--- a/src/HOL/Analysis/Cartesian_Euclidean_Space.thy Mon Apr 09 16:20:23 2018 +0200
+++ b/src/HOL/Analysis/Cartesian_Euclidean_Space.thy Mon Apr 09 17:21:10 2018 +0100
@@ -297,7 +297,7 @@
lemma norm_le_l1_cart: "norm x <= sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
by (simp add: norm_vec_def L2_set_le_sum)
-lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
+lemma scalar_mult_eq_scaleR [simp]: "c *s x = c *\<^sub>R x"
unfolding scaleR_vec_def vector_scalar_mult_def by simp
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
--- a/src/HOL/Analysis/Determinants.thy Mon Apr 09 16:20:23 2018 +0200
+++ b/src/HOL/Analysis/Determinants.thy Mon Apr 09 17:21:10 2018 +0100
@@ -197,7 +197,7 @@
by (simp add: det_diagonal mat_def)
lemma det_0 [simp]: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0"
- by (simp add: det_def prod_zero)
+ by (simp add: det_def prod_zero power_0_left)
lemma det_permute_rows:
fixes A :: "'a::comm_ring_1^'n^'n"
@@ -815,20 +815,16 @@
apply (simp only: ab_left_minus add.assoc[symmetric])
apply simp
done
- from c ci
have thr0: "- row i A = sum (\<lambda>j. (1/ c i) *s (c j *s row j A)) (?U - {i})"
- unfolding sum.remove[OF fU iU] sum_cmul
- apply -
apply (rule vector_mul_lcancel_imp[OF ci])
- apply (auto simp add: field_simps)
- unfolding *
- apply rule
+ using c ci unfolding sum.remove[OF fU iU] sum_cmul
+ apply (auto simp add: field_simps *)
done
have thr: "- row i A \<in> span {row j A| j. j \<noteq> i}"
unfolding thr0
apply (rule span_sum)
apply simp
- apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])+
+ apply (rule span_mul [where 'a="real^'n"])
apply (rule span_superset)
apply auto
done
@@ -869,7 +865,7 @@
have thd0: "det (\<chi> i. if i = k then row k A + (\<Sum>i \<in> ?Uk. x $ i *s row i A) else row i A) = det A"
apply (rule det_row_span)
apply (rule span_sum)
- apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])+
+ apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])
apply (rule span_superset)
apply auto
done
--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Mon Apr 09 16:20:23 2018 +0200
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Mon Apr 09 17:21:10 2018 +0100
@@ -1289,7 +1289,8 @@
have "prj1 (vf X - uf X) ^ DIM('M) = (\<Prod>i::'M \<in> Basis. prj1 (vf X - uf X))"
by (rule prod_constant [symmetric])
also have "\<dots> = (\<Prod>i\<in>Basis. vf X \<bullet> i - uf X \<bullet> i)"
- using prj1_idem [OF \<open>X \<in> \<D>\<close>] by (auto simp: algebra_simps intro: prod.cong)
+ apply (rule prod.cong [OF refl])
+ by (simp add: \<open>X \<in> \<D>\<close> inner_diff_left prj1_idem)
finally have prj1_eq: "prj1 (vf X - uf X) ^ DIM('M) = (\<Prod>i\<in>Basis. vf X \<bullet> i - uf X \<bullet> i)" .
have "uf X \<in> cbox (uf X) (vf X)" "vf X \<in> cbox (uf X) (vf X)"
using uvz [OF \<open>X \<in> \<D>\<close>] by (force simp: mem_box)+
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Mon Apr 09 16:20:23 2018 +0200
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Mon Apr 09 17:21:10 2018 +0100
@@ -660,12 +660,18 @@
lemma integrable_cmul: "f integrable_on S \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on S"
unfolding integrable_on_def by(auto intro: has_integral_cmul)
-lemma integrable_on_cmult_iff:
+lemma integrable_on_scaleR_iff [simp]:
+ fixes c :: real
+ assumes "c \<noteq> 0"
+ shows "(\<lambda>x. c *\<^sub>R f x) integrable_on S \<longleftrightarrow> f integrable_on S"
+ using integrable_cmul[of "\<lambda>x. c *\<^sub>R f x" S "1 / c"] integrable_cmul[of f S c] \<open>c \<noteq> 0\<close>
+ by auto
+
+lemma integrable_on_cmult_iff [simp]:
fixes c :: real
assumes "c \<noteq> 0"
shows "(\<lambda>x. c * f x) integrable_on S \<longleftrightarrow> f integrable_on S"
- using integrable_cmul[of "\<lambda>x. c * f x" S "1 / c"] integrable_cmul[of f S c] \<open>c \<noteq> 0\<close>
- by auto
+ using integrable_on_scaleR_iff [of c f] assms by simp
lemma integrable_on_cmult_left:
assumes "f integrable_on S"
@@ -676,6 +682,9 @@
lemma integrable_neg: "f integrable_on S \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on S"
unfolding integrable_on_def by(auto intro: has_integral_neg)
+lemma integrable_neg_iff: "(\<lambda>x. -f(x)) integrable_on S \<longleftrightarrow> f integrable_on S"
+ using integrable_neg by fastforce
+
lemma integrable_diff:
"f integrable_on S \<Longrightarrow> g integrable_on S \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on S"
unfolding integrable_on_def by(auto intro: has_integral_diff)
@@ -2295,20 +2304,20 @@
using assms negligible_subset by force
lemma negligible_Un:
- assumes "negligible s"
- and "negligible t"
- shows "negligible (s \<union> t)"
- unfolding negligible_def
-proof (safe, goal_cases)
- case (1 a b)
- note assms[unfolded negligible_def,rule_format,of a b]
- then show ?case
- apply (subst has_integral_spike_eq[OF assms(2)])
- defer
- apply assumption
- unfolding indicator_def
- apply auto
- done
+ assumes "negligible S" and T: "negligible T"
+ shows "negligible (S \<union> T)"
+proof -
+ have "(indicat_real (S \<union> T) has_integral 0) (cbox a b)"
+ if S0: "(indicat_real S has_integral 0) (cbox a b)"
+ and "(indicat_real T has_integral 0) (cbox a b)" for a b
+ proof (subst has_integral_spike_eq[OF T])
+ show "indicat_real S x = indicat_real (S \<union> T) x" if "x \<in> cbox a b - T" for x
+ by (metis Diff_iff Un_iff indicator_def that)
+ show "(indicat_real S has_integral 0) (cbox a b)"
+ by (simp add: S0)
+ qed
+ with assms show ?thesis
+ unfolding negligible_def by blast
qed
lemma negligible_Un_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> negligible s \<and> negligible t"
@@ -3430,7 +3439,7 @@
by (simp add: inner_simps field_simps)
ultimately show ?thesis using False
by (simp add: image_affinity_cbox content_cbox'
- prod.distrib[symmetric] prod_constant[symmetric] inner_diff_left)
+ prod.distrib[symmetric] prod_constant[symmetric] inner_diff_left del: prod_constant)
qed
qed
@@ -5235,19 +5244,21 @@
lemma has_integral_Un:
fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
- assumes f: "(f has_integral i) s" "(f has_integral j) t"
- and neg: "negligible (s \<inter> t)"
- shows "(f has_integral (i + j)) (s \<union> t)"
-proof -
- note * = has_integral_restrict_UNIV[symmetric, of f]
- show ?thesis
- unfolding *
- apply (rule has_integral_spike[OF assms(3)])
- defer
- apply (rule has_integral_add[OF f[unfolded *]])
- apply auto
- done
-qed
+ assumes f: "(f has_integral i) S" "(f has_integral j) T"
+ and neg: "negligible (S \<inter> T)"
+ shows "(f has_integral (i + j)) (S \<union> T)"
+ unfolding has_integral_restrict_UNIV[symmetric, of f]
+proof (rule has_integral_spike[OF neg])
+ let ?f = "\<lambda>x. (if x \<in> S then f x else 0) + (if x \<in> T then f x else 0)"
+ show "(?f has_integral i + j) UNIV"
+ by (simp add: f has_integral_add)
+qed auto
+
+lemma integral_Un [simp]:
+ fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
+ assumes "f integrable_on S" "f integrable_on T" "negligible (S \<inter> T)"
+ shows "integral (S \<union> T) f = integral S f + integral T f"
+ using has_integral_Un by (simp add: has_integral_Un assms integrable_integral integral_unique)
lemma integrable_Un:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b :: banach"
--- a/src/HOL/Computational_Algebra/Formal_Power_Series.thy Mon Apr 09 16:20:23 2018 +0200
+++ b/src/HOL/Computational_Algebra/Formal_Power_Series.thy Mon Apr 09 17:21:10 2018 +0100
@@ -3434,10 +3434,12 @@
then show ?thesis by simp
next
case (Suc m)
- have th0: "a^n = prod (\<lambda>k. a) {0..m}" "(a oo c) ^ n = prod (\<lambda>k. a oo c) {0..m}"
+ have "(\<Prod>n = 0..m. a) oo c = (\<Prod>n = 0..m. a oo c)"
+ using c0 fps_compose_prod_distrib by blast
+ moreover have th0: "a^n = prod (\<lambda>k. a) {0..m}" "(a oo c) ^ n = prod (\<lambda>k. a oo c) {0..m}"
by (simp_all add: prod_constant Suc)
- then show ?thesis
- by (simp add: fps_compose_prod_distrib[OF c0])
+ ultimately show ?thesis
+ by presburger
qed
lemma fps_compose_uminus: "- (a::'a::ring_1 fps) oo c = - (a oo c)"
--- a/src/HOL/Factorial.thy Mon Apr 09 16:20:23 2018 +0200
+++ b/src/HOL/Factorial.thy Mon Apr 09 17:21:10 2018 +0100
@@ -290,17 +290,13 @@
using prod_constant [where A="{0.. h}" and y="- 1 :: 'a"]
by auto
with Suc show ?thesis
- using pochhammer_Suc_prod_rev [of "b - of_nat k + 1"]
- by (auto simp add: pochhammer_Suc_prod prod.distrib [symmetric] eq of_nat_diff)
+ using pochhammer_Suc_prod_rev [of "b - of_nat k + 1"]
+ by (auto simp add: pochhammer_Suc_prod prod.distrib [symmetric] eq of_nat_diff simp del: prod_constant)
qed
lemma pochhammer_minus':
"pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
- apply (simp only: pochhammer_minus [where b = b])
- apply (simp only: mult.assoc [symmetric])
- apply (simp only: power_add [symmetric])
- apply simp
- done
+ by (simp add: pochhammer_minus)
lemma pochhammer_same: "pochhammer (- of_nat n) n =
((- 1) ^ n :: 'a::{semiring_char_0,comm_ring_1,semiring_no_zero_divisors}) * fact n"
--- a/src/HOL/Fields.thy Mon Apr 09 16:20:23 2018 +0200
+++ b/src/HOL/Fields.thy Mon Apr 09 17:21:10 2018 +0100
@@ -46,6 +46,14 @@
lemmas [arith_split] = nat_diff_split split_min split_max
+context linordered_nonzero_semiring
+begin
+lemma of_nat_max: "of_nat (max x y) = max (of_nat x) (of_nat y)"
+ by (auto simp: max_def)
+
+lemma of_nat_min: "of_nat (min x y) = min (of_nat x) (of_nat y)"
+ by (auto simp: min_def)
+end
text\<open>Lemmas \<open>divide_simps\<close> move division to the outside and eliminates them on (in)equalities.\<close>
--- a/src/HOL/Groups_Big.thy Mon Apr 09 16:20:23 2018 +0200
+++ b/src/HOL/Groups_Big.thy Mon Apr 09 17:21:10 2018 +0100
@@ -1335,7 +1335,7 @@
for f :: "'a \<Rightarrow> nat"
using prod_zero_iff by (simp del: neq0_conv add: zero_less_iff_neq_zero)
-lemma prod_constant: "(\<Prod>x\<in> A. y) = y ^ card A"
+lemma prod_constant [simp]: "(\<Prod>x\<in> A. y) = y ^ card A"
for y :: "'a::comm_monoid_mult"
by (induct A rule: infinite_finite_induct) simp_all
--- a/src/HOL/Int.thy Mon Apr 09 16:20:23 2018 +0200
+++ b/src/HOL/Int.thy Mon Apr 09 17:21:10 2018 +0100
@@ -111,7 +111,6 @@
end
-
subsection \<open>Ordering properties of arithmetic operations\<close>
instance int :: ordered_cancel_ab_semigroup_add
@@ -423,6 +422,12 @@
lemma of_int_power_less_of_int_cancel_iff[simp]: "of_int x < (of_int b) ^ w\<longleftrightarrow> x < b ^ w"
by (metis (mono_tags) of_int_less_iff of_int_power)
+lemma of_int_max: "of_int (max x y) = max (of_int x) (of_int y)"
+ by (auto simp: max_def)
+
+lemma of_int_min: "of_int (min x y) = min (of_int x) (of_int y)"
+ by (auto simp: min_def)
+
end
text \<open>Comparisons involving @{term of_int}.\<close>
--- a/src/HOL/Lattices_Big.thy Mon Apr 09 16:20:23 2018 +0200
+++ b/src/HOL/Lattices_Big.thy Mon Apr 09 17:21:10 2018 +0100
@@ -462,8 +462,47 @@
defines
Min = Min.F and Max = Max.F ..
+abbreviation MINIMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
+ where "MINIMUM A f \<equiv> Min(f ` A)"
+abbreviation MAXIMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
+ where "MAXIMUM A f \<equiv> Max(f ` A)"
+
end
+
+syntax (ASCII)
+ "_MIN1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3MIN _./ _)" [0, 10] 10)
+ "_MIN" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3MIN _:_./ _)" [0, 0, 10] 10)
+ "_MAX1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3MAX _./ _)" [0, 10] 10)
+ "_MAX" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3MAX _:_./ _)" [0, 0, 10] 10)
+
+syntax (output)
+ "_MIN1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3MIN _./ _)" [0, 10] 10)
+ "_MIN" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3MIN _:_./ _)" [0, 0, 10] 10)
+ "_MAX1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3MAX _./ _)" [0, 10] 10)
+ "_MAX" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3MAX _:_./ _)" [0, 0, 10] 10)
+
+syntax
+ "_MIN1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3MIN _./ _)" [0, 10] 10)
+ "_MIN" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3MIN _\<in>_./ _)" [0, 0, 10] 10)
+ "_MAX1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3MAX _./ _)" [0, 10] 10)
+ "_MAX" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3MAX _\<in>_./ _)" [0, 0, 10] 10)
+
+translations
+ "MIN x y. B" \<rightleftharpoons> "MIN x. MIN y. B"
+ "MIN x. B" \<rightleftharpoons> "CONST MINIMUM CONST UNIV (\<lambda>x. B)"
+ "MIN x. B" \<rightleftharpoons> "MIN x \<in> CONST UNIV. B"
+ "MIN x\<in>A. B" \<rightleftharpoons> "CONST MINIMUM A (\<lambda>x. B)"
+ "MAX x y. B" \<rightleftharpoons> "MAX x. MAX y. B"
+ "MAX x. B" \<rightleftharpoons> "CONST MAXIMUM CONST UNIV (\<lambda>x. B)"
+ "MAX x. B" \<rightleftharpoons> "MAX x \<in> CONST UNIV. B"
+ "MAX x\<in>A. B" \<rightleftharpoons> "CONST MAXIMUM A (\<lambda>x. B)"
+
+print_translation \<open>
+ [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFIMUM} @{syntax_const "_INF"},
+ Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPREMUM} @{syntax_const "_SUP"}]
+\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
+
text \<open>An aside: @{const Min}/@{const Max} on linear orders as special case of @{const Inf_fin}/@{const Sup_fin}\<close>
lemma Inf_fin_Min: