# HG changeset patch
# User wenzelm
# Date 1163069929 -3600
# Node ID de65ce2bfb32bf14d26f17159297068ddeb37a0f
# Parent  a2bd14226f9a1898c779df5883c1c6a35ce031f5
tuned;

diff -r a2bd14226f9a -r de65ce2bfb32 src/HOL/Library/Binomial.thy
--- a/src/HOL/Library/Binomial.thy	Thu Nov 09 11:58:47 2006 +0100
+++ b/src/HOL/Library/Binomial.thy	Thu Nov 09 11:58:49 2006 +0100
@@ -4,87 +4,82 @@
     Copyright   1997  University of Cambridge
 *)
 
-header{*Binomial Coefficients*}
+header {* Binomial Coefficients *}
 
 theory Binomial
 imports Main
 begin
 
-text{*This development is based on the work of Andy Gordon and
-Florian Kammueller*}
+text {* This development is based on the work of Andy Gordon and
+  Florian Kammueller. *}
 
 consts
   binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat"      (infixl "choose" 65)
-
 primrec
-  binomial_0:   "(0     choose k) = (if k = 0 then 1 else 0)"
-
+  binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)"
   binomial_Suc: "(Suc n choose k) =
                  (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
 
 lemma binomial_n_0 [simp]: "(n choose 0) = 1"
-by (cases n) simp_all
+  by (cases n) simp_all
 
 lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
-by simp
+  by simp
 
 lemma binomial_Suc_Suc [simp]:
-     "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
-by simp
+    "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
+  by simp
 
-lemma binomial_eq_0 [rule_format]: "\<forall>k. n < k --> (n choose k) = 0"
-apply (induct "n")
-apply auto
-done
+lemma binomial_eq_0: "!!k. n < k ==> (n choose k) = 0"
+  by (induct n) auto
 
 declare binomial_0 [simp del] binomial_Suc [simp del]
 
 lemma binomial_n_n [simp]: "(n choose n) = 1"
-apply (induct "n")
-apply (simp_all add: binomial_eq_0)
-done
+  by (induct n) (simp_all add: binomial_eq_0)
 
 lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n"
-by (induct "n", simp_all)
+  by (induct n) simp_all
 
 lemma binomial_1 [simp]: "(n choose Suc 0) = n"
-by (induct "n", simp_all)
+  by (induct n) simp_all
 
-lemma zero_less_binomial [rule_format]: "k \<le> n --> 0 < (n choose k)"
-by (rule_tac m = n and n = k in diff_induct, simp_all)
+lemma zero_less_binomial: "k \<le> n ==> 0 < (n choose k)"
+  by (induct n k rule: diff_induct) simp_all
 
 lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"
-apply (safe intro!: binomial_eq_0)
-apply (erule contrapos_pp)
-apply (simp add: zero_less_binomial)
-done
+  apply (safe intro!: binomial_eq_0)
+  apply (erule contrapos_pp)
+  apply (simp add: zero_less_binomial)
+  done
 
 lemma zero_less_binomial_iff: "(0 < n choose k) = (k\<le>n)"
-by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric])
+  by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric])
 
 (*Might be more useful if re-oriented*)
-lemma Suc_times_binomial_eq [rule_format]:
-     "\<forall>k. k \<le> n --> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
-apply (induct "n")
-apply (simp add: binomial_0, clarify)
-apply (case_tac "k")
-apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq
-                      binomial_eq_0)
-done
+lemma Suc_times_binomial_eq:
+    "!!k. k \<le> n ==> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
+  apply (induct n)
+  apply (simp add: binomial_0)
+  apply (case_tac k)
+  apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq
+    binomial_eq_0)
+  done
 
 text{*This is the well-known version, but it's harder to use because of the
   need to reason about division.*}
 lemma binomial_Suc_Suc_eq_times:
-     "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
-by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc
-        del: mult_Suc mult_Suc_right)
+    "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
+  by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc
+    del: mult_Suc mult_Suc_right)
 
 text{*Another version, with -1 instead of Suc.*}
 lemma times_binomial_minus1_eq:
-     "[|k \<le> n;  0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"
-apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)
-apply (simp split add: nat_diff_split, auto)
-done
+    "[|k \<le> n;  0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"
+  apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)
+  apply (simp split add: nat_diff_split, auto)
+  done
+
 
 subsubsection {* Theorems about @{text "choose"} *}
 
@@ -132,7 +127,7 @@
 *}
 
 lemma n_sub_lemma:
-  "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
+    "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
   apply (induct k)
    apply (simp add: card_s_0_eq_empty, atomize)
   apply (rotate_tac -1, erule finite_induct)
@@ -166,10 +161,10 @@
     using Suc by simp
   also have "\<dots> =  a*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k)) +
                    b*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
-    by(rule nat_distrib)
+    by (rule nat_distrib)
   also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^(k+1) * b^(n-k)) +
                   (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k+1))"
-    by(simp add: setsum_right_distrib mult_ac)
+    by (simp add: setsum_right_distrib mult_ac)
   also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^k * b^(n+1-k)) +
                   (\<Sum>k=1..n+1. (n choose (k - 1)) * a^k * b^(n+1-k))"
     by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le
@@ -177,10 +172,10 @@
   also have "\<dots> = a^(n+1) + b^(n+1) +
                   (\<Sum>k=1..n. (n choose (k - 1)) * a^k * b^(n+1-k)) +
                   (\<Sum>k=1..n. (n choose k) * a^k * b^(n+1-k))"
-    by(simp add: decomp2)
+    by (simp add: decomp2)
   also have
-    "\<dots> = a^(n+1) + b^(n+1) + (\<Sum>k=1..n. (n+1 choose k) * a^k * b^(n+1-k))"
-    by(simp add: nat_distrib setsum_addf binomial.simps)
+      "\<dots> = a^(n+1) + b^(n+1) + (\<Sum>k=1..n. (n+1 choose k) * a^k * b^(n+1-k))"
+    by (simp add: nat_distrib setsum_addf binomial.simps)
   also have "\<dots> = (\<Sum>k=0..n+1. (n+1 choose k) * a^k * b^(n+1-k))"
     using decomp by simp
   finally show ?case by simp
diff -r a2bd14226f9a -r de65ce2bfb32 src/HOL/Library/GCD.thy
--- a/src/HOL/Library/GCD.thy	Thu Nov 09 11:58:47 2006 +0100
+++ b/src/HOL/Library/GCD.thy	Thu Nov 09 11:58:49 2006 +0100
@@ -21,10 +21,10 @@
 recdef gcd  "measure ((\<lambda>(m, n). n) :: nat \<times> nat => nat)"
   "gcd (m, n) = (if n = 0 then m else gcd (n, m mod n))"
 
-constdefs
+definition
   is_gcd :: "nat => nat => nat => bool"  -- {* @{term gcd} as a relation *}
-  "is_gcd p m n == p dvd m \<and> p dvd n \<and>
-    (\<forall>d. d dvd m \<and> d dvd n --> d dvd p)"
+  "is_gcd p m n = (p dvd m \<and> p dvd n \<and>
+    (\<forall>d. d dvd m \<and> d dvd n --> d dvd p))"
 
 
 lemma gcd_induct:
@@ -38,18 +38,15 @@
 
 
 lemma gcd_0 [simp]: "gcd (m, 0) = m"
-  apply simp
-  done
+  by simp
 
 lemma gcd_non_0: "0 < n ==> gcd (m, n) = gcd (n, m mod n)"
-  apply simp
-  done
+  by simp
 
 declare gcd.simps [simp del]
 
 lemma gcd_1 [simp]: "gcd (m, Suc 0) = 1"
-  apply (simp add: gcd_non_0)
-  done
+  by (simp add: gcd_non_0)
 
 text {*
   \medskip @{term "gcd (m, n)"} divides @{text m} and @{text n}.  The
@@ -59,7 +56,7 @@
 lemma gcd_dvd1 [iff]: "gcd (m, n) dvd m"
   and gcd_dvd2 [iff]: "gcd (m, n) dvd n"
   apply (induct m n rule: gcd_induct)
-   apply (simp_all add: gcd_non_0)
+     apply (simp_all add: gcd_non_0)
   apply (blast dest: dvd_mod_imp_dvd)
   done
 
@@ -70,16 +67,13 @@
 *}
 
 lemma gcd_greatest: "k dvd m ==> k dvd n ==> k dvd gcd (m, n)"
-  apply (induct m n rule: gcd_induct)
-   apply (simp_all add: gcd_non_0 dvd_mod)
-  done
+  by (induct m n rule: gcd_induct) (simp_all add: gcd_non_0 dvd_mod)
 
 lemma gcd_greatest_iff [iff]: "(k dvd gcd (m, n)) = (k dvd m \<and> k dvd n)"
-  apply (blast intro!: gcd_greatest intro: dvd_trans)
-  done
+  by (blast intro!: gcd_greatest intro: dvd_trans)
 
 lemma gcd_zero: "(gcd (m, n) = 0) = (m = 0 \<and> n = 0)"
-  by (simp only: dvd_0_left_iff [THEN sym] gcd_greatest_iff)
+  by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff)
 
 
 text {*
@@ -199,8 +193,6 @@
   done
 
 lemma gcd_add_mult: "gcd (m, k * m + n) = gcd (m, n)"
-  apply (induct k)
-   apply (simp_all add: add_assoc)
-  done
+  by (induct k) (simp_all add: add_assoc)
 
 end
diff -r a2bd14226f9a -r de65ce2bfb32 src/HOL/Library/Parity.thy
--- a/src/HOL/Library/Parity.thy	Thu Nov 09 11:58:47 2006 +0100
+++ b/src/HOL/Library/Parity.thy	Thu Nov 09 11:58:49 2006 +0100
@@ -1,4 +1,4 @@
-(*  Title:      Parity.thy
+(*  Title:      HOL/Library/Parity.thy
     ID:         $Id$
     Author:     Jeremy Avigad
 *)
@@ -28,14 +28,17 @@
 
 subsection {* Even and odd are mutually exclusive *}
 
-lemma int_pos_lt_two_imp_zero_or_one: 
+lemma int_pos_lt_two_imp_zero_or_one:
     "0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1"
   by auto
 
 lemma neq_one_mod_two [simp]: "((x::int) mod 2 ~= 0) = (x mod 2 = 1)"
-  apply (subgoal_tac "x mod 2 = 0 | x mod 2 = 1", force)
-  apply (rule int_pos_lt_two_imp_zero_or_one, auto)
-  done
+proof -
+  have "x mod 2 = 0 | x mod 2 = 1"
+    by (rule int_pos_lt_two_imp_zero_or_one) auto
+  then show ?thesis by force
+qed
+
 
 subsection {* Behavior under integer arithmetic operations *}
 
@@ -49,7 +52,7 @@
   by (simp add: even_def zmod_zmult1_eq)
 
 lemma even_product: "even((x::int) * y) = (even x | even y)"
-  apply (auto simp add: even_times_anything anything_times_even) 
+  apply (auto simp add: even_times_anything anything_times_even)
   apply (rule ccontr)
   apply (auto simp add: odd_times_odd)
   done
@@ -75,24 +78,22 @@
 lemma even_neg: "even (-(x::int)) = even x"
   by (auto simp add: even_def zmod_zminus1_eq_if)
 
-lemma even_difference: 
-  "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))"
+lemma even_difference:
+    "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))"
   by (simp only: diff_minus even_sum even_neg)
 
-lemma even_pow_gt_zero [rule_format]: 
-    "even (x::int) ==> 0 < n --> even (x^n)"
-  apply (induct n)
-  apply (auto simp add: even_product)
-  done
+lemma even_pow_gt_zero:
+    "even (x::int) ==> 0 < n ==> even (x^n)"
+  by (induct n) (auto simp add: even_product)
 
 lemma odd_pow: "odd x ==> odd((x::int)^n)"
   apply (induct n)
-  apply (simp add: even_def)
+   apply (simp add: even_def)
   apply (simp add: even_product)
   done
 
 lemma even_power: "even ((x::int)^n) = (even x & 0 < n)"
-  apply (auto simp add: even_pow_gt_zero) 
+  apply (auto simp add: even_pow_gt_zero)
   apply (erule contrapos_pp, erule odd_pow)
   apply (erule contrapos_pp, simp add: even_def)
   done
@@ -103,29 +104,32 @@
 lemma odd_one: "odd (1::int)"
   by (simp add: even_def)
 
-lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero 
+lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero
   odd_one even_product even_sum even_neg even_difference even_power
 
 
 subsection {* Equivalent definitions *}
 
-lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" 
+lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x"
   by (auto simp add: even_def)
 
-lemma two_times_odd_div_two_plus_one: "odd (x::int) ==> 
+lemma two_times_odd_div_two_plus_one: "odd (x::int) ==>
     2 * (x div 2) + 1 = x"
-  apply (insert zmod_zdiv_equality [of x 2, THEN sym])
-  by (simp add: even_def)
+  apply (insert zmod_zdiv_equality [of x 2, symmetric])
+  apply (simp add: even_def)
+  done
 
 lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)"
   apply auto
   apply (rule exI)
-  by (erule two_times_even_div_two [THEN sym])
+  apply (erule two_times_even_div_two [symmetric])
+  done
 
 lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)"
   apply auto
   apply (rule exI)
-  by (erule two_times_odd_div_two_plus_one [THEN sym])
+  apply (erule two_times_odd_div_two_plus_one [symmetric])
+  done
 
 
 subsection {* even and odd for nats *}
@@ -136,15 +140,15 @@
 lemma even_nat_product: "even((x::nat) * y) = (even x | even y)"
   by (simp add: even_nat_def int_mult)
 
-lemma even_nat_sum: "even ((x::nat) + y) = 
+lemma even_nat_sum: "even ((x::nat) + y) =
     ((even x & even y) | (odd x & odd y))"
   by (unfold even_nat_def, simp)
 
-lemma even_nat_difference: 
+lemma even_nat_difference:
     "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
-  apply (auto simp add: even_nat_def zdiff_int [THEN sym])
-  apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym])
-  apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym])
+  apply (auto simp add: even_nat_def zdiff_int [symmetric])
+  apply (case_tac "x < y", auto simp add: zdiff_int [symmetric])
+  apply (case_tac "x < y", auto simp add: zdiff_int [symmetric])
   done
 
 lemma even_nat_Suc: "even (Suc x) = odd x"
@@ -156,18 +160,18 @@
 lemma even_nat_zero: "even (0::nat)"
   by (simp add: even_nat_def)
 
-lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard] 
+lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard]
   even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power
 
 
 subsection {* Equivalent definitions *}
 
-lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==> 
+lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==>
     x = 0 | x = Suc 0"
   by auto
 
 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
-  apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
+  apply (insert mod_div_equality [of x "Suc (Suc 0)", symmetric])
   apply (drule subst, assumption)
   apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0")
   apply force
@@ -177,16 +181,16 @@
   done
 
 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
-  apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
+  apply (insert mod_div_equality [of x "Suc (Suc 0)", symmetric])
   apply (drule subst, assumption)
   apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0")
-  apply force 
+  apply force
   apply (subgoal_tac "0 < Suc (Suc 0)")
   apply (frule mod_less_divisor [of "Suc (Suc 0)" x])
   apply (erule nat_lt_two_imp_zero_or_one, auto)
   done
 
-lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)" 
+lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
   apply (rule iffI)
   apply (erule even_nat_mod_two_eq_zero)
   apply (insert odd_nat_mod_two_eq_one [of x], auto)
@@ -198,69 +202,71 @@
   apply (frule nat_lt_two_imp_zero_or_one, auto)
   done
 
-lemma even_nat_div_two_times_two: "even (x::nat) ==> 
+lemma even_nat_div_two_times_two: "even (x::nat) ==>
     Suc (Suc 0) * (x div Suc (Suc 0)) = x"
-  apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
+  apply (insert mod_div_equality [of x "Suc (Suc 0)", symmetric])
   apply (drule even_nat_mod_two_eq_zero, simp)
   done
 
-lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==> 
-    Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x"  
-  apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
+lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>
+    Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x"
+  apply (insert mod_div_equality [of x "Suc (Suc 0)", symmetric])
   apply (drule odd_nat_mod_two_eq_one, simp)
   done
 
 lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
   apply (rule iffI, rule exI)
-  apply (erule even_nat_div_two_times_two [THEN sym], auto)
+  apply (erule even_nat_div_two_times_two [symmetric], auto)
   done
 
 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
   apply (rule iffI, rule exI)
-  apply (erule odd_nat_div_two_times_two_plus_one [THEN sym], auto)
+  apply (erule odd_nat_div_two_times_two_plus_one [symmetric], auto)
   done
 
 subsection {* Parity and powers *}
 
-lemma minus_one_even_odd_power:
-     "(even x --> (- 1::'a::{comm_ring_1,recpower})^x = 1) & 
+lemma  minus_one_even_odd_power:
+     "(even x --> (- 1::'a::{comm_ring_1,recpower})^x = 1) &
       (odd x --> (- 1::'a)^x = - 1)"
   apply (induct x)
   apply (rule conjI)
   apply simp
   apply (insert even_nat_zero, blast)
   apply (simp add: power_Suc)
-done
+  done
 
 lemma minus_one_even_power [simp]:
-     "even x ==> (- 1::'a::{comm_ring_1,recpower})^x = 1"
-  by (rule minus_one_even_odd_power [THEN conjunct1, THEN mp], assumption)
+    "even x ==> (- 1::'a::{comm_ring_1,recpower})^x = 1"
+  using minus_one_even_odd_power by blast
 
 lemma minus_one_odd_power [simp]:
-     "odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1"
-  by (rule minus_one_even_odd_power [THEN conjunct2, THEN mp], assumption)
+    "odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1"
+  using minus_one_even_odd_power by blast
 
 lemma neg_one_even_odd_power:
-     "(even x --> (-1::'a::{number_ring,recpower})^x = 1) & 
+     "(even x --> (-1::'a::{number_ring,recpower})^x = 1) &
       (odd x --> (-1::'a)^x = -1)"
   apply (induct x)
   apply (simp, simp add: power_Suc)
   done
 
 lemma neg_one_even_power [simp]:
-     "even x ==> (-1::'a::{number_ring,recpower})^x = 1"
-  by (rule neg_one_even_odd_power [THEN conjunct1, THEN mp], assumption)
+    "even x ==> (-1::'a::{number_ring,recpower})^x = 1"
+  using neg_one_even_odd_power by blast
 
 lemma neg_one_odd_power [simp]:
-     "odd x ==> (-1::'a::{number_ring,recpower})^x = -1"
-  by (rule neg_one_even_odd_power [THEN conjunct2, THEN mp], assumption)
+    "odd x ==> (-1::'a::{number_ring,recpower})^x = -1"
+  using neg_one_even_odd_power by blast
 
 lemma neg_power_if:
-     "(-x::'a::{comm_ring_1,recpower}) ^ n = 
+     "(-x::'a::{comm_ring_1,recpower}) ^ n =
       (if even n then (x ^ n) else -(x ^ n))"
-  by (induct n, simp_all split: split_if_asm add: power_Suc) 
+  apply (induct n)
+  apply (simp_all split: split_if_asm add: power_Suc)
+  done
 
-lemma zero_le_even_power: "even n ==> 
+lemma zero_le_even_power: "even n ==>
     0 <= (x::'a::{recpower,ordered_ring_strict}) ^ n"
   apply (simp add: even_nat_equiv_def2)
   apply (erule exE)
@@ -269,7 +275,7 @@
   apply (rule zero_le_square)
   done
 
-lemma zero_le_odd_power: "odd n ==> 
+lemma zero_le_odd_power: "odd n ==>
     (0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (0 <= x)"
   apply (simp add: odd_nat_equiv_def2)
   apply (erule exE)
@@ -280,23 +286,23 @@
   apply auto
   apply (subgoal_tac "x = 0 & 0 < y")
   apply (erule conjE, assumption)
-  apply (subst power_eq_0_iff [THEN sym])
+  apply (subst power_eq_0_iff [symmetric])
   apply (subgoal_tac "0 <= x^y * x^y")
   apply simp
   apply (rule zero_le_square)+
-done
+  done
 
-lemma zero_le_power_eq: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) = 
+lemma zero_le_power_eq: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) =
     (even n | (odd n & 0 <= x))"
   apply auto
-  apply (subst zero_le_odd_power [THEN sym])
+  apply (subst zero_le_odd_power [symmetric])
   apply assumption+
   apply (erule zero_le_even_power)
-  apply (subst zero_le_odd_power) 
+  apply (subst zero_le_odd_power)
   apply assumption+
-done
+  done
 
-lemma zero_less_power_eq: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) = 
+lemma zero_less_power_eq: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) =
     (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
   apply (rule iffI)
   apply clarsimp
@@ -306,7 +312,7 @@
   apply (subgoal_tac "~ (0 <= x^n)")
   apply simp
   apply (subst zero_le_odd_power)
-  apply assumption 
+  apply assumption
   apply simp
   apply (rule notI)
   apply (simp add: power_0_left)
@@ -323,99 +329,91 @@
   apply (subst zero_le_odd_power)
   apply assumption
   apply (erule order_less_imp_le)
-done
+  done
 
 lemma power_less_zero_eq: "((x::'a::{recpower,ordered_idom}) ^ n < 0) =
-    (odd n & x < 0)" 
-  apply (subst linorder_not_le [THEN sym])+
+    (odd n & x < 0)"
+  apply (subst linorder_not_le [symmetric])+
   apply (subst zero_le_power_eq)
   apply auto
-done
+  done
 
 lemma power_le_zero_eq: "((x::'a::{recpower,ordered_idom}) ^ n <= 0) =
     (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
-  apply (subst linorder_not_less [THEN sym])+
+  apply (subst linorder_not_less [symmetric])+
   apply (subst zero_less_power_eq)
   apply auto
-done
+  done
 
-lemma power_even_abs: "even n ==> 
+lemma power_even_abs: "even n ==>
     (abs (x::'a::{recpower,ordered_idom}))^n = x^n"
-  apply (subst power_abs [THEN sym])
+  apply (subst power_abs [symmetric])
   apply (simp add: zero_le_even_power)
-done
+  done
 
 lemma zero_less_power_nat_eq: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)"
-  by (induct n, auto)
+  by (induct n) auto
 
-lemma power_minus_even [simp]: "even n ==> 
+lemma power_minus_even [simp]: "even n ==>
     (- x)^n = (x^n::'a::{recpower,comm_ring_1})"
   apply (subst power_minus)
   apply simp
-done
+  done
 
-lemma power_minus_odd [simp]: "odd n ==> 
+lemma power_minus_odd [simp]: "odd n ==>
     (- x)^n = - (x^n::'a::{recpower,comm_ring_1})"
   apply (subst power_minus)
   apply simp
-done
+  done
 
-(* Simplify, when the exponent is a numeral *)
+
+text {* Simplify, when the exponent is a numeral *}
 
 lemmas power_0_left_number_of = power_0_left [of "number_of w", standard]
 declare power_0_left_number_of [simp]
 
-lemmas zero_le_power_eq_number_of =
+lemmas zero_le_power_eq_number_of [simp] =
     zero_le_power_eq [of _ "number_of w", standard]
-declare zero_le_power_eq_number_of [simp]
 
-lemmas zero_less_power_eq_number_of =
+lemmas zero_less_power_eq_number_of [simp] =
     zero_less_power_eq [of _ "number_of w", standard]
-declare zero_less_power_eq_number_of [simp]
 
-lemmas power_le_zero_eq_number_of =
+lemmas power_le_zero_eq_number_of [simp] =
     power_le_zero_eq [of _ "number_of w", standard]
-declare power_le_zero_eq_number_of [simp]
 
-lemmas power_less_zero_eq_number_of =
+lemmas power_less_zero_eq_number_of [simp] =
     power_less_zero_eq [of _ "number_of w", standard]
-declare power_less_zero_eq_number_of [simp]
 
-lemmas zero_less_power_nat_eq_number_of =
+lemmas zero_less_power_nat_eq_number_of [simp] =
     zero_less_power_nat_eq [of _ "number_of w", standard]
-declare zero_less_power_nat_eq_number_of [simp]
 
-lemmas power_eq_0_iff_number_of = power_eq_0_iff [of _ "number_of w", standard]
-declare power_eq_0_iff_number_of [simp]
+lemmas power_eq_0_iff_number_of [simp] = power_eq_0_iff [of _ "number_of w", standard]
 
-lemmas power_even_abs_number_of = power_even_abs [of "number_of w" _, standard]
-declare power_even_abs_number_of [simp]
+lemmas power_even_abs_number_of [simp] = power_even_abs [of "number_of w" _, standard]
 
 
 subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
 
 lemma even_power_le_0_imp_0:
-     "a ^ (2*k) \<le> (0::'a::{ordered_idom,recpower}) ==> a=0"
-apply (induct k) 
-apply (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc)  
-done
+    "a ^ (2*k) \<le> (0::'a::{ordered_idom,recpower}) ==> a=0"
+  by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc)
 
 lemma zero_le_power_iff:
-     "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)"
-      (is "?P n")
+  "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)"
 proof cases
   assume even: "even n"
   then obtain k where "n = 2*k"
     by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)
-  thus ?thesis by (simp add: zero_le_even_power even) 
+  thus ?thesis by (simp add: zero_le_even_power even)
 next
   assume odd: "odd n"
   then obtain k where "n = Suc(2*k)"
     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
   thus ?thesis
-    by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power 
-             dest!: even_power_le_0_imp_0) 
-qed 
+    by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power
+             dest!: even_power_le_0_imp_0)
+qed
+
 
 subsection {* Miscellaneous *}
 
@@ -429,20 +427,20 @@
   apply (simp add: even_def)
   done
 
-lemma div_Suc: "Suc a div c = a div c + Suc 0 div c + 
+lemma div_Suc: "Suc a div c = a div c + Suc 0 div c +
     (a mod c + Suc 0 mod c) div c"
   apply (subgoal_tac "Suc a = a + Suc 0")
   apply (erule ssubst)
   apply (rule div_add1_eq, simp)
   done
 
-lemma even_nat_plus_one_div_two: "even (x::nat) ==> 
-   (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)"
+lemma even_nat_plus_one_div_two: "even (x::nat) ==>
+    (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)"
   apply (subst div_Suc)
   apply (simp add: even_nat_equiv_def)
   done
 
-lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> 
+lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>
     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))"
   apply (subst div_Suc)
   apply (simp add: odd_nat_equiv_def)