author | nipkow |
Fri, 21 Feb 2020 17:51:56 +0100 | |
changeset 71463 | a31a9da43694 |
parent 70755 | 3fb16bed5d6c |
child 71830 | 7a997ead54b0 |
permissions | -rw-r--r-- |
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(* Author: Tobias Nipkow *) |
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section \<open>Red-Black Tree Implementation of Maps\<close> |
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theory RBT_Map |
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imports |
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RBT_Set |
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Lookup2 |
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begin |
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fun upd :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where |
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"upd x y Leaf = R Leaf (x,y) Leaf" | |
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"upd x y (B l (a,b) r) = (case cmp x a of |
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LT \<Rightarrow> baliL (upd x y l) (a,b) r | |
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GT \<Rightarrow> baliR l (a,b) (upd x y r) | |
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EQ \<Rightarrow> B l (x,y) r)" | |
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"upd x y (R l (a,b) r) = (case cmp x a of |
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LT \<Rightarrow> R (upd x y l) (a,b) r | |
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GT \<Rightarrow> R l (a,b) (upd x y r) | |
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EQ \<Rightarrow> R l (x,y) r)" |
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definition update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where |
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"update x y t = paint Black (upd x y t)" |
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fun del :: "'a::linorder \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt" where |
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"del x Leaf = Leaf" | |
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"del x (Node l ((a,b), c) r) = (case cmp x a of |
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LT \<Rightarrow> if l \<noteq> Leaf \<and> color l = Black |
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then baldL (del x l) (a,b) r else R (del x l) (a,b) r | |
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GT \<Rightarrow> if r \<noteq> Leaf\<and> color r = Black |
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then baldR l (a,b) (del x r) else R l (a,b) (del x r) | |
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EQ \<Rightarrow> app l r)" |
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definition delete :: "'a::linorder \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where |
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"delete x t = paint Black (del x t)" |
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subsection "Functional Correctness Proofs" |
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lemma inorder_upd: |
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"sorted1(inorder t) \<Longrightarrow> inorder(upd x y t) = upd_list x y (inorder t)" |
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by(induction x y t rule: upd.induct) |
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(auto simp: upd_list_simps inorder_baliL inorder_baliR) |
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lemma inorder_update: |
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"sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)" |
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by(simp add: update_def inorder_upd inorder_paint) |
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lemma inorder_del: |
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"sorted1(inorder t) \<Longrightarrow> inorder(del x t) = del_list x (inorder t)" |
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by(induction x t rule: del.induct) |
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(auto simp: del_list_simps inorder_app inorder_baldL inorder_baldR) |
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lemma inorder_delete: |
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"sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)" |
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by(simp add: delete_def inorder_del inorder_paint) |
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subsection \<open>Structural invariants\<close> |
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subsubsection \<open>Update\<close> |
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lemma invc_upd: assumes "invc t" |
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shows "color t = Black \<Longrightarrow> invc (upd x y t)" "invc2 (upd x y t)" |
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using assms |
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by (induct x y t rule: upd.induct) (auto simp: invc_baliL invc_baliR invc2I) |
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lemma invh_upd: assumes "invh t" |
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shows "invh (upd x y t)" "bheight (upd x y t) = bheight t" |
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using assms |
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by(induct x y t rule: upd.induct) |
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(auto simp: invh_baliL invh_baliR bheight_baliL bheight_baliR) |
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theorem rbt_update: "rbt t \<Longrightarrow> rbt (update x y t)" |
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by (simp add: invc_upd(2) invh_upd(1) color_paint_Black invh_paint rbt_def update_def) |
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subsubsection \<open>Deletion\<close> |
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lemma del_invc_invh: "invh t \<Longrightarrow> invc t \<Longrightarrow> invh (del x t) \<and> |
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(color t = Red \<and> bheight (del x t) = bheight t \<and> invc (del x t) \<or> |
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color t = Black \<and> bheight (del x t) = bheight t - 1 \<and> invc2 (del x t))" |
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proof (induct x t rule: del.induct) |
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case (2 x _ y _ c) |
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have "x = y \<or> x < y \<or> x > y" by auto |
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thus ?case proof (elim disjE) |
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assume "x = y" |
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with 2 show ?thesis |
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by (cases c) (simp_all add: invh_app invc_app) |
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next |
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assume "x < y" |
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with 2 show ?thesis |
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by(cases c) |
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(auto simp: invh_baldL_invc invc_baldL invc2_baldL dest: neq_LeafD) |
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next |
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assume "y < x" |
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with 2 show ?thesis |
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by(cases c) |
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(auto simp: invh_baldR_invc invc_baldR invc2_baldR dest: neq_LeafD) |
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qed |
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qed auto |
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theorem rbt_delete: "rbt t \<Longrightarrow> rbt (delete k t)" |
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by (metis delete_def rbt_def color_paint_Black del_invc_invh invc2I invh_paint) |
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interpretation M: Map_by_Ordered |
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where empty = empty and lookup = lookup and update = update and delete = delete |
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and inorder = inorder and inv = rbt |
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proof (standard, goal_cases) |
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case 1 show ?case by (simp add: empty_def) |
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case 2 thus ?case by(simp add: lookup_map_of) |
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next |
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case 3 thus ?case by(simp add: inorder_update) |
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next |
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case 4 thus ?case by(simp add: inorder_delete) |
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next |
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case 5 thus ?case by (simp add: rbt_def empty_def) |
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next |
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case 6 thus ?case by (simp add: rbt_update) |
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next |
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case 7 thus ?case by (simp add: rbt_delete) |
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qed |
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end |