src/HOL/Data_Structures/RBT_Map.thy
 author nipkow Wed Jun 13 15:24:20 2018 +0200 (10 months ago) changeset 68440 6826718f732d parent 68431 b294e095f64c permissions -rw-r--r--
qualify interpretations to avoid clashes
```     1 (* Author: Tobias Nipkow *)
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```     2
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```     3 section \<open>Red-Black Tree Implementation of Maps\<close>
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```     4
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```     5 theory RBT_Map
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```     6 imports
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```     7   RBT_Set
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```     8   Lookup2
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```     9 begin
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```    10
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```    11 fun upd :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
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```    12 "upd x y Leaf = R Leaf (x,y) Leaf" |
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```    13 "upd x y (B l (a,b) r) = (case cmp x a of
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```    14   LT \<Rightarrow> baliL (upd x y l) (a,b) r |
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```    15   GT \<Rightarrow> baliR l (a,b) (upd x y r) |
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```    16   EQ \<Rightarrow> B l (x,y) r)" |
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```    17 "upd x y (R l (a,b) r) = (case cmp x a of
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```    18   LT \<Rightarrow> R (upd x y l) (a,b) r |
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```    19   GT \<Rightarrow> R l (a,b) (upd x y r) |
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```    20   EQ \<Rightarrow> R l (x,y) r)"
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```    21
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```    22 definition update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
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```    23 "update x y t = paint Black (upd x y t)"
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```    24
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```    25 fun del :: "'a::linorder \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt" where
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```    26 "del x Leaf = Leaf" |
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```    27 "del x (Node l (a,b) c r) = (case cmp x a of
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```    28      LT \<Rightarrow> if l \<noteq> Leaf \<and> color l = Black
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```    29            then baldL (del x l) (a,b) r else R (del x l) (a,b) r |
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```    30      GT \<Rightarrow> if r \<noteq> Leaf\<and> color r = Black
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```    31            then baldR l (a,b) (del x r) else R l (a,b) (del x r) |
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```    32   EQ \<Rightarrow> combine l r)"
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```    33
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```    34 definition delete :: "'a::linorder \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
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```    35 "delete x t = paint Black (del x t)"
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```    36
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```    37
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```    38 subsection "Functional Correctness Proofs"
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```    39
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```    40 lemma inorder_upd:
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```    41   "sorted1(inorder t) \<Longrightarrow> inorder(upd x y t) = upd_list x y (inorder t)"
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```    42 by(induction x y t rule: upd.induct)
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```    43   (auto simp: upd_list_simps inorder_baliL inorder_baliR)
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```    44
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```    45 lemma inorder_update:
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```    46   "sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
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```    47 by(simp add: update_def inorder_upd inorder_paint)
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```    48
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```    49 lemma inorder_del:
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```    50  "sorted1(inorder t) \<Longrightarrow>  inorder(del x t) = del_list x (inorder t)"
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```    51 by(induction x t rule: del.induct)
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```    52   (auto simp: del_list_simps inorder_combine inorder_baldL inorder_baldR)
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```    53
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```    54 lemma inorder_delete:
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```    55   "sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
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```    56 by(simp add: delete_def inorder_del inorder_paint)
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```    57
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```    58
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```    59 subsection \<open>Structural invariants\<close>
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```    60
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```    61 subsubsection \<open>Update\<close>
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```    62
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```    63 lemma invc_upd: assumes "invc t"
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```    64   shows "color t = Black \<Longrightarrow> invc (upd x y t)" "invc2 (upd x y t)"
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```    65 using assms
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```    66 by (induct x y t rule: upd.induct) (auto simp: invc_baliL invc_baliR invc2I)
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```    67
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```    68 lemma invh_upd: assumes "invh t"
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```    69   shows "invh (upd x y t)" "bheight (upd x y t) = bheight t"
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```    70 using assms
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```    71 by(induct x y t rule: upd.induct)
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```    72   (auto simp: invh_baliL invh_baliR bheight_baliL bheight_baliR)
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```    73
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```    74 theorem rbt_update: "rbt t \<Longrightarrow> rbt (update x y t)"
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```    75 by (simp add: invc_upd(2) invh_upd(1) color_paint_Black invc_paint_Black invh_paint
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```    76   rbt_def update_def)
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```    77
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```    78
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```    79 subsubsection \<open>Deletion\<close>
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```    80
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```    81 lemma del_invc_invh: "invh t \<Longrightarrow> invc t \<Longrightarrow> invh (del x t) \<and>
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```    82    (color t = Red \<and> bheight (del x t) = bheight t \<and> invc (del x t) \<or>
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```    83     color t = Black \<and> bheight (del x t) = bheight t - 1 \<and> invc2 (del x t))"
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```    84 proof (induct x t rule: del.induct)
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```    85 case (2 x _ y _ c)
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```    86   have "x = y \<or> x < y \<or> x > y" by auto
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```    87   thus ?case proof (elim disjE)
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```    88     assume "x = y"
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```    89     with 2 show ?thesis
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```    90     by (cases c) (simp_all add: invh_combine invc_combine)
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```    91   next
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```    92     assume "x < y"
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```    93     with 2 show ?thesis
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```    94       by(cases c)
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```    95         (auto simp: invh_baldL_invc invc_baldL invc2_baldL dest: neq_LeafD)
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```    96   next
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```    97     assume "y < x"
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```    98     with 2 show ?thesis
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```    99       by(cases c)
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```   100         (auto simp: invh_baldR_invc invc_baldR invc2_baldR dest: neq_LeafD)
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```   101   qed
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```   102 qed auto
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```   103
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```   104 theorem rbt_delete: "rbt t \<Longrightarrow> rbt (delete k t)"
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```   105 by (metis delete_def rbt_def color_paint_Black del_invc_invh invc_paint_Black invc2I invh_paint)
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```   106
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```   107 interpretation M: Map_by_Ordered
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```   108 where empty = empty and lookup = lookup and update = update and delete = delete
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```   109 and inorder = inorder and inv = rbt
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```   110 proof (standard, goal_cases)
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```   111   case 1 show ?case by (simp add: empty_def)
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```   112 next
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```   113   case 2 thus ?case by(simp add: lookup_map_of)
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```   114 next
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```   115   case 3 thus ?case by(simp add: inorder_update)
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```   116 next
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```   117   case 4 thus ?case by(simp add: inorder_delete)
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```   118 next
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```   119   case 5 thus ?case by (simp add: rbt_def empty_def)
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```   120 next
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```   121   case 6 thus ?case by (simp add: rbt_update)
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```   122 next
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```   123   case 7 thus ?case by (simp add: rbt_delete)
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```   124 qed
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```   125
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```   126 end
```