author nipkow Fri Nov 10 22:05:30 2017 +0100 (19 months ago) changeset 67040 c1b87d15774a parent 67039 690b4b334889 child 67041 f8b0367046bd child 67071 a462583f0a37
replaced raw proof blocks by local lemmas
```     1.1 --- a/src/HOL/Data_Structures/AA_Map.thy	Thu Nov 09 10:24:00 2017 +0100
1.2 +++ b/src/HOL/Data_Structures/AA_Map.thy	Fri Nov 10 22:05:30 2017 +0100
1.3 @@ -72,64 +72,62 @@
1.4
1.5  lemma invar_update: "invar t \<Longrightarrow> invar(update a b t)"
1.6  proof(induction t)
1.7 -  case (Node n l xy r)
1.8 +  case N: (Node n l xy r)
1.9    hence il: "invar l" and ir: "invar r" by auto
1.10 +  note iil = N.IH(1)[OF il]
1.11 +  note iir = N.IH(2)[OF ir]
1.12    obtain x y where [simp]: "xy = (x,y)" by fastforce
1.13 -  note N = Node
1.14    let ?t = "Node n l xy r"
1.15    have "a < x \<or> a = x \<or> x < a" by auto
1.16    moreover
1.17 -  { assume "a < x"
1.18 -    note iil = Node.IH(1)[OF il]
1.19 -    have ?case
1.20 -    proof (cases rule: lvl_update[of a b l])
1.21 -      case (Same) thus ?thesis
1.22 -        using \<open>a<x\<close> invar_NodeL[OF Node.prems iil Same]
1.23 -        by (simp add: skew_invar split_invar del: invar.simps)
1.24 +  have ?case if "a < x"
1.25 +  proof (cases rule: lvl_update[of a b l])
1.26 +    case (Same) thus ?thesis
1.27 +      using \<open>a<x\<close> invar_NodeL[OF N.prems iil Same]
1.28 +      by (simp add: skew_invar split_invar del: invar.simps)
1.29 +  next
1.30 +    case (Incr)
1.31 +    then obtain t1 w t2 where ial[simp]: "update a b l = Node n t1 w t2"
1.32 +      using N.prems by (auto simp: lvl_Suc_iff)
1.33 +    have l12: "lvl t1 = lvl t2"
1.34 +      by (metis Incr(1) ial lvl_update_incr_iff tree.inject)
1.35 +    have "update a b ?t = split(skew(Node n (update a b l) xy r))"
1.37 +    also have "skew(Node n (update a b l) xy r) = Node n t1 w (Node n t2 xy r)"
1.38 +      by(simp)
1.39 +    also have "invar(split \<dots>)"
1.40 +    proof (cases r)
1.41 +      case Leaf
1.42 +      hence "l = Leaf" using N.prems by(auto simp: lvl_0_iff)
1.43 +      thus ?thesis using Leaf ial by simp
1.44      next
1.45 -      case (Incr)
1.46 -      then obtain t1 w t2 where ial[simp]: "update a b l = Node n t1 w t2"
1.47 -        using Node.prems by (auto simp: lvl_Suc_iff)
1.48 -      have l12: "lvl t1 = lvl t2"
1.49 -        by (metis Incr(1) ial lvl_update_incr_iff tree.inject)
1.50 -      have "update a b ?t = split(skew(Node n (update a b l) xy r))"
1.52 -      also have "skew(Node n (update a b l) xy r) = Node n t1 w (Node n t2 xy r)"
1.53 -        by(simp)
1.54 -      also have "invar(split \<dots>)"
1.55 -      proof (cases r)
1.56 -        case Leaf
1.57 -        hence "l = Leaf" using Node.prems by(auto simp: lvl_0_iff)
1.58 -        thus ?thesis using Leaf ial by simp
1.59 +      case [simp]: (Node m t3 y t4)
1.60 +      show ?thesis (*using N(3) iil l12 by(auto)*)
1.61 +      proof cases
1.62 +        assume "m = n" thus ?thesis using N(3) iil by(auto)
1.63        next
1.64 -        case [simp]: (Node m t3 y t4)
1.65 -        show ?thesis (*using N(3) iil l12 by(auto)*)
1.66 -        proof cases
1.67 -          assume "m = n" thus ?thesis using N(3) iil by(auto)
1.68 -        next
1.69 -          assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
1.70 -        qed
1.71 +        assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
1.72        qed
1.73 -      finally show ?thesis .
1.74      qed
1.75 -  }
1.76 +    finally show ?thesis .
1.77 +  qed
1.78    moreover
1.79 -  { assume "x < a"
1.80 -    note iir = Node.IH(2)[OF ir]
1.81 +  have ?case if "x < a"
1.82 +  proof -
1.83      from \<open>invar ?t\<close> have "n = lvl r \<or> n = lvl r + 1" by auto
1.84 -    hence ?case
1.85 +    thus ?case
1.86      proof
1.87        assume 0: "n = lvl r"
1.88        have "update a b ?t = split(skew(Node n l xy (update a b r)))"
1.89          using \<open>a>x\<close> by(auto)
1.90        also have "skew(Node n l xy (update a b r)) = Node n l xy (update a b r)"
1.91 -        using Node.prems by(simp add: skew_case split: tree.split)
1.92 +        using N.prems by(simp add: skew_case split: tree.split)
1.93        also have "invar(split \<dots>)"
1.94        proof -
1.95          from lvl_update_sngl[OF ir sngl_if_invar[OF \<open>invar ?t\<close> 0], of a b]
1.96          obtain t1 p t2 where iar: "update a b r = Node n t1 p t2"
1.97 -          using Node.prems 0 by (auto simp: lvl_Suc_iff)
1.98 -        from Node.prems iar 0 iir
1.99 +          using N.prems 0 by (auto simp: lvl_Suc_iff)
1.100 +        from N.prems iar 0 iir
1.101          show ?thesis by (auto simp: split_case split: tree.splits)
1.102        qed
1.103        finally show ?thesis .
1.104 @@ -139,7 +137,7 @@
1.105        show ?thesis
1.106        proof (cases rule: lvl_update[of a b r])
1.107          case (Same)
1.108 -        show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF Node.prems 1 iir Same]
1.109 +        show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF N.prems 1 iir Same]
1.110            by (auto simp add: skew_invar split_invar)
1.111        next
1.112          case (Incr)
1.113 @@ -147,8 +145,9 @@
1.114            by (auto simp add: skew_invar split_invar split: if_splits)
1.115        qed
1.116      qed
1.117 -  }
1.118 -  moreover { assume "a = x" hence ?case using Node.prems by auto }
1.119 +  qed
1.120 +  moreover
1.121 +  have "a = x \<Longrightarrow> ?case" using N.prems by auto
1.122    ultimately show ?case by blast
1.123  qed simp
1.124
```
```     2.1 --- a/src/HOL/Data_Structures/AA_Set.thy	Thu Nov 09 10:24:00 2017 +0100
2.2 +++ b/src/HOL/Data_Structures/AA_Set.thy	Fri Nov 10 22:05:30 2017 +0100
2.3 @@ -201,63 +201,61 @@
2.4
2.5  lemma invar_insert: "invar t \<Longrightarrow> invar(insert a t)"
2.6  proof(induction t)
2.7 -  case (Node n l x r)
2.8 +  case N: (Node n l x r)
2.9    hence il: "invar l" and ir: "invar r" by auto
2.10 -  note N = Node
2.11 +  note iil = N.IH(1)[OF il]
2.12 +  note iir = N.IH(2)[OF ir]
2.13    let ?t = "Node n l x r"
2.14    have "a < x \<or> a = x \<or> x < a" by auto
2.15    moreover
2.16 -  { assume "a < x"
2.17 -    note iil = Node.IH(1)[OF il]
2.18 -    have ?case
2.19 -    proof (cases rule: lvl_insert[of a l])
2.20 -      case (Same) thus ?thesis
2.21 -        using \<open>a<x\<close> invar_NodeL[OF Node.prems iil Same]
2.22 -        by (simp add: skew_invar split_invar del: invar.simps)
2.23 +  have ?case if "a < x"
2.24 +  proof (cases rule: lvl_insert[of a l])
2.25 +    case (Same) thus ?thesis
2.26 +      using \<open>a<x\<close> invar_NodeL[OF N.prems iil Same]
2.27 +      by (simp add: skew_invar split_invar del: invar.simps)
2.28 +  next
2.29 +    case (Incr)
2.30 +    then obtain t1 w t2 where ial[simp]: "insert a l = Node n t1 w t2"
2.31 +      using N.prems by (auto simp: lvl_Suc_iff)
2.32 +    have l12: "lvl t1 = lvl t2"
2.33 +      by (metis Incr(1) ial lvl_insert_incr_iff tree.inject)
2.34 +    have "insert a ?t = split(skew(Node n (insert a l) x r))"
2.36 +    also have "skew(Node n (insert a l) x r) = Node n t1 w (Node n t2 x r)"
2.37 +      by(simp)
2.38 +    also have "invar(split \<dots>)"
2.39 +    proof (cases r)
2.40 +      case Leaf
2.41 +      hence "l = Leaf" using N.prems by(auto simp: lvl_0_iff)
2.42 +      thus ?thesis using Leaf ial by simp
2.43      next
2.44 -      case (Incr)
2.45 -      then obtain t1 w t2 where ial[simp]: "insert a l = Node n t1 w t2"
2.46 -        using Node.prems by (auto simp: lvl_Suc_iff)
2.47 -      have l12: "lvl t1 = lvl t2"
2.48 -        by (metis Incr(1) ial lvl_insert_incr_iff tree.inject)
2.49 -      have "insert a ?t = split(skew(Node n (insert a l) x r))"
2.51 -      also have "skew(Node n (insert a l) x r) = Node n t1 w (Node n t2 x r)"
2.52 -        by(simp)
2.53 -      also have "invar(split \<dots>)"
2.54 -      proof (cases r)
2.55 -        case Leaf
2.56 -        hence "l = Leaf" using Node.prems by(auto simp: lvl_0_iff)
2.57 -        thus ?thesis using Leaf ial by simp
2.58 +      case [simp]: (Node m t3 y t4)
2.59 +      show ?thesis (*using N(3) iil l12 by(auto)*)
2.60 +      proof cases
2.61 +        assume "m = n" thus ?thesis using N(3) iil by(auto)
2.62        next
2.63 -        case [simp]: (Node m t3 y t4)
2.64 -        show ?thesis (*using N(3) iil l12 by(auto)*)
2.65 -        proof cases
2.66 -          assume "m = n" thus ?thesis using N(3) iil by(auto)
2.67 -        next
2.68 -          assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
2.69 -        qed
2.70 +        assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
2.71        qed
2.72 -      finally show ?thesis .
2.73      qed
2.74 -  }
2.75 +    finally show ?thesis .
2.76 +  qed
2.77    moreover
2.78 -  { assume "x < a"
2.79 -    note iir = Node.IH(2)[OF ir]
2.80 +  have ?case if "x < a"
2.81 +  proof -
2.82      from \<open>invar ?t\<close> have "n = lvl r \<or> n = lvl r + 1" by auto
2.83 -    hence ?case
2.84 +    thus ?case
2.85      proof
2.86        assume 0: "n = lvl r"
2.87        have "insert a ?t = split(skew(Node n l x (insert a r)))"
2.88          using \<open>a>x\<close> by(auto)
2.89        also have "skew(Node n l x (insert a r)) = Node n l x (insert a r)"
2.90 -        using Node.prems by(simp add: skew_case split: tree.split)
2.91 +        using N.prems by(simp add: skew_case split: tree.split)
2.92        also have "invar(split \<dots>)"
2.93        proof -
2.94          from lvl_insert_sngl[OF ir sngl_if_invar[OF \<open>invar ?t\<close> 0], of a]
2.95          obtain t1 y t2 where iar: "insert a r = Node n t1 y t2"
2.96 -          using Node.prems 0 by (auto simp: lvl_Suc_iff)
2.97 -        from Node.prems iar 0 iir
2.98 +          using N.prems 0 by (auto simp: lvl_Suc_iff)
2.99 +        from N.prems iar 0 iir
2.100          show ?thesis by (auto simp: split_case split: tree.splits)
2.101        qed
2.102        finally show ?thesis .
2.103 @@ -267,7 +265,7 @@
2.104        show ?thesis
2.105        proof (cases rule: lvl_insert[of a r])
2.106          case (Same)
2.107 -        show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF Node.prems 1 iir Same]
2.108 +        show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF N.prems 1 iir Same]
2.109            by (auto simp add: skew_invar split_invar)
2.110        next
2.111          case (Incr)
2.112 @@ -275,8 +273,9 @@
2.113            by (auto simp add: skew_invar split_invar split: if_splits)
2.114        qed
2.115      qed
2.116 -  }
2.117 -  moreover { assume "a = x" hence ?case using Node.prems by auto }
2.118 +  qed
2.119 +  moreover
2.120 +  have "a = x \<Longrightarrow> ?case" using N.prems by auto
2.121    ultimately show ?case by blast
2.122  qed simp
2.123
```
```     3.1 --- a/src/HOL/Data_Structures/Brother12_Map.thy	Thu Nov 09 10:24:00 2017 +0100
3.2 +++ b/src/HOL/Data_Structures/Brother12_Map.thy	Fri Nov 10 22:05:30 2017 +0100
3.3 @@ -126,61 +126,58 @@
3.4    { case 1
3.5      then obtain l a b r where [simp]: "t = N2 l (a,b) r" and
3.6        lr: "l \<in> T h" "r \<in> T h" "l \<in> B h \<or> r \<in> B h" by auto
3.7 -    { assume "x < a"
3.8 -      have ?case
3.9 -      proof cases
3.10 -        assume "l \<in> B h"
3.11 -        from n2_type3[OF Suc.IH(1)[OF this] lr(2)]
3.12 -        show ?thesis using `x<a` by(simp)
3.13 -      next
3.14 -        assume "l \<notin> B h"
3.15 -        hence "l \<in> U h" "r \<in> B h" using lr by auto
3.16 -        from n2_type1[OF Suc.IH(2)[OF this(1)] this(2)]
3.17 -        show ?thesis using `x<a` by(simp)
3.18 -      qed
3.19 -    } moreover
3.20 -    { assume "x > a"
3.21 -      have ?case
3.22 +    have ?case if "x < a"
3.23 +    proof cases
3.24 +      assume "l \<in> B h"
3.25 +      from n2_type3[OF Suc.IH(1)[OF this] lr(2)]
3.26 +      show ?thesis using `x<a` by(simp)
3.27 +    next
3.28 +      assume "l \<notin> B h"
3.29 +      hence "l \<in> U h" "r \<in> B h" using lr by auto
3.30 +      from n2_type1[OF Suc.IH(2)[OF this(1)] this(2)]
3.31 +      show ?thesis using `x<a` by(simp)
3.32 +    qed
3.33 +    moreover
3.34 +    have ?case if "x > a"
3.35 +    proof cases
3.36 +      assume "r \<in> B h"
3.37 +      from n2_type3[OF lr(1) Suc.IH(1)[OF this]]
3.38 +      show ?thesis using `x>a` by(simp)
3.39 +    next
3.40 +      assume "r \<notin> B h"
3.41 +      hence "l \<in> B h" "r \<in> U h" using lr by auto
3.42 +      from n2_type2[OF this(1) Suc.IH(2)[OF this(2)]]
3.43 +      show ?thesis using `x>a` by(simp)
3.44 +    qed
3.45 +    moreover
3.46 +    have ?case if [simp]: "x=a"
3.47 +    proof (cases "del_min r")
3.48 +      case None
3.49 +      show ?thesis
3.50        proof cases
3.51          assume "r \<in> B h"
3.52 -        from n2_type3[OF lr(1) Suc.IH(1)[OF this]]
3.53 -        show ?thesis using `x>a` by(simp)
3.54 +        with del_minNoneN0[OF this None] lr show ?thesis by(simp)
3.55        next
3.56          assume "r \<notin> B h"
3.57 -        hence "l \<in> B h" "r \<in> U h" using lr by auto
3.58 -        from n2_type2[OF this(1) Suc.IH(2)[OF this(2)]]
3.59 -        show ?thesis using `x>a` by(simp)
3.60 +        hence "r \<in> U h" using lr by auto
3.61 +        with del_minNoneN1[OF this None] lr(3) show ?thesis by (simp)
3.62        qed
3.63 -    } moreover
3.64 -    { assume [simp]: "x=a"
3.65 -      have ?case
3.66 -      proof (cases "del_min r")
3.67 -        case None
3.68 -        show ?thesis
3.69 -        proof cases
3.70 -          assume "r \<in> B h"
3.71 -          with del_minNoneN0[OF this None] lr show ?thesis by(simp)
3.72 -        next
3.73 -          assume "r \<notin> B h"
3.74 -          hence "r \<in> U h" using lr by auto
3.75 -          with del_minNoneN1[OF this None] lr(3) show ?thesis by (simp)
3.76 -        qed
3.77 +    next
3.78 +      case [simp]: (Some br')
3.79 +      obtain b r' where [simp]: "br' = (b,r')" by fastforce
3.80 +      show ?thesis
3.81 +      proof cases
3.82 +        assume "r \<in> B h"
3.83 +        from del_min_type(1)[OF this] n2_type3[OF lr(1)]
3.84 +        show ?thesis by simp
3.85        next
3.86 -        case [simp]: (Some br')
3.87 -        obtain b r' where [simp]: "br' = (b,r')" by fastforce
3.88 -        show ?thesis
3.89 -        proof cases
3.90 -          assume "r \<in> B h"
3.91 -          from del_min_type(1)[OF this] n2_type3[OF lr(1)]
3.92 -          show ?thesis by simp
3.93 -        next
3.94 -          assume "r \<notin> B h"
3.95 -          hence "l \<in> B h" and "r \<in> U h" using lr by auto
3.96 -          from del_min_type(2)[OF this(2)] n2_type2[OF this(1)]
3.97 -          show ?thesis by simp
3.98 -        qed
3.99 +        assume "r \<notin> B h"
3.100 +        hence "l \<in> B h" and "r \<in> U h" using lr by auto
3.101 +        from del_min_type(2)[OF this(2)] n2_type2[OF this(1)]
3.102 +        show ?thesis by simp
3.103        qed
3.104 -    } ultimately show ?case by auto
3.105 +    qed
3.106 +    ultimately show ?case by auto
3.107    }
3.108    { case 2 with Suc.IH(1) show ?case by auto }
3.109  qed auto
```
```     4.1 --- a/src/HOL/Data_Structures/Brother12_Set.thy	Thu Nov 09 10:24:00 2017 +0100
4.2 +++ b/src/HOL/Data_Structures/Brother12_Set.thy	Fri Nov 10 22:05:30 2017 +0100
4.3 @@ -258,9 +258,9 @@
4.4      then obtain t1 a t2 where [simp]: "t = N2 t1 a t2" and
4.5        t1: "t1 \<in> T h" and t2: "t2 \<in> T h" and t12: "t1 \<in> B h \<or> t2 \<in> B h"
4.6        by auto
4.7 -    { assume "x < a"
4.8 -      hence "?case \<longleftrightarrow> n2 (ins x t1) a t2 \<in> Bp (Suc h)" by simp
4.9 -      also have "\<dots>"
4.10 +    have ?case if "x < a"
4.11 +    proof -
4.12 +      have "n2 (ins x t1) a t2 \<in> Bp (Suc h)"
4.13        proof cases
4.14          assume "t1 \<in> B h"
4.15          with t2 show ?thesis by (simp add: Suc.IH(1) n2_type)
4.16 @@ -269,12 +269,12 @@
4.17          hence 1: "t1 \<in> U h" and 2: "t2 \<in> B h" using t1 t12 by auto
4.18          show ?thesis by (metis Suc.IH(2)[OF 1] Bp_if_B[OF 2] n2_type)
4.19        qed
4.20 -      finally have ?case .
4.21 -    }
4.22 +      with `x < a` show ?case by simp
4.23 +    qed
4.24      moreover
4.25 -    { assume "a < x"
4.26 -      hence "?case \<longleftrightarrow> n2 t1 a (ins x t2) \<in> Bp (Suc h)" by simp
4.27 -      also have "\<dots>"
4.28 +    have ?case if "a < x"
4.29 +    proof -
4.30 +      have "n2 t1 a (ins x t2) \<in> Bp (Suc h)"
4.31        proof cases
4.32          assume "t2 \<in> B h"
4.33          with t1 show ?thesis by (simp add: Suc.IH(1) n2_type)
4.34 @@ -283,12 +283,14 @@
4.35          hence 1: "t1 \<in> B h" and 2: "t2 \<in> U h" using t2 t12 by auto
4.36          show ?thesis by (metis Bp_if_B[OF 1] Suc.IH(2)[OF 2] n2_type)
4.37        qed
4.38 -    }
4.39 -    moreover
4.40 -    { assume "x = a"
4.41 +      with `a < x` show ?case by simp
4.42 +    qed
4.43 +    moreover
4.44 +    have ?case if "x = a"
4.45 +    proof -
4.46        from 1 have "t \<in> Bp (Suc h)" by(rule Bp_if_B)
4.47 -      hence "?case" using `x = a` by simp
4.48 -    }
4.49 +      thus "?case" using `x = a` by simp
4.50 +    qed
4.51      ultimately show ?case by auto
4.52    next
4.53      case 2 thus ?case using Suc(1) n1_type by fastforce }
4.54 @@ -398,61 +400,58 @@
4.55    { case 1
4.56      then obtain l a r where [simp]: "t = N2 l a r" and
4.57        lr: "l \<in> T h" "r \<in> T h" "l \<in> B h \<or> r \<in> B h" by auto
4.58 -    { assume "x < a"
4.59 -      have ?case
4.60 -      proof cases
4.61 -        assume "l \<in> B h"
4.62 -        from n2_type3[OF Suc.IH(1)[OF this] lr(2)]
4.63 -        show ?thesis using `x<a` by(simp)
4.64 -      next
4.65 -        assume "l \<notin> B h"
4.66 -        hence "l \<in> U h" "r \<in> B h" using lr by auto
4.67 -        from n2_type1[OF Suc.IH(2)[OF this(1)] this(2)]
4.68 -        show ?thesis using `x<a` by(simp)
4.69 -      qed
4.70 -    } moreover
4.71 -    { assume "x > a"
4.72 -      have ?case
4.73 +    have ?case if "x < a"
4.74 +    proof cases
4.75 +      assume "l \<in> B h"
4.76 +      from n2_type3[OF Suc.IH(1)[OF this] lr(2)]
4.77 +      show ?thesis using `x<a` by(simp)
4.78 +    next
4.79 +      assume "l \<notin> B h"
4.80 +      hence "l \<in> U h" "r \<in> B h" using lr by auto
4.81 +      from n2_type1[OF Suc.IH(2)[OF this(1)] this(2)]
4.82 +      show ?thesis using `x<a` by(simp)
4.83 +    qed
4.84 +    moreover
4.85 +    have ?case if "x > a"
4.86 +    proof cases
4.87 +      assume "r \<in> B h"
4.88 +      from n2_type3[OF lr(1) Suc.IH(1)[OF this]]
4.89 +      show ?thesis using `x>a` by(simp)
4.90 +    next
4.91 +      assume "r \<notin> B h"
4.92 +      hence "l \<in> B h" "r \<in> U h" using lr by auto
4.93 +      from n2_type2[OF this(1) Suc.IH(2)[OF this(2)]]
4.94 +      show ?thesis using `x>a` by(simp)
4.95 +    qed
4.96 +    moreover
4.97 +    have ?case if [simp]: "x=a"
4.98 +    proof (cases "del_min r")
4.99 +      case None
4.100 +      show ?thesis
4.101        proof cases
4.102          assume "r \<in> B h"
4.103 -        from n2_type3[OF lr(1) Suc.IH(1)[OF this]]
4.104 -        show ?thesis using `x>a` by(simp)
4.105 +        with del_minNoneN0[OF this None] lr show ?thesis by(simp)
4.106        next
4.107          assume "r \<notin> B h"
4.108 -        hence "l \<in> B h" "r \<in> U h" using lr by auto
4.109 -        from n2_type2[OF this(1) Suc.IH(2)[OF this(2)]]
4.110 -        show ?thesis using `x>a` by(simp)
4.111 +        hence "r \<in> U h" using lr by auto
4.112 +        with del_minNoneN1[OF this None] lr(3) show ?thesis by (simp)
4.113        qed
4.114 -    } moreover
4.115 -    { assume [simp]: "x=a"
4.116 -      have ?case
4.117 -      proof (cases "del_min r")
4.118 -        case None
4.119 -        show ?thesis
4.120 -        proof cases
4.121 -          assume "r \<in> B h"
4.122 -          with del_minNoneN0[OF this None] lr show ?thesis by(simp)
4.123 -        next
4.124 -          assume "r \<notin> B h"
4.125 -          hence "r \<in> U h" using lr by auto
4.126 -          with del_minNoneN1[OF this None] lr(3) show ?thesis by (simp)
4.127 -        qed
4.128 +    next
4.129 +      case [simp]: (Some br')
4.130 +      obtain b r' where [simp]: "br' = (b,r')" by fastforce
4.131 +      show ?thesis
4.132 +      proof cases
4.133 +        assume "r \<in> B h"
4.134 +        from del_min_type(1)[OF this] n2_type3[OF lr(1)]
4.135 +        show ?thesis by simp
4.136        next
4.137 -        case [simp]: (Some br')
4.138 -        obtain b r' where [simp]: "br' = (b,r')" by fastforce
4.139 -        show ?thesis
4.140 -        proof cases
4.141 -          assume "r \<in> B h"
4.142 -          from del_min_type(1)[OF this] n2_type3[OF lr(1)]
4.143 -          show ?thesis by simp
4.144 -        next
4.145 -          assume "r \<notin> B h"
4.146 -          hence "l \<in> B h" and "r \<in> U h" using lr by auto
4.147 -          from del_min_type(2)[OF this(2)] n2_type2[OF this(1)]
4.148 -          show ?thesis by simp
4.149 -        qed
4.150 +        assume "r \<notin> B h"
4.151 +        hence "l \<in> B h" and "r \<in> U h" using lr by auto
4.152 +        from del_min_type(2)[OF this(2)] n2_type2[OF this(1)]
4.153 +        show ?thesis by simp
4.154        qed
4.155 -    } ultimately show ?case by auto
4.156 +    qed
4.157 +    ultimately show ?case by auto
4.158    }
4.159    { case 2 with Suc.IH(1) show ?case by auto }
4.160  qed auto
```