--- a/src/HOL/Analysis/Cartesian_Euclidean_Space.thy Sun Apr 08 12:31:08 2018 +0200
+++ b/src/HOL/Analysis/Cartesian_Euclidean_Space.thy Mon Apr 09 15:20:11 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/Factorial.thy Sun Apr 08 12:31:08 2018 +0200
+++ b/src/HOL/Factorial.thy Mon Apr 09 15:20:11 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 Sun Apr 08 12:31:08 2018 +0200
+++ b/src/HOL/Fields.thy Mon Apr 09 15:20:11 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 Sun Apr 08 12:31:08 2018 +0200
+++ b/src/HOL/Groups_Big.thy Mon Apr 09 15:20:11 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 Sun Apr 08 12:31:08 2018 +0200
+++ b/src/HOL/Int.thy Mon Apr 09 15:20:11 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 Sun Apr 08 12:31:08 2018 +0200
+++ b/src/HOL/Lattices_Big.thy Mon Apr 09 15:20:11 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: