# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1396565512 -3600
# Node ID 36489d77c4840735a168eb43cd049e33819101b1
# Parent  3610e0a7693cfc59c5405755fb5f5c8ad2fb1c96
removing simprule status for divide_minus_left and divide_minus_right

diff -r 3610e0a7693c -r 36489d77c484 src/HOL/Complex.thy
--- a/src/HOL/Complex.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/Complex.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -145,7 +145,7 @@
   by (simp add: complex_inverse_def)
 
 instance
-  by intro_classes (simp_all add: complex_mult_def
+  by intro_classes (simp_all add: complex_mult_def divide_minus_left
     distrib_left distrib_right right_diff_distrib left_diff_distrib
     complex_inverse_def complex_divide_def
     power2_eq_square add_divide_distrib [symmetric]
@@ -656,7 +656,7 @@
     moreover have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)"
       by (metis add_divide_distrib)
     ultimately show ?thesis using Complex False `0 < x\<^sup>2 + y\<^sup>2`
-      apply (simp add: complex_divide_def  zero_less_divide_iff less_divide_eq)
+      apply (simp add: complex_divide_def divide_minus_left zero_less_divide_iff less_divide_eq)
       apply (metis less_divide_eq mult_zero_left diff_conv_add_uminus minus_divide_left)
       done
   qed
@@ -844,7 +844,7 @@
     real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"
       apply (induct n)
       apply (simp add: cos_coeff_def sin_coeff_def)
-      apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)
+      apply (simp add: sin_coeff_Suc cos_coeff_Suc divide_minus_left del: mult_Suc)
       done } note * = this
   show "Re (cis b) = Re (exp (Complex 0 b))"
     unfolding exp_def cis_def cos_def
diff -r 3610e0a7693c -r 36489d77c484 src/HOL/Deriv.thy
--- a/src/HOL/Deriv.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/Deriv.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -825,7 +825,7 @@
 
 lemma DERIV_mirror:
   "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
-  by (simp add: DERIV_def filterlim_at_split filterlim_at_left_to_right
+  by (simp add: DERIV_def filterlim_at_split filterlim_at_left_to_right divide_minus_right
                 tendsto_minus_cancel_left field_simps conj_commute)
 
 text {* Caratheodory formulation of derivative at a point *}
@@ -908,8 +908,8 @@
     fix h::real
     assume "0 < h" "h < s"
     with all [of "-h"] show "f x < f (x-h)"
-    proof (simp add: abs_if pos_less_divide_eq split add: split_if_asm)
-      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
+    proof (simp add: abs_if pos_less_divide_eq divide_minus_right split add: split_if_asm)
+      assume "- ((f (x-h) - f x) / h) < l" and h: "0 < h"
       with l
       have "0 < (f (x-h) - f x) / h" by arith
       thus "f x < f (x-h)"
@@ -1628,7 +1628,8 @@
     ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
   ((\<lambda> x. f x / g x) ---> y) (at_left x)"
   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
-  by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
+  by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) 
+     (auto simp: DERIV_mirror divide_minus_left divide_minus_right)
 
 lemma lhopital:
   "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
@@ -1739,7 +1740,8 @@
     ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
     ((\<lambda> x. f x / g x) ---> y) (at_left x)"
   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
-  by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
+  by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"])
+     (auto simp: divide_minus_left divide_minus_right DERIV_mirror)
 
 lemma lhopital_at_top:
   "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
@@ -1796,7 +1798,7 @@
     unfolding filterlim_at_right_to_top
     apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
     using eventually_ge_at_top[where c="1::real"]
-    by eventually_elim simp
+    by eventually_elim (simp add: divide_minus_left divide_minus_right)
 qed
 
 end
diff -r 3610e0a7693c -r 36489d77c484 src/HOL/Fields.thy
--- a/src/HOL/Fields.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/Fields.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -152,11 +152,11 @@
 lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"
   by (simp add: divide_inverse nonzero_inverse_minus_eq)
 
-lemma divide_minus_left [simp]: "(-a) / b = - (a / b)"
+lemma divide_minus_left: "(-a) / b = - (a / b)"
   by (simp add: divide_inverse)
 
 lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
-  using add_divide_distrib [of a "- b" c] by simp
+  using add_divide_distrib [of a "- b" c] by (simp add: divide_inverse)
 
 lemma nonzero_eq_divide_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
 proof -
@@ -408,7 +408,7 @@
   "- (a / b) = a / - b"
   by (simp add: divide_inverse)
 
-lemma divide_minus_right [simp]:
+lemma divide_minus_right:
   "a / - b = - (a / b)"
   by (simp add: divide_inverse)
 
@@ -1045,13 +1045,13 @@
 lemma divide_right_mono_neg: "a <= b 
     ==> c <= 0 ==> b / c <= a / c"
 apply (drule divide_right_mono [of _ _ "- c"])
-apply auto
+apply (auto simp: divide_minus_right)
 done
 
 lemma divide_left_mono_neg: "a <= b 
     ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
   apply (drule divide_left_mono [of _ _ "- c"])
-  apply (auto simp add: mult_commute)
+  apply (auto simp add: divide_minus_left mult_commute)
 done
 
 lemma inverse_le_iff:
diff -r 3610e0a7693c -r 36489d77c484 src/HOL/Library/Convex.thy
--- a/src/HOL/Library/Convex.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/Library/Convex.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -656,7 +656,7 @@
 proof -
   have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)" using DERIV_log by auto
   then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
-    by (auto simp: DERIV_minus)
+    by (auto simp: divide_minus_left DERIV_minus)
   have "\<And>z :: real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
     using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
   from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
@@ -664,7 +664,7 @@
     DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
     by auto
   then have f''0: "\<And>z :: real. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
-    unfolding inverse_eq_divide by (auto simp add: mult_assoc)
+    by (auto simp add: inverse_eq_divide divide_minus_left mult_assoc)
   have f''_ge0: "\<And>z :: real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
     using `b > 1` by (auto intro!:less_imp_le simp add: divide_pos_pos[of 1] mult_pos_pos)
   from f''_ge0_imp_convex[OF pos_is_convex,
diff -r 3610e0a7693c -r 36489d77c484 src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy
--- a/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -998,7 +998,7 @@
            f (Suc n) u * (z-u) ^ n / of_nat (fact n) +
            f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / of_nat (fact (Suc n)) -
            f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / of_nat (fact (Suc n))"
-        using Suc by simp
+        using Suc by (simp add: divide_minus_left)
       also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / of_nat (fact (Suc n))"
       proof -
         have "of_nat(fact(Suc n)) *
diff -r 3610e0a7693c -r 36489d77c484 src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -2277,9 +2277,7 @@
       using Min_ge_iff[of i 0 ] and obt(1)
       unfolding t_def i_def
       using obt(4)[unfolded le_less]
-      apply auto
-      unfolding divide_le_0_iff
-      apply auto
+      apply (auto simp: divide_le_0_iff divide_minus_right)
       done
     have t: "\<forall>v\<in>s. u v + t * w v \<ge> 0"
     proof
@@ -2316,7 +2314,7 @@
 
     obtain a where "a \<in> s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
       using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
-    then have a: "a \<in> s" "u a + t * w a = 0" by auto
+    then have a: "a \<in> s" "u a + t * w a = 0" by (auto simp: divide_minus_right)
     have *: "\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
       unfolding setsum_diff1'[OF obt(1) `a\<in>s`] by auto
     have "(\<Sum>v\<in>s. u v + t * w v) = 1"
diff -r 3610e0a7693c -r 36489d77c484 src/HOL/Multivariate_Analysis/Linear_Algebra.thy
--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -1139,7 +1139,7 @@
       setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
       using fS vS uv by (simp add: setsum_diff1 divide_inverse field_simps)
     also have "\<dots> = ?a"
-      unfolding scaleR_right.setsum [symmetric] u using uv by simp
+      unfolding scaleR_right.setsum [symmetric] u using uv by (simp add: divide_minus_left)
     finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
     with th0 have ?lhs
       unfolding dependent_def span_explicit
@@ -2131,7 +2131,7 @@
     case False
     with span_mul[OF th, of "- 1/ k"]
     have th1: "f a \<in> span (f ` b)"
-      by auto
+      by (auto simp: divide_minus_left)
     from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
     have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
     from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
diff -r 3610e0a7693c -r 36489d77c484 src/HOL/NSA/HDeriv.thy
--- a/src/HOL/NSA/HDeriv.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/NSA/HDeriv.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -359,7 +359,7 @@
       have "inverse (- (h * star_of x) + - (star_of x * star_of x)) =
         (inverse (star_of x + h) - inverse (star_of x)) / h"
       apply (simp add: division_ring_inverse_diff nonzero_inverse_mult_distrib [symmetric]
-        nonzero_inverse_minus_eq [symmetric] ac_simps ring_distribs)
+        nonzero_inverse_minus_eq [symmetric] ac_simps ring_distribs divide_minus_left)
       apply (subst nonzero_inverse_minus_eq [symmetric])
       using distrib_right [symmetric, of h "star_of x" "star_of x"] apply simp
       apply (simp add: field_simps)
diff -r 3610e0a7693c -r 36489d77c484 src/HOL/Probability/Information.thy
--- a/src/HOL/Probability/Information.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/Probability/Information.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -945,7 +945,7 @@
   show "- (\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) =
     log b (measure MX A)"
     unfolding eq using uniform_distributed_params[OF X]
-    by (subst lebesgue_integral_cmult) (auto simp: measure_def)
+    by (subst lebesgue_integral_cmult) (auto simp: divide_minus_left measure_def)
 qed
 
 lemma (in information_space) entropy_simple_distributed:
diff -r 3610e0a7693c -r 36489d77c484 src/HOL/Probability/Radon_Nikodym.thy
--- a/src/HOL/Probability/Radon_Nikodym.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/Probability/Radon_Nikodym.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -241,7 +241,7 @@
       by (auto simp: finite_measure_restricted N.finite_measure_restricted sets_eq)
     from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X .. note X = this
     with S have "?P (S \<inter> X) S n"
-      by (simp add: measure_restricted sets_eq sets.Int) (metis inf_absorb2)
+      by (simp add: divide_minus_left measure_restricted sets_eq sets.Int) (metis inf_absorb2)
     hence "\<exists>A. ?P A S n" .. }
   note Ex_P = this
   def A \<equiv> "rec_nat (space M) (\<lambda>n A. SOME B. ?P B A n)"
@@ -280,7 +280,7 @@
       hence "0 < - ?d B" by auto
       from ex_inverse_of_nat_Suc_less[OF this]
       obtain n where *: "?d B < - 1 / real (Suc n)"
-        by (auto simp: real_eq_of_nat inverse_eq_divide field_simps)
+        by (auto simp: divide_minus_left real_eq_of_nat inverse_eq_divide field_simps)
       have "B \<subseteq> A (Suc n)" using B by (auto simp del: nat.rec(2))
       from epsilon[OF B(1) this] *
       show False by auto
diff -r 3610e0a7693c -r 36489d77c484 src/HOL/Rat.thy
--- a/src/HOL/Rat.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/Rat.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -665,7 +665,7 @@
   by transfer (simp add: add_frac_eq)
 
 lemma of_rat_minus: "of_rat (- a) = - of_rat a"
-  by transfer simp
+  by transfer (simp add: divide_minus_left)
 
 lemma of_rat_neg_one [simp]:
   "of_rat (- 1) = - 1"
diff -r 3610e0a7693c -r 36489d77c484 src/HOL/Real_Vector_Spaces.thy
--- a/src/HOL/Real_Vector_Spaces.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/Real_Vector_Spaces.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -1116,10 +1116,10 @@
 by (simp add: sgn_div_norm divide_inverse)
 
 lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
-unfolding real_sgn_eq by simp
+  by (rule sgn_pos)
 
 lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
-unfolding real_sgn_eq by simp
+  by (rule sgn_neg)
 
 lemma norm_conv_dist: "norm x = dist x 0"
   unfolding dist_norm by simp
diff -r 3610e0a7693c -r 36489d77c484 src/HOL/Transcendental.thy
--- a/src/HOL/Transcendental.thy	Fri Apr 04 16:43:47 2014 +0200
+++ b/src/HOL/Transcendental.thy	Thu Apr 03 23:51:52 2014 +0100
@@ -2145,7 +2145,7 @@
 
 lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
   unfolding cos_coeff_def sin_coeff_def
-  by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)
+  by (simp del: mult_Suc, auto simp add: divide_minus_left odd_Suc_mult_two_ex)
 
 lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
   unfolding sin_coeff_def
@@ -2169,7 +2169,7 @@
   by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
 
 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
-  by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
+  by (simp add: divide_minus_left diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
 
 text{*Now at last we can get the derivatives of exp, sin and cos*}
 
@@ -3239,7 +3239,7 @@
     assume "x \<in> {-1..1}"
     then show "x \<in> sin ` {- pi / 2..pi / 2}"
       using arcsin_lbound arcsin_ubound
-      by (intro image_eqI[where x="arcsin x"]) auto
+      by (intro image_eqI[where x="arcsin x"]) (auto simp: divide_minus_left)
   qed simp
   finally show ?thesis .
 qed
@@ -3338,12 +3338,14 @@
 
 lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- pi/2))"
   by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. - pi/2 < x \<and> x < pi/2" and P="\<lambda>x. True" and g=arctan])
-     (auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
+     (auto simp: le_less eventually_at dist_real_def divide_minus_left 
+           simp del: less_divide_eq_numeral1
            intro!: tan_monotone exI[of _ "pi/2"])
 
 lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"
   by (rule filterlim_at_top_at_left[where Q="\<lambda>x. - pi/2 < x \<and> x < pi/2" and P="\<lambda>x. True" and g=arctan])
-     (auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
+     (auto simp: le_less eventually_at dist_real_def divide_minus_left 
+           simp del: less_divide_eq_numeral1
            intro!: tan_monotone exI[of _ "pi/2"])
 
 lemma tendsto_arctan_at_top: "(arctan ---> (pi/2)) at_top"
@@ -3965,7 +3967,7 @@
   show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
     unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
     unfolding sgn_real_def
-    by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
+    by (simp add: divide_minus_left tan_def cos_arctan sin_arctan sin_diff cos_diff)
 qed
 
 theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")