959 |
qed "order_mult"; |
|
960 |
||
961 |
(* FIXME: too too long! *) |
|
962 |
Goal "ALL p q a. 0 < n & \ |
|
963 |
\ poly (pderiv p) ~= poly [] & \ |
|
964 |
\ poly p = poly ([- a, 1] %^ n *** q) & ~ [- a, 1] divides q \ |
|
965 |
\ --> n = Suc (order a (pderiv p))"; |
|
966 |
by (induct_tac "n" 1); |
|
967 |
by (Step_tac 1); |
|
968 |
by (rtac order_unique_lemma 1 THEN rtac conjI 1); |
|
969 |
by (assume_tac 1); |
|
970 |
by (subgoal_tac "ALL r. r divides (pderiv p) = \ |
|
971 |
\ r divides (pderiv ([-a, 1] %^ Suc n *** q))" 1); |
|
972 |
by (dtac poly_pderiv_welldef 2); |
|
973 |
by (asm_full_simp_tac (simpset() addsimps [divides_def] delsimps [pmult_Cons, |
|
974 |
pexp_Suc]) 2); |
|
975 |
by (asm_full_simp_tac (simpset() delsimps [pmult_Cons,pexp_Suc]) 1); |
|
976 |
by (rtac conjI 1); |
|
977 |
by (asm_full_simp_tac (simpset() addsimps [divides_def,fun_eq] |
|
978 |
delsimps [pmult_Cons,pexp_Suc]) 1); |
|
979 |
by (res_inst_tac |
|
980 |
[("x","[-a, 1] *** (pderiv q) +++ real (Suc n) %* q")] exI 1); |
|
981 |
by (asm_full_simp_tac (simpset() addsimps [poly_pderiv_mult, |
|
982 |
poly_pderiv_exp_prime,poly_add,poly_mult,poly_cmult, |
|
983 |
real_add_mult_distrib2] @ real_mult_ac |
|
984 |
delsimps [pmult_Cons,pexp_Suc]) 1); |
|
985 |
by (asm_full_simp_tac (simpset() addsimps [poly_mult,real_add_mult_distrib2, |
|
986 |
real_add_mult_distrib] @ real_mult_ac delsimps [pmult_Cons]) 1); |
|
987 |
by (thin_tac "ALL r. \ |
|
988 |
\ r divides pderiv p = \ |
|
989 |
\ r divides pderiv ([- a, 1] %^ Suc n *** q)" 1); |
|
990 |
by (rewtac divides_def); |
|
991 |
by (simp_tac (simpset() addsimps [poly_pderiv_mult, |
|
992 |
poly_pderiv_exp_prime,fun_eq,poly_add,poly_mult] |
|
993 |
delsimps [pmult_Cons,pexp_Suc]) 1); |
|
994 |
by (rtac swap 1 THEN assume_tac 1); |
|
995 |
by (rotate_tac 3 1 THEN etac swap 1); |
|
996 |
by (asm_full_simp_tac (simpset() delsimps [pmult_Cons,pexp_Suc]) 1); |
|
997 |
by (Step_tac 1); |
|
998 |
by (res_inst_tac [("x","inverse(real (Suc n)) %* (qa +++ --(pderiv q))")] |
|
999 |
exI 1); |
|
1000 |
by (subgoal_tac "poly ([-a, 1] %^ n *** q) = \ |
|
1001 |
\ poly ([-a, 1] %^ n *** ([-a, 1] *** (inverse (real (Suc n)) %* \ |
|
1002 |
\ (qa +++ -- (pderiv q)))))" 1); |
|
1003 |
by (dtac (poly_mult_left_cancel RS iffD1) 1); |
|
1004 |
by (Asm_full_simp_tac 1); |
|
1005 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq,poly_mult,poly_add,poly_cmult, |
|
1006 |
poly_minus] delsimps [pmult_Cons]) 1); |
|
1007 |
by (Step_tac 1); |
|
1008 |
by (res_inst_tac [("c1","real (Suc n)")] (real_mult_left_cancel |
|
1009 |
RS iffD1) 1); |
|
1010 |
by (Simp_tac 1); |
|
1011 |
by (rtac ((CLAIM_SIMP |
|
1012 |
"a * (b * (c * (d * e))) = e * (b * (c * (d * (a::real))))" |
|
1013 |
real_mult_ac) RS ssubst) 1); |
|
1014 |
by (rotate_tac 2 1); |
|
1015 |
by (dres_inst_tac [("x","xa")] spec 1); |
|
12481
ea5d6da573c5
mods due to reorienting and renaming of real_minus_mult_eq1/2
nipkow
parents:
12224
diff
changeset
|
1016 |
by (asm_full_simp_tac (simpset() |
ea5d6da573c5
mods due to reorienting and renaming of real_minus_mult_eq1/2
nipkow
parents:
12224
diff
changeset
|
1017 |
addsimps [real_add_mult_distrib] @ real_mult_ac |
12224 | 1018 |
delsimps [pmult_Cons]) 1); |
1019 |
qed_spec_mp "lemma_order_pderiv"; |
|
1020 |
||
1021 |
Goal "[| poly (pderiv p) ~= poly []; order a p ~= 0 |] \ |
|
1022 |
\ ==> (order a p = Suc (order a (pderiv p)))"; |
|
1023 |
by (case_tac "poly p = poly []" 1); |
|
1024 |
by (auto_tac (claset() addDs [pderiv_zero],simpset())); |
|
1025 |
by (dres_inst_tac [("a","a"),("p","p")] order_decomp 1); |
|
1026 |
by (blast_tac (claset() addIs [lemma_order_pderiv]) 1); |
|
1027 |
qed "order_pderiv"; |
|
1028 |
||
1029 |
(* ------------------------------------------------------------------------- *) |
|
1030 |
(* Now justify the standard squarefree decomposition, i.e. f / gcd(f,f'). *) |
|
1031 |
(* `a la Harrison *) |
|
1032 |
(* ------------------------------------------------------------------------- *) |
|
1033 |
||
1034 |
Goal "[| poly (pderiv p) ~= poly []; \ |
|
1035 |
\ poly p = poly (q *** d); \ |
|
1036 |
\ poly (pderiv p) = poly (e *** d); \ |
|
1037 |
\ poly d = poly (r *** p +++ s *** pderiv p) \ |
|
1038 |
\ |] ==> order a q = (if order a p = 0 then 0 else 1)"; |
|
1039 |
by (subgoal_tac "order a p = order a q + order a d" 1); |
|
1040 |
by (res_inst_tac [("s","order a (q *** d)")] trans 2); |
|
1041 |
by (blast_tac (claset() addIs [order_poly]) 2); |
|
1042 |
by (rtac order_mult 2); |
|
1043 |
by (rtac notI 2 THEN Asm_full_simp_tac 2); |
|
1044 |
by (case_tac "order a p = 0" 1); |
|
1045 |
by (Asm_full_simp_tac 1); |
|
1046 |
by (subgoal_tac "order a (pderiv p) = order a e + order a d" 1); |
|
1047 |
by (res_inst_tac [("s","order a (e *** d)")] trans 2); |
|
1048 |
by (blast_tac (claset() addIs [order_poly]) 2); |
|
1049 |
by (rtac order_mult 2); |
|
1050 |
by (rtac notI 2 THEN Asm_full_simp_tac 2); |
|
1051 |
by (case_tac "poly p = poly []" 1); |
|
1052 |
by (dres_inst_tac [("p","p")] pderiv_zero 1); |
|
1053 |
by (Asm_full_simp_tac 1); |
|
1054 |
by (dtac order_pderiv 1 THEN assume_tac 1); |
|
1055 |
by (subgoal_tac "order a (pderiv p) <= order a d" 1); |
|
1056 |
by (subgoal_tac "([-a, 1] %^ (order a (pderiv p))) divides d" 2); |
|
1057 |
by (asm_full_simp_tac (simpset() addsimps [poly_entire,order_divides]) 2); |
|
1058 |
by (subgoal_tac "([-a, 1] %^ (order a (pderiv p))) divides p & \ |
|
1059 |
\ ([-a, 1] %^ (order a (pderiv p))) divides (pderiv p)" 2); |
|
1060 |
by (asm_simp_tac (simpset() addsimps [order_divides]) 3); |
|
1061 |
by (asm_full_simp_tac (simpset() addsimps [divides_def] |
|
1062 |
delsimps [pexp_Suc,pmult_Cons]) 2); |
|
1063 |
by (Step_tac 2); |
|
1064 |
by (res_inst_tac [("x","r *** qa +++ s *** qaa")] exI 2); |
|
1065 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq,poly_add,poly_mult, |
|
1066 |
real_add_mult_distrib, real_add_mult_distrib2] @ real_mult_ac |
|
1067 |
delsimps [pexp_Suc,pmult_Cons]) 2); |
|
1068 |
by Auto_tac; |
|
1069 |
qed "poly_squarefree_decomp_order"; |
|
1070 |
||
1071 |
||
1072 |
Goal "[| poly (pderiv p) ~= poly []; \ |
|
1073 |
\ poly p = poly (q *** d); \ |
|
1074 |
\ poly (pderiv p) = poly (e *** d); \ |
|
1075 |
\ poly d = poly (r *** p +++ s *** pderiv p) \ |
|
1076 |
\ |] ==> ALL a. order a q = (if order a p = 0 then 0 else 1)"; |
|
1077 |
by (blast_tac (claset() addIs [poly_squarefree_decomp_order]) 1); |
|
1078 |
qed "poly_squarefree_decomp_order2"; |
|
1079 |
||
1080 |
Goal "poly p ~= poly [] ==> (poly p a = 0) = (order a p ~= 0)"; |
|
1081 |
by (rtac (order_root RS ssubst) 1); |
|
1082 |
by Auto_tac; |
|
1083 |
qed "order_root2"; |
|
1084 |
||
1085 |
Goal "[| poly (pderiv p) ~= poly []; order a p ~= 0 |] \ |
|
1086 |
\ ==> (order a (pderiv p) = n) = (order a p = Suc n)"; |
|
1087 |
by (auto_tac (claset() addDs [order_pderiv],simpset())); |
|
1088 |
qed "order_pderiv2"; |
|
1089 |
||
1090 |
Goalw [rsquarefree_def] |
|
1091 |
"rsquarefree p = (ALL a. ~(poly p a = 0 & poly (pderiv p) a = 0))"; |
|
1092 |
by (case_tac "poly p = poly []" 1); |
|
1093 |
by (Asm_full_simp_tac 1); |
|
1094 |
by (Asm_full_simp_tac 1); |
|
1095 |
by (case_tac "poly (pderiv p) = poly []" 1); |
|
1096 |
by (Asm_full_simp_tac 1); |
|
1097 |
by (dtac pderiv_iszero 1 THEN Clarify_tac 1); |
|
1098 |
by (subgoal_tac "ALL a. order a p = order a [h]" 1); |
|
1099 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq]) 1); |
|
1100 |
by (rtac allI 1); |
|
1101 |
by (cut_inst_tac [("p","[h]"),("a","a")] order_root 1); |
|
1102 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq]) 1); |
|
1103 |
by (blast_tac (claset() addIs [order_poly]) 1); |
|
1104 |
by (auto_tac (claset(),simpset() addsimps [order_root,order_pderiv2])); |
|
1105 |
by (dtac spec 1 THEN Auto_tac); |
|
1106 |
qed "rsquarefree_roots"; |
|
1107 |
||
1108 |
Goal "[1] *** p = p"; |
|
1109 |
by Auto_tac; |
|
1110 |
qed "pmult_one"; |
|
1111 |
Addsimps [pmult_one]; |
|
1112 |
||
1113 |
Goal "poly [] = poly [0]"; |
|
1114 |
by (simp_tac (simpset() addsimps [fun_eq]) 1); |
|
1115 |
qed "poly_Nil_zero"; |
|
1116 |
||
1117 |
Goalw [rsquarefree_def] |
|
1118 |
"[| rsquarefree p; poly p a = 0 |] \ |
|
1119 |
\ ==> EX q. (poly p = poly ([-a, 1] *** q)) & poly q a ~= 0"; |
|
1120 |
by (Step_tac 1); |
|
1121 |
by (forw_inst_tac [("a","a")] order_decomp 1); |
|
1122 |
by (dres_inst_tac [("x","a")] spec 1); |
|
1123 |
by (dres_inst_tac [("a1","a")] (order_root2 RS sym) 1); |
|
1124 |
by (auto_tac (claset(),simpset() delsimps [pmult_Cons])); |
|
1125 |
by (res_inst_tac [("x","q")] exI 1 THEN Step_tac 1); |
|
1126 |
by (asm_full_simp_tac (simpset() addsimps [poly_mult,fun_eq]) 1); |
|
1127 |
by (dres_inst_tac [("p1","q")] (poly_linear_divides RS iffD1) 1); |
|
1128 |
by (asm_full_simp_tac (simpset() addsimps [divides_def] |
|
1129 |
delsimps [pmult_Cons]) 1); |
|
1130 |
by (Step_tac 1); |
|
1131 |
by (dres_inst_tac [("x","[]")] spec 1); |
|
1132 |
by (auto_tac (claset(),simpset() addsimps [fun_eq])); |
|
1133 |
qed "rsquarefree_decomp"; |
|
1134 |
||
1135 |
Goal "[| poly (pderiv p) ~= poly []; \ |
|
1136 |
\ poly p = poly (q *** d); \ |
|
1137 |
\ poly (pderiv p) = poly (e *** d); \ |
|
1138 |
\ poly d = poly (r *** p +++ s *** pderiv p) \ |
|
1139 |
\ |] ==> rsquarefree q & (ALL a. (poly q a = 0) = (poly p a = 0))"; |
|
1140 |
by (ftac poly_squarefree_decomp_order2 1); |
|
1141 |
by (TRYALL(assume_tac)); |
|
1142 |
by (case_tac "poly p = poly []" 1); |
|
1143 |
by (blast_tac (claset() addDs [pderiv_zero]) 1); |
|
1144 |
by (simp_tac (simpset() addsimps [rsquarefree_def, |
|
1145 |
order_root] delsimps [pmult_Cons]) 1); |
|
1146 |
by (asm_full_simp_tac (simpset() addsimps [poly_entire] |
|
1147 |
delsimps [pmult_Cons]) 1); |
|
1148 |
qed "poly_squarefree_decomp"; |
|
1149 |
||
1150 |
||
1151 |
(* ------------------------------------------------------------------------- *) |
|
1152 |
(* Normalization of a polynomial. *) |
|
1153 |
(* ------------------------------------------------------------------------- *) |
|
1154 |
||
1155 |
Goal "poly (pnormalize p) = poly p"; |
|
1156 |
by (induct_tac "p" 1); |
|
1157 |
by (auto_tac (claset(),simpset() addsimps [fun_eq])); |
|
1158 |
qed "poly_normalize"; |
|
1159 |
Addsimps [poly_normalize]; |
|
1160 |
||
1161 |
||
1162 |
(* ------------------------------------------------------------------------- *) |
|
1163 |
(* The degree of a polynomial. *) |
|
1164 |
(* ------------------------------------------------------------------------- *) |
|
1165 |
||
1166 |
Goal "list_all (%c. c = 0) p --> pnormalize p = []"; |
|
1167 |
by (induct_tac "p" 1); |
|
1168 |
by Auto_tac; |
|
1169 |
qed_spec_mp "lemma_degree_zero"; |
|
1170 |
||
1171 |
Goalw [degree_def] "poly p = poly [] ==> degree p = 0"; |
|
1172 |
by (case_tac "pnormalize p = []" 1); |
|
1173 |
by (auto_tac (claset() addDs [lemma_degree_zero],simpset() |
|
1174 |
addsimps [poly_zero])); |
|
1175 |
qed "degree_zero"; |
|
1176 |
||
1177 |
(* ------------------------------------------------------------------------- *) |
|
1178 |
(* Tidier versions of finiteness of roots. *) |
|
1179 |
(* ------------------------------------------------------------------------- *) |
|
1180 |
||
1181 |
Goal "poly p ~= poly [] ==> finite {x. poly p x = 0}"; |
|
1182 |
by (auto_tac (claset(),simpset() addsimps [poly_roots_finite])); |
|
1183 |
by (res_inst_tac [("B","{x::real. EX n. (n::nat) < N & (x = j n)}")] |
|
1184 |
finite_subset 1); |
|
1185 |
by (induct_tac "N" 2); |
|
1186 |
by Auto_tac; |
|
1187 |
by (subgoal_tac "{x::real. EX na. na < Suc n & (x = j na)} = \ |
|
1188 |
\ {(j n)} Un {x. EX na. na < n & (x = j na)}" 1); |
|
1189 |
by (auto_tac (claset(),simpset() addsimps [ARITH_PROVE |
|
1190 |
"(na < Suc n) = (na = n | na < n)"])); |
|
1191 |
qed "poly_roots_finite_set"; |
|
1192 |
||
1193 |
(* ------------------------------------------------------------------------- *) |
|
1194 |
(* bound for polynomial. *) |
|
1195 |
(* ------------------------------------------------------------------------- *) |
|
1196 |
||
1197 |
Goal "abs(x) <= k --> abs(poly p x) <= poly (map abs p) k"; |
|
1198 |
by (induct_tac "p" 1); |
|
1199 |
by Auto_tac; |
|
1200 |
by (res_inst_tac [("j","abs a + abs(x * poly list x)")] real_le_trans 1); |
|
1201 |
by (rtac abs_triangle_ineq 1); |
|
1202 |
by (auto_tac (claset() addSIs [real_mult_le_mono],simpset() |
|
1203 |
addsimps [abs_mult])); |
|
1204 |
by (arith_tac 1); |
|
1205 |
qed_spec_mp "poly_mono"; |