Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
to their abstract counterparts, while other binary numerals work correctly.
(* Title: HOL/ex/AVL.ML
ID: $Id$
Author: Cornelia Pusch and Tobias Nipkow
Copyright 1998 TUM
*)
(****************************** isbal **********************************)
Addsimps[Let_def];
(* rotations preserve isbal property *)
Goalw [bal_def]
"height l = Suc(Suc(height r)) --> bal l = Right --> isbal l --> isbal r \
\ --> isbal (lr_rot (n, l, r))";
by (case_tac "l" 1);
by (Asm_simp_tac 1);
by (case_tac "tree2" 1);
by (Asm_simp_tac 1);
by (asm_simp_tac (simpset() addsimps [max_def]) 1);
by (arith_tac 1);
qed_spec_mp "isbal_lr_rot";
Goalw [bal_def]
"height l = Suc(Suc(height r)) --> bal l ~= Right --> isbal l --> isbal r \
\ --> isbal (r_rot (n, l, r))";
by (case_tac "l" 1);
by (Asm_simp_tac 1);
by (asm_simp_tac (simpset() addsimps [max_def]) 1);
qed_spec_mp "isbal_r_rot";
Goalw [bal_def]
"height r = Suc(Suc(height l)) --> bal r = Left --> isbal l --> isbal r \
\ --> isbal (rl_rot (n, l, r))";
by (case_tac "r" 1);
by (Asm_simp_tac 1);
by (case_tac "tree1" 1);
by (asm_simp_tac (simpset() addsimps [max_def]) 1);
by (asm_simp_tac (simpset() addsimps [max_def]) 1);
by (arith_tac 1);
qed_spec_mp "isbal_rl_rot";
Goalw [bal_def]
"height r = Suc(Suc(height l)) --> bal r ~= Left --> isbal l --> isbal r \
\ --> isbal (l_rot (n, l, r))";
by (case_tac "r" 1);
by (Asm_simp_tac 1);
by (asm_simp_tac (simpset() addsimps [max_def]) 1);
qed_spec_mp "isbal_l_rot";
(* lemmas about height after rotation *)
Goalw [bal_def]
"bal l = Right --> height l = Suc(Suc(height r)) \
\ --> Suc(height (lr_rot (n, l, r))) = height (MKT n l r) ";
by (case_tac "l" 1);
by (Asm_simp_tac 1);
by (case_tac "tree2" 1);
by (Asm_simp_tac 1);
by (asm_simp_tac (simpset() addsimps [max_def]) 1);
qed_spec_mp "height_lr_rot";
Goalw [bal_def]
"height l = Suc(Suc(height r)) --> bal l ~= Right \
\ --> Suc(height (r_rot (n, l, r))) = height (MKT n l r) | \
\ height (r_rot (n, l, r)) = height (MKT n l r)";
by (case_tac "l" 1);
by (Asm_simp_tac 1);
by (asm_simp_tac (simpset() addsimps [max_def]) 1);
qed_spec_mp "height_r_rot";
Goalw [l_bal_def]
"height l = Suc(Suc(height r)) \
\ --> Suc(height (l_bal n l r)) = height (MKT n l r) | \
\ height (l_bal n l r) = height (MKT n l r)";
by (case_tac "bal l = Right" 1);
by (fast_tac (claset() addDs [height_lr_rot] addss simpset()) 1);
by (fast_tac (claset() addDs [height_r_rot] addss simpset()) 1);
qed_spec_mp "height_l_bal";
Goalw [bal_def]
"height r = Suc(Suc(height l)) --> bal r = Left \
\ --> Suc(height (rl_rot (n, l, r))) = height (MKT n l r)";
by (case_tac "r" 1);
by (Asm_simp_tac 1);
by (case_tac "tree1" 1);
by (Asm_simp_tac 1);
by (asm_simp_tac (simpset() addsimps [max_def]) 1);
by (arith_tac 1);
qed_spec_mp "height_rl_rot";
Goalw [bal_def]
"height r = Suc(Suc(height l)) --> bal r ~= Left \
\ --> Suc(height (l_rot (n, l, r))) = height (MKT n l r) | \
\ height (l_rot (n, l, r)) = height (MKT n l r)";
by (case_tac "r" 1);
by (Asm_simp_tac 1);
by (asm_simp_tac (simpset() addsimps [max_def]) 1);
qed_spec_mp "height_l_rot";
Goalw [r_bal_def]
"height r = Suc(Suc(height l)) \
\ --> Suc(height (r_bal n l r)) = height (MKT n l r) | \
\ height (r_bal n l r) = height (MKT n l r)";
by (case_tac "bal r = Left" 1);
by (fast_tac (claset() addDs [height_rl_rot] addss simpset()) 1);
by (fast_tac (claset() addDs [height_l_rot] addss simpset()) 1);
qed_spec_mp "height_r_bal";
(* lemma about height after insert *)
Goal
"isbal t \
\ --> height (insert x t) = height t | height (insert x t) = Suc(height t)";
by (induct_tac "t" 1);
by (Simp_tac 1);
by (case_tac "x=nat" 1);
by (Asm_simp_tac 1);
by (case_tac "x<nat" 1);
by (case_tac "height (insert x tree1) = Suc (Suc (height tree2))" 1);
by (forw_inst_tac [("n","nat")] height_l_bal 1);
by (asm_full_simp_tac (simpset() addsimps [max_def]) 1);
by(fast_tac (claset() addss simpset()) 1);
by (asm_full_simp_tac (simpset() addsimps [max_def]) 1);
by(fast_tac (claset() addss simpset()) 1);
by (case_tac "height (insert x tree2) = Suc (Suc (height tree1))" 1);
by (forw_inst_tac [("n","nat")] height_r_bal 1);
by (asm_full_simp_tac (simpset() addsimps [max_def]) 1);
by(fast_tac (claset() addss simpset()) 1);
by (asm_full_simp_tac (simpset() addsimps [max_def]) 1);
by(fast_tac (claset() addss simpset()) 1);
qed_spec_mp "height_insert";
Goal
"!!x. [| height (insert x l) ~= Suc(Suc(height r)); isbal (insert x l); isbal (MKT n l r) |] \
\ ==> isbal (MKT n (insert x l) r)";
by (cut_inst_tac [("x","x"),("t","l")] height_insert 1);
by (Asm_full_simp_tac 1);
by (fast_tac (claset() addss simpset()) 1);
qed "isbal_insert_left";
Goal
"!!x. [| height (insert x r) ~= Suc(Suc(height l)); isbal (insert x r); isbal (MKT n l r) |] \
\ ==> isbal (MKT n l (insert x r))";
by (cut_inst_tac [("x","x"),("t","r")] height_insert 1);
by (Asm_full_simp_tac 1);
by (fast_tac (claset() addss simpset()) 1);
qed "isbal_insert_right";
(* insert-operation preserves isbal property *)
Goal
"isbal t --> isbal(insert x t)";
by (induct_tac "t" 1);
by (Simp_tac 1);
by (case_tac "x=nat" 1);
by (Asm_simp_tac 1);
by (case_tac "x<nat" 1);
by (case_tac "height (insert x tree1) = Suc (Suc (height tree2))" 1);
by (case_tac "bal (insert x tree1) = Right" 1);
by (fast_tac (claset() addIs [isbal_lr_rot] addss (simpset()
addsimps [l_bal_def])) 1);
by (fast_tac (claset() addIs [isbal_r_rot] addss (simpset()
addsimps [l_bal_def])) 1);
by (Clarify_tac 1);
by (forward_tac [isbal_insert_left] 1);
by (Asm_full_simp_tac 1);
ba 1;
by (Asm_full_simp_tac 1);
by (case_tac "height (insert x tree2) = Suc (Suc (height tree1))" 1);
by (case_tac "bal (insert x tree2) = Left" 1);
by (fast_tac (claset() addIs [isbal_rl_rot] addss (simpset()
addsimps [r_bal_def])) 1);
by (fast_tac (claset() addIs [isbal_l_rot] addss (simpset()
addsimps [r_bal_def])) 1);
by (Clarify_tac 1);
by (forward_tac [isbal_insert_right] 1);
by (Asm_full_simp_tac 1);
ba 1;
by (Asm_full_simp_tac 1);
qed_spec_mp "isbal_insert";
(****************************** isin **********************************)
(* rotations preserve isin property *)
Goalw [bal_def]
"height l = Suc(Suc(height r)) --> bal l = Right \
\ --> isin x (lr_rot (n, l, r)) = isin x (MKT n l r)";
by (case_tac "l" 1);
by (Asm_simp_tac 1);
by (case_tac "tree2" 1);
by (Asm_simp_tac 1);
by (fast_tac (claset() addss simpset()) 1);
qed_spec_mp "isin_lr_rot";
Goalw [bal_def]
"height l = Suc(Suc(height r)) --> bal l ~= Right \
\ --> isin x (r_rot (n, l, r)) = isin x (MKT n l r)";
by (case_tac "l" 1);
by (Asm_simp_tac 1);
by (fast_tac (claset() addss simpset()) 1);
qed_spec_mp "isin_r_rot";
Goalw [bal_def]
"height r = Suc(Suc(height l)) --> bal r = Left \
\ --> isin x (rl_rot (n, l, r)) = isin x (MKT n l r)";
by (case_tac "r" 1);
by (Asm_simp_tac 1);
by (case_tac "tree1" 1);
by (asm_simp_tac (simpset() addsimps [max_def,le_def]) 1);
by (fast_tac (claset() addss simpset()) 1);
qed_spec_mp "isin_rl_rot";
Goalw [bal_def]
"height r = Suc(Suc(height l)) --> bal r ~= Left \
\ --> isin x (l_rot (n, l, r)) = isin x (MKT n l r)";
by (case_tac "r" 1);
by (Asm_simp_tac 1);
by (fast_tac (claset() addss simpset()) 1);
qed_spec_mp "isin_l_rot";
(* isin insert *)
Goal
"isin y (insert x t) = (y=x | isin y t)";
by (induct_tac "t" 1);
by (Asm_simp_tac 1);
by(asm_simp_tac (simpset() addsimps [l_bal_def,isin_lr_rot,isin_r_rot,
r_bal_def,isin_rl_rot,isin_l_rot]) 1);
by(Blast_tac 1);
qed "isin_insert";
(****************************** isord **********************************)
(* rotations preserve isord property *)
Goalw [bal_def]
"height l = Suc(Suc(height r)) --> bal l = Right --> isord (MKT n l r) \
\ --> isord (lr_rot (n, l, r))";
by (case_tac "l" 1);
by (Asm_simp_tac 1);
by (case_tac "tree2" 1);
by (Asm_simp_tac 1);
by (Asm_simp_tac 1);
by(blast_tac (claset() addIs [less_trans])1);
qed_spec_mp "isord_lr_rot";
Goalw [bal_def]
"height l = Suc(Suc(height r)) --> bal l ~= Right --> isord (MKT n l r) \
\ --> isord (r_rot (n, l, r))";
by (case_tac "l" 1);
by (Asm_simp_tac 1);
by (auto_tac (claset() addIs [less_trans],simpset()));
qed_spec_mp "isord_r_rot";
Goalw [bal_def]
"height r = Suc(Suc(height l)) --> bal r = Left --> isord (MKT n l r) \
\ --> isord (rl_rot (n, l, r))";
by (case_tac "r" 1);
by (Asm_simp_tac 1);
by (case_tac "tree1" 1);
by (asm_simp_tac (simpset() addsimps [le_def]) 1);
by (Asm_simp_tac 1);
by(blast_tac (claset() addIs [less_trans])1);
qed_spec_mp "isord_rl_rot";
Goalw [bal_def]
"height r = Suc(Suc(height l)) --> bal r ~= Left --> isord (MKT n l r) \
\ --> isord (l_rot (n, l, r))";
by (case_tac "r" 1);
by (Asm_simp_tac 1);
by (Asm_simp_tac 1);
by(blast_tac (claset() addIs [less_trans])1);
qed_spec_mp "isord_l_rot";
(* insert operation presreves isord property *)
Goal
"isord t --> isord(insert x t)";
by (induct_tac "t" 1);
by (Simp_tac 1);
by (cut_inst_tac [("m","x"),("n","nat")] less_linear 1);
by (fast_tac (claset() addss (simpset() addsimps [l_bal_def,isord_lr_rot,
isord_r_rot,r_bal_def,isord_l_rot,isord_rl_rot,isin_insert])) 1);
qed_spec_mp "isord_insert";