--- a/src/HOL/Analysis/Affine.thy Fri Sep 03 18:20:13 2021 +0100
+++ b/src/HOL/Analysis/Affine.thy Fri Sep 03 22:52:51 2021 +0100
@@ -798,7 +798,7 @@
have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
unfolding * by (simp add: card_image inj_on_def)
also have "\<dots> > DIM('a)" using assms(2)
- unfolding card_Diff_singleton[OF assms(1) \<open>a\<in>s\<close>] by auto
+ unfolding card_Diff_singleton[OF \<open>a\<in>s\<close>] by auto
finally show ?thesis
apply (subst insert_Diff[OF \<open>a\<in>s\<close>, symmetric])
apply (rule dependent_imp_affine_dependent)
@@ -821,9 +821,7 @@
using \<open>a\<in>S\<close> by (auto simp: span_base span_diff intro: subset_le_dim)
also have "\<dots> < dim S + 1" by auto
also have "\<dots> \<le> card (S - {a})"
- using assms
- using card_Diff_singleton[OF assms(1) \<open>a\<in>S\<close>]
- by auto
+ using assms card_Diff_singleton[OF \<open>a\<in>S\<close>] by auto
finally show ?thesis
apply (subst insert_Diff[OF \<open>a\<in>S\<close>, symmetric])
apply (rule dependent_imp_affine_dependent)
--- a/src/HOL/Quotient_Examples/Quotient_FSet.thy Fri Sep 03 18:20:13 2021 +0100
+++ b/src/HOL/Quotient_Examples/Quotient_FSet.thy Fri Sep 03 22:52:51 2021 +0100
@@ -819,8 +819,7 @@
lemma card_remove_fset_le1:
shows "card_fset (remove_fset x xs) \<le> card_fset xs"
- unfolding remove_fset card_fset
- by (rule card_Diff1_le[OF finite_fset])
+ by simp
lemma card_psubset_fset:
shows "ys |\<subseteq>| xs \<Longrightarrow> card_fset ys < card_fset xs \<Longrightarrow> ys |\<subset>| xs"
@@ -835,21 +834,17 @@
lemma card_minus_insert_fset[simp]:
assumes "a |\<in>| A" and "a |\<notin>| B"
shows "card_fset (A - insert_fset a B) = card_fset (A - B) - 1"
- using assms
- unfolding in_fset card_fset minus_fset
- by (simp add: card_Diff_insert[OF finite_fset])
+ using assms by (simp add: in_fset card_fset)
lemma card_minus_subset_fset:
assumes "B |\<subseteq>| A"
shows "card_fset (A - B) = card_fset A - card_fset B"
using assms
- unfolding subset_fset card_fset minus_fset
- by (rule card_Diff_subset[OF finite_fset])
+ by (simp add: subset_fset card_fset card_Diff_subset)
lemma card_minus_fset:
shows "card_fset (A - B) = card_fset A - card_fset (A |\<inter>| B)"
- unfolding inter_fset card_fset minus_fset
- by (rule card_Diff_subset_Int) (simp)
+ by (simp add: card_fset card_Diff_subset_Int)
subsection \<open>concat_fset\<close>