# HG changeset patch
# User nipkow
# Date 1510347930 -3600
# Node ID c1b87d15774a0527c7831f543047320e30b7c89c
# Parent  690b4b3348892ff8767f928c73223ef04abe05fe
replaced raw proof blocks by local lemmas

diff -r 690b4b334889 -r c1b87d15774a src/HOL/Data_Structures/AA_Map.thy
--- a/src/HOL/Data_Structures/AA_Map.thy	Thu Nov 09 10:24:00 2017 +0100
+++ b/src/HOL/Data_Structures/AA_Map.thy	Fri Nov 10 22:05:30 2017 +0100
@@ -72,64 +72,62 @@
 
 lemma invar_update: "invar t \<Longrightarrow> invar(update a b t)"
 proof(induction t)
-  case (Node n l xy r)
+  case N: (Node n l xy r)
   hence il: "invar l" and ir: "invar r" by auto
+  note iil = N.IH(1)[OF il]
+  note iir = N.IH(2)[OF ir]
   obtain x y where [simp]: "xy = (x,y)" by fastforce
-  note N = Node
   let ?t = "Node n l xy r"
   have "a < x \<or> a = x \<or> x < a" by auto
   moreover
-  { assume "a < x"
-    note iil = Node.IH(1)[OF il]
-    have ?case
-    proof (cases rule: lvl_update[of a b l])
-      case (Same) thus ?thesis
-        using \<open>a<x\<close> invar_NodeL[OF Node.prems iil Same]
-        by (simp add: skew_invar split_invar del: invar.simps)
+  have ?case if "a < x"
+  proof (cases rule: lvl_update[of a b l])
+    case (Same) thus ?thesis
+      using \<open>a<x\<close> invar_NodeL[OF N.prems iil Same]
+      by (simp add: skew_invar split_invar del: invar.simps)
+  next
+    case (Incr)
+    then obtain t1 w t2 where ial[simp]: "update a b l = Node n t1 w t2"
+      using N.prems by (auto simp: lvl_Suc_iff)
+    have l12: "lvl t1 = lvl t2"
+      by (metis Incr(1) ial lvl_update_incr_iff tree.inject)
+    have "update a b ?t = split(skew(Node n (update a b l) xy r))"
+      by(simp add: \<open>a<x\<close>)
+    also have "skew(Node n (update a b l) xy r) = Node n t1 w (Node n t2 xy r)"
+      by(simp)
+    also have "invar(split \<dots>)"
+    proof (cases r)
+      case Leaf
+      hence "l = Leaf" using N.prems by(auto simp: lvl_0_iff)
+      thus ?thesis using Leaf ial by simp
     next
-      case (Incr)
-      then obtain t1 w t2 where ial[simp]: "update a b l = Node n t1 w t2"
-        using Node.prems by (auto simp: lvl_Suc_iff)
-      have l12: "lvl t1 = lvl t2"
-        by (metis Incr(1) ial lvl_update_incr_iff tree.inject)
-      have "update a b ?t = split(skew(Node n (update a b l) xy r))"
-        by(simp add: \<open>a<x\<close>)
-      also have "skew(Node n (update a b l) xy r) = Node n t1 w (Node n t2 xy r)"
-        by(simp)
-      also have "invar(split \<dots>)"
-      proof (cases r)
-        case Leaf
-        hence "l = Leaf" using Node.prems by(auto simp: lvl_0_iff)
-        thus ?thesis using Leaf ial by simp
+      case [simp]: (Node m t3 y t4)
+      show ?thesis (*using N(3) iil l12 by(auto)*)
+      proof cases
+        assume "m = n" thus ?thesis using N(3) iil by(auto)
       next
-        case [simp]: (Node m t3 y t4)
-        show ?thesis (*using N(3) iil l12 by(auto)*)
-        proof cases
-          assume "m = n" thus ?thesis using N(3) iil by(auto)
-        next
-          assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
-        qed
+        assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
       qed
-      finally show ?thesis .
     qed
-  }
+    finally show ?thesis .
+  qed
   moreover
-  { assume "x < a"
-    note iir = Node.IH(2)[OF ir]
+  have ?case if "x < a"
+  proof -
     from \<open>invar ?t\<close> have "n = lvl r \<or> n = lvl r + 1" by auto
-    hence ?case
+    thus ?case
     proof
       assume 0: "n = lvl r"
       have "update a b ?t = split(skew(Node n l xy (update a b r)))"
         using \<open>a>x\<close> by(auto)
       also have "skew(Node n l xy (update a b r)) = Node n l xy (update a b r)"
-        using Node.prems by(simp add: skew_case split: tree.split)
+        using N.prems by(simp add: skew_case split: tree.split)
       also have "invar(split \<dots>)"
       proof -
         from lvl_update_sngl[OF ir sngl_if_invar[OF \<open>invar ?t\<close> 0], of a b]
         obtain t1 p t2 where iar: "update a b r = Node n t1 p t2"
-          using Node.prems 0 by (auto simp: lvl_Suc_iff)
-        from Node.prems iar 0 iir
+          using N.prems 0 by (auto simp: lvl_Suc_iff)
+        from N.prems iar 0 iir
         show ?thesis by (auto simp: split_case split: tree.splits)
       qed
       finally show ?thesis .
@@ -139,7 +137,7 @@
       show ?thesis
       proof (cases rule: lvl_update[of a b r])
         case (Same)
-        show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF Node.prems 1 iir Same]
+        show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF N.prems 1 iir Same]
           by (auto simp add: skew_invar split_invar)
       next
         case (Incr)
@@ -147,8 +145,9 @@
           by (auto simp add: skew_invar split_invar split: if_splits)
       qed
     qed
-  }
-  moreover { assume "a = x" hence ?case using Node.prems by auto }
+  qed
+  moreover
+  have "a = x \<Longrightarrow> ?case" using N.prems by auto
   ultimately show ?case by blast
 qed simp
 
diff -r 690b4b334889 -r c1b87d15774a src/HOL/Data_Structures/AA_Set.thy
--- a/src/HOL/Data_Structures/AA_Set.thy	Thu Nov 09 10:24:00 2017 +0100
+++ b/src/HOL/Data_Structures/AA_Set.thy	Fri Nov 10 22:05:30 2017 +0100
@@ -201,63 +201,61 @@
 
 lemma invar_insert: "invar t \<Longrightarrow> invar(insert a t)"
 proof(induction t)
-  case (Node n l x r)
+  case N: (Node n l x r)
   hence il: "invar l" and ir: "invar r" by auto
-  note N = Node
+  note iil = N.IH(1)[OF il]
+  note iir = N.IH(2)[OF ir]
   let ?t = "Node n l x r"
   have "a < x \<or> a = x \<or> x < a" by auto
   moreover
-  { assume "a < x"
-    note iil = Node.IH(1)[OF il]
-    have ?case
-    proof (cases rule: lvl_insert[of a l])
-      case (Same) thus ?thesis
-        using \<open>a<x\<close> invar_NodeL[OF Node.prems iil Same]
-        by (simp add: skew_invar split_invar del: invar.simps)
+  have ?case if "a < x"
+  proof (cases rule: lvl_insert[of a l])
+    case (Same) thus ?thesis
+      using \<open>a<x\<close> invar_NodeL[OF N.prems iil Same]
+      by (simp add: skew_invar split_invar del: invar.simps)
+  next
+    case (Incr)
+    then obtain t1 w t2 where ial[simp]: "insert a l = Node n t1 w t2"
+      using N.prems by (auto simp: lvl_Suc_iff)
+    have l12: "lvl t1 = lvl t2"
+      by (metis Incr(1) ial lvl_insert_incr_iff tree.inject)
+    have "insert a ?t = split(skew(Node n (insert a l) x r))"
+      by(simp add: \<open>a<x\<close>)
+    also have "skew(Node n (insert a l) x r) = Node n t1 w (Node n t2 x r)"
+      by(simp)
+    also have "invar(split \<dots>)"
+    proof (cases r)
+      case Leaf
+      hence "l = Leaf" using N.prems by(auto simp: lvl_0_iff)
+      thus ?thesis using Leaf ial by simp
     next
-      case (Incr)
-      then obtain t1 w t2 where ial[simp]: "insert a l = Node n t1 w t2"
-        using Node.prems by (auto simp: lvl_Suc_iff)
-      have l12: "lvl t1 = lvl t2"
-        by (metis Incr(1) ial lvl_insert_incr_iff tree.inject)
-      have "insert a ?t = split(skew(Node n (insert a l) x r))"
-        by(simp add: \<open>a<x\<close>)
-      also have "skew(Node n (insert a l) x r) = Node n t1 w (Node n t2 x r)"
-        by(simp)
-      also have "invar(split \<dots>)"
-      proof (cases r)
-        case Leaf
-        hence "l = Leaf" using Node.prems by(auto simp: lvl_0_iff)
-        thus ?thesis using Leaf ial by simp
+      case [simp]: (Node m t3 y t4)
+      show ?thesis (*using N(3) iil l12 by(auto)*)
+      proof cases
+        assume "m = n" thus ?thesis using N(3) iil by(auto)
       next
-        case [simp]: (Node m t3 y t4)
-        show ?thesis (*using N(3) iil l12 by(auto)*)
-        proof cases
-          assume "m = n" thus ?thesis using N(3) iil by(auto)
-        next
-          assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
-        qed
+        assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
       qed
-      finally show ?thesis .
     qed
-  }
+    finally show ?thesis .
+  qed
   moreover
-  { assume "x < a"
-    note iir = Node.IH(2)[OF ir]
+  have ?case if "x < a"
+  proof -
     from \<open>invar ?t\<close> have "n = lvl r \<or> n = lvl r + 1" by auto
-    hence ?case
+    thus ?case
     proof
       assume 0: "n = lvl r"
       have "insert a ?t = split(skew(Node n l x (insert a r)))"
         using \<open>a>x\<close> by(auto)
       also have "skew(Node n l x (insert a r)) = Node n l x (insert a r)"
-        using Node.prems by(simp add: skew_case split: tree.split)
+        using N.prems by(simp add: skew_case split: tree.split)
       also have "invar(split \<dots>)"
       proof -
         from lvl_insert_sngl[OF ir sngl_if_invar[OF \<open>invar ?t\<close> 0], of a]
         obtain t1 y t2 where iar: "insert a r = Node n t1 y t2"
-          using Node.prems 0 by (auto simp: lvl_Suc_iff)
-        from Node.prems iar 0 iir
+          using N.prems 0 by (auto simp: lvl_Suc_iff)
+        from N.prems iar 0 iir
         show ?thesis by (auto simp: split_case split: tree.splits)
       qed
       finally show ?thesis .
@@ -267,7 +265,7 @@
       show ?thesis
       proof (cases rule: lvl_insert[of a r])
         case (Same)
-        show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF Node.prems 1 iir Same]
+        show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF N.prems 1 iir Same]
           by (auto simp add: skew_invar split_invar)
       next
         case (Incr)
@@ -275,8 +273,9 @@
           by (auto simp add: skew_invar split_invar split: if_splits)
       qed
     qed
-  }
-  moreover { assume "a = x" hence ?case using Node.prems by auto }
+  qed
+  moreover
+  have "a = x \<Longrightarrow> ?case" using N.prems by auto
   ultimately show ?case by blast
 qed simp
 
diff -r 690b4b334889 -r c1b87d15774a src/HOL/Data_Structures/Brother12_Map.thy
--- a/src/HOL/Data_Structures/Brother12_Map.thy	Thu Nov 09 10:24:00 2017 +0100
+++ b/src/HOL/Data_Structures/Brother12_Map.thy	Fri Nov 10 22:05:30 2017 +0100
@@ -126,61 +126,58 @@
   { case 1
     then obtain l a b r where [simp]: "t = N2 l (a,b) r" and
       lr: "l \<in> T h" "r \<in> T h" "l \<in> B h \<or> r \<in> B h" by auto
-    { assume "x < a"
-      have ?case
-      proof cases
-        assume "l \<in> B h"
-        from n2_type3[OF Suc.IH(1)[OF this] lr(2)]
-        show ?thesis using `x<a` by(simp)
-      next
-        assume "l \<notin> B h"
-        hence "l \<in> U h" "r \<in> B h" using lr by auto
-        from n2_type1[OF Suc.IH(2)[OF this(1)] this(2)]
-        show ?thesis using `x<a` by(simp)
-      qed
-    } moreover
-    { assume "x > a"
-      have ?case
+    have ?case if "x < a"
+    proof cases
+      assume "l \<in> B h"
+      from n2_type3[OF Suc.IH(1)[OF this] lr(2)]
+      show ?thesis using `x<a` by(simp)
+    next
+      assume "l \<notin> B h"
+      hence "l \<in> U h" "r \<in> B h" using lr by auto
+      from n2_type1[OF Suc.IH(2)[OF this(1)] this(2)]
+      show ?thesis using `x<a` by(simp)
+    qed
+    moreover
+    have ?case if "x > a"
+    proof cases
+      assume "r \<in> B h"
+      from n2_type3[OF lr(1) Suc.IH(1)[OF this]]
+      show ?thesis using `x>a` by(simp)
+    next
+      assume "r \<notin> B h"
+      hence "l \<in> B h" "r \<in> U h" using lr by auto
+      from n2_type2[OF this(1) Suc.IH(2)[OF this(2)]]
+      show ?thesis using `x>a` by(simp)
+    qed
+    moreover
+    have ?case if [simp]: "x=a"
+    proof (cases "del_min r")
+      case None
+      show ?thesis
       proof cases
         assume "r \<in> B h"
-        from n2_type3[OF lr(1) Suc.IH(1)[OF this]]
-        show ?thesis using `x>a` by(simp)
+        with del_minNoneN0[OF this None] lr show ?thesis by(simp)
       next
         assume "r \<notin> B h"
-        hence "l \<in> B h" "r \<in> U h" using lr by auto
-        from n2_type2[OF this(1) Suc.IH(2)[OF this(2)]]
-        show ?thesis using `x>a` by(simp)
+        hence "r \<in> U h" using lr by auto
+        with del_minNoneN1[OF this None] lr(3) show ?thesis by (simp)
       qed
-    } moreover
-    { assume [simp]: "x=a"
-      have ?case
-      proof (cases "del_min r")
-        case None
-        show ?thesis
-        proof cases
-          assume "r \<in> B h"
-          with del_minNoneN0[OF this None] lr show ?thesis by(simp)
-        next
-          assume "r \<notin> B h"
-          hence "r \<in> U h" using lr by auto
-          with del_minNoneN1[OF this None] lr(3) show ?thesis by (simp)
-        qed
+    next
+      case [simp]: (Some br')
+      obtain b r' where [simp]: "br' = (b,r')" by fastforce
+      show ?thesis
+      proof cases
+        assume "r \<in> B h"
+        from del_min_type(1)[OF this] n2_type3[OF lr(1)]
+        show ?thesis by simp
       next
-        case [simp]: (Some br')
-        obtain b r' where [simp]: "br' = (b,r')" by fastforce
-        show ?thesis
-        proof cases
-          assume "r \<in> B h"
-          from del_min_type(1)[OF this] n2_type3[OF lr(1)]
-          show ?thesis by simp
-        next
-          assume "r \<notin> B h"
-          hence "l \<in> B h" and "r \<in> U h" using lr by auto
-          from del_min_type(2)[OF this(2)] n2_type2[OF this(1)]
-          show ?thesis by simp
-        qed
+        assume "r \<notin> B h"
+        hence "l \<in> B h" and "r \<in> U h" using lr by auto
+        from del_min_type(2)[OF this(2)] n2_type2[OF this(1)]
+        show ?thesis by simp
       qed
-    } ultimately show ?case by auto                         
+    qed
+    ultimately show ?case by auto                         
   }
   { case 2 with Suc.IH(1) show ?case by auto }
 qed auto
diff -r 690b4b334889 -r c1b87d15774a src/HOL/Data_Structures/Brother12_Set.thy
--- a/src/HOL/Data_Structures/Brother12_Set.thy	Thu Nov 09 10:24:00 2017 +0100
+++ b/src/HOL/Data_Structures/Brother12_Set.thy	Fri Nov 10 22:05:30 2017 +0100
@@ -258,9 +258,9 @@
     then obtain t1 a t2 where [simp]: "t = N2 t1 a t2" and
       t1: "t1 \<in> T h" and t2: "t2 \<in> T h" and t12: "t1 \<in> B h \<or> t2 \<in> B h"
       by auto
-    { assume "x < a"
-      hence "?case \<longleftrightarrow> n2 (ins x t1) a t2 \<in> Bp (Suc h)" by simp
-      also have "\<dots>"
+    have ?case if "x < a"
+    proof -
+      have "n2 (ins x t1) a t2 \<in> Bp (Suc h)"
       proof cases
         assume "t1 \<in> B h"
         with t2 show ?thesis by (simp add: Suc.IH(1) n2_type)
@@ -269,12 +269,12 @@
         hence 1: "t1 \<in> U h" and 2: "t2 \<in> B h" using t1 t12 by auto
         show ?thesis by (metis Suc.IH(2)[OF 1] Bp_if_B[OF 2] n2_type)
       qed
-      finally have ?case .
-    }
+      with `x < a` show ?case by simp
+    qed
     moreover
-    { assume "a < x"
-      hence "?case \<longleftrightarrow> n2 t1 a (ins x t2) \<in> Bp (Suc h)" by simp
-      also have "\<dots>"
+    have ?case if "a < x"
+    proof -
+      have "n2 t1 a (ins x t2) \<in> Bp (Suc h)"
       proof cases
         assume "t2 \<in> B h"
         with t1 show ?thesis by (simp add: Suc.IH(1) n2_type)
@@ -283,12 +283,14 @@
         hence 1: "t1 \<in> B h" and 2: "t2 \<in> U h" using t2 t12 by auto
         show ?thesis by (metis Bp_if_B[OF 1] Suc.IH(2)[OF 2] n2_type)
       qed
-    }
-    moreover 
-    { assume "x = a"
+      with `a < x` show ?case by simp
+    qed
+    moreover
+    have ?case if "x = a"
+    proof -
       from 1 have "t \<in> Bp (Suc h)" by(rule Bp_if_B)
-      hence "?case" using `x = a` by simp
-    }
+      thus "?case" using `x = a` by simp
+    qed
     ultimately show ?case by auto
   next
     case 2 thus ?case using Suc(1) n1_type by fastforce }
@@ -398,61 +400,58 @@
   { case 1
     then obtain l a r where [simp]: "t = N2 l a r" and
       lr: "l \<in> T h" "r \<in> T h" "l \<in> B h \<or> r \<in> B h" by auto
-    { assume "x < a"
-      have ?case
-      proof cases
-        assume "l \<in> B h"
-        from n2_type3[OF Suc.IH(1)[OF this] lr(2)]
-        show ?thesis using `x<a` by(simp)
-      next
-        assume "l \<notin> B h"
-        hence "l \<in> U h" "r \<in> B h" using lr by auto
-        from n2_type1[OF Suc.IH(2)[OF this(1)] this(2)]
-        show ?thesis using `x<a` by(simp)
-      qed
-    } moreover
-    { assume "x > a"
-      have ?case
+    have ?case if "x < a"
+    proof cases
+      assume "l \<in> B h"
+      from n2_type3[OF Suc.IH(1)[OF this] lr(2)]
+      show ?thesis using `x<a` by(simp)
+    next
+      assume "l \<notin> B h"
+      hence "l \<in> U h" "r \<in> B h" using lr by auto
+      from n2_type1[OF Suc.IH(2)[OF this(1)] this(2)]
+      show ?thesis using `x<a` by(simp)
+    qed
+    moreover
+    have ?case if "x > a"
+    proof cases
+      assume "r \<in> B h"
+      from n2_type3[OF lr(1) Suc.IH(1)[OF this]]
+      show ?thesis using `x>a` by(simp)
+    next
+      assume "r \<notin> B h"
+      hence "l \<in> B h" "r \<in> U h" using lr by auto
+      from n2_type2[OF this(1) Suc.IH(2)[OF this(2)]]
+      show ?thesis using `x>a` by(simp)
+    qed
+    moreover
+    have ?case if [simp]: "x=a"
+    proof (cases "del_min r")
+      case None
+      show ?thesis
       proof cases
         assume "r \<in> B h"
-        from n2_type3[OF lr(1) Suc.IH(1)[OF this]]
-        show ?thesis using `x>a` by(simp)
+        with del_minNoneN0[OF this None] lr show ?thesis by(simp)
       next
         assume "r \<notin> B h"
-        hence "l \<in> B h" "r \<in> U h" using lr by auto
-        from n2_type2[OF this(1) Suc.IH(2)[OF this(2)]]
-        show ?thesis using `x>a` by(simp)
+        hence "r \<in> U h" using lr by auto
+        with del_minNoneN1[OF this None] lr(3) show ?thesis by (simp)
       qed
-    } moreover
-    { assume [simp]: "x=a"
-      have ?case
-      proof (cases "del_min r")
-        case None
-        show ?thesis
-        proof cases
-          assume "r \<in> B h"
-          with del_minNoneN0[OF this None] lr show ?thesis by(simp)
-        next
-          assume "r \<notin> B h"
-          hence "r \<in> U h" using lr by auto
-          with del_minNoneN1[OF this None] lr(3) show ?thesis by (simp)
-        qed
+    next
+      case [simp]: (Some br')
+      obtain b r' where [simp]: "br' = (b,r')" by fastforce
+      show ?thesis
+      proof cases
+        assume "r \<in> B h"
+        from del_min_type(1)[OF this] n2_type3[OF lr(1)]
+        show ?thesis by simp
       next
-        case [simp]: (Some br')
-        obtain b r' where [simp]: "br' = (b,r')" by fastforce
-        show ?thesis
-        proof cases
-          assume "r \<in> B h"
-          from del_min_type(1)[OF this] n2_type3[OF lr(1)]
-          show ?thesis by simp
-        next
-          assume "r \<notin> B h"
-          hence "l \<in> B h" and "r \<in> U h" using lr by auto
-          from del_min_type(2)[OF this(2)] n2_type2[OF this(1)]
-          show ?thesis by simp
-        qed
+        assume "r \<notin> B h"
+        hence "l \<in> B h" and "r \<in> U h" using lr by auto
+        from del_min_type(2)[OF this(2)] n2_type2[OF this(1)]
+        show ?thesis by simp
       qed
-    } ultimately show ?case by auto
+    qed
+    ultimately show ?case by auto
   }
   { case 2 with Suc.IH(1) show ?case by auto }
 qed auto