author Andreas Lochbihler Tue Dec 01 17:18:34 2015 +0100 (2015-12-01) changeset 61767 f58d75535f66 parent 61766 507b39df1a57 parent 61763 96d2c1b9a30a child 61768 99f1eaf70c3d
merged
```     1.1 --- a/src/HOL/Complex.thy	Tue Dec 01 12:35:11 2015 +0100
1.2 +++ b/src/HOL/Complex.thy	Tue Dec 01 17:18:34 2015 +0100
1.3 @@ -748,9 +748,6 @@
1.4
1.5  subsubsection \<open>Complex exponential\<close>
1.6
1.7 -abbreviation Exp :: "complex \<Rightarrow> complex"
1.8 -  where "Exp \<equiv> exp"
1.9 -
1.10  lemma cis_conv_exp: "cis b = exp (\<i> * b)"
1.11  proof -
1.12    { fix n :: nat
1.13 @@ -766,29 +763,29 @@
1.14               intro!: sums_unique sums_add sums_mult sums_of_real)
1.15  qed
1.16
1.17 -lemma Exp_eq_polar: "Exp z = exp (Re z) * cis (Im z)"
1.18 +lemma exp_eq_polar: "exp z = exp (Re z) * cis (Im z)"
1.19    unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp
1.20
1.21  lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
1.22 -  unfolding Exp_eq_polar by simp
1.23 +  unfolding exp_eq_polar by simp
1.24
1.25  lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
1.26 -  unfolding Exp_eq_polar by simp
1.27 +  unfolding exp_eq_polar by simp
1.28
1.29  lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1"
1.30    by (simp add: norm_complex_def)
1.31
1.32  lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)"
1.33 -  by (simp add: cis.code cmod_complex_polar Exp_eq_polar)
1.34 +  by (simp add: cis.code cmod_complex_polar exp_eq_polar)
1.35
1.36 -lemma complex_Exp_Ex: "\<exists>a r. z = complex_of_real r * Exp a"
1.37 +lemma complex_exp_exists: "\<exists>a r. z = complex_of_real r * exp a"
1.38    apply (insert rcis_Ex [of z])
1.39 -  apply (auto simp add: Exp_eq_polar rcis_def mult.assoc [symmetric])
1.40 +  apply (auto simp add: exp_eq_polar rcis_def mult.assoc [symmetric])
1.41    apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
1.42    done
1.43
1.44 -lemma Exp_two_pi_i [simp]: "Exp((2::complex) * complex_of_real pi * ii) = 1"
1.45 -  by (simp add: Exp_eq_polar complex_eq_iff)
1.46 +lemma exp_two_pi_i [simp]: "exp(2 * complex_of_real pi * ii) = 1"
1.47 +  by (simp add: exp_eq_polar complex_eq_iff)
1.48
1.49  subsubsection \<open>Complex argument\<close>
1.50
```
```     2.1 --- a/src/HOL/Decision_Procs/MIR.thy	Tue Dec 01 12:35:11 2015 +0100
2.2 +++ b/src/HOL/Decision_Procs/MIR.thy	Tue Dec 01 17:18:34 2015 +0100
2.3 @@ -1644,9 +1644,8 @@
2.4    "(real_of_int (a::int) + b > 0) = (real_of_int a + real_of_int (floor b) > 0 \<or> (real_of_int a + real_of_int (floor b) = 0 \<and> real_of_int (floor b) - b < 0))"
2.5  proof-
2.6    have th: "(real_of_int a + b >0) = (real_of_int (-a) + (-b)< 0)" by arith
2.7 -  show ?thesis using myless[of _ "real_of_int (floor b)"]
2.8 -    by (simp only:th split_int_less_real'[where a="-a" and b="-b"])
2.9 -    (simp add: algebra_simps,arith)
2.10 +  show ?thesis
2.11 +    by (simp only:th split_int_less_real'[where a="-a" and b="-b"]) (auto simp add: algebra_simps)
2.12  qed
2.13
2.14  lemma split_int_le_real:
2.15 @@ -3765,8 +3764,7 @@
2.16  proof-
2.17    let ?ss = "s - real_of_int (floor s)"
2.18    from real_of_int_floor_add_one_gt[where r="s", simplified myless[of "s"]]
2.19 -    of_int_floor_le  have ss0:"?ss \<ge> 0" and ss1:"?ss < 1"
2.20 -    by (auto simp add: myle[of _ "s", symmetric] myless[of "?ss"])
2.21 +    of_int_floor_le  have ss0:"?ss \<ge> 0" and ss1:"?ss < 1" by (auto simp: floor_less_cancel)
2.22    from np have n0: "real_of_int n \<ge> 0" by simp
2.23    from mult_left_mono[OF up n0] mult_strict_left_mono[OF u1 np]
2.24    have nu0:"real_of_int n * u - s \<ge> -s" and nun:"real_of_int n * u -s < real_of_int n - s" by auto
2.25 @@ -4807,7 +4805,7 @@
2.26    shows "(Ifm bs (E p)) = (\<exists> (i::int). Ifm (real_of_int i#bs) (E (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (exsplit p))))" (is "?lhs = ?rhs")
2.27  proof-
2.28    have "?rhs = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm (x#(real_of_int i)#bs) (exsplit p))"
2.29 -    by (simp add: myless[of _ "1"] myless[of _ "0"] ac_simps)
2.30 +    by auto
2.31    also have "\<dots> = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm ((real_of_int i + x) #bs) p)"
2.32      by (simp only: exsplit[OF qf] ac_simps)
2.33    also have "\<dots> = (\<exists> x. Ifm (x#bs) p)"
```
```     3.1 --- a/src/HOL/Finite_Set.thy	Tue Dec 01 12:35:11 2015 +0100
3.2 +++ b/src/HOL/Finite_Set.thy	Tue Dec 01 17:18:34 2015 +0100
3.3 @@ -319,6 +319,16 @@
3.4    apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
3.5    done
3.6
3.7 +lemma finite_finite_vimage_IntI:
3.8 +  assumes "finite F" and "\<And>y. y \<in> F \<Longrightarrow> finite ((h -` {y}) \<inter> A)"
3.9 +  shows "finite (h -` F \<inter> A)"
3.10 +proof -
3.11 +  have *: "h -` F \<inter> A = (\<Union> y\<in>F. (h -` {y}) \<inter> A)"
3.12 +    by blast
3.13 +  show ?thesis
3.14 +    by (simp only: * assms finite_UN_I)
3.15 +qed
3.16 +
3.17  lemma finite_vimageI:
3.18    "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
3.19    using finite_vimage_IntI[of F h UNIV] by auto
```
```     4.1 --- a/src/HOL/Groups.thy	Tue Dec 01 12:35:11 2015 +0100
4.2 +++ b/src/HOL/Groups.thy	Tue Dec 01 17:18:34 2015 +0100
4.3 @@ -999,6 +999,9 @@
4.4  apply (simp add: algebra_simps)
4.5  done
4.6
4.7 +lemma diff_gt_0_iff_gt [simp]: "a - b > 0 \<longleftrightarrow> a > b"
4.8 +by (simp add: less_diff_eq)
4.9 +
4.10  lemma diff_le_eq[algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
4.11  by (auto simp add: le_less diff_less_eq )
4.12
```
```     5.1 --- a/src/HOL/Inequalities.thy	Tue Dec 01 12:35:11 2015 +0100
5.2 +++ b/src/HOL/Inequalities.thy	Tue Dec 01 17:18:34 2015 +0100
5.3 @@ -66,7 +66,6 @@
5.4        using assms by (cases "i \<le> j") (auto simp: algebra_simps)
5.5    } hence "?S \<le> 0"
5.6      by (auto intro!: setsum_nonpos simp: mult_le_0_iff)
5.7 -       (auto simp: field_simps)
5.8    finally show ?thesis by (simp add: algebra_simps)
5.9  qed
5.10
```
```     6.1 --- a/src/HOL/Library/BigO.thy	Tue Dec 01 12:35:11 2015 +0100
6.2 +++ b/src/HOL/Library/BigO.thy	Tue Dec 01 17:18:34 2015 +0100
6.3 @@ -200,8 +200,6 @@
6.4    apply (auto simp add: fun_Compl_def func_plus)
6.5    apply (drule_tac x = x in spec)+
6.6    apply force
6.7 -  apply (drule_tac x = x in spec)+
6.8 -  apply force
6.9    done
6.10
6.11  lemma bigo_abs: "(\<lambda>x. abs (f x)) =o O(f)"
```
```     7.1 --- a/src/HOL/Library/Float.thy	Tue Dec 01 12:35:11 2015 +0100
7.2 +++ b/src/HOL/Library/Float.thy	Tue Dec 01 17:18:34 2015 +0100
7.3 @@ -1116,10 +1116,11 @@
7.4  proof -
7.5    have "0 \<le> log 2 x - real_of_int \<lfloor>log 2 x\<rfloor>"
7.6      by (simp add: algebra_simps)
7.7 -  from this assms
7.8 +  with assms
7.9    show ?thesis
7.10 -    by (auto simp: truncate_down_def round_down_def mult_powr_eq
7.11 +    apply (auto simp: truncate_down_def round_down_def mult_powr_eq
7.12        intro!: ge_one_powr_ge_zero mult_pos_pos)
7.13 +    by linarith
7.14  qed
7.15
7.16  lemma truncate_down_nonneg: "0 \<le> y \<Longrightarrow> 0 \<le> truncate_down prec y"
```
```     8.1 --- a/src/HOL/Library/Infinite_Set.thy	Tue Dec 01 12:35:11 2015 +0100
8.2 +++ b/src/HOL/Library/Infinite_Set.thy	Tue Dec 01 17:18:34 2015 +0100
8.3 @@ -72,10 +72,7 @@
8.4      by(induction rule: finite_psubset_induct)(meson Diff_subset card_Diff1_less card_psubset finite_Diff step)
8.5  qed
8.6
8.7 -text \<open>
8.8 -  As a concrete example, we prove that the set of natural numbers is
8.9 -  infinite.
8.10 -\<close>
8.11 +text \<open>As a concrete example, we prove that the set of natural numbers is infinite.\<close>
8.12
8.13  lemma infinite_nat_iff_unbounded_le: "infinite (S::nat set) \<longleftrightarrow> (\<forall>m. \<exists>n\<ge>m. n \<in> S)"
8.14    using frequently_cofinite[of "\<lambda>x. x \<in> S"]
8.15 @@ -94,6 +91,7 @@
8.16  lemma finite_nat_bounded: "finite (S::nat set) \<Longrightarrow> \<exists>k. S \<subseteq> {..<k}"
8.17    by (simp add: finite_nat_iff_bounded)
8.18
8.19 +
8.20  text \<open>
8.21    For a set of natural numbers to be infinite, it is enough to know
8.22    that for any number larger than some \<open>k\<close>, there is some larger
8.23 @@ -150,6 +148,32 @@
8.24    obtains y where "y \<in> f`A" and "infinite (f -` {y})"
8.25    using assms by (blast dest: inf_img_fin_dom)
8.26
8.27 +proposition finite_image_absD:
8.28 +    fixes S :: "'a::linordered_ring set"
8.29 +    shows "finite (abs ` S) \<Longrightarrow> finite S"
8.30 +  by (rule ccontr) (auto simp: abs_eq_iff vimage_def dest: inf_img_fin_dom)
8.31 +
8.32 +text \<open>The set of integers is also infinite.\<close>
8.33 +
8.34 +lemma infinite_int_iff_infinite_nat_abs: "infinite (S::int set) \<longleftrightarrow> infinite ((nat o abs) ` S)"
8.35 +  by (auto simp: transfer_nat_int_set_relations o_def image_comp dest: finite_image_absD)
8.36 +
8.37 +proposition infinite_int_iff_unbounded_le: "infinite (S::int set) \<longleftrightarrow> (\<forall>m. \<exists>n. abs n \<ge> m \<and> n \<in> S)"
8.38 +  apply (simp add: infinite_int_iff_infinite_nat_abs infinite_nat_iff_unbounded_le o_def image_def)
8.39 +  apply (metis abs_ge_zero nat_le_eq_zle le_nat_iff)
8.40 +  done
8.41 +
8.42 +proposition infinite_int_iff_unbounded: "infinite (S::int set) \<longleftrightarrow> (\<forall>m. \<exists>n. abs n > m \<and> n \<in> S)"
8.43 +  apply (simp add: infinite_int_iff_infinite_nat_abs infinite_nat_iff_unbounded o_def image_def)
8.44 +  apply (metis (full_types) nat_le_iff nat_mono not_le)
8.45 +  done
8.46 +
8.47 +proposition finite_int_iff_bounded: "finite (S::int set) \<longleftrightarrow> (\<exists>k. abs ` S \<subseteq> {..<k})"
8.48 +  using infinite_int_iff_unbounded_le[of S] by (simp add: subset_eq) (metis not_le)
8.49 +
8.50 +proposition finite_int_iff_bounded_le: "finite (S::int set) \<longleftrightarrow> (\<exists>k. abs ` S \<subseteq> {.. k})"
8.51 +  using infinite_int_iff_unbounded[of S] by (simp add: subset_eq) (metis not_le)
8.52 +
8.53  subsection "Infinitely Many and Almost All"
8.54
8.55  text \<open>
8.56 @@ -385,24 +409,5 @@
8.57      unfolding bij_betw_def by (auto intro: enumerate_in_set)
8.58  qed
8.59
8.60 -subsection "Miscellaneous"
8.61 -
8.62 -text \<open>
8.63 -  A few trivial lemmas about sets that contain at most one element.
8.64 -  These simplify the reasoning about deterministic automata.
8.65 -\<close>
8.66 -
8.67 -definition atmost_one :: "'a set \<Rightarrow> bool"
8.68 -  where "atmost_one S \<longleftrightarrow> (\<forall>x y. x\<in>S \<and> y\<in>S \<longrightarrow> x = y)"
8.69 -
8.70 -lemma atmost_one_empty: "S = {} \<Longrightarrow> atmost_one S"
8.71 -  by (simp add: atmost_one_def)
8.72 -
8.73 -lemma atmost_one_singleton: "S = {x} \<Longrightarrow> atmost_one S"
8.74 -  by (simp add: atmost_one_def)
8.75 -
8.76 -lemma atmost_one_unique [elim]: "atmost_one S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> y = x"
8.77 -  by (simp add: atmost_one_def)
8.78 -
8.79  end
8.80
```
```     9.1 --- a/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Tue Dec 01 12:35:11 2015 +0100
9.2 +++ b/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Tue Dec 01 17:18:34 2015 +0100
9.3 @@ -3012,7 +3012,7 @@
9.4                   \<subseteq> ball (p t) (ee (p t))"
9.5              apply (intro subset_path_image_join pi_hgn pi_ghn')
9.6              using \<open>N>0\<close> Suc.prems
9.7 -            apply (auto simp: dist_norm field_simps closed_segment_eq_real_ivl ptgh_ee)
9.8 +            apply (auto simp: path_image_subpath dist_norm field_simps closed_segment_eq_real_ivl ptgh_ee)
9.9              done
9.10            have pi0: "(f has_contour_integral 0)
9.11                         (subpath (n/ N) ((Suc n)/N) g +++ linepath(g ((Suc n) / N)) (h((Suc n) / N)) +++
9.12 @@ -3492,7 +3492,7 @@
9.13    by (simp add: winding_number_valid_path)
9.14
9.15  lemma winding_number_subpath_trivial [simp]: "z \<noteq> g x \<Longrightarrow> winding_number (subpath x x g) z = 0"
9.16 -  by (simp add: winding_number_valid_path)
9.17 +  by (simp add: path_image_subpath winding_number_valid_path)
9.18
9.19  lemma winding_number_join:
9.20    assumes g1: "path g1" "z \<notin> path_image g1"
9.21 @@ -3742,7 +3742,7 @@
9.22          by (rule continuous_at_imp_continuous_within)
9.23        have gdx: "\<gamma> differentiable at x"
9.24          using x by (simp add: g_diff_at)
9.25 -      have "((\<lambda>c. Exp (- integral {a..c} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))) * (\<gamma> c - z)) has_derivative (\<lambda>h. 0))
9.26 +      have "((\<lambda>c. exp (- integral {a..c} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))) * (\<gamma> c - z)) has_derivative (\<lambda>h. 0))
9.27            (at x within {a..b})"
9.28          using x gdx t
9.29          apply (clarsimp simp add: differentiable_iff_scaleR)
9.30 @@ -3781,7 +3781,7 @@
9.31                      "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
9.32                      "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
9.33      using winding_number [OF assms, of 1] by auto
9.34 -  have [simp]: "(winding_number \<gamma> z \<in> \<int>) = (Exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)"
9.35 +  have [simp]: "(winding_number \<gamma> z \<in> \<int>) = (exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)"
9.36        using p by (simp add: exp_eq_1 complex_is_Int_iff)
9.37    have "winding_number p z \<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p"
9.38      using p z
9.39 @@ -3840,7 +3840,7 @@
9.40      using eqArg by (simp add: i_def)
9.41    have gpdt: "\<gamma> piecewise_C1_differentiable_on {0..t}"
9.42      by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl piecewise_C1_differentiable_on_subset gpd t)
9.43 -  have "Exp (- i) * (\<gamma> t - z) = \<gamma> 0 - z"
9.44 +  have "exp (- i) * (\<gamma> t - z) = \<gamma> 0 - z"
9.45      unfolding i_def
9.46      apply (rule winding_number_exp_integral [OF gpdt])
9.47      using t z unfolding path_image_def
9.48 @@ -3855,7 +3855,7 @@
9.49      apply (subst Complex_Transcendental.Arg_eq [of r])
9.50      apply (simp add: iArg)
9.51      using *
9.52 -    apply (simp add: Exp_eq_polar field_simps)
9.53 +    apply (simp add: exp_eq_polar field_simps)
9.54      done
9.55    with t show ?thesis
9.56      by (rule_tac x="exp(Re i) / norm r" in exI) (auto simp: path_image_def)
9.57 @@ -4225,8 +4225,8 @@
9.58      also have "... = winding_number (subpath 0 x \<gamma>) z"
9.59        apply (subst winding_number_valid_path)
9.60        using assms x
9.61 -      apply (simp_all add: valid_path_subpath)
9.62 -      by (force simp: closed_segment_eq_real_ivl path_image_def)
9.63 +      apply (simp_all add: path_image_subpath valid_path_subpath)
9.64 +      by (force simp: path_image_def)
9.65      finally show ?thesis .
9.66    qed
9.67    show ?thesis
9.68 @@ -4277,7 +4277,7 @@
9.69      have gt: "\<gamma> t - z = - (of_real (exp (- (2 * pi * Im (winding_number (subpath 0 t \<gamma>) z)))) * (\<gamma> 0 - z))"
9.70        using winding_number_exp_2pi [of "subpath 0 t \<gamma>" z]
9.71        apply (simp add: t \<gamma> valid_path_imp_path)
9.72 -      using closed_segment_eq_real_ivl path_image_def t z by (fastforce simp add: Euler sub12)
9.73 +      using closed_segment_eq_real_ivl path_image_def t z by (fastforce simp: path_image_subpath Euler sub12)
9.74      have "b < a \<bullet> \<gamma> 0"
9.75      proof -
9.76        have "\<gamma> 0 \<in> {c. b < a \<bullet> c}"
9.77 @@ -4321,7 +4321,7 @@
9.78      have "isCont (winding_number \<gamma>) z"
9.79        by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z)
9.80      then obtain d where "d>0" and d: "\<And>x'. dist x' z < d \<Longrightarrow> dist (winding_number \<gamma> x') (winding_number \<gamma> z) < abs(Re(winding_number \<gamma> z)) - 1/2"
9.81 -      using continuous_at_eps_delta wnz_12 diff_less_iff(1) by blast
9.82 +      using continuous_at_eps_delta wnz_12 diff_gt_0_iff_gt by blast
9.83      def z' \<equiv> "z - (d / (2 * cmod a)) *\<^sub>R a"
9.84      have *: "a \<bullet> z' \<le> b - d / 3 * cmod a"
9.85        unfolding z'_def inner_mult_right' divide_inverse
```
```    10.1 --- a/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Tue Dec 01 12:35:11 2015 +0100
10.2 +++ b/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Tue Dec 01 17:18:34 2015 +0100
10.3 @@ -654,7 +654,6 @@
10.4      done
10.5  qed
10.6
10.7 -
10.8  corollary
10.9    shows Arg_ge_0: "0 \<le> Arg z"
10.10      and Arg_lt_2pi: "Arg z < 2*pi"
10.11 @@ -772,7 +771,7 @@
10.12  lemma Arg_inverse: "Arg(inverse z) = (if z \<in> \<real> \<and> 0 \<le> Re z then Arg z else 2*pi - Arg z)"
10.13    apply (cases "z=0", simp)
10.14    apply (rule Arg_unique [of "inverse (norm z)"])
10.15 -  using Arg_ge_0 [of z] Arg_lt_2pi [of z] Arg_eq [of z] Arg_eq_0 [of z] Exp_two_pi_i
10.16 +  using Arg_ge_0 [of z] Arg_lt_2pi [of z] Arg_eq [of z] Arg_eq_0 [of z] exp_two_pi_i
10.17    apply (auto simp: of_real_numeral algebra_simps exp_diff divide_simps)
10.18    done
10.19
10.20 @@ -849,8 +848,11 @@
10.21    by auto
10.22
10.23  lemma Arg_exp: "0 \<le> Im z \<Longrightarrow> Im z < 2*pi \<Longrightarrow> Arg(exp z) = Im z"
10.24 -  by (rule Arg_unique [of  "exp(Re z)"]) (auto simp: Exp_eq_polar)
10.25 -
10.26 +  by (rule Arg_unique [of  "exp(Re z)"]) (auto simp: exp_eq_polar)
10.27 +
10.28 +lemma complex_split_polar:
10.29 +  obtains r a::real where "z = complex_of_real r * (cos a + \<i> * sin a)" "0 \<le> r" "0 \<le> a" "a < 2*pi"
10.30 +  using Arg cis.ctr cis_conv_exp by fastforce
10.31
10.32  subsection\<open>Analytic properties of tangent function\<close>
10.33
10.34 @@ -898,7 +900,7 @@
10.35    have "exp(ln z) = z & -pi < Im(ln z) & Im(ln z) \<le> pi" unfolding ln_complex_def
10.36      apply (rule theI [where a = "Complex (ln(norm z)) \<phi>"])
10.37      using z assms \<phi>
10.38 -    apply (auto simp: field_simps exp_complex_eqI Exp_eq_polar cis.code)
10.39 +    apply (auto simp: field_simps exp_complex_eqI exp_eq_polar cis.code)
10.40      done
10.41    then show "exp(ln z) = z" "-pi < Im(ln z)" "Im(ln z) \<le> pi"
10.42      by auto
10.43 @@ -1516,14 +1518,14 @@
10.44    shows "((\<lambda>z. z powr r :: complex) has_field_derivative r * z powr (r - 1)) (at z)"
10.45  proof (subst DERIV_cong_ev[OF refl _ refl])
10.46    from assms have "eventually (\<lambda>z. z \<noteq> 0) (nhds z)" by (intro t1_space_nhds) auto
10.47 -  thus "eventually (\<lambda>z. z powr r = Exp (r * Ln z)) (nhds z)"
10.48 +  thus "eventually (\<lambda>z. z powr r = exp (r * Ln z)) (nhds z)"
10.49      unfolding powr_def by eventually_elim simp
10.50
10.51 -  have "((\<lambda>z. Exp (r * Ln z)) has_field_derivative Exp (r * Ln z) * (inverse z * r)) (at z)"
10.52 +  have "((\<lambda>z. exp (r * Ln z)) has_field_derivative exp (r * Ln z) * (inverse z * r)) (at z)"
10.53      using assms by (auto intro!: derivative_eq_intros has_field_derivative_powr)
10.54 -  also have "Exp (r * Ln z) * (inverse z * r) = r * z powr (r - 1)"
10.55 +  also have "exp (r * Ln z) * (inverse z * r) = r * z powr (r - 1)"
10.56      unfolding powr_def by (simp add: assms exp_diff field_simps)
10.57 -  finally show "((\<lambda>z. Exp (r * Ln z)) has_field_derivative r * z powr (r - 1)) (at z)"
10.58 +  finally show "((\<lambda>z. exp (r * Ln z)) has_field_derivative r * z powr (r - 1)) (at z)"
10.59      by simp
10.60  qed
10.61
10.62 @@ -2405,7 +2407,7 @@
10.63
10.64  lemma Re_Arcsin_bound: "abs(Re(Arcsin z)) \<le> pi"
10.65    by (meson Re_Arcsin_bounds abs_le_iff less_eq_real_def minus_less_iff)
10.66 -
10.67 +
10.68
10.69  subsection\<open>Interrelations between Arcsin and Arccos\<close>
10.70
10.71 @@ -2481,7 +2483,6 @@
10.72    apply (simp add: cos_squared_eq)
10.73    using assms
10.74    apply (auto simp: Re_cos Im_cos add_pos_pos mult_le_0_iff zero_le_mult_iff)
10.75 -  apply (auto simp: algebra_simps)
10.76    done
10.77
10.78  lemma sin_cos_csqrt:
10.79 @@ -2491,7 +2492,6 @@
10.80    apply (simp add: sin_squared_eq)
10.81    using assms
10.82    apply (auto simp: Re_sin Im_sin add_pos_pos mult_le_0_iff zero_le_mult_iff)
10.83 -  apply (auto simp: algebra_simps)
10.84    done
10.85
10.86  lemma Arcsin_Arccos_csqrt_pos:
10.87 @@ -2661,7 +2661,7 @@
10.88    by ( simp add: of_real_sqrt del: csqrt_of_real_nonneg)
10.89
10.90  lemma arcsin_arccos_sqrt_pos: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin x = arccos(sqrt(1 - x\<^sup>2))"
10.91 -  apply (simp add: abs_square_le_1 diff_le_iff arcsin_eq_Re_Arcsin arccos_eq_Re_Arccos)
10.92 +  apply (simp add: abs_square_le_1 arcsin_eq_Re_Arcsin arccos_eq_Re_Arccos)
10.93    apply (subst Arcsin_Arccos_csqrt_pos)
10.94    apply (auto simp: power_le_one csqrt_1_diff_eq)
10.95    done
10.96 @@ -2671,7 +2671,7 @@
10.97    by (simp add: arcsin_minus)
10.98
10.99  lemma arccos_arcsin_sqrt_pos: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos x = arcsin(sqrt(1 - x\<^sup>2))"
10.100 -  apply (simp add: abs_square_le_1 diff_le_iff arcsin_eq_Re_Arcsin arccos_eq_Re_Arccos)
10.101 +  apply (simp add: abs_square_le_1 arcsin_eq_Re_Arcsin arccos_eq_Re_Arccos)
10.102    apply (subst Arccos_Arcsin_csqrt_pos)
10.103    apply (auto simp: power_le_one csqrt_1_diff_eq)
10.104    done
```
```    11.1 --- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Tue Dec 01 12:35:11 2015 +0100
11.2 +++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Tue Dec 01 17:18:34 2015 +0100
11.3 @@ -4397,7 +4397,7 @@
11.4      using \<open>y \<in> s\<close>
11.5    proof -
11.6      show "inner (y - z) z < inner (y - z) y"
11.7 -      apply (subst diff_less_iff(1)[symmetric])
11.8 +      apply (subst diff_gt_0_iff_gt [symmetric])
11.9        unfolding inner_diff_right[symmetric] and inner_gt_zero_iff
11.10        using \<open>y\<in>s\<close> \<open>z\<notin>s\<close>
11.11        apply auto
```
```    12.1 --- a/src/HOL/Multivariate_Analysis/Derivative.thy	Tue Dec 01 12:35:11 2015 +0100
12.2 +++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Tue Dec 01 17:18:34 2015 +0100
12.3 @@ -688,7 +688,7 @@
12.4      "x \<in> {a <..< b}"
12.5      "(\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)" ..
12.6    then show ?thesis
12.7 -    by (metis (hide_lams) assms(1) diff_less_iff(1) eq_iff_diff_eq_0
12.8 +    by (metis (hide_lams) assms(1) diff_gt_0_iff_gt eq_iff_diff_eq_0
12.9        zero_less_mult_iff nonzero_mult_divide_cancel_right not_real_square_gt_zero
12.10        times_divide_eq_left)
12.11  qed
```
```    13.1 --- a/src/HOL/Multivariate_Analysis/Integration.thy	Tue Dec 01 12:35:11 2015 +0100
13.2 +++ b/src/HOL/Multivariate_Analysis/Integration.thy	Tue Dec 01 17:18:34 2015 +0100
13.3 @@ -599,7 +599,7 @@
13.4    then have cd_ne: "\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i"
13.5      using assms unfolding box_ne_empty by auto
13.6    have "\<And>i. i \<in> Basis \<Longrightarrow> 0 \<le> b \<bullet> i - a \<bullet> i"
13.7 -    using ab_ne by (metis diff_le_iff(1))
13.8 +    using ab_ne by auto
13.9    moreover
13.10    have "\<And>i. i \<in> Basis \<Longrightarrow> b \<bullet> i - a \<bullet> i \<le> d \<bullet> i - c \<bullet> i"
13.11      using assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(2)]
```
```    14.1 --- a/src/HOL/Multivariate_Analysis/Path_Connected.thy	Tue Dec 01 12:35:11 2015 +0100
14.2 +++ b/src/HOL/Multivariate_Analysis/Path_Connected.thy	Tue Dec 01 17:18:34 2015 +0100
14.3 @@ -375,7 +375,7 @@
14.4  lemma path_compose_reversepath: "f o reversepath p = reversepath(f o p)"
14.5    by (rule ext) (simp add: reversepath_def)
14.6
14.7 -lemma join_paths_eq:
14.8 +lemma joinpaths_eq:
14.9    "(\<And>t. t \<in> {0..1} \<Longrightarrow> p t = p' t) \<Longrightarrow>
14.10     (\<And>t. t \<in> {0..1} \<Longrightarrow> q t = q' t)
14.11     \<Longrightarrow>  t \<in> {0..1} \<Longrightarrow> (p +++ q) t = (p' +++ q') t"
14.12 @@ -453,8 +453,6 @@
14.13  lemma path_join_imp: "\<lbrakk>path g1; path g2; pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> path(g1 +++ g2)"
14.14    by (simp add: path_join)
14.15
14.16 -lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join
14.17 -
14.18  lemma simple_path_join_loop:
14.19    assumes "arc g1" "arc g2"
14.20            "pathfinish g1 = pathstart g2"  "pathfinish g2 = pathstart g1"
14.21 @@ -563,18 +561,18 @@
14.22  definition subpath :: "real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a::real_normed_vector"
14.23    where "subpath a b g \<equiv> \<lambda>x. g((b - a) * x + a)"
14.24
14.25 -lemma path_image_subpath_gen [simp]:
14.26 -  fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
14.27 +lemma path_image_subpath_gen:
14.28 +  fixes g :: "_ \<Rightarrow> 'a::real_normed_vector"
14.29    shows "path_image(subpath u v g) = g ` (closed_segment u v)"
14.30    apply (simp add: closed_segment_real_eq path_image_def subpath_def)
14.31    apply (subst o_def [of g, symmetric])
14.32    apply (simp add: image_comp [symmetric])
14.33    done
14.34
14.35 -lemma path_image_subpath [simp]:
14.36 +lemma path_image_subpath:
14.37    fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
14.38    shows "path_image(subpath u v g) = (if u \<le> v then g ` {u..v} else g ` {v..u})"
14.39 -  by (simp add: closed_segment_eq_real_ivl)
14.40 +  by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl)
14.41
14.42  lemma path_subpath [simp]:
14.43    fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
14.44 @@ -614,7 +612,7 @@
14.45
14.46  lemma affine_ineq:
14.47    fixes x :: "'a::linordered_idom"
14.48 -  assumes "x \<le> 1" "v < u"
14.49 +  assumes "x \<le> 1" "v \<le> u"
14.50      shows "v + x * u \<le> u + x * v"
14.51  proof -
14.52    have "(1-x)*(u-v) \<ge> 0"
14.53 @@ -726,7 +724,7 @@
14.54
14.55  lemma path_image_subpath_subset:
14.56      "\<lbrakk>path g; u \<in> {0..1}; v \<in> {0..1}\<rbrakk> \<Longrightarrow> path_image(subpath u v g) \<subseteq> path_image g"
14.57 -  apply (simp add: closed_segment_real_eq image_affinity_atLeastAtMost)
14.58 +  apply (simp add: closed_segment_real_eq image_affinity_atLeastAtMost path_image_subpath)
14.59    apply (auto simp: path_image_def)
14.60    done
14.61
14.62 @@ -805,7 +803,7 @@
14.63      apply (rule that [OF `0 \<le> u` `u \<le> 1`])
14.64      apply (metis DiffI disj frontier_def g0 notin pathstart_def)
14.65      using `0 \<le> u` g0 disj
14.66 -    apply (simp add:)
14.67 +    apply (simp add: path_image_subpath_gen)
14.68      apply (auto simp: closed_segment_eq_real_ivl pathstart_def pathfinish_def subpath_def)
14.69      apply (rename_tac y)
14.70      apply (drule_tac x="y/u" in spec)
14.71 @@ -825,7 +823,7 @@
14.72    show ?thesis
14.73      apply (rule that [of "subpath 0 u g"])
14.74      using assms u
14.75 -    apply simp_all
14.76 +    apply (simp_all add: path_image_subpath)
14.77      apply (simp add: pathstart_def)
14.78      apply (force simp: closed_segment_eq_real_ivl path_image_def)
14.79      done
14.80 @@ -966,7 +964,7 @@
14.81    unfolding linepath_def
14.82    by (intro continuous_intros)
14.83
14.84 -lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)"
14.85 +lemma continuous_on_linepath [intro,continuous_intros]: "continuous_on s (linepath a b)"
14.86    using continuous_linepath_at
14.87    by (auto intro!: continuous_at_imp_continuous_on)
14.88
14.89 @@ -982,6 +980,9 @@
14.90    unfolding reversepath_def linepath_def
14.91    by auto
14.92
14.93 +lemma linepath_0 [simp]: "linepath 0 b x = x *\<^sub>R b"
14.94 +  by (simp add: linepath_def)
14.95 +
14.96  lemma arc_linepath:
14.97    assumes "a \<noteq> b"
14.98    shows "arc (linepath a b)"
14.99 @@ -1566,7 +1567,7 @@
14.100        have CC: "1 \<le> 1 + (C - 1) * u"
14.101          using `x \<noteq> a` `0 \<le> u`
14.102          apply (simp add: C_def divide_simps norm_minus_commute)
14.103 -        by (metis Bx diff_le_iff(1) less_eq_real_def mult_nonneg_nonneg)
14.104 +        using Bx by auto
14.105        have *: "\<And>v. (1 - u) *\<^sub>R x + u *\<^sub>R (a + v *\<^sub>R (x - a)) = a + (1 + (v - 1) * u) *\<^sub>R (x - a)"
14.106          by (simp add: algebra_simps)
14.107        have "a + ((1 / (1 + C * u - u)) *\<^sub>R x + ((u / (1 + C * u - u)) *\<^sub>R a + (C * u / (1 + C * u - u)) *\<^sub>R x)) =
14.108 @@ -1601,7 +1602,7 @@
14.109        have DD: "1 \<le> 1 + (D - 1) * u"
14.110          using `y \<noteq> a` `0 \<le> u`
14.111          apply (simp add: D_def divide_simps norm_minus_commute)
14.112 -        by (metis By diff_le_iff(1) less_eq_real_def mult_nonneg_nonneg)
14.113 +        using By by auto
14.114        have *: "\<And>v. (1 - u) *\<^sub>R y + u *\<^sub>R (a + v *\<^sub>R (y - a)) = a + (1 + (v - 1) * u) *\<^sub>R (y - a)"
14.115          by (simp add: algebra_simps)
14.116        have "a + ((1 / (1 + D * u - u)) *\<^sub>R y + ((u / (1 + D * u - u)) *\<^sub>R a + (D * u / (1 + D * u - u)) *\<^sub>R y)) =
14.117 @@ -2793,7 +2794,7 @@
14.118  proposition homotopic_paths_sym_eq: "homotopic_paths s p q \<longleftrightarrow> homotopic_paths s q p"
14.119    by (metis homotopic_paths_sym)
14.120
14.121 -proposition homotopic_paths_trans:
14.122 +proposition homotopic_paths_trans [trans]:
14.123       "\<lbrakk>homotopic_paths s p q; homotopic_paths s q r\<rbrakk> \<Longrightarrow> homotopic_paths s p r"
14.124    apply (simp add: homotopic_paths_def)
14.125    apply (rule homotopic_with_trans, assumption)
14.126 @@ -3262,4 +3263,83 @@
14.127      by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
14.128  qed
14.129
14.130 +subsection\<open> Homotopy and subpaths\<close>
14.131 +
14.132 +lemma homotopic_join_subpaths1:
14.133 +  assumes "path g" and pag: "path_image g \<subseteq> s"
14.134 +      and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}" "u \<le> v" "v \<le> w"
14.135 +    shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
14.136 +proof -
14.137 +  have 1: "t * 2 \<le> 1 \<Longrightarrow> u + t * (v * 2) \<le> v + t * (u * 2)" for t
14.138 +    using affine_ineq \<open>u \<le> v\<close> by fastforce
14.139 +  have 2: "t * 2 > 1 \<Longrightarrow> u + (2*t - 1) * v \<le> v + (2*t - 1) * w" for t
14.140 +    by (metis add_mono_thms_linordered_semiring(1) diff_gt_0_iff_gt less_eq_real_def mult.commute mult_right_mono \<open>u \<le> v\<close> \<open>v \<le> w\<close>)
14.141 +  have t2: "\<And>t::real. t*2 = 1 \<Longrightarrow> t = 1/2" by auto
14.142 +  show ?thesis
14.143 +    apply (rule homotopic_paths_subset [OF _ pag])
14.144 +    using assms
14.145 +    apply (cases "w = u")
14.146 +    using homotopic_paths_rinv [of "subpath u v g" "path_image g"]
14.147 +    apply (force simp: closed_segment_eq_real_ivl image_mono path_image_def subpath_refl)
14.148 +      apply (rule homotopic_paths_sym)
14.149 +      apply (rule homotopic_paths_reparametrize
14.150 +             [where f = "\<lambda>t. if  t \<le> 1 / 2
14.151 +                             then inverse((w - u)) *\<^sub>R (2 * (v - u)) *\<^sub>R t
14.152 +                             else inverse((w - u)) *\<^sub>R ((v - u) + (w - v) *\<^sub>R (2 *\<^sub>R t - 1))"])
14.153 +      using \<open>path g\<close> path_subpath u w apply blast
14.154 +      using \<open>path g\<close> path_image_subpath_subset u w(1) apply blast
14.155 +      apply simp_all
14.156 +      apply (subst split_01)
14.157 +      apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
14.158 +      apply (simp_all add: field_simps not_le)
14.159 +      apply (force dest!: t2)
14.160 +      apply (force simp: algebra_simps mult_left_mono affine_ineq dest!: 1 2)
14.161 +      apply (simp add: joinpaths_def subpath_def)
14.162 +      apply (force simp: algebra_simps)
14.163 +      done
14.164 +qed
14.165 +
14.166 +lemma homotopic_join_subpaths2:
14.167 +  assumes "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
14.168 +    shows "homotopic_paths s (subpath w v g +++ subpath v u g) (subpath w u g)"
14.169 +by (metis assms homotopic_paths_reversepath_D pathfinish_subpath pathstart_subpath reversepath_joinpaths reversepath_subpath)
14.170 +
14.171 +lemma homotopic_join_subpaths3:
14.172 +  assumes hom: "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
14.173 +      and "path g" and pag: "path_image g \<subseteq> s"
14.174 +      and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}"
14.175 +    shows "homotopic_paths s (subpath v w g +++ subpath w u g) (subpath v u g)"
14.176 +proof -
14.177 +  have "homotopic_paths s (subpath u w g +++ subpath w v g) ((subpath u v g +++ subpath v w g) +++ subpath w v g)"
14.178 +    apply (rule homotopic_paths_join)
14.179 +    using hom homotopic_paths_sym_eq apply blast
14.180 +    apply (metis \<open>path g\<close> homotopic_paths_eq pag path_image_subpath_subset path_subpath subset_trans v w)
14.181 +    apply (simp add:)
14.182 +    done
14.183 +  also have "homotopic_paths s ((subpath u v g +++ subpath v w g) +++ subpath w v g) (subpath u v g +++ subpath v w g +++ subpath w v g)"
14.184 +    apply (rule homotopic_paths_sym [OF homotopic_paths_assoc])
14.185 +    using assms by (simp_all add: path_image_subpath_subset [THEN order_trans])
14.186 +  also have "homotopic_paths s (subpath u v g +++ subpath v w g +++ subpath w v g)
14.187 +                               (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))"
14.188 +    apply (rule homotopic_paths_join)
14.189 +    apply (metis \<open>path g\<close> homotopic_paths_eq order.trans pag path_image_subpath_subset path_subpath u v)
14.190 +    apply (metis (no_types, lifting) \<open>path g\<close> homotopic_paths_linv order_trans pag path_image_subpath_subset path_subpath pathfinish_subpath reversepath_subpath v w)
14.191 +    apply (simp add:)
14.192 +    done
14.193 +  also have "homotopic_paths s (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)"
14.194 +    apply (rule homotopic_paths_rid)
14.195 +    using \<open>path g\<close> path_subpath u v apply blast
14.196 +    apply (meson \<open>path g\<close> order.trans pag path_image_subpath_subset u v)
14.197 +    done
14.198 +  finally have "homotopic_paths s (subpath u w g +++ subpath w v g) (subpath u v g)" .
14.199 +  then show ?thesis
14.200 +    using homotopic_join_subpaths2 by blast
14.201 +qed
14.202 +
14.203 +proposition homotopic_join_subpaths:
14.204 +   "\<lbrakk>path g; path_image g \<subseteq> s; u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
14.205 +    \<Longrightarrow> homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
14.206 +apply (rule le_cases3 [of u v w])
14.207 +using homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3 by metis+
14.208 +
14.209  end
```
```    15.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Tue Dec 01 12:35:11 2015 +0100
15.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Tue Dec 01 17:18:34 2015 +0100
15.3 @@ -817,6 +817,9 @@
15.4  definition cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
15.5    where "cball x e = {y. dist x y \<le> e}"
15.6
15.7 +definition sphere :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
15.8 +  where "sphere x e = {y. dist x y = e}"
15.9 +
15.10  lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
15.11    by (simp add: ball_def)
15.12
15.13 @@ -863,19 +866,6 @@
15.14  lemma cball_diff_eq_sphere: "cball a r - ball a r =  {x. dist x a = r}"
15.15    by (auto simp: cball_def ball_def dist_commute)
15.16
15.17 -lemma diff_less_iff:
15.18 -  "(a::real) - b > 0 \<longleftrightarrow> a > b"
15.19 -  "(a::real) - b < 0 \<longleftrightarrow> a < b"
15.20 -  "a - b < c \<longleftrightarrow> a < c + b" "a - b > c \<longleftrightarrow> a > c + b"
15.21 -  by arith+
15.22 -
15.23 -lemma diff_le_iff:
15.24 -  "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b"
15.25 -  "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
15.26 -  "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
15.27 -  "a - b \<ge> c \<longleftrightarrow> a \<ge> c + b"
15.28 -  by arith+
15.29 -
15.30  lemma open_ball [intro, simp]: "open (ball x e)"
15.31  proof -
15.32    have "open (dist x -` {..<e})"
15.33 @@ -7347,7 +7337,7 @@
15.34      then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
15.35        unfolding image_iff Bex_def mem_box
15.36        apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
15.37 -      apply (auto simp add: pos_le_divide_eq pos_divide_le_eq mult.commute diff_le_iff inner_distrib inner_diff_left)
15.38 +      apply (auto simp add: pos_le_divide_eq pos_divide_le_eq mult.commute inner_distrib inner_diff_left)
15.39        done
15.40    }
15.41    moreover
15.42 @@ -7357,7 +7347,7 @@
15.43      then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
15.44        unfolding image_iff Bex_def mem_box
15.45        apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
15.46 -      apply (auto simp add: neg_le_divide_eq neg_divide_le_eq mult.commute diff_le_iff inner_distrib inner_diff_left)
15.47 +      apply (auto simp add: neg_le_divide_eq neg_divide_le_eq mult.commute inner_distrib inner_diff_left)
15.48        done
15.49    }
15.50    ultimately show ?thesis using False by (auto simp: cbox_def)
15.51 @@ -8187,8 +8177,8 @@
15.52    shows "cball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r < 0"
15.53          (is "?lhs = ?rhs")
15.54  proof
15.55 -  assume ?lhs
15.56 -  then show ?rhs
15.57 +  assume ?lhs
15.58 +  then show ?rhs
15.59    proof (cases "r < 0")
15.60      case True then show ?rhs by simp
15.61    next
15.62 @@ -8209,13 +8199,13 @@
15.63          using  \<open>a \<noteq> a'\<close> by (simp add: abs_mult_pos field_simps)
15.64        finally have [simp]: "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = \<bar>norm (a - a') + r\<bar>" by linarith
15.65        show ?thesis
15.66 -        using subsetD [where c = "a' + (1 + r / norm(a - a')) *\<^sub>R (a - a')", OF \<open>?lhs\<close>] \<open>a \<noteq> a'\<close>
15.67 +        using subsetD [where c = "a' + (1 + r / norm(a - a')) *\<^sub>R (a - a')", OF \<open>?lhs\<close>] \<open>a \<noteq> a'\<close>
15.69      qed
15.70      then show ?rhs by (simp add: dist_norm)
15.71    qed
15.72  next
15.73 -  assume ?rhs then show ?lhs
15.74 +  assume ?rhs then show ?lhs
15.75      apply (auto simp: ball_def dist_norm )
15.76      apply (metis add.commute add_le_cancel_right dist_norm dist_triangle_alt order_trans)
15.77      using le_less_trans apply fastforce
15.78 @@ -8227,8 +8217,8 @@
15.79    shows "cball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r < r' \<or> r < 0"
15.80          (is "?lhs = ?rhs")
15.81  proof
15.82 -  assume ?lhs
15.83 -  then show ?rhs
15.84 +  assume ?lhs
15.85 +  then show ?rhs
15.86    proof (cases "r < 0")
15.87      case True then show ?rhs by simp
15.88    next
15.89 @@ -8256,7 +8246,7 @@
15.90      then show ?rhs by (simp add: dist_norm)
15.91    qed
15.92  next
15.93 -  assume ?rhs then show ?lhs
15.94 +  assume ?rhs then show ?lhs
15.95      apply (auto simp: ball_def dist_norm )
15.96      apply (metis add.commute add_le_cancel_right dist_norm dist_triangle_alt le_less_trans)
15.97      using le_less_trans apply fastforce
15.98 @@ -8268,10 +8258,10 @@
15.99    shows "ball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"
15.100          (is "?lhs = ?rhs")
15.101  proof (cases "r \<le> 0")
15.102 -  case True then show ?thesis
15.103 +  case True then show ?thesis
15.104      using dist_not_less_zero less_le_trans by force
15.105  next
15.106 -  case False show ?thesis
15.107 +  case False show ?thesis
15.108    proof
15.109      assume ?lhs
15.110      then have "(cball a r \<subseteq> cball a' r')"
15.111 @@ -8280,7 +8270,7 @@
15.112        using False cball_subset_cball_iff by fastforce
15.113    next
15.114      assume ?rhs with False show ?lhs
15.115 -      using ball_subset_cball cball_subset_cball_iff by blast
15.116 +      using ball_subset_cball cball_subset_cball_iff by blast
15.117    qed
15.118  qed
15.119
15.120 @@ -8289,10 +8279,10 @@
15.121    shows "ball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"
15.122          (is "?lhs = ?rhs")
15.123  proof (cases "r \<le> 0")
15.124 -  case True then show ?thesis
15.125 +  case True then show ?thesis
15.126      using dist_not_less_zero less_le_trans by force
15.127  next
15.128 -  case False show ?thesis
15.129 +  case False show ?thesis
15.130    proof
15.131      assume ?lhs
15.132      then have "0 < r'"
15.133 @@ -8316,22 +8306,22 @@
15.134    shows "ball x d = ball y e \<longleftrightarrow> d \<le> 0 \<and> e \<le> 0 \<or> x=y \<and> d=e"
15.135          (is "?lhs = ?rhs")
15.136  proof
15.137 -  assume ?lhs
15.138 -  then show ?rhs
15.139 +  assume ?lhs
15.140 +  then show ?rhs
15.141    proof (cases "d \<le> 0 \<or> e \<le> 0")
15.142 -    case True
15.143 +    case True
15.144        with \<open>?lhs\<close> show ?rhs
15.145          by safe (simp_all only: ball_eq_empty [of y e, symmetric] ball_eq_empty [of x d, symmetric])
15.146    next
15.147      case False
15.148 -    with \<open>?lhs\<close> show ?rhs
15.149 +    with \<open>?lhs\<close> show ?rhs
15.150        apply (auto simp add: set_eq_subset ball_subset_ball_iff dist_norm norm_minus_commute algebra_simps)
15.151        apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
15.152        apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
15.153        done
15.154    qed
15.155  next
15.156 -  assume ?rhs then show ?lhs
15.157 +  assume ?rhs then show ?lhs
15.158      by (auto simp add: set_eq_subset ball_subset_ball_iff)
15.159  qed
15.160
15.161 @@ -8340,22 +8330,22 @@
15.162    shows "cball x d = cball y e \<longleftrightarrow> d < 0 \<and> e < 0 \<or> x=y \<and> d=e"
15.163          (is "?lhs = ?rhs")
15.164  proof
15.165 -  assume ?lhs
15.166 -  then show ?rhs
15.167 +  assume ?lhs
15.168 +  then show ?rhs
15.169    proof (cases "d < 0 \<or> e < 0")
15.170 -    case True
15.171 +    case True
15.172        with \<open>?lhs\<close> show ?rhs
15.173          by safe (simp_all only: cball_eq_empty [of y e, symmetric] cball_eq_empty [of x d, symmetric])
15.174    next
15.175      case False
15.176 -    with \<open>?lhs\<close> show ?rhs
15.177 +    with \<open>?lhs\<close> show ?rhs
15.178        apply (auto simp add: set_eq_subset cball_subset_cball_iff dist_norm norm_minus_commute algebra_simps)
15.179        apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
15.180        apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
15.181        done
15.182    qed
15.183  next
15.184 -  assume ?rhs then show ?lhs
15.185 +  assume ?rhs then show ?lhs
15.186      by (auto simp add: set_eq_subset cball_subset_cball_iff)
15.187  qed
15.188
15.189 @@ -8363,7 +8353,7 @@
15.190    fixes x :: "'a :: euclidean_space"
15.191    shows "ball x d = cball y e \<longleftrightarrow> d \<le> 0 \<and> e < 0" (is "?lhs = ?rhs")
15.192  proof
15.193 -  assume ?lhs
15.194 +  assume ?lhs
15.195    then show ?rhs
15.196      apply (auto simp add: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff algebra_simps)
15.198 @@ -8371,7 +8361,7 @@
15.199      using \<open>?lhs\<close> ball_eq_empty cball_eq_empty apply blast+
15.200      done
15.201  next
15.202 -  assume ?rhs then show ?lhs
15.203 +  assume ?rhs then show ?lhs
15.204      by (auto simp add: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff)
15.205  qed
15.206
15.207 @@ -8379,7 +8369,7 @@
15.208    fixes x :: "'a :: euclidean_space"
15.209    shows "cball x d = ball y e \<longleftrightarrow> d < 0 \<and> e \<le> 0" (is "?lhs = ?rhs")
15.210  proof
15.211 -  assume ?lhs
15.212 +  assume ?lhs
15.213    then show ?rhs
15.214      apply (auto simp add: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff algebra_simps)
15.216 @@ -8387,7 +8377,7 @@
15.217      using \<open>?lhs\<close> ball_eq_empty cball_eq_empty apply blast+
15.218      done
15.219  next
15.220 -  assume ?rhs then show ?lhs
15.221 +  assume ?rhs then show ?lhs
15.222      by (auto simp add: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff)
15.223  qed
15.224
```
```    16.1 --- a/src/HOL/Multivariate_Analysis/Weierstrass.thy	Tue Dec 01 12:35:11 2015 +0100
16.2 +++ b/src/HOL/Multivariate_Analysis/Weierstrass.thy	Tue Dec 01 17:18:34 2015 +0100
16.3 @@ -380,9 +380,7 @@
16.4        by (simp add: algebra_simps power_mult power2_eq_square power_mult_distrib [symmetric])
16.5      also have "... \<le> (1/(k * (p t))^n) * 1"
16.6        apply (rule mult_left_mono [OF power_le_one])
16.7 -      apply (metis diff_le_iff(1) less_eq_real_def mult.commute power_le_one power_mult ptn_pos ptn_le)
16.8 -      using pt_pos [OF t] \<open>k>0\<close>
16.9 -      apply auto
16.10 +      using pt_pos \<open>k>0\<close> p01 power_le_one t apply auto
16.11        done
16.12      also have "... \<le> (1 / (k*\<delta>))^n"
16.13        using \<open>k>0\<close> \<delta>01  power_mono pt_\<delta> t
```
```    17.1 --- a/src/HOL/NSA/Examples/NSPrimes.thy	Tue Dec 01 12:35:11 2015 +0100
17.2 +++ b/src/HOL/NSA/Examples/NSPrimes.thy	Tue Dec 01 17:18:34 2015 +0100
17.3 @@ -61,7 +61,7 @@
17.4  (* Goldblatt: Exercise 5.11(3a) - p 57  *)
17.5  lemma starprime:
17.6    "starprime = {p. 1 < p & (\<forall>m. m dvd p --> m = 1 | m = p)}"
17.7 -by (transfer, auto simp add: prime_nat_def)
17.8 +by (transfer, auto simp add: prime_def)
17.9
17.10  (* Goldblatt Exercise 5.11(3b) - p 57  *)
17.11  lemma hyperprime_factor_exists [rule_format]:
17.12 @@ -262,17 +262,14 @@
17.13  text{*Already proved as @{text primes_infinite}, but now using non-standard naturals.*}
17.14  theorem not_finite_prime: "~ finite {p::nat. prime p}"
17.15  apply (rule hypnat_infinite_has_nonstandard_iff [THEN iffD2])
17.16 -apply (cut_tac hypnat_dvd_all_hypnat_of_nat)
17.17 -apply (erule exE)
17.18 -apply (erule conjE)
17.19 -apply (subgoal_tac "1 < N + 1")
17.20 -prefer 2 apply (blast intro: hypnat_add_one_gt_one)
17.21 +using hypnat_dvd_all_hypnat_of_nat
17.22 +apply clarify
17.23 +apply (drule hypnat_add_one_gt_one)
17.24  apply (drule hyperprime_factor_exists)
17.25 -apply auto
17.26 +apply clarify
17.27  apply (subgoal_tac "k \<notin> hypnat_of_nat ` {p. prime p}")
17.28 -apply (force simp add: starprime_def, safe)
17.29 -apply (drule_tac x = x in bspec, auto)
17.30 -apply (metis add.commute hdvd_diff hdvd_one_eq_one hypnat_diff_add_inverse2 hypnat_one_not_prime)
17.31 +apply (force simp add: starprime_def)
17.32 +apply (metis Compl_iff add.commute dvd_add_left_iff empty_iff hdvd_one_eq_one hypnat_one_not_prime imageE insert_iff mem_Collect_eq zero_not_prime_nat)
17.33  done
17.34
17.35  end
```
```    18.1 --- a/src/HOL/Number_Theory/Eratosthenes.thy	Tue Dec 01 12:35:11 2015 +0100
18.2 +++ b/src/HOL/Number_Theory/Eratosthenes.thy	Tue Dec 01 17:18:34 2015 +0100
18.3 @@ -294,8 +294,8 @@
18.4      from 2 show ?thesis
18.5        apply (auto simp add: numbers_of_marks_sieve numeral_2_eq_2 set_primes_upto
18.6          dest: prime_gt_Suc_0_nat)
18.7 -      apply (metis One_nat_def Suc_le_eq less_not_refl prime_nat_def)
18.8 -      apply (metis One_nat_def Suc_le_eq aux prime_nat_def)
18.9 +      apply (metis One_nat_def Suc_le_eq less_not_refl prime_def)
18.10 +      apply (metis One_nat_def Suc_le_eq aux prime_def)
18.11        done
18.12    qed
18.13  qed
```
```    19.1 --- a/src/HOL/Number_Theory/Pocklington.thy	Tue Dec 01 12:35:11 2015 +0100
19.2 +++ b/src/HOL/Number_Theory/Pocklington.thy	Tue Dec 01 17:18:34 2015 +0100
19.3 @@ -457,11 +457,11 @@
19.4    proof
19.5      assume "prime n"
19.6      then show ?rhs
19.7 -      by (metis one_not_prime_nat prime_nat_def)
19.8 +      by (metis one_not_prime_nat prime_def)
19.9    next
19.10      assume ?rhs
19.11      with False show "prime n"
19.12 -      by (auto simp: prime_def) (metis One_nat_def prime_factor_nat prime_nat_def)
19.13 +      by (auto simp: prime_def) (metis One_nat_def prime_factor_nat prime_def)
19.14    qed
19.15  qed
19.16
19.17 @@ -538,7 +538,7 @@
19.18    and pp: "prime p" and pn: "p dvd n"
19.19    shows "[p = 1] (mod q)"
19.20  proof -
19.21 -  have p01: "p \<noteq> 0" "p \<noteq> 1" using pp one_not_prime_nat zero_not_prime_nat by auto
19.22 +  have p01: "p \<noteq> 0" "p \<noteq> 1" using pp one_not_prime_nat zero_not_prime_nat by (auto intro: prime_gt_0_nat)
19.23    obtain k where k: "a ^ (q * r) - 1 = n*k"
19.24      by (metis an cong_to_1_nat dvd_def nqr)
19.25    from pn[unfolded dvd_def] obtain l where l: "n = p*l" by blast
19.26 @@ -689,7 +689,7 @@
19.27      from p(2) obtain m where m: "n = p*m" unfolding dvd_def by blast
19.28      from n m have m0: "m > 0" "m\<noteq>0" by auto
19.29      have "1 < p"
19.30 -      by (metis p(1) prime_nat_def)
19.31 +      by (metis p(1) prime_def)
19.32      with m0 m have mn: "m < n" by auto
19.33      from H[rule_format, OF mn m0(2)] obtain ps where ps: "primefact ps m" ..
19.34      from ps m p(1) have "primefact (p#ps) n" by (simp add: primefact_def)
```
```    20.1 --- a/src/HOL/Number_Theory/Primes.thy	Tue Dec 01 12:35:11 2015 +0100
20.2 +++ b/src/HOL/Number_Theory/Primes.thy	Tue Dec 01 17:18:34 2015 +0100
20.3 @@ -37,38 +37,33 @@
20.4  definition prime :: "nat \<Rightarrow> bool"
20.5    where "prime p = (1 < p \<and> (\<forall>m. m dvd p \<longrightarrow> m = 1 \<or> m = p))"
20.6
20.7 -lemmas prime_nat_def = prime_def
20.8 -
20.9 -
20.10  subsection \<open>Primes\<close>
20.11
20.12  lemma prime_odd_nat: "prime p \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
20.13 -  apply (auto simp add: prime_nat_def even_iff_mod_2_eq_zero dvd_eq_mod_eq_0)
20.14 +  apply (auto simp add: prime_def even_iff_mod_2_eq_zero dvd_eq_mod_eq_0)
20.15    apply (metis dvd_eq_mod_eq_0 even_Suc mod_by_1 nat_dvd_not_less not_mod_2_eq_0_eq_1 zero_less_numeral)
20.16    done
20.17
20.18 -(* FIXME Is there a better way to handle these, rather than making them elim rules? *)
20.19 +lemma prime_gt_0_nat: "prime p \<Longrightarrow> p > 0"
20.20 +  unfolding prime_def by auto
20.21
20.22 -lemma prime_gt_0_nat [elim]: "prime p \<Longrightarrow> p > 0"
20.23 -  unfolding prime_nat_def by auto
20.24 +lemma prime_ge_1_nat: "prime p \<Longrightarrow> p >= 1"
20.25 +  unfolding prime_def by auto
20.26
20.27 -lemma prime_ge_1_nat [elim]: "prime p \<Longrightarrow> p >= 1"
20.28 -  unfolding prime_nat_def by auto
20.29 +lemma prime_gt_1_nat: "prime p \<Longrightarrow> p > 1"
20.30 +  unfolding prime_def by auto
20.31
20.32 -lemma prime_gt_1_nat [elim]: "prime p \<Longrightarrow> p > 1"
20.33 -  unfolding prime_nat_def by auto
20.34 -
20.35 -lemma prime_ge_Suc_0_nat [elim]: "prime p \<Longrightarrow> p >= Suc 0"
20.36 -  unfolding prime_nat_def by auto
20.37 +lemma prime_ge_Suc_0_nat: "prime p \<Longrightarrow> p >= Suc 0"
20.38 +  unfolding prime_def by auto
20.39
20.40 -lemma prime_gt_Suc_0_nat [elim]: "prime p \<Longrightarrow> p > Suc 0"
20.41 -  unfolding prime_nat_def by auto
20.42 +lemma prime_gt_Suc_0_nat: "prime p \<Longrightarrow> p > Suc 0"
20.43 +  unfolding prime_def by auto
20.44
20.45 -lemma prime_ge_2_nat [elim]: "prime p \<Longrightarrow> p >= 2"
20.46 -  unfolding prime_nat_def by auto
20.47 +lemma prime_ge_2_nat: "prime p \<Longrightarrow> p >= 2"
20.48 +  unfolding prime_def by auto
20.49
20.50  lemma prime_imp_coprime_nat: "prime p \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
20.51 -  apply (unfold prime_nat_def)
20.52 +  apply (unfold prime_def)
20.53    apply (metis gcd_dvd1_nat gcd_dvd2_nat)
20.54    done
20.55
20.56 @@ -105,7 +100,7 @@
20.57
20.58  lemma not_prime_eq_prod_nat: "(n::nat) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
20.59      EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
20.60 -  unfolding prime_nat_def dvd_def apply auto
20.61 +  unfolding prime_def dvd_def apply auto
20.62    by (metis mult.commute linorder_neq_iff linorder_not_le mult_1
20.63        n_less_n_mult_m one_le_mult_iff less_imp_le_nat)
20.64
20.65 @@ -129,18 +124,18 @@
20.66  subsubsection \<open>Make prime naively executable\<close>
20.67
20.68  lemma zero_not_prime_nat [simp]: "~prime (0::nat)"
20.69 -  by (simp add: prime_nat_def)
20.70 +  by (simp add: prime_def)
20.71
20.72  lemma one_not_prime_nat [simp]: "~prime (1::nat)"
20.73 -  by (simp add: prime_nat_def)
20.74 +  by (simp add: prime_def)
20.75
20.76  lemma Suc_0_not_prime_nat [simp]: "~prime (Suc 0)"
20.77 -  by (simp add: prime_nat_def)
20.78 +  by (simp add: prime_def)
20.79
20.80  lemma prime_nat_code [code]:
20.81      "prime p \<longleftrightarrow> p > 1 \<and> (\<forall>n \<in> {1<..<p}. ~ n dvd p)"
20.82    apply (simp add: Ball_def)
20.83 -  apply (metis One_nat_def less_not_refl prime_nat_def dvd_triv_right not_prime_eq_prod_nat)
20.84 +  apply (metis One_nat_def less_not_refl prime_def dvd_triv_right not_prime_eq_prod_nat)
20.85    done
20.86
20.87  lemma prime_nat_simp:
20.88 @@ -178,7 +173,7 @@
20.89    using two_is_prime_nat
20.90    apply blast
20.91    apply (metis One_nat_def dvd.order_trans dvd_refl less_Suc0 linorder_neqE_nat
20.92 -    nat_dvd_not_less neq0_conv prime_nat_def)
20.93 +    nat_dvd_not_less neq0_conv prime_def)
20.94    done
20.95
20.96  text \<open>One property of coprimality is easier to prove via prime factors.\<close>
20.97 @@ -239,7 +234,8 @@
20.98        by (rule dvd_diff_nat)
20.99      then have "p dvd 1" by simp
20.100      then have "p <= 1" by auto
20.101 -    moreover from \<open>prime p\<close> have "p > 1" by auto
20.102 +    moreover from \<open>prime p\<close> have "p > 1"
20.103 +      using prime_def by blast
20.104      ultimately have False by auto}
20.105    then have "n < p" by presburger
20.106    with \<open>prime p\<close> and \<open>p <= fact n + 1\<close> show ?thesis by auto
20.107 @@ -270,7 +266,7 @@
20.108  proof -
20.109    from assms have
20.110      "1 < p * q" and P: "\<And>m. m dvd p * q \<Longrightarrow> m = 1 \<or> m = p * q"
20.111 -    unfolding prime_nat_def by auto
20.112 +    unfolding prime_def by auto
20.113    from \<open>1 < p * q\<close> have "p \<noteq> 0" by (cases p) auto
20.114    then have Q: "p = p * q \<longleftrightarrow> q = 1" by auto
20.115    have "p dvd p * q" by simp
```
```    21.1 --- a/src/HOL/Number_Theory/UniqueFactorization.thy	Tue Dec 01 12:35:11 2015 +0100
21.2 +++ b/src/HOL/Number_Theory/UniqueFactorization.thy	Tue Dec 01 17:18:34 2015 +0100
21.3 @@ -107,9 +107,7 @@
21.4    ultimately have "a ^ count M a dvd a ^ count N a"
21.5      by (elim coprime_dvd_mult_nat)
21.6    with a show ?thesis
21.7 -    apply (intro power_dvd_imp_le)
21.8 -    apply auto
21.9 -    done
21.10 +    using power_dvd_imp_le prime_def by blast
21.11  next
21.12    case False
21.13    then show ?thesis
21.14 @@ -247,14 +245,14 @@
21.15    using assms apply auto
21.16    done
21.17
21.18 -lemma prime_factors_gt_0_nat [elim]:
21.19 +lemma prime_factors_gt_0_nat:
21.20    fixes p :: nat
21.21    shows "p \<in> prime_factors x \<Longrightarrow> p > 0"
21.22 -  by (auto dest!: prime_factors_prime_nat)
21.23 +    using prime_factors_prime_nat by force
21.24
21.25 -lemma prime_factors_gt_0_int [elim]:
21.26 +lemma prime_factors_gt_0_int:
21.27    shows "x \<ge> 0 \<Longrightarrow> p \<in> prime_factors x \<Longrightarrow> int p > (0::int)"
21.28 -  by auto
21.29 +    by (simp add: prime_factors_gt_0_nat)
21.30
21.31  lemma prime_factors_finite_nat [iff]:
21.32    fixes n :: nat
21.33 @@ -303,7 +301,8 @@
21.34  proof -
21.35    from assms have "f \<in> multiset"
21.36      by (auto simp add: multiset_def)
21.37 -  moreover from assms have "n > 0" by force
21.38 +  moreover from assms have "n > 0"
21.39 +    by (auto intro: prime_gt_0_nat)
21.40    ultimately have "multiset_prime_factorization n = Abs_multiset f"
21.41      apply (unfold multiset_prime_factorization_def)
21.42      apply (subst if_P, assumption)
21.43 @@ -723,16 +722,16 @@
21.44      (\<Prod>p \<in> prime_factors x \<union> prime_factors y. p ^ min (multiplicity p x) (multiplicity p y))"
21.45    (is "_ = ?z")
21.46  proof -
21.47 -  have [arith]: "?z > 0"
21.48 -    by auto
21.49 +  have [arith]: "?z > 0"
21.50 +    using prime_factors_gt_0_nat by auto
21.51    have aux: "\<And>p. prime p \<Longrightarrow> multiplicity p ?z = min (multiplicity p x) (multiplicity p y)"
21.52      apply (subst multiplicity_prod_prime_powers_nat)
21.53      apply auto
21.54      done
21.55    have "?z dvd x"
21.56 -    by (intro multiplicity_dvd'_nat) (auto simp add: aux)
21.57 +    by (intro multiplicity_dvd'_nat) (auto simp add: aux intro: prime_gt_0_nat)
21.58    moreover have "?z dvd y"
21.59 -    by (intro multiplicity_dvd'_nat) (auto simp add: aux)
21.60 +    by (intro multiplicity_dvd'_nat) (auto simp add: aux intro: prime_gt_0_nat)
21.61    moreover have "w dvd x \<and> w dvd y \<longrightarrow> w dvd ?z" for w
21.62    proof (cases "w = 0")
21.63      case True
21.64 @@ -758,7 +757,7 @@
21.65    (is "_ = ?z")
21.66  proof -
21.67    have [arith]: "?z > 0"
21.68 -    by auto
21.69 +    by (auto intro: prime_gt_0_nat)
21.70    have aux: "\<And>p. prime p \<Longrightarrow> multiplicity p ?z = max (multiplicity p x) (multiplicity p y)"
21.71      apply (subst multiplicity_prod_prime_powers_nat)
21.72      apply auto
21.73 @@ -776,7 +775,7 @@
21.74      then show ?thesis
21.75        apply auto
21.76        apply (rule multiplicity_dvd'_nat)
21.77 -      apply (auto intro: dvd_multiplicity_nat simp add: aux)
21.78 +      apply (auto intro: prime_gt_0_nat dvd_multiplicity_nat simp add: aux)
21.79        done
21.80    qed
21.81    ultimately have "?z = lcm x y"
```
```    22.1 --- a/src/HOL/Old_Number_Theory/Legacy_GCD.thy	Tue Dec 01 12:35:11 2015 +0100
22.2 +++ b/src/HOL/Old_Number_Theory/Legacy_GCD.thy	Tue Dec 01 17:18:34 2015 +0100
22.3 @@ -665,7 +665,8 @@
22.4    apply (simp del: pos_mod_sign add: zgcd_def abs_if nat_mod_distrib)
22.5    apply (auto simp add: gcd_non_0 nat_mod_distrib [symmetric] zmod_zminus1_eq_if)
22.6    apply (frule_tac a = m in pos_mod_bound)
22.7 -  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2 nat_le_eq_zle)
22.8 +  apply (simp del: pos_mod_bound add: algebra_simps nat_diff_distrib gcd_diff2 nat_le_eq_zle)
22.9 +  apply (metis dual_order.strict_implies_order gcd.simps gcd_0_left gcd_diff2 mod_by_0 nat_mono)
22.10    done
22.11
22.12  lemma zgcd_eq: "zgcd m n = zgcd n (m mod n)"
```
```    23.1 --- a/src/HOL/Orderings.thy	Tue Dec 01 12:35:11 2015 +0100
23.2 +++ b/src/HOL/Orderings.thy	Tue Dec 01 17:18:34 2015 +0100
23.3 @@ -310,6 +310,11 @@
23.4    "(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P"
23.5  using linear by blast
23.6
23.7 +lemma (in linorder) le_cases3:
23.8 +  "\<lbrakk>\<lbrakk>x \<le> y; y \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> x; x \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>x \<le> z; z \<le> y\<rbrakk> \<Longrightarrow> P;
23.9 +    \<lbrakk>z \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> z; z \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>z \<le> x; x \<le> y\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
23.10 +by (blast intro: le_cases)
23.11 +
23.12  lemma linorder_cases [case_names less equal greater]:
23.13    "(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P"
23.14  using less_linear by blast
```
```    24.1 --- a/src/HOL/Probability/Lebesgue_Measure.thy	Tue Dec 01 12:35:11 2015 +0100
24.2 +++ b/src/HOL/Probability/Lebesgue_Measure.thy	Tue Dec 01 17:18:34 2015 +0100
24.3 @@ -46,8 +46,8 @@
24.4    assume l_r[simp]: "\<And>n. l n \<le> r n" and "a \<le> b" and disj: "disjoint_family (\<lambda>n. {l n<..r n})"
24.5    assume lr_eq_ab: "(\<Union>i. {l i<..r i}) = {a<..b}"
24.6
24.7 -  have [intro, simp]: "\<And>a b. a \<le> b \<Longrightarrow> 0 \<le> F b - F a"
24.8 -    by (auto intro!: l_r mono_F simp: diff_le_iff)
24.9 +  have [intro, simp]: "\<And>a b. a \<le> b \<Longrightarrow> F a \<le> F b"
24.10 +    by (auto intro!: l_r mono_F)
24.11
24.12    { fix S :: "nat set" assume "finite S"
24.13      moreover note `a \<le> b`
24.14 @@ -92,7 +92,7 @@
24.15            by (auto simp add: Ioc_subset_iff intro!: mono_F)
24.16          finally show ?case
24.17            by (auto intro: add_mono)
24.18 -      qed (simp add: `a \<le> b` less_le)
24.19 +      qed (auto simp add: `a \<le> b` less_le)
24.20      qed }
24.21    note claim1 = this
24.22
24.23 @@ -280,7 +280,7 @@
24.24      by (auto simp add: claim1 intro!: suminf_bound)
24.25    ultimately show "(\<Sum>n. ereal (F (r n) - F (l n))) = ereal (F b - F a)"
24.26      by simp
24.27 -qed (auto simp: Ioc_inj diff_le_iff mono_F)
24.28 +qed (auto simp: Ioc_inj mono_F)
24.29
24.30  lemma measure_interval_measure_Ioc:
24.31    assumes "a \<le> b"
```
```    25.1 --- a/src/HOL/Proofs/Extraction/Euclid.thy	Tue Dec 01 12:35:11 2015 +0100
25.2 +++ b/src/HOL/Proofs/Extraction/Euclid.thy	Tue Dec 01 17:18:34 2015 +0100
25.3 @@ -29,7 +29,7 @@
25.4    by (induct m) auto
25.5
25.6  lemma prime_eq: "prime (p::nat) = (1 < p \<and> (\<forall>m. m dvd p \<longrightarrow> 1 < m \<longrightarrow> m = p))"
25.7 -  apply (simp add: prime_nat_def)
25.8 +  apply (simp add: prime_def)
25.9    apply (rule iffI)
25.10    apply blast
25.11    apply (erule conjE)
```
```    26.1 --- a/src/HOL/Real_Vector_Spaces.thy	Tue Dec 01 12:35:11 2015 +0100
26.2 +++ b/src/HOL/Real_Vector_Spaces.thy	Tue Dec 01 17:18:34 2015 +0100
26.3 @@ -9,6 +9,10 @@
26.4  imports Real Topological_Spaces
26.5  begin
26.6
26.7 +
26.8 +lemma (in ordered_ab_group_add) diff_ge_0_iff_ge [simp]: "a - b \<ge> 0 \<longleftrightarrow> a \<ge> b"
26.9 +  by (simp add: le_diff_eq)
26.10 +
26.11  subsection \<open>Locale for additive functions\<close>
26.12
26.13  locale additive =
26.14 @@ -777,6 +781,11 @@
26.15    thus ?thesis by simp
26.16  qed
26.17
26.19 +  fixes a b :: "'a::real_normed_vector"
26.20 +  shows "norm (a + b) \<le> c \<Longrightarrow> norm b \<le> norm a + c"
26.21 +    by (metis add.commute diff_le_eq norm_diff_ineq order.trans)
26.22 +
26.23  lemma norm_diff_triangle_ineq:
26.24    fixes a b c d :: "'a::real_normed_vector"
26.25    shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
```
```    27.1 --- a/src/HOL/Rings.thy	Tue Dec 01 12:35:11 2015 +0100
27.2 +++ b/src/HOL/Rings.thy	Tue Dec 01 17:18:34 2015 +0100
27.3 @@ -1559,6 +1559,9 @@
27.4  lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
27.5    by (simp add: not_less)
27.6
27.7 +proposition abs_eq_iff: "abs x = abs y \<longleftrightarrow> x = y \<or> x = -y"
27.8 +  by (auto simp add: abs_if split: split_if_asm)
27.9 +
27.10  end
27.11
27.12  class linordered_ring_strict = ring + linordered_semiring_strict
```
```    28.1 --- a/src/HOL/Tools/BNF/bnf_comp.ML	Tue Dec 01 12:35:11 2015 +0100
28.2 +++ b/src/HOL/Tools/BNF/bnf_comp.ML	Tue Dec 01 17:18:34 2015 +0100
28.3 @@ -181,7 +181,8 @@
28.4      val CCA = mk_T_of_bnf oDs CAs outer;
28.5      val CBs = @{map 3} mk_T_of_bnf Dss Bss_repl inners;
28.6      val outer_sets = mk_sets_of_bnf (replicate olive oDs) (replicate olive CAs) outer;
28.7 -    val inner_setss = @{map 3} mk_sets_of_bnf (map (replicate ilive) Dss) (replicate olive Ass) inners;
28.8 +    val inner_setss =
28.9 +      @{map 3} mk_sets_of_bnf (map (replicate ilive) Dss) (replicate olive Ass) inners;
28.10      val inner_bds = @{map 3} mk_bd_of_bnf Dss Ass_repl inners;
28.11      val outer_bd = mk_bd_of_bnf oDs CAs outer;
28.12
28.13 @@ -692,7 +693,8 @@
28.14      val ((kill_poss, As), (inners', ((cache', unfold_set'), lthy'))) =
28.15        normalize_bnfs qualify Ass Ds flatten_tyargs inners accum;
28.16
28.17 -    val Ds = oDs @ flat (@{map 3} (uncurry append oo curry swap oo map o nth) tfreess kill_poss Dss);
28.18 +    val Ds =
28.19 +      oDs @ flat (@{map 3} (uncurry append oo curry swap oo map o nth) tfreess kill_poss Dss);
28.20      val As = map TFree As;
28.21    in
28.22      apfst (rpair (Ds, As))
```
```    29.1 --- a/src/HOL/Tools/BNF/bnf_comp_tactics.ML	Tue Dec 01 12:35:11 2015 +0100
29.2 +++ b/src/HOL/Tools/BNF/bnf_comp_tactics.ML	Tue Dec 01 17:18:34 2015 +0100
29.3 @@ -74,14 +74,16 @@
29.4       map (fn thm => rtac ctxt (thm RS fun_cong)) set_map0s @
29.5       [rtac ctxt (Gset_map0 RS comp_eq_dest_lhs), rtac ctxt sym, rtac ctxt trans_o_apply,
29.6       rtac ctxt trans_image_cong_o_apply, rtac ctxt trans_image_cong_o_apply,
29.7 -     rtac ctxt (@{thm image_cong} OF [Gset_map0 RS comp_eq_dest_lhs RS arg_cong_Union, refl] RS trans),
29.8 +     rtac ctxt (@{thm image_cong} OF [Gset_map0 RS comp_eq_dest_lhs RS arg_cong_Union, refl]
29.9 +       RS trans),
29.10       rtac ctxt @{thm trans[OF comp_eq_dest[OF Union_natural[symmetric]]]}, rtac ctxt arg_cong_Union,
29.11       rtac ctxt @{thm trans[OF comp_eq_dest_lhs[OF image_o_collect[symmetric]]]},
29.12       rtac ctxt @{thm fun_cong[OF arg_cong[of _ _ collect]]}] @
29.13       [REPEAT_DETERM_N (length set_map0s) o EVERY' [rtac ctxt @{thm trans[OF image_insert]},
29.14 -        rtac ctxt @{thm arg_cong2[of _ _ _ _ insert]}, rtac ctxt @{thm ext}, rtac ctxt trans_o_apply,
29.15 -        rtac ctxt trans_image_cong_o_apply, rtac ctxt @{thm trans[OF image_image]},
29.16 -        rtac ctxt @{thm sym[OF trans[OF o_apply]]}, rtac ctxt @{thm image_cong[OF refl o_apply]}],
29.17 +        rtac ctxt @{thm arg_cong2[of _ _ _ _ insert]}, rtac ctxt @{thm ext},
29.18 +        rtac ctxt trans_o_apply, rtac ctxt trans_image_cong_o_apply,
29.19 +        rtac ctxt @{thm trans[OF image_image]}, rtac ctxt @{thm sym[OF trans[OF o_apply]]},
29.20 +        rtac ctxt @{thm image_cong[OF refl o_apply]}],
29.21       rtac ctxt @{thm image_empty}]) 1;
29.22
29.23  fun mk_comp_map_cong0_tac ctxt set'_eq_sets comp_set_alts map_cong0 map_cong0s =
29.24 @@ -96,9 +98,9 @@
29.25            EVERY' [select_prem_tac ctxt n (dtac ctxt @{thm meta_spec}) (k + 1), etac ctxt meta_mp,
29.26              rtac ctxt (equalityD2 RS set_mp), rtac ctxt (set_alt RS fun_cong RS trans),
29.27              rtac ctxt trans_o_apply, rtac ctxt (@{thm collect_def} RS arg_cong_Union),
29.28 -            rtac ctxt @{thm UnionI}, rtac ctxt @{thm UN_I}, REPEAT_DETERM_N i o rtac ctxt @{thm insertI2},
29.29 -            rtac ctxt @{thm insertI1}, rtac ctxt (o_apply RS equalityD2 RS set_mp),
29.30 -            etac ctxt @{thm imageI}, assume_tac ctxt])
29.31 +            rtac ctxt @{thm UnionI}, rtac ctxt @{thm UN_I},
29.32 +            REPEAT_DETERM_N i o rtac ctxt @{thm insertI2}, rtac ctxt @{thm insertI1},
29.33 +            rtac ctxt (o_apply RS equalityD2 RS set_mp), etac ctxt @{thm imageI}, assume_tac ctxt])
29.34            comp_set_alts))
29.35        map_cong0s) 1)
29.36    end;
29.37 @@ -220,14 +222,15 @@
29.38
29.39  fun mk_permute_in_alt_tac ctxt src dest =
29.40    (rtac ctxt @{thm Collect_cong} THEN'
29.41 -  mk_rotate_eq_tac ctxt (rtac ctxt refl) trans @{thm conj_assoc} @{thm conj_commute} @{thm conj_cong}
29.42 -    dest src) 1;
29.43 +  mk_rotate_eq_tac ctxt (rtac ctxt refl) trans @{thm conj_assoc} @{thm conj_commute}
29.44 +    @{thm conj_cong} dest src) 1;
29.45
29.46
29.47  (* Miscellaneous *)
29.48
29.49  fun mk_le_rel_OO_tac ctxt outer_le_rel_OO outer_rel_mono inner_le_rel_OOs =
29.50 -  EVERY' (map (rtac ctxt) (@{thm order_trans} :: outer_le_rel_OO :: outer_rel_mono :: inner_le_rel_OOs)) 1;
29.51 +  EVERY' (map (rtac ctxt) (@{thm order_trans} :: outer_le_rel_OO :: outer_rel_mono ::
29.52 +    inner_le_rel_OOs)) 1;
29.53
29.54  fun mk_simple_rel_OO_Grp_tac ctxt rel_OO_Grp in_alt_thm =
29.55    rtac ctxt (trans OF [rel_OO_Grp, in_alt_thm RS @{thm OO_Grp_cong} RS sym]) 1;
```
```    30.1 --- a/src/HOL/Tools/BNF/bnf_def.ML	Tue Dec 01 12:35:11 2015 +0100
30.2 +++ b/src/HOL/Tools/BNF/bnf_def.ML	Tue Dec 01 17:18:34 2015 +0100
30.3 @@ -1300,10 +1300,12 @@
30.4                  val funTs = map (fn T => bdT --> T) ranTs;
30.5                  val ran_bnfT = mk_bnf_T ranTs Calpha;
30.6                  val (revTs, Ts) = `rev (bd_bnfT :: funTs);
30.7 -                val cTs = map (SOME o Thm.ctyp_of lthy) [ran_bnfT, Library.foldr1 HOLogic.mk_prodT Ts];
30.8 -                val tinst = fold (fn T => fn t => HOLogic.mk_case_prod (Term.absdummy T t)) (tl revTs)
30.9 -                  (Term.absdummy (hd revTs) (Term.list_comb (mk_bnf_map bdTs ranTs,
30.10 -                    map Bound (live - 1 downto 0)) \$ Bound live));
30.11 +                val cTs = map (SOME o Thm.ctyp_of lthy) [ran_bnfT,
30.12 +                  Library.foldr1 HOLogic.mk_prodT Ts];
30.13 +                val tinst = fold (fn T => fn t =>
30.14 +                  HOLogic.mk_case_prod (Term.absdummy T t)) (tl revTs)
30.15 +                    (Term.absdummy (hd revTs) (Term.list_comb (mk_bnf_map bdTs ranTs,
30.16 +                      map Bound (live - 1 downto 0)) \$ Bound live));
30.17                  val cts = [NONE, SOME (Thm.cterm_of lthy tinst)];
30.18                in
30.19                  Thm.instantiate' cTs cts @{thm surj_imp_ordLeq}
30.20 @@ -1420,7 +1422,8 @@
30.21          val in_rel = Lazy.lazy mk_in_rel;
30.22
30.23          fun mk_rel_flip () =
30.24 -          unfold_thms lthy @{thms conversep_iff} (Lazy.force rel_conversep RS @{thm predicate2_eqD});
30.25 +          unfold_thms lthy @{thms conversep_iff}
30.26 +            (Lazy.force rel_conversep RS @{thm predicate2_eqD});
30.27
30.28          val rel_flip = Lazy.lazy mk_rel_flip;
30.29
```
```    31.1 --- a/src/HOL/Tools/BNF/bnf_def_tactics.ML	Tue Dec 01 12:35:11 2015 +0100
31.2 +++ b/src/HOL/Tools/BNF/bnf_def_tactics.ML	Tue Dec 01 17:18:34 2015 +0100
31.3 @@ -97,18 +97,20 @@
31.4  fun mk_rel_Grp_tac ctxt rel_OO_Grps map_id0 map_cong0 map_id map_comp set_maps =
31.5    let
31.6      val n = length set_maps;
31.7 -    val rel_OO_Grps_tac = if null rel_OO_Grps then K all_tac else rtac ctxt (hd rel_OO_Grps RS trans);
31.8 +    val rel_OO_Grps_tac =
31.9 +      if null rel_OO_Grps then K all_tac else rtac ctxt (hd rel_OO_Grps RS trans);
31.10    in
31.11      if null set_maps then
31.12        unfold_thms_tac ctxt ((map_id0 RS @{thm Grp_UNIV_id}) :: rel_OO_Grps) THEN
31.13        rtac ctxt @{thm Grp_UNIV_idI[OF refl]} 1
31.14      else
31.15        EVERY' [rel_OO_Grps_tac, rtac ctxt @{thm antisym}, rtac ctxt @{thm predicate2I},
31.16 -        REPEAT_DETERM o
31.17 -          eresolve_tac ctxt [CollectE, exE, conjE, @{thm GrpE}, @{thm relcomppE}, @{thm conversepE}],
31.18 -        hyp_subst_tac ctxt, rtac ctxt @{thm GrpI}, rtac ctxt trans, rtac ctxt map_comp, rtac ctxt map_cong0,
31.19 -        REPEAT_DETERM_N n o
31.20 -          EVERY' [rtac ctxt @{thm Collect_case_prod_Grp_eqD}, etac ctxt @{thm set_mp}, assume_tac ctxt],
31.21 +        REPEAT_DETERM o eresolve_tac ctxt
31.22 +          [CollectE, exE, conjE, @{thm GrpE}, @{thm relcomppE}, @{thm conversepE}],
31.23 +        hyp_subst_tac ctxt, rtac ctxt @{thm GrpI}, rtac ctxt trans, rtac ctxt map_comp,
31.24 +          rtac ctxt map_cong0,
31.25 +        REPEAT_DETERM_N n o EVERY' [rtac ctxt @{thm Collect_case_prod_Grp_eqD},
31.26 +          etac ctxt @{thm set_mp}, assume_tac ctxt],
31.27          rtac ctxt CollectI,
31.28          CONJ_WRAP' (fn thm => EVERY' [rtac ctxt (thm RS ord_eq_le_trans),
31.29            rtac ctxt @{thm image_subsetI}, rtac ctxt @{thm Collect_case_prod_Grp_in},
31.30 @@ -151,8 +153,9 @@
31.31        unfold_thms_tac ctxt [rel_compp, rel_conversep, rel_Grp, @{thm vimage2p_Grp}] THEN
31.32        TRYALL (EVERY' [rtac ctxt iffI, rtac ctxt @{thm relcomppI}, rtac ctxt @{thm GrpI},
31.33          resolve_tac ctxt [map_id, refl], rtac ctxt CollectI,
31.34 -        CONJ_WRAP' (K (rtac ctxt @{thm subset_UNIV})) ks, rtac ctxt @{thm relcomppI}, assume_tac ctxt,
31.35 -        rtac ctxt @{thm conversepI}, rtac ctxt @{thm GrpI}, resolve_tac ctxt [map_id, refl], rtac ctxt CollectI,
31.36 +        CONJ_WRAP' (K (rtac ctxt @{thm subset_UNIV})) ks, rtac ctxt @{thm relcomppI},
31.37 +        assume_tac ctxt, rtac ctxt @{thm conversepI}, rtac ctxt @{thm GrpI},
31.38 +        resolve_tac ctxt [map_id, refl], rtac ctxt CollectI,
31.39          CONJ_WRAP' (K (rtac ctxt @{thm subset_UNIV})) ks,
31.40          REPEAT_DETERM o eresolve_tac ctxt @{thms relcomppE conversepE GrpE},
31.41          dtac ctxt (trans OF [sym, map_id]), hyp_subst_tac ctxt, assume_tac ctxt])
```
```    32.1 --- a/src/HOL/Tools/BNF/bnf_fp_def_sugar.ML	Tue Dec 01 12:35:11 2015 +0100
32.2 +++ b/src/HOL/Tools/BNF/bnf_fp_def_sugar.ML	Tue Dec 01 17:18:34 2015 +0100
32.3 @@ -75,8 +75,8 @@
32.4    val fp_sugar_of_global: theory -> string -> fp_sugar option
32.5    val fp_sugars_of: Proof.context -> fp_sugar list
32.6    val fp_sugars_of_global: theory -> fp_sugar list
32.7 -  val fp_sugars_interpretation: string ->
32.8 -    (fp_sugar list -> local_theory -> local_theory)-> theory -> theory
32.9 +  val fp_sugars_interpretation: string -> (fp_sugar list -> local_theory -> local_theory) ->
32.10 +    theory -> theory
32.11    val interpret_fp_sugars: (string -> bool) -> fp_sugar list -> local_theory -> local_theory
32.12    val register_fp_sugars_raw: fp_sugar list -> local_theory -> local_theory
32.13    val register_fp_sugars: (string -> bool) -> fp_sugar list -> local_theory -> local_theory
32.14 @@ -334,7 +334,7 @@
32.15  );
32.16
32.17  fun fp_sugar_of_generic context =
32.18 -  Option.map (transfer_fp_sugar (Context.theory_of context)) o Symtab.lookup (Data.get context)
32.19 +  Option.map (transfer_fp_sugar (Context.theory_of context)) o Symtab.lookup (Data.get context);
32.20
32.21  fun fp_sugars_of_generic context =
32.22    Symtab.fold (cons o transfer_fp_sugar (Context.theory_of context) o snd) (Data.get context) [];
32.23 @@ -1206,7 +1206,8 @@
32.24    let
32.25      val ctr_Tsss' = map repair_nullary_single_ctr ctr_Tsss;
32.26      val g_absTs = map range_type fun_Ts;
32.27 -    val g_Tsss = map repair_nullary_single_ctr (@{map 5} dest_absumprodT absTs repTs ns mss g_absTs);
32.28 +    val g_Tsss =
32.29 +      map repair_nullary_single_ctr (@{map 5} dest_absumprodT absTs repTs ns mss g_absTs);
32.30      val g_Tssss = @{map 3} (fn C => map2 (map2 (map (curry (op -->) C) oo unzip_corecT)))
32.31        Cs ctr_Tsss' g_Tsss;
32.32      val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) g_Tssss;
32.33 @@ -1312,7 +1313,8 @@
32.34             ctor_rec_absTs reps fss xssss)))
32.35    end;
32.36
32.37 -fun define_corec (_, cs, cpss, (((pgss, _, _, _), cqgsss), f_absTs)) mk_binding fpTs Cs abss dtor_corec =
32.38 +fun define_corec (_, cs, cpss, (((pgss, _, _, _), cqgsss), f_absTs)) mk_binding fpTs Cs abss
32.39 +    dtor_corec =
32.40    let
32.41      val nn = length fpTs;
32.42      val fpT = range_type (snd (strip_typeN nn (fastype_of dtor_corec)));
32.43 @@ -1371,8 +1373,8 @@
32.44
32.45      val rel_induct0_thm =
32.46        Goal.prove_sorry lthy vars prems goal (fn {context = ctxt, prems} =>
32.47 -        mk_rel_induct0_tac ctxt ctor_rel_induct prems (map (Thm.cterm_of ctxt) IRs) exhausts ctor_defss
32.48 -          ctor_injects pre_rel_defs abs_inverses live_nesting_rel_eqs)
32.49 +        mk_rel_induct0_tac ctxt ctor_rel_induct prems (map (Thm.cterm_of ctxt) IRs) exhausts
32.50 +          ctor_defss ctor_injects pre_rel_defs abs_inverses live_nesting_rel_eqs)
32.51        |> Thm.close_derivation;
32.52    in
32.53      (postproc_co_induct lthy (length fpA_Ts) @{thm predicate2D} @{thm predicate2D_conj}
32.54 @@ -1713,8 +1715,8 @@
32.55          val thm =
32.56            Goal.prove_sorry lthy vars prems (HOLogic.mk_Trueprop concl)
32.57              (fn {context = ctxt, prems} =>
32.58 -               mk_set_induct0_tac ctxt (map (Thm.cterm_of ctxt'') Ps) prems dtor_set_inducts exhausts
32.59 -                 set_pre_defs ctor_defs dtor_ctors Abs_pre_inverses)
32.60 +               mk_set_induct0_tac ctxt (map (Thm.cterm_of ctxt'') Ps) prems dtor_set_inducts
32.61 +                 exhausts set_pre_defs ctor_defs dtor_ctors Abs_pre_inverses)
32.62            |> Thm.close_derivation;
32.63
32.64          val case_names_attr =
32.65 @@ -1811,7 +1813,8 @@
32.66               []
32.67             else
32.68               [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
32.69 -                Library.foldr1 HOLogic.mk_conj (@{map 3} (build_rel_app rs') usels vsels ctrXs_Ts))]);
32.70 +                Library.foldr1 HOLogic.mk_conj
32.71 +                  (@{map 3} (build_rel_app rs') usels vsels ctrXs_Ts))]);
32.72
32.73          fun mk_prem_concl rs' n udiscs uselss vdiscs vselss ctrXs_Tss =
32.74            Library.foldr1 HOLogic.mk_conj (flat (@{map 6} (mk_prem_ctr_concls rs' n)
32.75 @@ -2323,8 +2326,9 @@
32.76        in
32.77          Goal.prove_sorry lthy vars [] goal
32.78            (fn {context = ctxt, prems = _} =>
32.79 -             mk_rec_transfer_tac ctxt nn ns (map (Thm.cterm_of ctxt) Ss) (map (Thm.cterm_of ctxt) Rs) xsssss
32.80 -               rec_defs xtor_co_rec_transfers pre_rel_defs live_nesting_rel_eqs)
32.81 +             mk_rec_transfer_tac ctxt nn ns (map (Thm.cterm_of ctxt) Ss)
32.82 +               (map (Thm.cterm_of ctxt) Rs) xsssss rec_defs xtor_co_rec_transfers pre_rel_defs
32.83 +               live_nesting_rel_eqs)
32.84          |> Thm.close_derivation
32.85          |> Conjunction.elim_balanced nn
32.86        end;
32.87 @@ -2431,7 +2435,8 @@
32.88          val rec_o_map_thmss = derive_rec_o_map_thmss lthy recs rec_defs;
32.89
32.90          val simp_thmss =
32.91 -          @{map 6} mk_simp_thms ctr_sugars rec_thmss map_thmss rel_injectss rel_distinctss set_thmss;
32.92 +          @{map 6} mk_simp_thms ctr_sugars rec_thmss map_thmss rel_injectss rel_distinctss
32.93 +            set_thmss;
32.94
32.95          val common_notes =
32.96            (if nn > 1 then
```
```    33.1 --- a/src/HOL/Tools/BNF/bnf_fp_def_sugar_tactics.ML	Tue Dec 01 12:35:11 2015 +0100
33.2 +++ b/src/HOL/Tools/BNF/bnf_fp_def_sugar_tactics.ML	Tue Dec 01 17:18:34 2015 +0100
33.3 @@ -305,13 +305,15 @@
33.4    end;
33.5
33.6  fun solve_prem_prem_tac ctxt =
33.7 -  REPEAT o (eresolve_tac ctxt @{thms bexE rev_bexI} ORELSE' rtac ctxt @{thm rev_bexI[OF UNIV_I]} ORELSE'
33.8 -    hyp_subst_tac ctxt ORELSE' resolve_tac ctxt @{thms disjI1 disjI2}) THEN'
33.9 +  REPEAT o (eresolve_tac ctxt @{thms bexE rev_bexI} ORELSE'
33.10 +    rtac ctxt @{thm rev_bexI[OF UNIV_I]} ORELSE' hyp_subst_tac ctxt ORELSE'
33.11 +    resolve_tac ctxt @{thms disjI1 disjI2}) THEN'
33.12    (rtac ctxt refl ORELSE' assume_tac ctxt ORELSE' rtac ctxt @{thm singletonI});
33.13
33.14  fun mk_induct_leverage_prem_prems_tac ctxt nn kks fp_abs_inverses abs_inverses set_maps
33.15      pre_set_defs =
33.16 -  HEADGOAL (EVERY' (maps (fn kk => [select_prem_tac ctxt nn (dtac ctxt meta_spec) kk, etac ctxt meta_mp,
33.17 +  HEADGOAL (EVERY' (maps (fn kk => [select_prem_tac ctxt nn (dtac ctxt meta_spec) kk,
33.18 +    etac ctxt meta_mp,
33.19      SELECT_GOAL (unfold_thms_tac ctxt (pre_set_defs @ fp_abs_inverses @ abs_inverses @ set_maps @
33.20        sumprod_thms_set)),
33.21      solve_prem_prem_tac ctxt]) (rev kks)));
33.22 @@ -366,9 +368,10 @@
33.23  fun mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn kk n pre_rel_def fp_abs_inverse abs_inverse
33.24      dtor_ctor exhaust ctr_defs discss selss =
33.25    let val ks = 1 upto n in
33.26 -    EVERY' ([rtac ctxt allI, rtac ctxt allI, rtac ctxt impI, select_prem_tac ctxt nn (dtac ctxt meta_spec) kk,
33.27 -        dtac ctxt meta_spec, dtac ctxt meta_mp, assume_tac ctxt, rtac ctxt exhaust,
33.28 -        K (co_induct_inst_as_projs_tac ctxt 0), hyp_subst_tac ctxt] @
33.29 +    EVERY' ([rtac ctxt allI, rtac ctxt allI, rtac ctxt impI,
33.30 +        select_prem_tac ctxt nn (dtac ctxt meta_spec) kk, dtac ctxt meta_spec, dtac ctxt meta_mp,
33.31 +        assume_tac ctxt, rtac ctxt exhaust, K (co_induct_inst_as_projs_tac ctxt 0),
33.32 +        hyp_subst_tac ctxt] @
33.33        @{map 4} (fn k => fn ctr_def => fn discs => fn sels =>
33.34          EVERY' ([rtac ctxt exhaust, K (co_induct_inst_as_projs_tac ctxt 1)] @
33.35            map2 (fn k' => fn discs' =>
33.36 @@ -435,8 +438,8 @@
33.37          abs_inject :: ctor_defs @ nesting_rel_eqs @ simp_thms' @
33.38          @{thms id_bnf_def rel_sum_simps rel_prod_apply vimage2p_def Inl_Inr_False
33.39            iffD2[OF eq_False Inr_not_Inl] sum.inject prod.inject}) THEN
33.40 -      REPEAT_DETERM (HEADGOAL ((REPEAT_DETERM o etac ctxt conjE) THEN' (REPEAT_DETERM o rtac ctxt conjI) THEN'
33.41 -        (rtac ctxt refl ORELSE' assume_tac ctxt))))
33.42 +      REPEAT_DETERM (HEADGOAL ((REPEAT_DETERM o etac ctxt conjE) THEN'
33.43 +        (REPEAT_DETERM o rtac ctxt conjI) THEN' (rtac ctxt refl ORELSE' assume_tac ctxt))))
33.44      cts assms exhausts discss selss ctor_defss dtor_ctors ctor_injects abs_injects rel_pre_defs
33.45        abs_inverses);
33.46
33.47 @@ -445,7 +448,8 @@
33.48    rtac ctxt ctor_rel_induct 1 THEN EVERY (@{map 6} (fn cterm => fn exhaust => fn ctor_defs =>
33.49        fn ctor_inject => fn rel_pre_list_def => fn Abs_inverse =>
33.50          HEADGOAL (rtac ctxt exhaust THEN_ALL_NEW (rtac ctxt exhaust THEN_ALL_NEW
33.51 -          (rtac ctxt (infer_instantiate' ctxt (replicate 4 NONE @ [SOME cterm]) @{thm arg_cong2} RS iffD2)
33.52 +          (rtac ctxt (infer_instantiate' ctxt (replicate 4 NONE @ [SOME cterm]) @{thm arg_cong2}
33.53 +              RS iffD2)
33.54              THEN' assume_tac ctxt THEN' assume_tac ctxt THEN' TRY o resolve_tac ctxt assms))) THEN
33.55          unfold_thms_tac ctxt (ctor_inject :: rel_pre_list_def :: ctor_defs @ nesting_rel_eqs @
33.56            @{thms id_bnf_def vimage2p_def}) THEN
33.57 @@ -485,12 +489,14 @@
33.58        (rtac ctxt @{thm singletonI} ORELSE' assume_tac ctxt));
33.59
33.60  fun mk_set_cases_tac ctxt ct assms exhaust sets =
33.61 -  HEADGOAL (rtac ctxt (infer_instantiate' ctxt [SOME ct] exhaust) THEN_ALL_NEW hyp_subst_tac ctxt) THEN
33.62 +  HEADGOAL (rtac ctxt (infer_instantiate' ctxt [SOME ct] exhaust)
33.63 +    THEN_ALL_NEW hyp_subst_tac ctxt) THEN
33.64    unfold_thms_tac ctxt sets THEN
33.66      (eresolve_tac ctxt @{thms FalseE emptyE singletonE UnE UN_E insertE} ORELSE'
33.67       hyp_subst_tac ctxt ORELSE'
33.68 -     SELECT_GOAL (SOLVE (HEADGOAL (eresolve_tac ctxt assms THEN' REPEAT_DETERM o assume_tac ctxt)))));
33.69 +     SELECT_GOAL (SOLVE (HEADGOAL (eresolve_tac ctxt assms THEN' REPEAT_DETERM o
33.70 +       assume_tac ctxt)))));
33.71
33.72  fun mk_set_intros_tac ctxt sets =
33.73    TRYALL Goal.conjunction_tac THEN unfold_thms_tac ctxt sets THEN
33.74 @@ -505,7 +511,8 @@
33.75      val assms_tac =
33.76        let val assms' = map (unfold_thms ctxt (@{thm id_bnf_def} :: ctor_defs)) assms in
33.77          fold (curry (op ORELSE')) (map (fn thm =>
33.78 -            funpow (length (Thm.prems_of thm)) (fn tac => tac THEN' assume_tac ctxt) (rtac ctxt thm)) assms')
33.79 +            funpow (length (Thm.prems_of thm)) (fn tac => tac THEN' assume_tac ctxt)
33.80 +              (rtac ctxt thm)) assms')
33.81            (etac ctxt FalseE)
33.82        end;
33.83      val exhausts' = map (fn thm => thm RS @{thm asm_rl[of "P x y" for P x y]}) exhausts
33.84 @@ -519,8 +526,8 @@
33.85      unfold_thms_tac ctxt (Abs_pre_inverses @ dtor_ctors @ set_pre_defs @ ctor_defs @
33.86        @{thms id_bnf_def o_apply sum_set_simps prod_set_simps UN_empty UN_insert Un_empty_left
33.87          Un_empty_right empty_iff singleton_iff}) THEN
33.88 -    REPEAT (HEADGOAL (hyp_subst_tac ctxt ORELSE' eresolve_tac ctxt @{thms UN_E UnE singletonE} ORELSE'
33.89 -      assms_tac))
33.90 +    REPEAT (HEADGOAL (hyp_subst_tac ctxt ORELSE'
33.91 +      eresolve_tac ctxt @{thms UN_E UnE singletonE} ORELSE' assms_tac))
33.92    end;
33.93
33.94  end;
```
```    34.1 --- a/src/HOL/Tools/BNF/bnf_gfp_rec_sugar.ML	Tue Dec 01 12:35:11 2015 +0100
34.2 +++ b/src/HOL/Tools/BNF/bnf_gfp_rec_sugar.ML	Tue Dec 01 17:18:34 2015 +0100
34.3 @@ -1009,8 +1009,8 @@
34.4      |> map2 abs_curried_balanced arg_Tss
34.5      |> (fn ts => Syntax.check_terms ctxt ts
34.6        handle ERROR _ => nonprimitive_corec ctxt [])
34.7 -    |> @{map 3} (fn b => fn mx => fn t => ((b, mx), ((Binding.concealed (Thm.def_binding b), []), t)))
34.8 -      bs mxs
34.9 +    |> @{map 3} (fn b => fn mx => fn t =>
34.10 +      ((b, mx), ((Binding.concealed (Thm.def_binding b), []), t))) bs mxs
34.11      |> rpair excludess'
34.12    end;
34.13
34.14 @@ -1037,7 +1037,8 @@
34.15          val prems = maps (s_not_conj o #prems) disc_eqns;
34.16          val ctr_rhs_opt = Option.map #ctr_rhs_opt sel_eqn_opt |> the_default NONE;
34.17          val code_rhs_opt = Option.map #code_rhs_opt sel_eqn_opt |> the_default NONE;
34.18 -        val eqn_pos = Option.map (curry (op +) 1 o #eqn_pos) sel_eqn_opt |> the_default 100000; (* FIXME *)
34.19 +        val eqn_pos = Option.map (curry (op +) 1 o #eqn_pos) sel_eqn_opt
34.20 +          |> the_default 100000; (* FIXME *)
34.21
34.22          val extra_disc_eqn =
34.23            {fun_name = fun_name, fun_T = fun_T, fun_args = fun_args, ctr = ctr, ctr_no = ctr_no,
34.24 @@ -1307,7 +1308,8 @@
34.25              Goal.prove_sorry lthy [] [] goal
34.26                (fn {context = ctxt, prems = _} =>
34.27                  mk_primcorec_sel_tac ctxt def_thms distincts split_sels split_sel_asms
34.28 -                fp_nesting_maps fp_nesting_map_ident0s fp_nesting_map_comps corec_sel k m excludesss)
34.29 +                  fp_nesting_maps fp_nesting_map_ident0s fp_nesting_map_comps corec_sel k m
34.30 +                  excludesss)
34.31              |> Thm.close_derivation
34.32              |> `(is_some code_rhs_opt ? fold_thms lthy sel_defs) (*mildly too aggressive*)
34.33              |> pair sel
34.34 @@ -1444,7 +1446,8 @@
34.35                        Goal.prove_sorry lthy [] [] raw_goal
34.36                          (fn {context = ctxt, prems = _} =>
34.37                            mk_primcorec_raw_code_tac ctxt distincts discIs split_sels split_sel_asms
34.38 -                            ms ctr_thms (if exhaustive_code then try the_single nchotomys else NONE))
34.39 +                            ms ctr_thms
34.40 +                            (if exhaustive_code then try the_single nchotomys else NONE))
34.41                        |> Thm.close_derivation;
34.42                    in
34.43                      Goal.prove_sorry lthy [] [] goal
```
```    35.1 --- a/src/HOL/Tools/BNF/bnf_gfp_rec_sugar_tactics.ML	Tue Dec 01 12:35:11 2015 +0100
35.2 +++ b/src/HOL/Tools/BNF/bnf_gfp_rec_sugar_tactics.ML	Tue Dec 01 17:18:34 2015 +0100
35.3 @@ -72,7 +72,8 @@
35.4  fun mk_primcorec_assumption_tac ctxt discIs =
35.5    SELECT_GOAL (unfold_thms_tac ctxt @{thms fst_conv snd_conv not_not not_False_eq_True
35.6        not_True_eq_False de_Morgan_conj de_Morgan_disj} THEN
35.7 -    SOLVE (HEADGOAL (REPEAT o (rtac ctxt refl ORELSE' assume_tac ctxt ORELSE' etac ctxt conjE ORELSE'
35.8 +    SOLVE (HEADGOAL (REPEAT o (rtac ctxt refl ORELSE' assume_tac ctxt ORELSE'
35.9 +    etac ctxt conjE ORELSE'
35.10      eresolve_tac ctxt falseEs ORELSE'
35.11      resolve_tac ctxt @{thms TrueI conjI disjI1 disjI2} ORELSE'
35.12      dresolve_tac ctxt discIs THEN' assume_tac ctxt ORELSE'
35.13 @@ -137,7 +138,8 @@
35.14      resolve_tac ctxt split_connectI ORELSE'
35.15      Splitter.split_asm_tac ctxt (split_if_asm :: split_asms) ORELSE'
35.16      Splitter.split_tac ctxt (split_if :: splits) ORELSE'
35.17 -    eresolve_tac ctxt (map (fn thm => thm RS neq_eq_eq_contradict) distincts) THEN' assume_tac ctxt ORELSE'
35.18 +    eresolve_tac ctxt (map (fn thm => thm RS neq_eq_eq_contradict) distincts) THEN'
35.19 +    assume_tac ctxt ORELSE'
35.20      etac ctxt notE THEN' assume_tac ctxt ORELSE'
35.21      (CHANGED o SELECT_GOAL (unfold_thms_tac ctxt (@{thms fst_conv snd_conv id_def comp_def split_def
35.22           sum.case sum.sel sum.distinct[THEN eq_False[THEN iffD2]]} @
35.23 @@ -148,7 +150,8 @@
35.24
35.25  fun mk_primcorec_ctr_tac ctxt m collapse disc_fun_opt sel_funs =
35.26    HEADGOAL (rtac ctxt ((if null sel_funs then collapse else collapse RS sym) RS trans) THEN'
35.27 -    (the_default (K all_tac) (Option.map (rtac ctxt) disc_fun_opt)) THEN' REPEAT_DETERM_N m o assume_tac ctxt) THEN
35.28 +    (the_default (K all_tac) (Option.map (rtac ctxt) disc_fun_opt)) THEN'
35.29 +    REPEAT_DETERM_N m o assume_tac ctxt) THEN
35.30    unfold_thms_tac ctxt (@{thm split_def} :: unfold_lets @ sel_funs) THEN HEADGOAL (rtac ctxt refl);
35.31
35.32  fun inst_split_eq ctxt split =
```
```    36.1 --- a/src/HOL/Tools/BNF/bnf_lfp_rec_sugar.ML	Tue Dec 01 12:35:11 2015 +0100
36.2 +++ b/src/HOL/Tools/BNF/bnf_lfp_rec_sugar.ML	Tue Dec 01 17:18:34 2015 +0100
36.3 @@ -458,7 +458,8 @@
36.4      (recs, ctr_poss)
36.5      |-> map2 (fn recx => fn ctr_pos => list_comb (recx, rec_args) |> permute_args ctr_pos)
36.6      |> Syntax.check_terms ctxt
36.7 -    |> @{map 3} (fn b => fn mx => fn t => ((b, mx), ((Binding.concealed (Thm.def_binding b), []), t)))
36.8 +    |> @{map 3} (fn b => fn mx => fn t =>
36.9 +        ((b, mx), ((Binding.concealed (Thm.def_binding b), []), t)))
36.10        bs mxs
36.11    end;
36.12
```
```    37.1 --- a/src/HOL/Tools/BNF/bnf_lfp_rec_sugar_more.ML	Tue Dec 01 12:35:11 2015 +0100
37.2 +++ b/src/HOL/Tools/BNF/bnf_lfp_rec_sugar_more.ML	Tue Dec 01 17:18:34 2015 +0100
37.3 @@ -40,7 +40,8 @@
37.4    | basic_lfp_sugars_of bs arg_Ts callers callssss0 lthy0 =
37.5      let
37.6        val ((missing_arg_Ts, perm0_kks,
37.7 -            fp_sugars as {fp_nesting_bnfs, fp_co_induct_sugar = {common_co_inducts = [common_induct], ...}, ...} :: _,
37.8 +            fp_sugars as {fp_nesting_bnfs,
37.9 +              fp_co_induct_sugar = {common_co_inducts = [common_induct], ...}, ...} :: _,
37.10              (lfp_sugar_thms, _)), lthy) =
37.11          nested_to_mutual_fps (K true) Least_FP bs arg_Ts callers callssss0 lthy0;
37.12
```
```    38.1 --- a/src/HOL/Tools/BNF/bnf_lfp_size.ML	Tue Dec 01 12:35:11 2015 +0100
38.2 +++ b/src/HOL/Tools/BNF/bnf_lfp_size.ML	Tue Dec 01 17:18:34 2015 +0100
38.3 @@ -42,7 +42,8 @@
38.4    Context.proof_map (Data.map (Symtab.update (T_name, (size_name, (size_simps, size_gen_o_maps)))));
38.5
38.6  fun register_size_global T_name size_name size_simps size_gen_o_maps =
38.7 -  Context.theory_map (Data.map (Symtab.update (T_name, (size_name, (size_simps, size_gen_o_maps)))));
38.8 +  Context.theory_map
38.9 +    (Data.map (Symtab.update (T_name, (size_name, (size_simps, size_gen_o_maps)))));
38.10
38.11  val size_of = Symtab.lookup o Data.get o Context.Proof;
38.12  val size_of_global = Symtab.lookup o Data.get o Context.Theory;
38.13 @@ -70,8 +71,9 @@
38.14  fun mk_size_neq ctxt cts exhaust sizes =
38.15    HEADGOAL (rtac ctxt (infer_instantiate' ctxt (map SOME cts) exhaust)) THEN
38.16    ALLGOALS (hyp_subst_tac ctxt) THEN
38.17 -  Ctr_Sugar_Tactics.unfold_thms_tac ctxt (@{thm neq0_conv} :: sizes) THEN
38.18 -  ALLGOALS (REPEAT_DETERM o (rtac ctxt @{thm zero_less_Suc} ORELSE' rtac ctxt @{thm trans_less_add2}));
38.19 +  unfold_thms_tac ctxt (@{thm neq0_conv} :: sizes) THEN
38.20 +  ALLGOALS (REPEAT_DETERM o (rtac ctxt @{thm zero_less_Suc} ORELSE'
38.21 +    rtac ctxt @{thm trans_less_add2}));
38.22
38.23  fun generate_datatype_size (fp_sugars as ({T = Type (_, As), BT = Type (_, Bs), fp = Least_FP,
38.24          fp_res = {bnfs = fp_bnfs, ...}, fp_nesting_bnfs, live_nesting_bnfs, ...} : fp_sugar) :: _)
38.25 @@ -236,7 +238,8 @@
38.26          (Spec_Rules.retrieve lthy0 @{const size ('a)}
38.27           |> map_filter (try (fn (Spec_Rules.Equational, (_, [thm])) => thm)));
38.28
38.29 -      val nested_size_maps = map (mk_pointful lthy2) nested_size_gen_o_maps @ nested_size_gen_o_maps;
38.30 +      val nested_size_maps =
38.31 +        map (mk_pointful lthy2) nested_size_gen_o_maps @ nested_size_gen_o_maps;
38.32        val all_inj_maps =
38.33          @{thm prod.inj_map} :: map inj_map_of_bnf (fp_bnfs @ fp_nesting_bnfs @ live_nesting_bnfs)
38.34          |> distinct Thm.eq_thm_prop;
```
```    39.1 --- a/src/HOL/Tools/Ctr_Sugar/ctr_sugar.ML	Tue Dec 01 12:35:11 2015 +0100
39.2 +++ b/src/HOL/Tools/Ctr_Sugar/ctr_sugar.ML	Tue Dec 01 17:18:34 2015 +0100
39.3 @@ -50,8 +50,8 @@
39.4    val ctr_sugars_of_global: theory -> ctr_sugar list
39.5    val ctr_sugar_of_case: Proof.context -> string -> ctr_sugar option
39.6    val ctr_sugar_of_case_global: theory -> string -> ctr_sugar option
39.7 -  val ctr_sugar_interpretation: string ->
39.8 -    (ctr_sugar -> local_theory -> local_theory) -> theory -> theory
39.9 +  val ctr_sugar_interpretation: string -> (ctr_sugar -> local_theory -> local_theory) -> theory ->
39.10 +    theory
39.11    val interpret_ctr_sugar: (string -> bool) -> ctr_sugar -> local_theory -> local_theory
39.12    val register_ctr_sugar_raw: ctr_sugar -> local_theory -> local_theory
39.13    val register_ctr_sugar: (string -> bool) -> ctr_sugar -> local_theory -> local_theory
```
```    40.1 --- a/src/HOL/Tools/Ctr_Sugar/ctr_sugar_tactics.ML	Tue Dec 01 12:35:11 2015 +0100
40.2 +++ b/src/HOL/Tools/Ctr_Sugar/ctr_sugar_tactics.ML	Tue Dec 01 17:18:34 2015 +0100
40.3 @@ -54,7 +54,8 @@
40.4
40.5  fun mk_nchotomy_tac ctxt n exhaust =
40.6    HEADGOAL (rtac ctxt allI THEN' rtac ctxt exhaust THEN'
40.7 -    EVERY' (maps (fn k => [rtac ctxt (mk_disjIN n k), REPEAT_DETERM o rtac ctxt exI, assume_tac ctxt])
40.8 +    EVERY' (maps (fn k =>
40.9 +        [rtac ctxt (mk_disjIN n k), REPEAT_DETERM o rtac ctxt exI, assume_tac ctxt])
40.10        (1 upto n)));
40.11
40.12  fun mk_unique_disc_def_tac ctxt m uexhaust =
40.13 @@ -114,7 +115,8 @@
40.14    else
40.15      let val ks = 1 upto n in
40.16        HEADGOAL (rtac ctxt uexhaust_disc THEN'
40.17 -        EVERY' (@{map 5} (fn k => fn m => fn distinct_discss => fn distinct_discss' => fn uuncollapse =>
40.18 +        EVERY' (@{map 5} (fn k => fn m => fn distinct_discss => fn distinct_discss' =>
40.19 +            fn uuncollapse =>
40.20            EVERY' [rtac ctxt (uuncollapse RS trans) THEN'
40.21              TRY o assume_tac ctxt, rtac ctxt sym, rtac ctxt vexhaust_disc,
40.22              EVERY' (@{map 4} (fn k' => fn distinct_discs => fn distinct_discs' => fn vuncollapse =>
40.23 @@ -124,13 +126,17 @@
40.24                     (if m = 0 then
40.25                        [rtac ctxt refl]
40.26                      else
40.27 -                      [if n = 1 then K all_tac
40.28 -                       else EVERY' [dtac ctxt meta_mp, assume_tac ctxt, dtac ctxt meta_mp, assume_tac ctxt],
40.29 -                       REPEAT_DETERM_N (Int.max (0, m - 1)) o etac ctxt conjE,
40.30 -                       asm_simp_tac (ss_only [] ctxt)])
40.31 +                      [if n = 1 then
40.32 +                         K all_tac
40.33 +                       else
40.34 +                         EVERY' [dtac ctxt meta_mp, assume_tac ctxt, dtac ctxt meta_mp,
40.35 +                           assume_tac ctxt],
40.36 +                         REPEAT_DETERM_N (Int.max (0, m - 1)) o etac ctxt conjE,
40.37 +                         asm_simp_tac (ss_only [] ctxt)])
40.38                   else
40.39                     [dtac ctxt (the_single (if k = n then distinct_discs else distinct_discs')),
40.40 -                    etac ctxt (if k = n then @{thm iff_contradict(1)} else @{thm iff_contradict(2)}),
40.41 +                    etac ctxt (if k = n then @{thm iff_contradict(1)}
40.42 +                      else @{thm iff_contradict(2)}),
40.43                      assume_tac ctxt, assume_tac ctxt]))
40.44                ks distinct_discss distinct_discss' uncollapses)])
40.45            ks ms distinct_discsss distinct_discsss' uncollapses))
40.46 @@ -152,8 +158,8 @@
40.47      val ks = 1 upto n;
40.48    in
40.49      HEADGOAL (rtac ctxt (case_def' RS trans) THEN' rtac ctxt @{thm the_equality} THEN'
40.50 -      rtac ctxt (mk_disjIN n k) THEN' REPEAT_DETERM o rtac ctxt exI THEN' rtac ctxt conjI THEN' rtac ctxt refl THEN'
40.51 -      rtac ctxt refl THEN'
40.52 +      rtac ctxt (mk_disjIN n k) THEN' REPEAT_DETERM o rtac ctxt exI THEN' rtac ctxt conjI THEN'
40.53 +      rtac ctxt refl THEN' rtac ctxt refl THEN'
40.54        EVERY' (map2 (fn k' => fn distincts =>
40.55          (if k' < n then etac ctxt disjE else K all_tac) THEN'
40.56          (if k' = k then mk_case_same_ctr_tac ctxt injects
40.57 @@ -182,7 +188,8 @@
40.58         simp_tac (ss_only (@{thms simp_thms} @ cases @ nth selss (k - 1) @ nth injectss (k - 1) @
40.59           flat (nth distinctsss (k - 1))) ctxt)) k) THEN
40.60      ALLGOALS (etac ctxt thin_rl THEN' rtac ctxt iffI THEN'
40.61 -      REPEAT_DETERM o rtac ctxt allI THEN' rtac ctxt impI THEN' REPEAT_DETERM o etac ctxt conjE THEN'
40.62 +      REPEAT_DETERM o rtac ctxt allI THEN' rtac ctxt impI THEN'
40.63 +      REPEAT_DETERM o etac ctxt conjE THEN'
40.64        hyp_subst_tac ctxt THEN' assume_tac ctxt THEN'
40.65        REPEAT_DETERM o etac ctxt allE THEN' etac ctxt impE THEN'
40.66        REPEAT_DETERM o (rtac ctxt conjI THEN' rtac ctxt refl) THEN'
```
```    41.1 --- a/src/HOL/Tools/Transfer/transfer_bnf.ML	Tue Dec 01 12:35:11 2015 +0100
41.2 +++ b/src/HOL/Tools/Transfer/transfer_bnf.ML	Tue Dec 01 17:18:34 2015 +0100
41.3 @@ -66,11 +66,6 @@
41.4
41.5  fun bnf_of_fp_sugar (fp_sugar:fp_sugar) = nth (#bnfs (#fp_res fp_sugar)) (#fp_res_index fp_sugar)
41.6
41.7 -fun fp_sugar_only_type_ctr f fp_sugars =
41.8 -  (case filter (is_Type o T_of_bnf o bnf_of_fp_sugar) fp_sugars of
41.9 -    [] => I
41.10 -  | fp_sugars' => f fp_sugars')
41.11 -
41.12  (* relation constraints - bi_total & co. *)
41.13
41.14  fun mk_relation_constraint name arg =
41.15 @@ -410,7 +405,7 @@
41.16
41.17  fun transfer_fp_sugars_interpretation fp_sugar lthy =
41.18    let
41.19 -    val lthy = pred_injects fp_sugar lthy
41.20 +    val lthy = lthy |> (is_Type o T_of_bnf o bnf_of_fp_sugar) fp_sugar ? pred_injects fp_sugar
41.21      val transfer_rules_notes = fp_sugar_transfer_rules fp_sugar
41.22    in
41.23      lthy
41.24 @@ -419,7 +414,6 @@
41.25    end
41.26
41.27  val _ =
41.28 -  Theory.setup (fp_sugars_interpretation transfer_plugin
41.29 -    (fp_sugar_only_type_ctr (fold transfer_fp_sugars_interpretation)))
41.30 +  Theory.setup (fp_sugars_interpretation transfer_plugin (fold transfer_fp_sugars_interpretation))
41.31
41.32  end
```
```    42.1 --- a/src/HOL/Transcendental.thy	Tue Dec 01 12:35:11 2015 +0100
42.2 +++ b/src/HOL/Transcendental.thy	Tue Dec 01 17:18:34 2015 +0100
42.3 @@ -10,6 +10,16 @@
42.4  imports Binomial Series Deriv NthRoot
42.5  begin
42.6
42.7 +lemma of_int_leD:
42.8 +  fixes x :: "'a :: linordered_idom"
42.9 +  shows "\<bar>of_int n\<bar> \<le> x \<Longrightarrow> n=0 \<or> x\<ge>1"
42.10 +  by (metis (no_types) le_less_trans not_less of_int_abs of_int_less_1_iff zabs_less_one_iff)
42.11 +
42.12 +lemma of_int_lessD:
42.13 +  fixes x :: "'a :: linordered_idom"
42.14 +  shows "\<bar>of_int n\<bar> < x \<Longrightarrow> n=0 \<or> x>1"
42.15 +  by (metis less_le_trans not_less of_int_abs of_int_less_1_iff zabs_less_one_iff)
42.16 +
42.17  lemma fact_in_Reals: "fact n \<in> \<real>" by (induction n) auto
42.18
42.19  lemma pochhammer_of_real: "pochhammer (of_real x) n = of_real (pochhammer x n)"
42.20 @@ -1979,8 +1989,7 @@
42.21        assume "x \<le> y" "y \<le> a"
42.22        with \<open>0 < x\<close> \<open>a < 1\<close> have "0 < 1 / y - 1" "0 < y"
42.23          by (auto simp: field_simps)
42.24 -      with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
42.25 -        by auto
42.26 +      with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z" by blast
42.27      qed
42.28      also have "\<dots> \<le> 0"
42.29        using ln_le_minus_one \<open>0 < x\<close> \<open>x < a\<close> by (auto simp: field_simps)
42.30 @@ -3090,6 +3099,11 @@
42.31    using cos_add [where x=x and y=x]
42.32    by (simp add: power2_eq_square)
42.33
42.34 +lemma sin_cos_le1:
42.35 +  fixes x::real shows "abs (sin x * sin y + cos x * cos y) \<le> 1"
42.36 +  using cos_diff [of x y]
42.37 +  by (metis abs_cos_le_one add.commute)
42.38 +
42.39  lemma DERIV_fun_pow: "DERIV g x :> m ==>
42.40        DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
42.41    by (auto intro!: derivative_eq_intros simp:)
```
```    43.1 --- a/src/HOL/ex/Dedekind_Real.thy	Tue Dec 01 12:35:11 2015 +0100
43.2 +++ b/src/HOL/ex/Dedekind_Real.thy	Tue Dec 01 17:18:34 2015 +0100
43.3 @@ -784,8 +784,7 @@
43.4      qed
43.5      hence "y < r" by simp
43.6      with ypos have  dless: "?d < (r * ?d)/y"
43.7 -      by (simp add: pos_less_divide_eq mult.commute [of ?d]
43.8 -                    mult_strict_right_mono dpos)
43.9 +      using dpos less_divide_eq_1 by fastforce
43.10      have "r + ?d < r*x"
43.11      proof -
43.12        have "r + ?d < r + (r * ?d)/y" by (simp add: dless)
```
```    44.1 --- a/src/HOL/ex/Sqrt.thy	Tue Dec 01 12:35:11 2015 +0100
44.2 +++ b/src/HOL/ex/Sqrt.thy	Tue Dec 01 17:18:34 2015 +0100
44.3 @@ -14,7 +14,7 @@
44.4    assumes "prime (p::nat)"
44.5    shows "sqrt p \<notin> \<rat>"
44.6  proof
44.7 -  from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_nat_def)
44.8 +  from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_def)
44.9    assume "sqrt p \<in> \<rat>"
44.10    then obtain m n :: nat where
44.11        n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
44.12 @@ -59,7 +59,7 @@
44.13    assumes "prime (p::nat)"
44.14    shows "sqrt p \<notin> \<rat>"
44.15  proof
44.16 -  from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_nat_def)
44.17 +  from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_def)
44.18    assume "sqrt p \<in> \<rat>"
44.19    then obtain m n :: nat where
44.20        n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
```
```    45.1 --- a/src/HOL/ex/Sqrt_Script.thy	Tue Dec 01 12:35:11 2015 +0100
45.2 +++ b/src/HOL/ex/Sqrt_Script.thy	Tue Dec 01 17:18:34 2015 +0100
45.3 @@ -17,7 +17,7 @@
45.4  subsection \<open>Preliminaries\<close>
45.5
45.6  lemma prime_nonzero:  "prime (p::nat) \<Longrightarrow> p \<noteq> 0"
45.7 -  by (force simp add: prime_nat_def)
45.8 +  by (force simp add: prime_def)
45.9
45.10  lemma prime_dvd_other_side:
45.11      "(n::nat) * n = p * (k * k) \<Longrightarrow> prime p \<Longrightarrow> p dvd n"
45.12 @@ -32,7 +32,7 @@
45.13    apply (erule disjE)
45.14     apply (frule mult_le_mono, assumption)
45.15     apply auto
45.16 -  apply (force simp add: prime_nat_def)
45.17 +  apply (force simp add: prime_def)
45.18    done
45.19
45.20  lemma rearrange: "(j::nat) * (p * j) = k * k \<Longrightarrow> k * k = p * (j * j)"
```