diff -r 323b15b5af73 -r f31f7599ef55 src/HOL/Library/Fundamental_Theorem_Algebra.thy
--- a/src/HOL/Library/Fundamental_Theorem_Algebra.thy Sat Jun 13 19:38:26 2015 +0200
+++ b/src/HOL/Library/Fundamental_Theorem_Algebra.thy Sat Jun 13 19:53:53 2015 +0200
@@ -159,8 +159,9 @@
assume IH: "\m 0 \ (\z. cmod (1 + b * z ^ m) < 1)"
assume n: "n \ 0"
let ?P = "\z n. cmod (1 + b * z ^ n) < 1"
- {
- assume e: "even n"
+ show "\z. ?P z n"
+ proof cases
+ assume "even n"
then have "\m. n = 2 * m"
by presburger
then obtain m where m: "n = 2 * m"
@@ -170,19 +171,17 @@
with IH[rule_format, of m] obtain z where z: "?P z m"
by blast
from z have "?P (csqrt z) n"
- by (simp add: m power_mult power2_csqrt)
- then have "\z. ?P z n" ..
- }
- moreover
- {
- assume o: "odd n"
- have th0: "cmod (complex_of_real (cmod b) / b) = 1"
- using b by (simp add: norm_divide)
- from o have "\m. n = Suc (2 * m)"
+ by (simp add: m power_mult)
+ then show ?thesis ..
+ next
+ assume "odd n"
+ then have "\m. n = Suc (2 * m)"
by presburger+
then obtain m where m: "n = Suc (2 * m)"
by blast
- from unimodular_reduce_norm[OF th0] o
+ have th0: "cmod (complex_of_real (cmod b) / b) = 1"
+ using b by (simp add: norm_divide)
+ from unimodular_reduce_norm[OF th0] \odd n\
have "\v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1")
apply (rule_tac x="1" in exI)
@@ -206,7 +205,7 @@
then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1"
by blast
let ?w = "v / complex_of_real (root n (cmod b))"
- from odd_real_root_pow[OF o, of "cmod b"]
+ from odd_real_root_pow[OF \odd n\, of "cmod b"]
have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
by (simp add: power_divide of_real_power[symmetric])
have th2:"cmod (complex_of_real (cmod b) / b) = 1"
@@ -222,9 +221,8 @@
done
from mult_left_less_imp_less[OF th4 th3]
have "?P ?w n" unfolding th1 .
- then have "\z. ?P z n" ..
- }
- ultimately show "\z. ?P z n" by blast
+ then show ?thesis ..
+ qed
qed
text \Bolzano-Weierstrass type property for closed disc in complex plane.\
@@ -1020,51 +1018,44 @@
"(\x. poly p x = (0::complex) \ poly q x = 0) \
p dvd (q ^ (degree p)) \ (p = 0 \ q = 0)"
proof -
- show ?thesis
- proof (cases "p = 0")
- case True
+ consider "p = 0" | "p \ 0" "degree p = 0" | n where "p \ 0" "degree p = Suc n"
+ by (cases "degree p") auto
+ then show ?thesis
+ proof cases
+ case 1
then have eq: "(\x. poly p x = (0::complex) \ poly q x = 0) \ q = 0"
by (auto simp add: poly_all_0_iff_0)
{
assume "p dvd (q ^ (degree p))"
then obtain r where r: "q ^ (degree p) = p * r" ..
- from r True have False by simp
+ from r 1 have False by simp
}
- with eq True show ?thesis by blast
+ with eq 1 show ?thesis by blast
next
- case False
- show ?thesis
- proof (cases "degree p = 0")
- case True
- with \p \ 0\ obtain k where k: "p = [:k:]" "k \ 0"
- by (cases p) (simp split: if_splits)
- then have th1: "\x. poly p x \ 0"
+ case 2
+ then obtain k where k: "p = [:k:]" "k \ 0"
+ by (cases p) (simp split: if_splits)
+ then have th1: "\x. poly p x \ 0"
+ by simp
+ from k 2(2) have "q ^ (degree p) = p * [:1/k:]"
+ by (simp add: one_poly_def)
+ then have th2: "p dvd (q ^ (degree p))" ..
+ from 2(1) th1 th2 show ?thesis
+ by blast
+ next
+ case 3
+ {
+ assume "p dvd (q ^ (Suc n))"
+ then obtain u where u: "q ^ (Suc n) = p * u" ..
+ fix x
+ assume h: "poly p x = 0" "poly q x \ 0"
+ then have "poly (q ^ (Suc n)) x \ 0"
by simp
- from k True have "q ^ (degree p) = p * [:1/k:]"
- by (simp add: one_poly_def)
- then have th2: "p dvd (q ^ (degree p))" ..
- from False th1 th2 show ?thesis
- by blast
- next
- case False
- assume dp: "degree p \ 0"
- then obtain n where n: "degree p = Suc n "
- by (cases "degree p") auto
- {
- assume "p dvd (q ^ (Suc n))"
- then obtain u where u: "q ^ (Suc n) = p * u" ..
- {
- fix x
- assume h: "poly p x = 0" "poly q x \ 0"
- then have "poly (q ^ (Suc n)) x \ 0"
- by simp
- then have False using u h(1)
- by (simp only: poly_mult) simp
- }
- }
- with n nullstellensatz_lemma[of p q "degree p"] dp
- show ?thesis by auto
- qed
+ then have False using u h(1)
+ by (simp only: poly_mult) simp
+ }
+ with 3 nullstellensatz_lemma[of p q "degree p"]
+ show ?thesis by auto
qed
qed