# HG changeset patch
# User wenzelm
# Date 1730669347 -3600
# Node ID f94b30fa2b6c96eba92d7c34f391d7629140452f
# Parent  405f7fd15f4e771edb799e25fc094cd43c5edc24
tuned proofs;

diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/BNF_Corec.thy
--- a/src/HOL/Library/BNF_Corec.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/BNF_Corec.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -139,11 +139,9 @@
 
 lemma cong_gen_cong: "cong (gen_cong R)"
 proof -
-  { fix R' x y
-    have "rel (gen_cong R) x y \<Longrightarrow> R \<le> R' \<Longrightarrow> cong R' \<Longrightarrow> R' (eval x) (eval y)"
-      by (force simp: rel_fun_def gen_cong_def cong_def dest: spec[of _ R']
+  have "rel (gen_cong R) x y \<Longrightarrow> R \<le> R' \<Longrightarrow> cong R' \<Longrightarrow> R' (eval x) (eval y)" for R' x y
+    by (force simp: rel_fun_def gen_cong_def cong_def dest: spec[of _ R']
         predicate2D[OF rel_mono, rotated -1, of _ _ _ R'])
-  }
   then show "cong (gen_cong R)" by (auto simp: equivp_gen_cong rel_fun_def gen_cong_def cong_def)
 qed
 
diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/Discrete.thy
--- a/src/HOL/Library/Discrete.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Discrete.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -37,7 +37,7 @@
     with \<open>n \<ge> 2\<close> show ?thesis by (rule double)
   qed
 qed
-  
+
 lemma log_zero [simp]: "log 0 = 0"
   by (simp add: log.simps)
 
@@ -177,11 +177,8 @@
   fixes n :: nat
   shows "finite {m. m\<^sup>2 \<le> n}" and "{m. m\<^sup>2 \<le> n} \<noteq> {}"
 proof -
-  { fix m
-    assume "m\<^sup>2 \<le> n"
-    then have "m \<le> n"
-      by (cases m) (simp_all add: power2_eq_square)
-  } note ** = this
+  have **: "m \<le> n" if "m\<^sup>2 \<le> n" for m
+    using that by (cases m) (simp_all add: power2_eq_square)
   then have "{m. m\<^sup>2 \<le> n} \<subseteq> {m. m \<le> n}" by auto
   then show "finite {m. m\<^sup>2 \<le> n}" by (rule finite_subset) rule
   have "0\<^sup>2 \<le> n" by simp
@@ -295,14 +292,14 @@
   using sqrt_aux [of n] by (auto simp add: sqrt_def intro: power2_nat_le_imp_le)
 
 text \<open>Additional facts about the discrete square root, thanks to Julian Biendarra, Manuel Eberl\<close>
-  
+
 lemma Suc_sqrt_power2_gt: "n < (Suc (Discrete.sqrt n))^2"
   using Max_ge[OF Discrete.sqrt_aux(1), of "Discrete.sqrt n + 1" n]
   by (cases "n < (Suc (Discrete.sqrt n))^2") (simp_all add: Discrete.sqrt_def)
 
 lemma le_sqrt_iff: "x \<le> Discrete.sqrt y \<longleftrightarrow> x^2 \<le> y"
 proof -
-  have "x \<le> Discrete.sqrt y \<longleftrightarrow> (\<exists>z. z\<^sup>2 \<le> y \<and> x \<le> z)"    
+  have "x \<le> Discrete.sqrt y \<longleftrightarrow> (\<exists>z. z\<^sup>2 \<le> y \<and> x \<le> z)"
     using Max_ge_iff[OF Discrete.sqrt_aux, of x y] by (simp add: Discrete.sqrt_def)
   also have "\<dots> \<longleftrightarrow> x^2 \<le> y"
   proof safe
@@ -311,7 +308,7 @@
   qed auto
   finally show ?thesis .
 qed
-  
+
 lemma le_sqrtI: "x^2 \<le> y \<Longrightarrow> x \<le> Discrete.sqrt y"
   by (simp add: le_sqrt_iff)
 
@@ -321,14 +318,14 @@
 lemma sqrt_leI:
   "(\<And>z. z^2 \<le> y \<Longrightarrow> z \<le> x) \<Longrightarrow> Discrete.sqrt y \<le> x"
   by (simp add: sqrt_le_iff)
-    
+
 lemma sqrt_Suc:
   "Discrete.sqrt (Suc n) = (if \<exists>m. Suc n = m^2 then Suc (Discrete.sqrt n) else Discrete.sqrt n)"
 proof cases
   assume "\<exists> m. Suc n = m^2"
   then obtain m where m_def: "Suc n = m^2" by blast
   then have lhs: "Discrete.sqrt (Suc n) = m" by simp
-  from m_def sqrt_power2_le[of n] 
+  from m_def sqrt_power2_le[of n]
     have "(Discrete.sqrt n)^2 < m^2" by linarith
   with power2_less_imp_less have lt_m: "Discrete.sqrt n < m" by blast
   from m_def Suc_sqrt_power2_gt[of "n"]
@@ -340,7 +337,7 @@
 next
   assume asm: "\<not> (\<exists> m. Suc n = m^2)"
   hence "Suc n \<noteq> (Discrete.sqrt (Suc n))^2" by simp
-  with sqrt_power2_le[of "Suc n"] 
+  with sqrt_power2_le[of "Suc n"]
     have "Discrete.sqrt (Suc n) \<le> Discrete.sqrt n" by (intro le_sqrtI) linarith
   moreover have "Discrete.sqrt (Suc n) \<ge> Discrete.sqrt n"
     by (intro monoD[OF mono_sqrt]) simp_all
diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/Extended_Nat.thy
--- a/src/HOL/Library/Extended_Nat.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Extended_Nat.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -656,20 +656,21 @@
 
 definition Sup_enat :: "enat set \<Rightarrow> enat" where
   "Sup_enat A = (if A = {} then 0 else if finite A then Max A else \<infinity>)"
+
 instance
 proof
   fix x :: "enat" and A :: "enat set"
-  { assume "x \<in> A" then show "Inf A \<le> x"
-      unfolding Inf_enat_def by (auto intro: Least_le) }
-  { assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" then show "x \<le> Inf A"
-      unfolding Inf_enat_def
-      by (cases "A = {}") (auto intro: LeastI2_ex) }
-  { assume "x \<in> A" then show "x \<le> Sup A"
-      unfolding Sup_enat_def by (cases "finite A") auto }
-  { assume "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" then show "Sup A \<le> x"
-      unfolding Sup_enat_def using finite_enat_bounded by auto }
-qed (simp_all add:
- inf_enat_def sup_enat_def bot_enat_def top_enat_def Inf_enat_def Sup_enat_def)
+  show "x \<in> A \<Longrightarrow> Inf A \<le> x"
+    unfolding Inf_enat_def by (auto intro: Least_le)
+  show "(\<And>y. y \<in> A \<Longrightarrow> x \<le> y) \<Longrightarrow> x \<le> Inf A"
+    unfolding Inf_enat_def
+    by (cases "A = {}") (auto intro: LeastI2_ex)
+  show "x \<in> A \<Longrightarrow> x \<le> Sup A"
+    unfolding Sup_enat_def by (cases "finite A") auto
+  show "(\<And>y. y \<in> A \<Longrightarrow> y \<le> x) \<Longrightarrow> Sup A \<le> x"
+    unfolding Sup_enat_def using finite_enat_bounded by auto
+qed (simp_all add: inf_enat_def sup_enat_def bot_enat_def top_enat_def Inf_enat_def Sup_enat_def)
+
 end
 
 instance enat :: complete_linorder ..
diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/Extended_Nonnegative_Real.thy
--- a/src/HOL/Library/Extended_Nonnegative_Real.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Extended_Nonnegative_Real.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -37,14 +37,15 @@
 proof  -
   have "(\<lambda>k. (\<Sum>n<k. f (n + i)) + (\<Sum>j<i. f j)) \<longlonglongrightarrow> l + (\<Sum>j<i. f j)"
     using assms by (auto intro!: tendsto_add simp: sums_def)
-  moreover
-  { fix k :: nat
+  moreover have "(\<Sum>j<k + i. f j) = (\<Sum>n<k. f (n + i)) + (\<Sum>j<i. f j)" for k :: nat
+  proof -
     have "(\<Sum>j<k + i. f j) = (\<Sum>j=i..<k + i. f j) + (\<Sum>j=0..<i. f j)"
       by (subst sum.union_disjoint[symmetric]) (auto intro!: sum.cong)
     also have "(\<Sum>j=i..<k + i. f j) = (\<Sum>j\<in>(\<lambda>n. n + i)`{0..<k}. f j)"
       unfolding image_add_atLeastLessThan by simp
-    finally have "(\<Sum>j<k + i. f j) = (\<Sum>n<k. f (n + i)) + (\<Sum>j<i. f j)"
-      by (auto simp: inj_on_def atLeast0LessThan sum.reindex) }
+    finally show ?thesis
+      by (auto simp: inj_on_def atLeast0LessThan sum.reindex)
+  qed
   ultimately have "(\<lambda>k. (\<Sum>n<k + i. f n)) \<longlonglongrightarrow> l + (\<Sum>j<i. f j)"
     by simp
   then show ?thesis
@@ -222,7 +223,7 @@
 
 instance
   by standard
-     (transfer ; auto simp: Inf_lower Inf_greatest Sup_upper Sup_least le_max_iff_disj max.absorb1)+
+     (transfer; auto simp: Inf_lower Inf_greatest Sup_upper Sup_least le_max_iff_disj max.absorb1)+
 
 end
 
@@ -236,8 +237,7 @@
 begin
 
 definition infinity_ennreal :: ennreal
-where
-  [simp]: "\<infinity> = (top::ennreal)"
+  where [simp]: "\<infinity> = (top::ennreal)"
 
 instance ..
 
@@ -293,7 +293,7 @@
 
 instance ennreal :: ordered_comm_semiring
   by standard
-     (transfer ; auto intro: add_mono mult_mono mult_ac ereal_left_distrib ereal_mult_left_mono)+
+     (transfer; auto intro: add_mono mult_mono mult_ac ereal_left_distrib ereal_mult_left_mono)+
 
 instance ennreal :: linordered_nonzero_semiring
 proof
@@ -788,7 +788,7 @@
 
 declare [[coercion ennreal]]
 
-lemma ennreal_cong: "x = y \<Longrightarrow> ennreal x = ennreal y" 
+lemma ennreal_cong: "x = y \<Longrightarrow> ennreal x = ennreal y"
   by simp
 
 lemma ennreal_cases[cases type: ennreal]:
@@ -975,7 +975,7 @@
   case (real r) then show ?thesis
   proof (cases "x = 0")
     case False then show ?thesis
-      by (smt (verit, best) ennreal_0 ennreal_power inverse_ennreal 
+      by (smt (verit, best) ennreal_0 ennreal_power inverse_ennreal
                inverse_nonnegative_iff_nonnegative power_inverse real zero_less_power)
   qed (simp add: top_power_ennreal)
 qed
@@ -1288,7 +1288,7 @@
   assumes *: "\<forall>\<^sub>F x in F. 0 \<le> f x" and x: "0 \<le> x"
   shows "(f \<longlongrightarrow> x) F"
 proof -
-  have "((\<lambda>x. enn2ereal (ennreal (f x))) \<longlongrightarrow> enn2ereal (ennreal x)) F 
+  have "((\<lambda>x. enn2ereal (ennreal (f x))) \<longlongrightarrow> enn2ereal (ennreal x)) F
     \<longleftrightarrow> (f \<longlongrightarrow> enn2ereal (ennreal x)) F"
     using "*" eventually_mono
     by (intro tendsto_cong) fastforce
@@ -1322,7 +1322,7 @@
 
 lemma ennreal_tendsto_0_iff: "(\<And>n. f n \<ge> 0) \<Longrightarrow> ((\<lambda>n. ennreal (f n)) \<longlonglongrightarrow> 0) \<longleftrightarrow> (f \<longlonglongrightarrow> 0)"
   by (metis (mono_tags) ennreal_0 eventuallyI order_refl tendsto_ennreal_iff)
-    
+
 lemma continuous_on_add_ennreal:
   fixes f g :: "'a::topological_space \<Rightarrow> ennreal"
   shows "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> continuous_on A (\<lambda>x. f x + g x)"
diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/Extended_Real.thy
--- a/src/HOL/Library/Extended_Real.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Extended_Real.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -1163,7 +1163,7 @@
   case False
   then obtain p q where "x = ereal p" "y = ereal q"
     by (metis MInfty_eq_minfinity ereal.distinct(3) uminus_ereal.elims)
-  then show ?thesis 
+  then show ?thesis
     by (metis assms field_le_epsilon ereal_less(2) ereal_less_eq(3) plus_ereal.simps(1))
 qed
 
@@ -1857,7 +1857,7 @@
     by (metis MInfty_eq_minfinity ereal.distinct(3) uminus_ereal.elims)
   moreover have "\<And>x. \<lbrakk>q < x; x < p\<rbrakk> \<Longrightarrow> x \<in> real_of_ereal ` {ereal q<..<ereal p}"
     by (metis greaterThanLessThan_iff imageI less_ereal.simps(1) real_of_ereal.simps(1))
-  ultimately show ?thesis 
+  ultimately show ?thesis
     by (auto elim!: less_ereal.elims)
 qed
 
@@ -2026,17 +2026,15 @@
 lemma tendsto_cmult_ereal[tendsto_intros, simp, intro]:
   assumes c: "\<bar>c\<bar> \<noteq> \<infinity>" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x::ereal) \<longlongrightarrow> c * x) F"
 proof -
-  { fix c :: ereal assume "0 < c" "c < \<infinity>"
-    then have "((\<lambda>x. c * f x::ereal) \<longlongrightarrow> c * x) F"
-      apply (intro tendsto_compose[OF _ f])
-      apply (auto intro!: order_tendstoI simp: eventually_at_topological)
-      apply (rule_tac x="{a/c <..}" in exI)
-      apply (auto split: ereal.split simp: ereal_divide_less_iff mult.commute) []
-      apply (rule_tac x="{..< a/c}" in exI)
-      apply (auto split: ereal.split simp: ereal_less_divide_iff mult.commute) []
-      done }
-  note * = this
-
+  have *: "((\<lambda>x. c * f x::ereal) \<longlongrightarrow> c * x) F" if "0 < c" "c < \<infinity>" for c :: ereal
+    using that
+    apply (intro tendsto_compose[OF _ f])
+    apply (auto intro!: order_tendstoI simp: eventually_at_topological)
+     apply (rule_tac x="{a/c <..}" in exI)
+     apply (auto split: ereal.split simp: ereal_divide_less_iff mult.commute) []
+    apply (rule_tac x="{..< a/c}" in exI)
+    apply (auto split: ereal.split simp: ereal_less_divide_iff mult.commute) []
+    done
   have "((0 < c \<and> c < \<infinity>) \<or> (-\<infinity> < c \<and> c < 0) \<or> c = 0)"
     using c by (cases c) auto
   then show ?thesis
@@ -2267,9 +2265,8 @@
   assumes "I \<noteq> {}" and "\<bar>c\<bar> \<noteq> \<infinity>"
   shows "(INF i\<in>I. c - f i) = c - (SUP i\<in>I. f i::ereal)"
 proof -
-  { fix b have "(-c) + b = - (c - b)"
-      using \<open>\<bar>c\<bar> \<noteq> \<infinity>\<close> by (cases c b rule: ereal2_cases) auto }
-  note * = this
+  have *: "(- c) + b = - (c - b)" for b
+    using \<open>\<bar>c\<bar> \<noteq> \<infinity>\<close> by (cases c b rule: ereal2_cases) auto
   show ?thesis
     using SUP_ereal_add_right[OF \<open>I \<noteq> {}\<close>, of "-c" f] \<open>\<bar>c\<bar> \<noteq> \<infinity>\<close>
     by (auto simp add: * ereal_SUP_uminus_eq)
@@ -2359,10 +2356,8 @@
 proof -
   have INF_less: "(INF i. f i) < \<infinity>" "(INF i. g i) < \<infinity>"
     using assms unfolding INF_less_iff by auto
-  { fix a b :: ereal assume "a \<noteq> \<infinity>" "b \<noteq> \<infinity>"
-    then have "- ((- a) + (- b)) = a + b"
-      by (cases a b rule: ereal2_cases) auto }
-  note * = this
+  have *: "- ((- a) + (- b)) = a + b" if "a \<noteq> \<infinity>" "b \<noteq> \<infinity>" for a b :: ereal
+    using that by (cases a b rule: ereal2_cases) auto
   have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))"
     by (simp add: fin *)
   also have "\<dots> = Inf (f ` UNIV) + Inf (g ` UNIV)"
@@ -2953,8 +2948,8 @@
   fixes x y :: "ereal"
   assumes "x \<noteq> -\<infinity>" "y \<noteq> -\<infinity>" "(f \<longlongrightarrow> x) F" "(g \<longlongrightarrow> y) F"
   shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
-proof cases
-  assume "x = \<infinity> \<or> y = \<infinity>"
+proof (cases "x = \<infinity> \<or> y = \<infinity>")
+  case True
   moreover
   { fix y :: ereal and f g :: "'a \<Rightarrow> ereal" assume "y \<noteq> -\<infinity>" "(f \<longlongrightarrow> \<infinity>) F" "(g \<longlongrightarrow> y) F"
     then obtain y' where "-\<infinity> < y'" "y' < y"
@@ -2974,7 +2969,7 @@
   ultimately show ?thesis
     using assms by (auto simp: add_ac)
 next
-  assume "\<not> (x = \<infinity> \<or> y = \<infinity>)"
+  case False
   with assms tendsto_add_ereal[of x y f F g]
   show ?thesis
     by auto
@@ -3939,7 +3934,7 @@
   fixes f :: "'a::t2_space \<Rightarrow> ereal"
   assumes "\<And>x. x \<in> A \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
   shows "continuous_on A f \<longleftrightarrow> continuous_on A (real_of_ereal \<circ> f)"
-proof 
+proof
   assume L: "continuous_on A f"
   have "f ` A \<subseteq> UNIV - {\<infinity>, -\<infinity>}"
     using assms by force
@@ -3950,7 +3945,7 @@
   then have "continuous_on A (ereal \<circ> (real_of_ereal \<circ> f))"
     by (meson continuous_at_iff_ereal continuous_on_eq_continuous_within)
   then show "continuous_on A f"
-    using assms ereal_real' by auto 
+    using assms ereal_real' by auto
 qed
 
 lemma continuous_uminus_ereal [continuous_intros]: "continuous_on (A :: ereal set) uminus"
diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/Landau_Symbols.thy
--- a/src/HOL/Library/Landau_Symbols.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Landau_Symbols.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -659,7 +659,7 @@
     using assms by (intro big_power_increasing[OF small_imp_big]) auto
   finally show ?thesis by simp
 qed
-  
+
 sublocale big: landau_symbol L L' Lr
 proof
   have L: "L = bigo \<or> L = bigomega"
@@ -1522,15 +1522,16 @@
   assumes "f \<in> o[F](h)" "g \<in> o[F](h)"
   shows   "(\<lambda>k. max (f k) (g k)) \<in> o[F](h)" "(\<lambda>k. min (f k) (g k)) \<in> o[F](h)"
 proof -
-  { fix c::real
-    assume "c>0"
-    with assms smallo_def
+  have "\<forall>\<^sub>F x in F. norm (max (f x) (g x)) \<le> c * norm(h x) \<and> norm (min (f x) (g x)) \<le> c * norm(h x)"
+    if "c>0" for c::real
+  proof -
+    from assms smallo_def that
     have "\<forall>\<^sub>F x in F. norm (f x) \<le> c * norm(h x)" "\<forall>\<^sub>F x in F. norm(g x) \<le> c * norm(h x)"
       by (auto simp: smallo_def)
-    then have "\<forall>\<^sub>F x in F. norm (max (f x) (g x)) \<le> c * norm(h x) \<and> norm (min (f x) (g x)) \<le> c * norm(h x)"
+    then show ?thesis
       by (smt (verit) eventually_elim2 max_def min_def)
-  } with assms   
-  show "(\<lambda>x. max (f x) (g x)) \<in> o[F](h)" "(\<lambda>x. min (f x) (g x)) \<in> o[F](h)"
+  qed
+  with assms show "(\<lambda>x. max (f x) (g x)) \<in> o[F](h)" "(\<lambda>x. min (f x) (g x)) \<in> o[F](h)"
     by (smt (verit) eventually_elim2 landau_o.smallI)+
 qed
 
diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/Liminf_Limsup.thy
--- a/src/HOL/Library/Liminf_Limsup.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Liminf_Limsup.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -431,16 +431,14 @@
   assumes c: "continuous_on UNIV f" and am: "mono f" and F: "F \<noteq> bot"
   shows "Liminf F (\<lambda>n. f (g n)) = f (Liminf F g)"
 proof -
-  { fix P assume "eventually P F"
-    have "\<exists>x. P x"
-    proof (rule ccontr)
-      assume "\<not> (\<exists>x. P x)" then have "P = (\<lambda>x. False)"
-        by auto
-      with \<open>eventually P F\<close> F show False
-        by auto
-    qed }
-  note * = this
-
+  have *: "\<exists>x. P x" if "eventually P F" for P
+  proof (rule ccontr)
+    assume "\<not> ?thesis"
+    then have "P = (\<lambda>x. False)"
+      by auto
+    with \<open>eventually P F\<close> F show False
+      by auto
+  qed
   have "f (SUP P\<in>{P. eventually P F}. Inf (g ` Collect P)) =
     Sup (f ` (\<lambda>P. Inf (g ` Collect P)) ` {P. eventually P F})"
     using am continuous_on_imp_continuous_within [OF c]
@@ -449,7 +447,7 @@
     by (simp add: Liminf_def image_comp)
   also have "\<dots> = (SUP P \<in> {P. eventually P F}. Inf (f ` (g ` Collect P)))"
     using * continuous_at_Inf_mono [OF am continuous_on_imp_continuous_within [OF c]]
-    by auto 
+    by auto
   finally show ?thesis by (auto simp: Liminf_def image_comp)
 qed
 
@@ -458,16 +456,14 @@
   assumes c: "continuous_on UNIV f" and am: "mono f" and F: "F \<noteq> bot"
   shows "Limsup F (\<lambda>n. f (g n)) = f (Limsup F g)"
 proof -
-  { fix P assume "eventually P F"
-    have "\<exists>x. P x"
-    proof (rule ccontr)
-      assume "\<not> (\<exists>x. P x)" then have "P = (\<lambda>x. False)"
-        by auto
-      with \<open>eventually P F\<close> F show False
-        by auto
-    qed }
-  note * = this
-
+  have *: "\<exists>x. P x" if "eventually P F" for P
+  proof (rule ccontr)
+    assume "\<not> ?thesis"
+    then have "P = (\<lambda>x. False)"
+      by auto
+    with \<open>eventually P F\<close> F show False
+      by auto
+  qed
   have "f (INF P\<in>{P. eventually P F}. Sup (g ` Collect P)) =
     Inf (f ` (\<lambda>P. Sup (g ` Collect P)) ` {P. eventually P F})"
     using am continuous_on_imp_continuous_within [OF c]
@@ -513,16 +509,13 @@
   assumes c: "continuous_on UNIV f" and am: "antimono f" and F: "F \<noteq> bot"
   shows "Limsup F (\<lambda>n. f (g n)) = f (Liminf F g)"
 proof -
-  { fix P assume "eventually P F"
-    have "\<exists>x. P x"
-    proof (rule ccontr)
-      assume "\<not> (\<exists>x. P x)" then have "P = (\<lambda>x. False)"
-        by auto
-      with \<open>eventually P F\<close> F show False
-        by auto
-    qed }
-  note * = this
-
+  have *: "\<exists>x. P x" if "eventually P F" for P
+  proof (rule ccontr)
+    assume "\<not> (\<exists>x. P x)" then have "P = (\<lambda>x. False)"
+      by auto
+    with \<open>eventually P F\<close> F show False
+      by auto
+  qed
   have "f (SUP P\<in>{P. eventually P F}. Inf (g ` Collect P)) =
     Inf (f ` (\<lambda>P. Inf (g ` Collect P)) ` {P. eventually P F})"
     using am continuous_on_imp_continuous_within [OF c]
diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/Multiset.thy
--- a/src/HOL/Library/Multiset.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Multiset.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -636,7 +636,7 @@
 lemma mset_subset_eq_insertD:
   assumes "add_mset x A \<subseteq># B"
   shows "x \<in># B \<and> A \<subset># B"
-proof 
+proof
   show "x \<in># B"
     using assms by (simp add: mset_subset_eqD)
   have "A \<subseteq># add_mset x A"
@@ -728,7 +728,7 @@
   by (simp add: union_mset_def)
 
 global_interpretation subset_mset: semilattice_neutr_order \<open>(\<union>#)\<close> \<open>{#}\<close> \<open>(\<supseteq>#)\<close> \<open>(\<supset>#)\<close>
-proof 
+proof
   show "\<And>a b. (b \<subseteq># a) = (a = a \<union># b)"
     by (simp add: Diff_eq_empty_iff_mset union_mset_def)
   show "\<And>a b. (b \<subset># a) = (a = a \<union># b \<and> a \<noteq> b)"
@@ -2224,7 +2224,7 @@
 lemma count_mset_set': "count (mset_set A) x = (if finite A \<and> x \<in> A then 1 else 0)"
   by auto
 
-lemma subset_imp_msubset_mset_set: 
+lemma subset_imp_msubset_mset_set:
   assumes "A \<subseteq> B" "finite B"
   shows   "mset_set A \<subseteq># mset_set B"
 proof (rule mset_subset_eqI)
@@ -2958,7 +2958,7 @@
   show ?case
   proof (cases "x \<in> set_mset M")
     case True
-    have "(\<Prod>y\<in>set_mset (add_mset x M). f y ^ count (add_mset x M) y) = 
+    have "(\<Prod>y\<in>set_mset (add_mset x M). f y ^ count (add_mset x M) y) =
             (\<Prod>y\<in>set_mset M. (if y = x then f x else 1) * f y ^ count M y)"
       using True add by (intro prod.cong) auto
     also have "\<dots> = f x * (\<Prod>y\<in>set_mset M. f y ^ count M y)"
@@ -2970,7 +2970,7 @@
     hence "(\<Prod>y\<in>set_mset (add_mset x M). f y ^ count (add_mset x M) y) =
               f x * (\<Prod>y\<in>set_mset M. f y ^ count (add_mset x M) y)"
       by (auto simp: not_in_iff)
-    also have "(\<Prod>y\<in>set_mset M. f y ^ count (add_mset x M) y) = 
+    also have "(\<Prod>y\<in>set_mset M. f y ^ count (add_mset x M) y) =
                  (\<Prod>y\<in>set_mset M. f y ^ count M y)"
       using False by (intro prod.cong) auto
     also note add.IH [symmetric]
@@ -3942,9 +3942,8 @@
       using one_step_implies_mult[of "M - M \<inter># N" "N - M \<inter># N" R "M \<inter># N"] *
       by (auto simp: multp_code_def Let_def)
   next
-    { fix I J K :: "'a multiset" assume "(I + J) \<inter># (I + K) = {#}"
-      then have "I = {#}" by (metis inter_union_distrib_right union_eq_empty)
-    } note [dest!] = this
+    have [dest!]: "I = {#}" if "(I + J) \<inter># (I + K) = {#}" for I J K
+      using that by (metis inter_union_distrib_right union_eq_empty)
     assume ?R thus ?L
       using mult_cancel_max
       using mult_implies_one_step[OF assms(2), of "N - M \<inter># N" "M - M \<inter># N"]
@@ -3975,11 +3974,14 @@
   assumes "irrefl_on (set_mset N \<inter> set_mset M) R" and "trans R" and "\<And>x y. P x y \<longleftrightarrow> (x, y) \<in> R"
   shows "multeqp_code P N M \<longleftrightarrow> (N, M) \<in> (mult R)\<^sup>="
 proof -
-  { assume "N \<noteq> M" "M - M \<inter># N = {#}"
-    then obtain y where "count N y \<noteq> count M y" by (auto simp flip: count_inject)
-    then have "\<exists>y. count M y < count N y" using \<open>M - M \<inter># N = {#}\<close>
+  have "\<exists>y. count M y < count N y" if "N \<noteq> M" "M - M \<inter># N = {#}"
+  proof -
+    from that obtain y where "count N y \<noteq> count M y"
+      by (auto simp flip: count_inject)
+    then show ?thesis
+      using \<open>M - M \<inter># N = {#}\<close>
       by (auto simp flip: count_inject dest!: le_neq_implies_less fun_cong[of _ _ y])
-  }
+  qed
   then have "multeqp_code P N M \<longleftrightarrow> multp_code P N M \<or> N = M"
     by (auto simp: multeqp_code_def multp_code_def Let_def in_diff_count)
   thus ?thesis
@@ -4538,7 +4540,7 @@
 lemma ex_mset: "\<exists>xs. mset xs = X"
   by (induct X) (simp, metis mset.simps(2))
 
-inductive pred_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> bool" 
+inductive pred_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> bool"
 where
   "pred_mset P {#}"
 | "\<lbrakk>P a; pred_mset P M\<rbrakk> \<Longrightarrow> pred_mset P (add_mset a M)"
diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/Order_Continuity.thy
--- a/src/HOL/Library/Order_Continuity.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Order_Continuity.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -133,12 +133,12 @@
   proof (rule lfp_lowerbound)
     have "mono (\<lambda>i::nat. (F ^^ i) bot)"
     proof -
-      { fix i::nat have "(F ^^ i) bot \<le> (F ^^ (Suc i)) bot"
-        proof (induct i)
-          case 0 show ?case by simp
-        next
-          case Suc thus ?case using monoD[OF mono Suc] by auto
-        qed }
+      have "(F ^^ i) bot \<le> (F ^^ (Suc i)) bot" for i::nat
+      proof (induct i)
+        case 0 show ?case by simp
+      next
+        case Suc thus ?case using monoD[OF mono Suc] by auto
+      qed
       thus ?thesis by (auto simp add: mono_iff_le_Suc)
     qed
     hence "F ?U = (SUP i. (F ^^ Suc i) bot)"
@@ -307,12 +307,14 @@
   proof (rule gfp_upperbound)
     have *: "antimono (\<lambda>i::nat. (F ^^ i) top)"
     proof -
-      { fix i::nat have "(F ^^ Suc i) top \<le> (F ^^ i) top"
-        proof (induct i)
-          case 0 show ?case by simp
-        next
-          case Suc thus ?case using monoD[OF mono Suc] by auto
-        qed }
+      have "(F ^^ Suc i) top \<le> (F ^^ i) top" for i::nat
+      proof (induct i)
+        case 0
+        show ?case by simp
+      next
+        case Suc
+        thus ?case using monoD[OF mono Suc] by auto
+      qed
       thus ?thesis by (auto simp add: antimono_iff_le_Suc)
     qed
     have "?U \<le> (INF i. (F ^^ Suc i) top)"
diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/Poly_Mapping.thy
--- a/src/HOL/Library/Poly_Mapping.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Poly_Mapping.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -471,11 +471,12 @@
 proof -
   have *: "finite {k. (\<exists>l. f1 l \<noteq> 0 \<and> (\<exists>q. f2 q \<noteq> 0 \<and> k = l + q))}"
     using assms by simp
-  { fix k l
+  have aux: "sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q} = (\<Sum>q. (f2 q when k = l + q))" for k l
+  proof -
     have "{q. (f2 q when k = l + q) \<noteq> 0} \<subseteq> {q. f2 q \<noteq> 0 \<and> k = l + q}" by auto
-    with fin2 have "sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q} = (\<Sum>q. (f2 q when k = l + q))"
-      by (simp add: Sum_any.expand_superset [of "{q. f2 q \<noteq> 0 \<and> k = l + q}"]) }
-  note aux = this
+    with fin2 show ?thesis
+      by (simp add: Sum_any.expand_superset [of "{q. f2 q \<noteq> 0 \<and> k = l + q}"])
+  qed
   have "{k. (\<Sum>l. f1 l * sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q}) \<noteq> 0}
     \<subseteq> {k. (\<exists>l. f1 l * sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q} \<noteq> 0)}"
     by (auto elim!: Sum_any.not_neutral_obtains_not_neutral)
@@ -526,7 +527,7 @@
       have "?lhs k = (\<Sum>(ab, c). (\<Sum>(a, b). f a * g b when ab = a + b) * h c when k = ab + c)"
         by (simp add: prod_fun_unfold_prod)
       also have "\<dots> = (\<Sum>(ab, c). (\<Sum>(a, b). f a * g b * h c when k = ab + c when ab = a + b))"
-        using fin_fg 
+        using fin_fg
         apply (simp add: Sum_any_left_distrib split_def flip: Sum_any_when_independent)
         apply (simp add: when_when when_mult mult_when conj_commute)
         done
@@ -635,7 +636,7 @@
     assume fin_g: "finite {k. g k \<noteq> 0}"
     assume fin_h: "finite {k. h k \<noteq> 0}"
     show "prod_fun (\<lambda>k. f k + g k) h = (\<lambda>k. prod_fun f h k + prod_fun g h k)"
-      by (auto simp: prod_fun_def fun_eq_iff algebra_simps 
+      by (auto simp: prod_fun_def fun_eq_iff algebra_simps
             Sum_any.distrib fin_f fin_g finite_mult_not_eq_zero_rightI)
   qed
 qed
@@ -921,14 +922,15 @@
 proof (intro_classes, transfer)
   fix f g h :: "'a \<Rightarrow> 'b"
   assume *: "less_fun f g \<or> f = g"
-  { assume "less_fun f g"
-    then obtain k where "f k < g k" "(\<And>k'. k' < k \<Longrightarrow> f k' = g k')"
+  have "less_fun (\<lambda>k. h k + f k) (\<lambda>k. h k + g k)" if "less_fun f g"
+  proof -
+    from that obtain k where "f k < g k" "(\<And>k'. k' < k \<Longrightarrow> f k' = g k')"
       by (blast elim!: less_funE)
     then have "h k + f k < h k + g k" "(\<And>k'. k' < k \<Longrightarrow> h k' + f k' = h k' + g k')"
       by simp_all
-    then have "less_fun (\<lambda>k. h k + f k) (\<lambda>k. h k + g k)"
+    then show ?thesis
       by (blast intro: less_funI)
-  }
+  qed
   with * show "less_fun (\<lambda>k. h k + f k) (\<lambda>k. h k + g k) \<or> (\<lambda>k. h k + f k) = (\<lambda>k. h k + g k)"
     by (auto simp: fun_eq_iff)
 qed
@@ -1273,12 +1275,14 @@
 proof(transfer fixing: s)
   fix p :: "'a \<Rightarrow> 'b"
   assume *: "finite {x. p x \<noteq> 0}"
-  { fix x
-    have "prod_fun (\<lambda>k'. s when 0 = k') p x =
-          (\<Sum>l :: 'a. if l = 0 then s * (\<Sum>q. p q when x = q) else 0)"
-      by(auto simp: prod_fun_def when_def intro: Sum_any.cong simp del: Sum_any.delta)
-    also have "\<dots> = (\<lambda>k. s * p k when p k \<noteq> 0) x" by(simp add: when_def)
-    also note calculation }
+  have "prod_fun (\<lambda>k'. s when 0 = k') p x = (\<lambda>k. s * p k when p k \<noteq> 0) x" (is "?lhs = ?rhs") for x
+  proof -
+    have "?lhs = (\<Sum>l :: 'a. if l = 0 then s * (\<Sum>q. p q when x = q) else 0)"
+      by (auto simp: prod_fun_def when_def intro: Sum_any.cong simp del: Sum_any.delta)
+    also have "\<dots> = ?rhs"
+      by (simp add: when_def)
+    finally show ?thesis .
+  qed
   then show "(\<lambda>k. s * p k when p k \<noteq> 0) = prod_fun (\<lambda>k'. s when 0 = k') p"
     by(simp add: fun_eq_iff)
 qed
@@ -1345,17 +1349,18 @@
   "keys (nth xs) =  fst ` {(n, v) \<in> set (enumerate 0 xs). v \<noteq> 0}"
 proof transfer
   fix xs :: "'a list"
-  { fix n
-    assume "nth_default 0 xs n \<noteq> 0"
-    then have "n < length xs" and "xs ! n \<noteq> 0"
+  have "n \<in> fst ` {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}"
+    if "nth_default 0 xs n \<noteq> 0" for n
+  proof -
+    from that have "n < length xs" and "xs ! n \<noteq> 0"
       by (auto simp: nth_default_def split: if_splits)
     then have "(n, xs ! n) \<in> {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}" (is "?x \<in> ?A")
       by (auto simp: in_set_conv_nth enumerate_eq_zip)
     then have "fst ?x \<in> fst ` ?A"
       by blast
-    then have "n \<in> fst ` {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}"
+    then show ?thesis
       by simp
-  }
+  qed
   then show "{k. nth_default 0 xs k \<noteq> 0} = fst ` {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}"
     by (auto simp: in_enumerate_iff_nth_default_eq)
 qed
@@ -1535,7 +1540,7 @@
   case False
   then show ?thesis
     by (simp add: Poly_Mapping.poly_mapping_size_def in_keys_iff)
-qed 
+qed
 
 lemma poly_mapping_size_estimation:
   "k \<in> keys m \<Longrightarrow> y \<le> f k + g (lookup m k) \<Longrightarrow> y < poly_mapping_size m"
@@ -1654,7 +1659,7 @@
 lemma keys_cmul: "Poly_Mapping.keys(frag_cmul c a) \<subseteq> Poly_Mapping.keys a"
   using finite_cmul_nonzero [of c a]
   by (metis lookup_frag_cmul mult_zero_right not_in_keys_iff_lookup_eq_zero subsetI)
-  
+
 
 lemma keys_cmul_iff [iff]: "i \<in> Poly_Mapping.keys (frag_cmul c x) \<longleftrightarrow> i \<in> Poly_Mapping.keys x \<and> c \<noteq> 0"
   by (metis in_keys_iff lookup_frag_cmul mult_eq_0_iff)
@@ -1662,7 +1667,7 @@
 lemma keys_minus [simp]: "Poly_Mapping.keys(-a) = Poly_Mapping.keys a"
   by (metis (no_types, opaque_lifting) in_keys_iff lookup_uminus neg_equal_0_iff_equal subsetI subset_antisym)
 
-lemma keys_diff: 
+lemma keys_diff:
   "Poly_Mapping.keys(a - b) \<subseteq> Poly_Mapping.keys a \<union> Poly_Mapping.keys b"
   by (auto simp: in_keys_iff lookup_minus)
 
@@ -1726,15 +1731,15 @@
 lemma frag_extend_add:
   "frag_extend f (a+b) = (frag_extend f a) + (frag_extend f b)"
 proof -
-  have *: "(\<Sum>i\<in>Poly_Mapping.keys a. frag_cmul (poly_mapping.lookup a i) (f i)) 
+  have *: "(\<Sum>i\<in>Poly_Mapping.keys a. frag_cmul (poly_mapping.lookup a i) (f i))
          = (\<Sum>i\<in>Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup a i) (f i))"
-          "(\<Sum>i\<in>Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup b i) (f i)) 
+          "(\<Sum>i\<in>Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup b i) (f i))
          = (\<Sum>i\<in>Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup b i) (f i))"
     by (auto simp: in_keys_iff intro: sum.mono_neutral_cong_left)
   have "frag_extend f (a+b) = (\<Sum>i\<in>Poly_Mapping.keys (a + b).
        frag_cmul (poly_mapping.lookup a i) (f i) + frag_cmul (poly_mapping.lookup b i) (f i)) "
     by (auto simp: frag_extend_def Poly_Mapping.lookup_add frag_cmul_distrib)
-  also have "... = (\<Sum>i \<in> Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup a i) (f i) 
+  also have "... = (\<Sum>i \<in> Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup a i) (f i)
                          + frag_cmul (poly_mapping.lookup b i) (f i))"
   proof (rule sum.mono_neutral_cong_left)
     show "\<forall>i\<in>keys a \<union> keys b - keys (a + b).
@@ -1830,7 +1835,7 @@
   fixes c :: "'a \<Rightarrow>\<^sub>0 int"
   assumes "Poly_Mapping.keys c \<subseteq> S \<union> T"
   obtains d e where "Poly_Mapping.keys d \<subseteq> S" "Poly_Mapping.keys e \<subseteq> T" "d + e = c"
-proof 
+proof
   let ?d = "frag_extend (\<lambda>f. if f \<in> S then frag_of f else 0) c"
   let ?e = "frag_extend (\<lambda>f. if f \<in> S then 0 else frag_of f) c"
   show "Poly_Mapping.keys ?d \<subseteq> S" "Poly_Mapping.keys ?e \<subseteq> T"
diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/Ramsey.thy
--- a/src/HOL/Library/Ramsey.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Ramsey.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -5,13 +5,13 @@
 section \<open>Ramsey's Theorem\<close>
 
 theory Ramsey
-  imports Infinite_Set Equipollence FuncSet 
+  imports Infinite_Set Equipollence FuncSet
 begin
 
 subsection \<open>Preliminary definitions\<close>
 
 abbreviation strict_sorted :: "'a::linorder list \<Rightarrow> bool" where
-  "strict_sorted \<equiv> sorted_wrt (<)" 
+  "strict_sorted \<equiv> sorted_wrt (<)"
 
 subsubsection \<open>The $n$-element subsets of a set $A$\<close>
 
@@ -89,12 +89,13 @@
   shows "[A]\<^bsup>4\<^esup> = {U. \<exists>u x y z. U = {u,x,y,z} \<and> u \<in> A \<and> x \<in> A \<and> y \<in> A \<and> z \<in> A \<and> u < x \<and> x < y \<and> y < z}"
     (is "_ = Collect ?RHS")
 proof -
-  { fix U
-    assume "U \<in> [A]\<^bsup>4\<^esup>"
-    then obtain l where "strict_sorted l" "List.set l = U" "length l = 4" "U \<subseteq> A"
+  have "?RHS U" if "U \<in> [A]\<^bsup>4\<^esup>" for U
+  proof -
+    from that obtain l where "strict_sorted l" "List.set l = U" "length l = 4" "U \<subseteq> A"
       by (simp add: nsets_def) (metis finite_set_strict_sorted)
-    then have "?RHS U"
-      unfolding numeral_nat length_Suc_conv by auto blast }
+    then show ?thesis
+      unfolding numeral_nat length_Suc_conv by auto blast
+  qed
   moreover
   have "Collect ?RHS \<subseteq> [A]\<^bsup>4\<^esup>"
     apply (clarsimp simp add: nsets_def eval_nat_numeral)
@@ -109,13 +110,14 @@
   shows "[A]\<^bsup>5\<^esup> = {U. \<exists>u v x y z. U = {u,v,x,y,z} \<and> u \<in> A \<and> v \<in> A \<and> x \<in> A \<and> y \<in> A \<and> z \<in> A \<and> u < v \<and> v < x \<and> x < y \<and> y < z}"
     (is "_ = Collect ?RHS")
 proof -
-  { fix U
-  assume "U \<in> [A]\<^bsup>5\<^esup>"
-  then obtain l where "strict_sorted l" "List.set l = U" "length l = 5" "U \<subseteq> A"
-    apply (simp add: nsets_def)
-    by (metis finite_set_strict_sorted)
-  then have "?RHS U"
-    unfolding numeral_nat length_Suc_conv by auto blast }
+  have "?RHS U" if "U \<in> [A]\<^bsup>5\<^esup>" for U
+  proof -
+    from that obtain l where "strict_sorted l" "List.set l = U" "length l = 5" "U \<subseteq> A"
+      apply (simp add: nsets_def)
+      by (metis finite_set_strict_sorted)
+    then show ?thesis
+      unfolding numeral_nat length_Suc_conv by auto blast
+  qed
   moreover
   have "Collect ?RHS \<subseteq> [A]\<^bsup>5\<^esup>"
     apply (clarsimp simp add: nsets_def eval_nat_numeral)
@@ -338,7 +340,7 @@
   using partn_lst_def monochromatic_def assms by metis
 
 lemma partn_lst_less:
-  assumes M: "partn_lst \<beta> \<alpha> n" and eq: "length \<alpha>' = length \<alpha>" 
+  assumes M: "partn_lst \<beta> \<alpha> n" and eq: "length \<alpha>' = length \<alpha>"
     and le: "\<And>i. i < length \<alpha> \<Longrightarrow> \<alpha>'!i \<le> \<alpha>!i "
   shows "partn_lst \<beta> \<alpha>' n"
 proof (clarsimp simp: partn_lst_def)
@@ -385,7 +387,7 @@
 text \<open>The Erdős--Szekeres bound, essentially extracted from the proof\<close>
 fun ES :: "[nat,nat,nat] \<Rightarrow> nat"
   where "ES 0 k l = max k l"
-  |     "ES (Suc r) k l = 
+  |     "ES (Suc r) k l =
             (if r=0 then k+l-1
              else if k=0 \<or> l=0 then 1 else Suc (ES r (ES (Suc r) (k-1) l) (ES (Suc r) k (l-1))))"
 
@@ -685,7 +687,7 @@
       using Suc.prems
     proof (induct k \<equiv> "q1 + q2" arbitrary: q1 q2)
       case 0
-      with partn_lst_0 show ?case by auto 
+      with partn_lst_0 show ?case by auto
     next
       case (Suc k)
       consider "q1 = 0 \<or> q2 = 0" | "q1 \<noteq> 0" "q2 \<noteq> 0" by auto
diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/Stream.thy
--- a/src/HOL/Library/Stream.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Stream.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -470,11 +470,12 @@
     next
       case False
       hence "x = flat (stl s) !! (y - length (shd s))" by (metis less(2,3) flat_snth)
-      moreover
-      { from less(2) have *: "length (shd s) > 0" by (cases s) simp_all
+      moreover have "y - length (shd s) < y"
+      proof -
+        from less(2) have *: "length (shd s) > 0" by (cases s) simp_all
         with False have "y > 0" by (cases y) simp_all
-        with * have "y - length (shd s) < y" by simp
-      }
+        with * show ?thesis by simp
+      qed
       moreover have "\<forall>xs \<in> sset (stl s). xs \<noteq> []" using less(2) by (cases s) auto
       ultimately have "\<exists>n m'. x = stl s !! n ! m' \<and> m' < length (stl s !! n)" by (intro less(1)) auto
       thus ?thesis by (metis snth.simps(2))
@@ -482,7 +483,9 @@
   qed
   thus "x \<in> ?R" by (auto simp: sset_range dest!: nth_mem)
 next
-  fix x xs assume "xs \<in> sset s" ?P "x \<in> set xs" thus "x \<in> ?L"
+  fix x xs
+  assume "xs \<in> sset s" ?P "x \<in> set xs"
+  thus "x \<in> ?L"
     by (induct rule: sset_induct)
       (metis UnI1 flat_unfold shift.simps(1) sset_shift,
        metis UnI2 flat_unfold shd_sset stl_sset sset_shift)
diff -r 405f7fd15f4e -r f94b30fa2b6c src/HOL/Library/Sublist.thy
--- a/src/HOL/Library/Sublist.thy	Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Sublist.thy	Sun Nov 03 22:29:07 2024 +0100
@@ -80,7 +80,7 @@
 next
   assume "xs = ys @ [y] \<or> prefix xs ys"
   then show "prefix xs (ys @ [y])"
-    by auto (metis append.assoc prefix_def) 
+    by auto (metis append.assoc prefix_def)
 qed
 
 lemma Cons_prefix_Cons [simp]: "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)"
@@ -96,7 +96,7 @@
   by (induct xs) simp_all
 
 lemma same_prefix_nil [simp]: "prefix (xs @ ys) xs = (ys = [])"
-  by (simp add: prefix_def) 
+  by (simp add: prefix_def)
 
 lemma prefix_prefix [simp]: "prefix xs ys \<Longrightarrow> prefix xs (ys @ zs)"
   unfolding prefix_def by fastforce
@@ -295,7 +295,7 @@
 
 lemma length_prefixes[simp]: "length (prefixes xs) = length xs+1"
   by (induction xs) auto
-    
+
 lemma distinct_prefixes [intro]: "distinct (prefixes xs)"
   by (induction xs) (auto simp: distinct_map)
 
@@ -310,8 +310,8 @@
 
 lemma last_prefixes [simp]: "last (prefixes xs) = xs"
   by (induction xs) (simp_all add: last_map)
-    
-lemma prefixes_append: 
+
+lemma prefixes_append:
   "prefixes (xs @ ys) = prefixes xs @ map (\<lambda>ys'. xs @ ys') (tl (prefixes ys))"
 proof (induction xs)
   case Nil
@@ -323,7 +323,7 @@
   (ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = zs@[z] \<and> xs = prefixes zs)) \<and> x = ys"
   by (cases ys rule: rev_cases) auto
 
-lemma prefixes_tailrec [code]: 
+lemma prefixes_tailrec [code]:
   "prefixes xs = rev (snd (foldl (\<lambda>(acc1, acc2) x. (x#acc1, rev (x#acc1)#acc2)) ([],[[]]) xs))"
 proof -
   have "foldl (\<lambda>(acc1, acc2) x. (x#acc1, rev (x#acc1)#acc2)) (ys, rev ys # zs) xs =
@@ -335,14 +335,14 @@
   qed simp_all
   from this [of "[]" "[]"] show ?thesis by simp
 qed
-  
+
 lemma set_prefixes_eq: "set (prefixes xs) = {ys. prefix ys xs}"
   by auto
 
 lemma card_set_prefixes [simp]: "card (set (prefixes xs)) = Suc (length xs)"
   by (subst distinct_card) auto
 
-lemma set_prefixes_append: 
+lemma set_prefixes_append:
   "set (prefixes (xs @ ys)) = set (prefixes xs) \<union> {xs @ ys' |ys'. ys' \<in> set (prefixes ys)}"
   by (subst prefixes_append, cases ys) auto
 
@@ -376,13 +376,14 @@
       by - (rule Least_equality, fastforce+)
     have 2: "?L \<noteq> {}" using \<open>x # xs \<in> L\<close> by auto
     from Suc.hyps(1)[OF 1[symmetric] 2] obtain ps where IH: "?P ?L ps" ..
-    { fix qs
-      assume "\<forall>qs. (\<forall>xa. x # xa \<in> L \<longrightarrow> prefix qs xa) \<longrightarrow> length qs \<le> length ps"
-      and "\<forall>xs\<in>L. prefix qs xs"
-      hence "length (tl qs) \<le> length ps"
-        by (metis Cons_prefix_Cons hd_Cons_tl list.sel(2) Nil_prefix) 
-      hence "length qs \<le> Suc (length ps)" by auto
-    }
+    have "length qs \<le> Suc (length ps)"
+      if "\<forall>qs. (\<forall>xa. x # xa \<in> L \<longrightarrow> prefix qs xa) \<longrightarrow> length qs \<le> length ps"
+      and "\<forall>xs\<in>L. prefix qs xs" for qs
+    proof -
+      from that have "length (tl qs) \<le> length ps"
+        by (metis Cons_prefix_Cons hd_Cons_tl list.sel(2) Nil_prefix)
+      thus ?thesis by auto
+    qed
     hence "?P L (x#ps)" using True IH by auto
     thus ?thesis ..
   next
@@ -544,7 +545,7 @@
   assumes "suffix xs ys"
   obtains zs where "ys = zs @ xs"
   using assms unfolding suffix_def by blast
-    
+
 lemma suffix_tl [simp]: "suffix (tl xs) xs"
   by (induct xs) (auto simp: suffix_def)
 
@@ -599,7 +600,7 @@
   then have "ys = rev zs @ xs" by simp
   then show "suffix xs ys" ..
 qed
-  
+
 lemma strict_suffix_to_prefix [code]: "strict_suffix xs ys \<longleftrightarrow> strict_prefix (rev xs) (rev ys)"
   by (auto simp: suffix_to_prefix strict_suffix_def strict_prefix_def)
 
@@ -645,7 +646,7 @@
 theorem suffix_Cons: "suffix xs (y # ys) \<longleftrightarrow> xs = y # ys \<or> suffix xs ys"
   unfolding suffix_def by (auto simp: Cons_eq_append_conv)
 
-theorem suffix_append: 
+theorem suffix_append:
   "suffix xs (ys @ zs) \<longleftrightarrow> suffix xs zs \<or> (\<exists>xs'. xs = xs' @ zs \<and> suffix xs' ys)"
   by (auto simp: suffix_def append_eq_append_conv2)
 
@@ -672,7 +673,7 @@
   then show ?case by (cases ys) simp_all
 next
   case (Suc n)
-  then show ?case 
+  then show ?case
     by (cases xs) (auto intro: Suc dest: suffix_ConsD' suffix_order.less_imp_le)
 qed
 
@@ -808,7 +809,7 @@
 lemma last_suffixes [simp]: "last (suffixes xs) = xs"
   by (cases xs) simp_all
 
-lemma suffixes_append: 
+lemma suffixes_append:
   "suffixes (xs @ ys) = suffixes ys @ map (\<lambda>xs'. xs' @ ys) (tl (suffixes xs))"
 proof (induction ys rule: rev_induct)
   case Nil
@@ -824,7 +825,7 @@
      (ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = z#zs \<and> xs = suffixes zs)) \<and> x = ys"
   by (cases ys) auto
 
-lemma suffixes_tailrec [code]: 
+lemma suffixes_tailrec [code]:
   "suffixes xs = rev (snd (foldl (\<lambda>(acc1, acc2) x. (x#acc1, (x#acc1)#acc2)) ([],[[]]) (rev xs)))"
 proof -
   have "foldl (\<lambda>(acc1, acc2) x. (x#acc1, (x#acc1)#acc2)) (ys, ys # zs) (rev xs) =
@@ -836,14 +837,14 @@
   qed simp_all
   from this [of "[]" "[]"] show ?thesis by simp
 qed
-  
+
 lemma set_suffixes_eq: "set (suffixes xs) = {ys. suffix ys xs}"
   by auto
-    
+
 lemma card_set_suffixes [simp]: "card (set (suffixes xs)) = Suc (length xs)"
   by (subst distinct_card) auto
-  
-lemma set_suffixes_append: 
+
+lemma set_suffixes_append:
   "set (suffixes (xs @ ys)) = set (suffixes ys) \<union> {xs' @ ys |xs'. xs' \<in> set (suffixes xs)}"
   by (subst suffixes_append, cases xs rule: rev_cases) auto
 
@@ -853,10 +854,10 @@
 
 lemma prefixes_conv_suffixes: "prefixes xs = map rev (suffixes (rev xs))"
   by (induction xs) auto
-    
+
 lemma prefixes_rev: "prefixes (rev xs) = map rev (suffixes xs)"
   by (induction xs) auto
-    
+
 lemma suffixes_rev: "suffixes (rev xs) = map rev (prefixes xs)"
   by (induction xs) auto
 
@@ -870,13 +871,13 @@
 | list_emb_Cons [intro] : "list_emb P xs ys \<Longrightarrow> list_emb P xs (y#ys)"
 | list_emb_Cons2 [intro]: "P x y \<Longrightarrow> list_emb P xs ys \<Longrightarrow> list_emb P (x#xs) (y#ys)"
 
-lemma list_emb_mono:                         
+lemma list_emb_mono:
   assumes "\<And>x y. P x y \<longrightarrow> Q x y"
   shows "list_emb P xs ys \<longrightarrow> list_emb Q xs ys"
-proof                                        
-  assume "list_emb P xs ys"                    
+proof
+  assume "list_emb P xs ys"
   then show "list_emb Q xs ys" by (induct) (auto simp: assms)
-qed 
+qed
 
 lemma list_emb_Nil2 [simp]:
   assumes "list_emb P xs []" shows "xs = []"
@@ -888,13 +889,11 @@
   using assms by (induct xs) auto
 
 lemma list_emb_Cons_Nil [simp]: "list_emb P (x#xs) [] = False"
-proof -
-  { assume "list_emb P (x#xs) []"
-    from list_emb_Nil2 [OF this] have False by simp
-  } moreover {
-    assume False
-    then have "list_emb P (x#xs) []" by simp
-  } ultimately show ?thesis by blast
+proof
+  show False if "list_emb P (x#xs) []"
+    using list_emb_Nil2 [OF that] by simp
+  show "list_emb P (x#xs) []" if False
+    using that ..
 qed
 
 lemma list_emb_append2 [intro]: "list_emb P xs ys \<Longrightarrow> list_emb P xs (zs @ ys)"
@@ -1002,13 +1001,13 @@
   "list_emb P (x#xs) [] \<longleftrightarrow> False"
   "list_emb P (x#xs) (y#ys) \<longleftrightarrow> (if P x y then list_emb P xs ys else list_emb P (x#xs) ys)"
   by simp_all
-    
+
 
 subsection \<open>Subsequences (special case of homeomorphic embedding)\<close>
 
 abbreviation subseq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
   where "subseq xs ys \<equiv> list_emb (=) xs ys"
-  
+
 definition strict_subseq where "strict_subseq xs ys \<longleftrightarrow> xs \<noteq> ys \<and> subseq xs ys"
 
 lemma subseq_Cons2: "subseq xs ys \<Longrightarrow> subseq (x#xs) (x#ys)" by auto
@@ -1073,7 +1072,7 @@
   thus "subseq xs ys"
     by (induction ys arbitrary: xs) (auto simp: Let_def)
 next
-  have [simp]: "[] \<in> set (subseqs ys)" for ys :: "'a list" 
+  have [simp]: "[] \<in> set (subseqs ys)" for ys :: "'a list"
     by (induction ys) (auto simp: Let_def)
   assume "subseq xs ys"
   thus "xs \<in> set (subseqs ys)"
@@ -1105,33 +1104,39 @@
 lemma subseq_append [simp]:
   "subseq (xs @ zs) (ys @ zs) \<longleftrightarrow> subseq xs ys" (is "?l = ?r")
 proof
-  { fix xs' ys' xs ys zs :: "'a list" assume "subseq xs' ys'"
-    then have "xs' = xs @ zs \<and> ys' = ys @ zs \<longrightarrow> subseq xs ys"
-    proof (induct arbitrary: xs ys zs)
-      case list_emb_Nil show ?case by simp
-    next
-      case (list_emb_Cons xs' ys' x)
-      { assume "ys=[]" then have ?case using list_emb_Cons(1) by auto }
-      moreover
-      { fix us assume "ys = x#us"
-        then have ?case using list_emb_Cons(2) by(simp add: list_emb.list_emb_Cons) }
-      ultimately show ?case by (auto simp:Cons_eq_append_conv)
-    next
-      case (list_emb_Cons2 x y xs' ys')
-      { assume "xs=[]" then have ?case using list_emb_Cons2(1) by auto }
-      moreover
-      { fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using list_emb_Cons2 by auto}
-      moreover
-      { fix us assume "xs=x#us" "ys=[]" then have ?case using list_emb_Cons2(2) by bestsimp }
-      ultimately show ?case using \<open>(=) x y\<close> by (auto simp: Cons_eq_append_conv)
-    qed }
-  moreover assume ?l
-  ultimately show ?r by blast
-next
-  assume ?r then show ?l by (metis list_emb_append_mono subseq_order.order_refl)
+  have "xs' = xs @ zs \<and> ys' = ys @ zs \<longrightarrow> subseq xs ys"
+    if "subseq xs' ys'" for xs' ys' xs ys zs :: "'a list"
+    using that
+  proof (induct arbitrary: xs ys zs)
+    case list_emb_Nil
+    show ?case by simp
+  next
+    case (list_emb_Cons xs' ys' x)
+    have ?case if "ys = []"
+      using list_emb_Cons(1) that by auto
+    moreover
+    have ?case if "ys = x#us" for us
+      using list_emb_Cons(2) that by (simp add: list_emb.list_emb_Cons)
+    ultimately show ?case
+      by (auto simp: Cons_eq_append_conv)
+  next
+    case (list_emb_Cons2 x y xs' ys')
+    have ?case if "xs = []"
+      using list_emb_Cons2(1) that by auto
+    moreover
+    have ?case if "xs = x#us" "ys = x#vs" for us vs
+      using list_emb_Cons2 that by auto
+    moreover
+    have ?case  if "xs = x#us" "ys = []" for us
+      using list_emb_Cons2(2) that by bestsimp
+    ultimately show ?case
+      using \<open>x = y\<close> by (auto simp: Cons_eq_append_conv)
+  qed
+  then show "?l \<Longrightarrow> ?r" by blast
+  show "?r \<Longrightarrow> ?l" by (metis list_emb_append_mono subseq_order.order_refl)
 qed
 
-lemma subseq_append_iff: 
+lemma subseq_append_iff:
   "subseq xs (ys @ zs) \<longleftrightarrow> (\<exists>xs1 xs2. xs = xs1 @ xs2 \<and> subseq xs1 ys \<and> subseq xs2 zs)"
   (is "?lhs = ?rhs")
 proof
@@ -1139,7 +1144,7 @@
   proof (induction xs "ys @ zs" arbitrary: ys zs rule: list_emb.induct)
     case (list_emb_Cons xs ws y ys zs)
     from list_emb_Cons(2)[of "tl ys" zs] and list_emb_Cons(2)[of "[]" "tl zs"] and list_emb_Cons(1,3)
-      show ?case by (cases ys) auto
+    show ?case by (cases ys) auto
   next
     case (list_emb_Cons2 x y xs ws ys zs)
     from list_emb_Cons2(3)[of "tl ys" zs] and list_emb_Cons2(3)[of "[]" "tl zs"]
@@ -1148,7 +1153,7 @@
   qed auto
 qed (auto intro: list_emb_append_mono)
 
-lemma subseq_appendE [case_names append]: 
+lemma subseq_appendE [case_names append]:
   assumes "subseq xs (ys @ zs)"
   obtains xs1 xs2 where "xs = xs1 @ xs2" "subseq xs1 ys" "subseq xs2 zs"
   using assms by (subst (asm) subseq_append_iff) auto
@@ -1173,13 +1178,13 @@
   assumes "subseq xs ys" shows "subseq (filter P xs) (filter P ys)"
   using assms by induct auto
 
-lemma subseq_conv_nths: 
-  "subseq xs ys \<longleftrightarrow> (\<exists>N. xs = nths ys N)" (is "?L = ?R")
+lemma subseq_conv_nths: "subseq xs ys \<longleftrightarrow> (\<exists>N. xs = nths ys N)"
+  (is "?L = ?R")
 proof
-  assume ?L
-  then show ?R
+  show ?R if ?L using that
   proof (induct)
-    case list_emb_Nil show ?case by (metis nths_empty)
+    case list_emb_Nil
+    show ?case by (metis nths_empty)
   next
     case (list_emb_Cons xs ys x)
     then obtain N where "xs = nths ys N" by blast
@@ -1194,27 +1199,30 @@
     moreover from list_emb_Cons2 have "x = y" by simp
     ultimately show ?case by blast
   qed
-next
-  assume ?R
-  then obtain N where "xs = nths ys N" ..
-  moreover have "subseq (nths ys N) ys"
-  proof (induct ys arbitrary: N)
-    case Nil show ?case by simp
-  next
-    case Cons then show ?case by (auto simp: nths_Cons)
+  show ?L if ?R
+  proof -
+    from that obtain N where "xs = nths ys N" ..
+    moreover have "subseq (nths ys N) ys"
+    proof (induct ys arbitrary: N)
+      case Nil
+      show ?case by simp
+    next
+      case Cons
+      then show ?case by (auto simp: nths_Cons)
+    qed
+    ultimately show ?thesis by simp
   qed
-  ultimately show ?L by simp
 qed
-  
-  
+
+
 subsection \<open>Contiguous sublists\<close>
 
 subsubsection \<open>\<open>sublist\<close>\<close>
 
-definition sublist :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where 
+definition sublist :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
   "sublist xs ys = (\<exists>ps ss. ys = ps @ xs @ ss)"
-  
-definition strict_sublist :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where 
+
+definition strict_sublist :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
   "strict_sublist xs ys \<longleftrightarrow> sublist xs ys \<and> xs \<noteq> ys"
 
 interpretation sublist_order: order sublist strict_sublist
@@ -1227,38 +1235,37 @@
   thus "sublist xs zs" unfolding sublist_def by blast
 next
   fix xs ys :: "'a list"
-  {
-    assume "sublist xs ys" "sublist ys xs"
-    then obtain as bs cs ds 
-      where xs: "xs = as @ ys @ bs" and ys: "ys = cs @ xs @ ds" 
+  show "xs = ys" if "sublist xs ys" "sublist ys xs"
+  proof -
+    from that obtain as bs cs ds where xs: "xs = as @ ys @ bs" and ys: "ys = cs @ xs @ ds"
       by (auto simp: sublist_def)
     have "xs = as @ cs @ xs @ ds @ bs" by (subst xs, subst ys) auto
-    also have "length \<dots> = length as + length cs + length xs + length bs + length ds" 
+    also have "length \<dots> = length as + length cs + length xs + length bs + length ds"
       by simp
     finally have "as = []" "bs = []" by simp_all
-    with xs show "xs = ys" by simp
-  }
-  thus "strict_sublist xs ys \<longleftrightarrow> (sublist xs ys \<and> \<not>sublist ys xs)"
+    with xs show ?thesis by simp
+  qed
+  thus "strict_sublist xs ys \<longleftrightarrow> (sublist xs ys \<and> \<not> sublist ys xs)"
     by (auto simp: strict_sublist_def)
 qed (auto simp: strict_sublist_def sublist_def intro: exI[of _ "[]"])
-  
+
 lemma sublist_Nil_left [simp, intro]: "sublist [] ys"
   by (auto simp: sublist_def)
-    
+
 lemma sublist_Cons_Nil [simp]: "\<not>sublist (x#xs) []"
   by (auto simp: sublist_def)
-    
+
 lemma sublist_Nil_right [simp]: "sublist xs [] \<longleftrightarrow> xs = []"
   by (cases xs) auto
-    
+
 lemma sublist_appendI [simp, intro]: "sublist xs (ps @ xs @ ss)"
   by (auto simp: sublist_def)
-    
+
 lemma sublist_append_leftI [simp, intro]: "sublist xs (ps @ xs)"
   by (auto simp: sublist_def intro: exI[of _ "[]"])
-    
+
 lemma sublist_append_rightI [simp, intro]: "sublist xs (xs @ ss)"
-  by (auto simp: sublist_def intro: exI[of _ "[]"]) 
+  by (auto simp: sublist_def intro: exI[of _ "[]"])
 
 lemma sublist_altdef: "sublist xs ys \<longleftrightarrow> (\<exists>ys'. prefix ys' ys \<and> suffix xs ys')"
 proof safe
@@ -1271,7 +1278,7 @@
   assume "prefix ys' ys" "suffix xs ys'"
   thus "sublist xs ys" by (auto simp: prefix_def suffix_def)
 qed
-  
+
 lemma sublist_altdef': "sublist xs ys \<longleftrightarrow> (\<exists>ys'. suffix ys' ys \<and> prefix xs ys')"
 proof safe
   assume "sublist xs ys"
@@ -1286,7 +1293,7 @@
 
 lemma sublist_Cons_right: "sublist xs (y # ys) \<longleftrightarrow> prefix xs (y # ys) \<or> sublist xs ys"
   by (auto simp: sublist_def prefix_def Cons_eq_append_conv)
-    
+
 lemma sublist_code [code]:
   "sublist [] ys \<longleftrightarrow> True"
   "sublist (x # xs) [] \<longleftrightarrow> False"
@@ -1294,7 +1301,7 @@
   by (simp_all add: sublist_Cons_right)
 
 lemma sublist_append:
-  "sublist xs (ys @ zs) \<longleftrightarrow> 
+  "sublist xs (ys @ zs) \<longleftrightarrow>
      sublist xs ys \<or> sublist xs zs \<or> (\<exists>xs1 xs2. xs = xs1 @ xs2 \<and> suffix xs1 ys \<and> prefix xs2 zs)"
 by (auto simp: sublist_altdef prefix_append suffix_append)
 
@@ -1315,10 +1322,10 @@
 
 lemma set_mono_sublist: "sublist xs ys \<Longrightarrow> set xs \<subseteq> set ys"
   by (auto simp add: sublist_def)
-    
+
 lemma prefix_imp_sublist [simp, intro]: "prefix xs ys \<Longrightarrow> sublist xs ys"
   by (auto simp: sublist_def prefix_def intro: exI[of _ "[]"])
-    
+
 lemma suffix_imp_sublist [simp, intro]: "suffix xs ys \<Longrightarrow> sublist xs ys"
   by (auto simp: sublist_def suffix_def intro: exI[of _ "[]"])
 
@@ -1333,13 +1340,13 @@
 
 lemma sublist_dropWhile [simp, intro]: "sublist (dropWhile P xs) xs"
   by (rule suffix_imp_sublist[OF suffix_dropWhile])
-    
+
 lemma sublist_tl [simp, intro]: "sublist (tl xs) xs"
   by (rule suffix_imp_sublist) (simp_all add: suffix_drop)
-    
+
 lemma sublist_butlast [simp, intro]: "sublist (butlast xs) xs"
   by (rule prefix_imp_sublist) (simp_all add: prefixeq_butlast)
-    
+
 lemma sublist_rev [simp]: "sublist (rev xs) (rev ys) = sublist xs ys"
 proof
   assume "sublist (rev xs) (rev ys)"
@@ -1354,15 +1361,15 @@
   also have "rev \<dots> = rev bs @ rev xs @ rev as" by simp
   finally show "sublist (rev xs) (rev ys)" by simp
 qed
-    
+
 lemma sublist_rev_left: "sublist (rev xs) ys = sublist xs (rev ys)"
   by (subst sublist_rev [symmetric]) (simp only: rev_rev_ident)
-    
+
 lemma sublist_rev_right: "sublist xs (rev ys) = sublist (rev xs) ys"
   by (subst sublist_rev [symmetric]) (simp only: rev_rev_ident)
 
-lemma snoc_sublist_snoc: 
-  "sublist (xs @ [x]) (ys @ [y]) \<longleftrightarrow> 
+lemma snoc_sublist_snoc:
+  "sublist (xs @ [x]) (ys @ [y]) \<longleftrightarrow>
      (x = y \<and> suffix xs ys \<or> sublist (xs @ [x]) ys) "
   by (subst (1 2) sublist_rev [symmetric])
      (simp del: sublist_rev add: sublist_Cons_right suffix_to_prefix)
@@ -1370,8 +1377,8 @@
 lemma sublist_snoc:
   "sublist xs (ys @ [y]) \<longleftrightarrow> suffix xs (ys @ [y]) \<or> sublist xs ys"
   by (subst (1 2) sublist_rev [symmetric])
-     (simp del: sublist_rev add: sublist_Cons_right suffix_to_prefix)     
-     
+     (simp del: sublist_rev add: sublist_Cons_right suffix_to_prefix)
+
 lemma sublist_imp_subseq [intro]: "sublist xs ys \<Longrightarrow> subseq xs ys"
   by (auto simp: sublist_def)
 
@@ -1415,7 +1422,7 @@
   "sublists [] = [[]]"
 | "sublists (x # xs) = sublists xs @ map ((#) x) (prefixes xs)"
 
-lemma in_set_sublists [simp]: "xs \<in> set (sublists ys) \<longleftrightarrow> sublist xs ys" 
+lemma in_set_sublists [simp]: "xs \<in> set (sublists ys) \<longleftrightarrow> sublist xs ys"
   by (induction ys arbitrary: xs) (auto simp: sublist_Cons_right prefix_Cons)
 
 lemma set_sublists_eq: "set (sublists xs) = {ys. sublist ys xs}"
@@ -1428,8 +1435,8 @@
 subsection \<open>Parametricity\<close>
 
 context includes lifting_syntax
-begin    
-  
+begin
+
 private lemma prefix_primrec:
   "prefix = rec_list (\<lambda>xs. True) (\<lambda>x xs xsa ys.
               case ys of [] \<Rightarrow> False | y # ys \<Rightarrow> x = y \<and> xsa ys)"
@@ -1446,7 +1453,7 @@
 qed
 
 private lemma list_emb_primrec:
-  "list_emb = (\<lambda>uu uua uuaa. rec_list (\<lambda>P xs. List.null xs) (\<lambda>y ys ysa P xs. case xs of [] \<Rightarrow> True 
+  "list_emb = (\<lambda>uu uua uuaa. rec_list (\<lambda>P xs. List.null xs) (\<lambda>y ys ysa P xs. case xs of [] \<Rightarrow> True
      | x # xs \<Rightarrow> if P x y then ysa P xs else ysa P (x # xs)) uuaa uu uua)"
 proof (intro ext, goal_cases)
   case (1 P xs ys)
@@ -1457,12 +1464,12 @@
 
 lemma prefix_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A"
-  shows   "(list_all2 A ===> list_all2 A ===> (=)) prefix prefix"  
+  shows   "(list_all2 A ===> list_all2 A ===> (=)) prefix prefix"
   unfolding prefix_primrec by transfer_prover
-    
+
 lemma suffix_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A"
-  shows   "(list_all2 A ===> list_all2 A ===> (=)) suffix suffix"  
+  shows   "(list_all2 A ===> list_all2 A ===> (=)) suffix suffix"
   unfolding suffix_to_prefix [abs_def] by transfer_prover
 
 lemma sublist_transfer [transfer_rule]:
@@ -1474,7 +1481,7 @@
   assumes [transfer_rule]: "bi_unique A"
   shows   "(list_all2 A ===> list_all2 A ===> (=)) parallel parallel"
   unfolding parallel_def by transfer_prover
-    
+
 
 
 lemma list_emb_transfer [transfer_rule]:
@@ -1483,34 +1490,34 @@
 
 lemma strict_prefix_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A"
-  shows   "(list_all2 A ===> list_all2 A ===> (=)) strict_prefix strict_prefix"  
+  shows   "(list_all2 A ===> list_all2 A ===> (=)) strict_prefix strict_prefix"
   unfolding strict_prefix_def by transfer_prover
-    
+
 lemma strict_suffix_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A"
-  shows   "(list_all2 A ===> list_all2 A ===> (=)) strict_suffix strict_suffix"  
+  shows   "(list_all2 A ===> list_all2 A ===> (=)) strict_suffix strict_suffix"
   unfolding strict_suffix_def by transfer_prover
-    
+
 lemma strict_subseq_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A"
-  shows   "(list_all2 A ===> list_all2 A ===> (=)) strict_subseq strict_subseq"  
+  shows   "(list_all2 A ===> list_all2 A ===> (=)) strict_subseq strict_subseq"
   unfolding strict_subseq_def by transfer_prover
-    
+
 lemma strict_sublist_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A"
-  shows   "(list_all2 A ===> list_all2 A ===> (=)) strict_sublist strict_sublist"  
+  shows   "(list_all2 A ===> list_all2 A ===> (=)) strict_sublist strict_sublist"
   unfolding strict_sublist_def by transfer_prover
 
 lemma prefixes_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A"
   shows   "(list_all2 A ===> list_all2 (list_all2 A)) prefixes prefixes"
   unfolding prefixes_def by transfer_prover
-    
+
 lemma suffixes_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A"
   shows   "(list_all2 A ===> list_all2 (list_all2 A)) suffixes suffixes"
   unfolding suffixes_def by transfer_prover
-    
+
 lemma sublists_transfer [transfer_rule]:
   assumes [transfer_rule]: "bi_unique A"
   shows   "(list_all2 A ===> list_all2 (list_all2 A)) sublists sublists"