From df9edfd02bda931a0633f0e66bf8e32d7347902b Mon Sep 17 00:00:00 2001 From: Amartya Das Sharma Date: Tue, 8 Sep 2020 21:00:41 +0200 Subject: [PATCH] commit to push --- 01/ex01.pdf | Bin 0 -> 132471 bytes 01/ex01.thy | 63 +++++++++ 01/ex01.thy~ | 34 +++++ 01/ex0104.thy | 11 ++ 01/hw01.thy | 45 +++++++ 01/hw01.thy~ | 45 +++++++ 02/ex02.pdf | Bin 0 -> 134572 bytes 02/ex02.thy | 67 ++++++++++ 02/ex0202.thy | 7 + 02/hw02.thy | 105 +++++++++++++++ 02/hw02.thy~ | 105 +++++++++++++++ 03/ex03.pdf | Bin 0 -> 118576 bytes 03/hw03.thy | 114 ++++++++++++++++ 03/hw03.thy~ | 125 ++++++++++++++++++ 04/BST_Demo.thy | 73 ++++++++++ 04/Homework04.thy | 71 ++++++++++ 04/ex04.pdf | Bin 0 -> 121511 bytes 04/ex04self.thy | 27 ++++ 04/hw04.thy | 101 ++++++++++++++ 04/hw04.thy~ | 101 ++++++++++++++ 05/ex05.pdf | Bin 0 -> 169299 bytes 05/hw05.thy | 49 +++++++ 05/hw05.thy~ | 44 +++++++ 05/tmpl05.thy | 183 +++++++++++++++++++++++++ 06/ex06.pdf | Bin 0 -> 134272 bytes 06/hw06.thy | 66 ++++++++++ 06/hw06.thy~ | 31 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index 0000000..632cb26 --- /dev/null +++ b/01/ex01.thy @@ -0,0 +1,63 @@ +theory ex01 + imports Main +begin + +term "op+" + +lemma "a + b = b + (a :: nat)" + by auto + +lemma "a + (b + c) = (a + b) + (c::nat)" + by auto + +term "Nil" +term "Cons a b" + +fun count :: "'a list \ 'a \ nat" where + "count [] _ = 0" +| "count (x#xs) y = (if x=y then count xs y + 1 else count xs y)" + +fun count' :: "'a list \ 'a \ nat" where + "count' [] _ = 0" +| "count' (x # xs) y = (if x = y then 1 else 0) + count' xs y" + +value "count [1,2,3,4,2,2,2::int] 2" + +find_theorems "length [] = _" +find_theorems "length (_ # _) = _" + +lemma "count xs a \ length xs" + apply(induction xs) + apply(simp) + apply(simp) + done + +fun snoc :: "'a list \ 'a \ 'a list" where + "snoc [] y = [y]" +| "snoc (x # xs) y = x # snoc xs y" + +lemma "snoc xs x = xs@[x]" + apply(induction xs) by auto + +fun reverse :: "'a list \ 'a list" where + "reverse [] = []" +| "reverse (x # xs) = snoc (reverse xs) x" + +lemma aux: "reverse (snoc xs x) = x # reverse xs" + apply(induction xs) by auto + +lemma rev_rev[simp]: "reverse (reverse xs) = xs" + apply (induction xs) + apply (auto simp:aux) + done + +lemma "reverse (reverse xs) = xs" + apply(induction xs) + apply(auto simp:) + apply(subst aux) + apply auto + done + + + +end \ No newline at end of file diff --git a/01/ex01.thy~ b/01/ex01.thy~ new file mode 100644 index 0000000..39b2149 --- /dev/null +++ b/01/ex01.thy~ @@ -0,0 +1,34 @@ +theory ex01 + imports Main +begin + +term "op+" + +lemma "a + b = b + (a :: nat)" + by auto + +lemma "a + (b + c) = (a + b) + (c::nat)" + by auto + +term "Nil" +term "Cons a b" + +fun count :: "'a list \ 'a \ nat" where + "count [] _ = 0" +| "count (x#xs) y = (if x=y then count xs y + 1 else count xs y)" + +fun count' :: "'a list \ 'a \ nat" where + "count' [] _ = 0" +| "count' (x # xs) y = (if x = y then 1 else 0) + count' xs y" + +value "count [1,2,3,4,2,2,2::int] 2" + +find_theorems "length [] = _" +find_theorems "length (_ # _) = _" + +lemma "count xs a \ length xs" + apply(induction xs) + apply(simp) + apply(simp) + done +end \ No newline at end of file diff --git a/01/ex0104.thy b/01/ex0104.thy new file mode 100644 index 0000000..a61880d --- /dev/null +++ b/01/ex0104.thy @@ -0,0 +1,11 @@ +theorem ex0104 + imports Main +begin + +term "op#" + +fun snoc :: "'a list \ 'a \ 'a list" where + "snoc [] y = [y]" +| "snoc (x # xs) y = x # snoc xs y" + +end \ No newline at end of file diff --git a/01/hw01.thy b/01/hw01.thy new file mode 100644 index 0000000..24ae11e --- /dev/null +++ b/01/hw01.thy @@ -0,0 +1,45 @@ +theory hw01 + imports Main +begin + +fun listsum:: "int list \ int" where + "listsum [] = 0" +| "listsum (x # xs) = listsum xs + x" + +value "listsum [1,2,3] = 6" +value "listsum [] = 0" +value "listsum [1,-2,3] = 2" + +lemma listsum_filter_x: "listsum (filter (\x. x\0) l) = listsum l" + apply(induction l) + apply(auto) + done + +lemma listsum_append: "listsum (xs @ ys) = listsum xs + listsum ys" + apply(induction xs) + apply(auto) + done + +lemma listsum_rev: "listsum (rev xs) = listsum xs" + apply(induction xs) + apply(auto simp:listsum_append) + done + +lemma listsum_noneg: "listsum (filter (\x. x>0) l) \ listsum l" + apply(induction l) + apply(auto) + done + +fun flatten :: "'a list list \ 'a list" where + "flatten [] = []" +| "flatten (l#ls) = l @ flatten ls" + +value "flatten [[1,2,3],[2]] = [1,2,3,2::int]" +value "flatten [[1,2,3],[],[2]] = [1,2,3,2::int]" + +lemma "listsum (flatten xs) = listsum(map listsum xs)" + apply(induction xs) + apply(auto simp:listsum_append) + done + +end \ No newline at end of file diff --git a/01/hw01.thy~ b/01/hw01.thy~ new file mode 100644 index 0000000..24ae11e --- /dev/null +++ b/01/hw01.thy~ @@ -0,0 +1,45 @@ +theory hw01 + imports Main +begin + +fun listsum:: "int list \ int" where + "listsum [] = 0" +| "listsum (x # xs) = listsum xs + x" + +value "listsum [1,2,3] = 6" +value "listsum [] = 0" +value "listsum [1,-2,3] = 2" + +lemma listsum_filter_x: "listsum (filter (\x. x\0) l) = listsum l" + apply(induction l) + apply(auto) + done + +lemma listsum_append: "listsum (xs @ ys) = listsum xs + listsum ys" + apply(induction xs) + apply(auto) + done + +lemma listsum_rev: "listsum (rev xs) = listsum xs" + apply(induction xs) + apply(auto simp:listsum_append) + done + +lemma listsum_noneg: "listsum (filter (\x. x>0) l) \ listsum l" + apply(induction l) + apply(auto) + done + +fun flatten :: "'a list list \ 'a list" where + "flatten [] = []" +| "flatten (l#ls) = l @ flatten ls" + +value "flatten [[1,2,3],[2]] = [1,2,3,2::int]" +value "flatten [[1,2,3],[],[2]] = [1,2,3,2::int]" + +lemma "listsum (flatten xs) = listsum(map listsum xs)" + apply(induction xs) + apply(auto simp:listsum_append) + done + +end \ No newline at end of file diff --git a/02/ex02.pdf b/02/ex02.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5256b595846c135cd99600e12c406471baa84ddd GIT binary patch literal 134572 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ltree = Leaf 'a | Node "'a ltree" "'a ltree" + +fun inorder:: "'a ltree \ 'a list" where + "inorder (Leaf x) = [x]" +| "inorder (Node l r) = inorder l @ inorder r" + +lemma + "fold f [] s = s" + "fold f (x # xs) s = fold f xs (f x s)" + by auto + +fun fold_tree:: "('b \ 'a \ 'a) \ 'b ltree \ 'a \ 'a" where + "fold_tree f (Leaf b) a = f b a" +| "fold_tree f (Node l r) a = fold_tree f r (fold_tree f l a)" + +value "fold_tree +(\x y. x + y) +(Node (Leaf (1::nat)) (Node (Leaf 2) (Leaf 4))) +0 +" + +lemma "fold_tree f t s = fold f (inorder t) s" + apply(induction t arbitrary: s) + apply(auto) + done + +fun mirror:: "'a ltree \ 'a ltree" where + "mirror (Leaf a) = Leaf a" +| "mirror (Node l r) = Node (mirror r) (mirror l)" + +lemma "inorder (mirror t) = rev (inorder t)" + by (induction t) auto + +fun shuffles:: "'a list \ 'a list \ 'a list list" where + "shuffles [] ys = [ys]" +| "shuffles xs [] = [xs]" +| "shuffles (x#xs) (y#ys) = + map(op # x) (shuffles xs (y#ys)) + @ map(op # y) (shuffles (x#xs) ys)" + +lemma "l\set (shuffles xs ys) \ length l = length xs + length ys" + apply(induction xs ys arbitrary: l rule: shuffles.induct) + apply(auto) + done + +fun list_sum :: "nat list \ nat" where + "list_sum [] = 0" +| "list_sum (x#xs) = x + list_sum xs" + +definition "list_sum' xs = fold (op +) xs 0" + +lemma auxi: "fold (op +) xs a = list_sum xs + a" + apply(induction xs arbitrary: a) + apply(auto) + done + +lemma "list_sum xs = list_sum' xs" + unfolding list_sum'_def + using auxi[where a=0] + apply auto + done + +end \ No newline at end of file diff --git a/02/ex0202.thy b/02/ex0202.thy new file mode 100644 index 0000000..f2c6c76 --- /dev/null +++ b/02/ex0202.thy @@ -0,0 +1,7 @@ +theory ex0202 + imports Main +begin + + + +end \ No newline at end of file diff --git a/02/hw02.thy b/02/hw02.thy new file mode 100644 index 0000000..20b6038 --- /dev/null +++ b/02/hw02.thy @@ -0,0 +1,105 @@ +theory hw02 + imports Main +begin + +fun collect:: "'a \ ('a \ 'b) list \ 'b list" where + "collect x [] = []" +| "collect x ((k,y)#xs) = (if x = k then y # collect x xs else collect x xs)" + +definition ctest :: " (int * int) list" where "ctest = [ +(2 ,3 ),(2 ,5 ),(2 ,7 ),(2 ,9 ), +(3 ,2 ),(3 ,4 ),(3 ,5 ),(3 ,7 ),(3 ,8 ), +(4 ,3 ),(4 ,5 ),(4 ,7 ),(4 ,9 ), +(5 ,2 ),(5 ,3 ),(5 ,4 ),(5 ,6 ),(5 ,7 ),(5 ,8 ),(5 ,9 ), +(6 ,5 ),(6 ,7 ), +(7 ,2 ),(7 ,3 ),(7 ,4 ),(7 ,5 ),(7 ,6 ),(7 ,8 ),(7 ,9 ), +(8 ,3 ),(8 ,5 ),(8 ,7 ),(8 ,9 ), +(9 ,2 ),(9 ,4 ),(9 ,5 ),(9 ,7 ),(9 ,8 ) +]" +value "collect 3 ctest = [2 ,4 ,5 ,7, 8 ]" +value "collect 1 ctest = []" + +lemma "collect x ys = map snd (filter (\kv. fst kv = x) ys)" + apply(induction ys) + apply(auto) + done + +fun collect_tr:: "'b list \ 'a \ ('a * 'b) list \ 'b list" where + "collect_tr acc x [] = rev acc" +| "collect_tr acc x ((k,v)#ys) = (if (x = k) then collect_tr (v # acc) x ys else collect_tr acc x ys)" + +lemma collect_gen:"collect_tr acc x ys = rev acc @ (collect x ys)" + apply(induction ys arbitrary:acc) + apply(auto) + done + +lemma "collect_tr [] x ys = collect x ys" + apply(induction ys) + apply(auto simp:collect_gen) + done + + + + + + +datatype 'a ltree = Leaf 'a | Node "'a ltree" "'a ltree" + +fun lheight:: "'a ltree \ nat" where + "lheight (Leaf x) = 0" +| "lheight (Node x y) = max (lheight x) (lheight y) + 1" + +value "lheight (Node (Leaf (1::nat)) (Node (Leaf 2) (Leaf 4)))" + +fun num_leafs:: "'a ltree \ nat" where + "num_leafs (Leaf x) = 1" +| "num_leafs (Node x y) = num_leafs x + num_leafs y" + +value "num_leafs (Node (Leaf (1::nat)) (Node (Leaf 2) (Leaf 4)))" + +fun balanced:: "'a ltree \ bool" where + "balanced (Leaf a) = True" +| "balanced (Node x y) = (op &)((op &)(lheight x = lheight y)(balanced x))(balanced y)" + +value "balanced (Node (Node (Leaf a\<^sub>1) (Node (Leaf a\<^sub>1) (Leaf a\<^sub>1))) (Node (Leaf a\<^sub>1) (Node (Leaf a\<^sub>1) (Leaf a\<^sub>1))))" + + +lemma "balanced t \ num_leafs t = 2 ^ lheight t" + apply(induction t) + apply(auto) + done + + + + + + +fun denc :: "int \ int list \ int list" where + "denc a [] = []" +| "denc a (x#xs) = (x-a) # denc x xs" + +value "denc 0 [1,2,4,8]" +value "denc 0 [3,4,5]" +value "denc 0 [5]" +value "denc 0 []" + +fun ddec :: "int \ int list \ int list" where + "ddec a [] = []" +| "ddec (a::int) (x#xs) = (x + a) # (ddec (x + a) xs)" + +value "ddec 0 [1,1,2,4]" +value "ddec 0 [3,1,1]" +value "ddec 0 [5]" +value "ddec 0 []" + +value "ddec 5 (denc 4 [1,2,3])" + +lemma encdecgen: "ddec n (denc n l) = l" + apply(induction l arbitrary:n) + by auto + +lemma "ddec 0 (denc 0 l) = l" + apply(auto simp:encdecgen) + done + +end \ No newline at end of file diff --git a/02/hw02.thy~ b/02/hw02.thy~ new file mode 100644 index 0000000..37a51fa --- /dev/null +++ b/02/hw02.thy~ @@ -0,0 +1,105 @@ +theory hw02 + imports Main +begin + +fun collect:: "'a \ ('a \ 'b) list \ 'b list" where + "collect x [] = []" +| "collect x ((k,y)#xs) = (if x = k then y # collect x xs else collect x xs)" + +definition ctest :: " (int * int) list" where "ctest = [ +(2 ,3 ),(2 ,5 ),(2 ,7 ),(2 ,9 ), +(3 ,2 ),(3 ,4 ),(3 ,5 ),(3 ,7 ),(3 ,8 ), +(4 ,3 ),(4 ,5 ),(4 ,7 ),(4 ,9 ), +(5 ,2 ),(5 ,3 ),(5 ,4 ),(5 ,6 ),(5 ,7 ),(5 ,8 ),(5 ,9 ), +(6 ,5 ),(6 ,7 ), +(7 ,2 ),(7 ,3 ),(7 ,4 ),(7 ,5 ),(7 ,6 ),(7 ,8 ),(7 ,9 ), +(8 ,3 ),(8 ,5 ),(8 ,7 ),(8 ,9 ), +(9 ,2 ),(9 ,4 ),(9 ,5 ),(9 ,7 ),(9 ,8 ) +]" +value "collect 3 ctest = [2 ,4 ,5 ,7, 8 ]" +value "collect 1 ctest = []" + +lemma "collect x ys = map snd (filter (\kv. fst kv = x) ys)" + apply(induction ys) + apply(auto) + done + +fun collect_tr:: "'b list \ 'a \ ('a * 'b) list \ 'b list" where + "collect_tr acc x [] = rev acc" +| "collect_tr acc x ((k,v)#ys) = (if (x = k) then collect_tr (v # acc) x ys else collect_tr acc x ys)" + +lemma collect_gen:"collect_tr acc x ys = rev acc @ (collect x ys)" + apply(induction ys arbitrary:acc) + apply(auto) + done + +lemma "collect_tr [] x ys = collect x ys" + apply(induction ys) + apply(auto simp:collect_gen) + done + + + + + + +datatype 'a ltree = Leaf 'a | Node "'a ltree" "'a ltree" + +fun lheight:: "'a ltree \ nat" where + "lheight (Leaf x) = 0" +| "lheight (Node x y) = max (lheight x) (lheight y) + 1" + +value "lheight (Node (Leaf (1::nat)) (Node (Leaf 2) (Leaf 4)))" + +fun num_leafs:: "'a ltree \ nat" where + "num_leafs (Leaf x) = 1" +| "num_leafs (Node x y) = num_leafs x + num_leafs y" + +value "num_leafs (Node (Leaf (1::nat)) (Node (Leaf 2) (Leaf 4)))" + +fun balanced:: "'a ltree \ bool" where + "balanced (Leaf a) = True" +| "balanced (Node x y) = (if (lheight x = lheight y) then (op &)(balanced x)(balanced y) else False)" + +value "balanced (Node (Node (Leaf a\<^sub>1) (Node (Leaf a\<^sub>1) (Leaf a\<^sub>1))) (Node (Leaf a\<^sub>1) (Node (Leaf a\<^sub>1) (Leaf a\<^sub>1))))" + + +lemma "balanced t \ num_leafs t = 2 ^ lheight t" + apply(induction t) + apply(auto) + done + + + + + + +fun denc :: "int \ int list \ int list" where + "denc a [] = []" +| "denc a (x#xs) = (x-a) # denc x xs" + +value "denc 0 [1,2,4,8]" +value "denc 0 [3,4,5]" +value "denc 0 [5]" +value "denc 0 []" + +fun ddec :: "int \ int list \ int list" where + "ddec a [] = []" +| "ddec (a::int) (x#xs) = (x + a) # (ddec (x + a) xs)" + +value "ddec 0 [1,1,2,4]" +value "ddec 0 [3,1,1]" +value "ddec 0 [5]" +value "ddec 0 []" + +value "ddec 5 (denc 4 [1,2,3])" + +lemma encdecgen: "ddec n (denc n l) = l" + apply(induction l arbitrary:n) + by auto + +lemma "ddec 0 (denc 0 l) = l" + apply(auto simp:encdecgen) + done + +end \ No newline at end of file diff --git a/03/ex03.pdf b/03/ex03.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2dc5a71f20ab6a45bad3ecf485f442b9b17afd63 GIT binary patch literal 118576 zcma&NQ>-vRv!=Uj+qP}nwr$&9>sz*M+qP}nwmJWtIs0nwWTul&s{5k*=1uykdaFql zM8s$r=~$si=a)y;p;!qR2<(llpm=zo=w(dp%v~%9n3&iY2>!nTMK5M)<6`PWKrd!v z=wd2jYHV*}3dP3<Jw`@FXHji}Mgiz}q8NH*=Dh)kT8a zwe*h!z8OSwTCzB<_}ZhC`R7vCAPt35I5p|Df}Ti%#tpqIYG+SK(gm4xLEapa_s`4q zb%#1vjK|P3@#)+O)sz#NfhogKEA!k6(>J`%%oA&CL?+ThH-#4N+J=r9^{!7@tJ_S|G?c2bvZ2p|?W48KAZD=(CeX 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"get _ _ = undefined " + +fun put:: "'a tree \ path \ 'a tree \ 'a tree" where + "put x [] y = y" +| "put Leaf _ _ = Leaf" +| "put (Node l a r) (x# xs) y = (if x = L then (Node (put l xs y) a r) else (Node l a (put r xs y)))" + +value "put (Node (Node (Leaf) b (Leaf)) d Leaf) [R,R] (Node Leaf q Leaf)" +value "valid (Node Leaf a Leaf) [R]" + +lemma put_invalid[simp]: "\valid t p \ put t p s = t" + apply(induction p rule:valid.induct) + apply(auto) + done + +lemma get_put[simp]: "put t p (get t p) = t" + apply(induction p rule:get.induct) + apply(auto) + done + + +lemma put_put[simp]: "valid t p \ put (put t p s) p s' = put t p s'" + apply(induction p arbitrary:s' rule: valid.induct) + apply(auto) + done + +lemma put_get[simp]: "valid t p \ get (put t p s) p = s" + apply(induction t p rule:get.induct) + apply(auto) + done + +lemma valid_put[simp]: "valid t p \ valid (put t p s) p" + apply(induction t p rule:valid.induct) + apply(auto) + done + +lemma valid_append[simp]: "valid t (p@q)\ valid t p \ valid (get t p) q" + apply(induction t p rule:valid.induct) + apply auto + done + +lemma get_append[simp]: "valid t p \ get t (p@q) = get (get t p) q" + apply(induction t p rule:get.induct) + apply auto + done + +lemma put_append[simp]: "put t (p@q) s = put t p (put (get t p) q s)" + apply(induction t p rule:get.induct) + apply auto + done + + +(*Part 2 - Incomplete, bst_remdups done, sublist done, first proof done with a generalization which I couldn't prove*) + + + +fun ins :: "'a::linorder \ 'a tree \ 'a tree" where +"ins x Leaf = Node Leaf x Leaf" | +"ins x (Node l a r) = + (if x < a then Node (ins x l) a r else + if x > a then Node l a (ins x r) + else Node l a r)" + +fun bst_remdups_aux:: "'a::linorder tree \ 'a list \ 'a list" where + "bst_remdups_aux t [] = inorder t" +| "bst_remdups_aux t (x # xs) = bst_remdups_aux (ins x t) xs" + +definition "bst_remdups xs \ bst_remdups_aux Leaf xs" + +value "bst_remdups [1::int,2,3,4,1,5,5,6,6,10,7]" + +value "bst_remdups_aux (Node Leaf (7::nat) (Node Leaf (1::nat) Leaf)) [2::nat]" + +lemma bst_gen[simp]:"set (bst_remdups_aux t s) = set(bst_remdups_aux t []) \ set s" + sorry + +value "filter (\x. x \ (2::nat)) [2::nat,3,4]" + +lemma "set (bst_remdups xs) = set xs" + unfolding bst_remdups_def + apply(induction xs) + apply(auto) + done + +fun remove:: "'a \ 'a list \ 'a list" where + "remove a [] = []" +| "remove a (x #xs) = (if (a = x) then xs else x #(remove a xs))" + +fun sublist:: "'a list \ 'a list \ bool" where + "sublist [] ys = True" +| "sublist xs [] = False" +| "sublist (x#xs) ys = (if x\ set ys then sublist xs (remove x ys) else False)" + + +end \ No newline at end of file diff --git a/03/hw03.thy~ b/03/hw03.thy~ new file mode 100644 index 0000000..6920130 --- /dev/null +++ b/03/hw03.thy~ @@ -0,0 +1,125 @@ +theory hw03 + imports "~~/src/HOL/Library/Tree" Main +begin + +declare Let_def [simp] + +datatype direction = L|R +type_synonym path = "direction list" + +fun valid:: "'a tree \ path \ bool" where + "valid _ [] = True" +| "valid Leaf _ = False" +| "valid (Node ltr a rtr) (x#xs) = (if x = L then (valid ltr xs) else (valid rtr xs))" + +fun get:: "'a tree \ path \ 'a tree" where + "get x [] = x" +| "get (Node ltr a rtr) (x# xs) = (if x = L then get ltr xs else get rtr xs)" +| "get _ _ = undefined " + +fun put:: "'a tree \ path \ 'a tree \ 'a tree" where + "put x [] y = y" +| "put Leaf _ _ = Leaf" +| "put (Node l a r) (x# xs) y = (if x = L then (Node (put l xs y) a r) else (Node l a (put r xs y)))" + +value "put (Node (Node (Leaf) b (Leaf)) d Leaf) [R,R] (Node Leaf q Leaf)" +value "valid (Node Leaf a Leaf) [R]" + +lemma put_invalid[simp]: "\valid t p \ put t p s = t" + apply(induction p rule:valid.induct) + apply(auto) + done + +lemma get_put[simp]: "put t p (get t p) = t" + apply(induction p rule:get.induct) + apply(auto) + done + + +lemma put_put[simp]: "valid t p \ put (put t p s) p s' = put t p s'" + apply(induction p arbitrary:s' rule: valid.induct) + apply(auto) + done + +lemma put_get[simp]: "valid t p \ get (put t p s) p = s" + apply(induction t p rule:get.induct) + apply(auto) + done + +lemma valid_put[simp]: "valid t p \ valid (put t p s) p" + apply(induction t p rule:valid.induct) + apply(auto) + done + +lemma valid_append[simp]: "valid t (p@q)\ valid t p \ valid (get t p) q" + apply(induction t p rule:valid.induct) + apply auto + done + +lemma get_append[simp]: "valid t p \ get t (p@q) = get (get t p) q" + apply(induction t p rule:get.induct) + apply auto + done + +lemma put_append[simp]: "put t (p@q) s = put t p (put (get t p) q s)" + apply(induction t p rule:get.induct) + apply auto + done + + +(*Part 2*) + + + +fun ins :: "'a::linorder \ 'a tree \ 'a tree" where +"ins x Leaf = Node Leaf x Leaf" | +"ins x (Node l a r) = + (if x < a then Node (ins x l) a r else + if x > a then Node l a (ins x r) + else Node l a r)" + +fun bst_remdups_aux:: "'a::linorder tree \ 'a list \ 'a list" where + "bst_remdups_aux t [] = inorder t" +| "bst_remdups_aux t (x # xs) = bst_remdups_aux (ins x t) xs" + +definition "bst_remdups xs \ bst_remdups_aux Leaf xs" + +value "bst_remdups [1::int,2,3,4,1,5,5,6,6,10]" + +fun unique::"'a list \ 'a list" where + "unique [] = []" +| "unique (x#xs) = (x # unique(filter(\x'. x \ x') xs))" + +value "bst_remdups_aux (Node Leaf (7::nat) (Node Leaf (1::nat) Leaf)) [2::nat]" + +lemma bst_gen[simp]:"bst_remdups xs = sort (unique xs)" + sorry + +value "filter (\x. x \ (2::nat)) [2::nat,3,4]" + +lemma "set (bst_remdups xs) = set xs" + apply(induction xs rule:bst_remdups_aux.induct) + apply(auto) + done + +lemma "distinct (bst_remdups xs)" + unfolding bst_remdups_def + apply(induction xs) + apply(auto simp:bst_gen) + +fun remove:: "'a \ 'a list \ 'a list" where + "remove a [] = []" +| "remove a (x #xs) = (if (a = x) then xs else x #(remove a xs))" + +fun sublist:: "'a list \ 'a list \ bool" where + "sublist [] ys = True" +| "sublist xs [] = False" +| "sublist (x#xs) ys = (if x\ set ys then sublist xs (remove x ys) else False)" + +lemma "sublist (bst_remdups xs) xs" + unfolding bst_remdups_def + apply(induction xs) + apply(auto simp:bst_gen) + done + +end \ No newline at end of file diff --git a/04/BST_Demo.thy b/04/BST_Demo.thy new file mode 100644 index 0000000..e60adf8 --- /dev/null +++ b/04/BST_Demo.thy @@ -0,0 +1,73 @@ +theory BST_Demo +imports "~~/src/HOL/Library/Tree" +begin + +(* useful most of the time: *) +declare Let_def [simp] + +section "BST Search and Insertion" + +fun isin :: "('a::linorder) tree \ 'a \ bool" where +"isin Leaf x = False" | +"isin (Node l a r) x = + (if x < a then isin l x else + if x > a then isin r x + else True)" + +fun ins :: "'a::linorder \ 'a tree \ 'a tree" where +"ins x Leaf = Node Leaf x Leaf" | +"ins x (Node l a r) = + (if x < a then Node (ins x l) a r else + if x > a then Node l a (ins x r) + else Node l a r)" + +subsection "Functional Correctness" + +lemma set_tree_isin: "bst t \ isin t x = (x \ set_tree t)" +apply(induction t) +apply auto +done + +lemma set_tree_ins: "set_tree (ins x t) = {x} \ set_tree t" +apply(induction t) +apply auto +done + +subsection "Preservation of Invariant" + +lemma bst_ins: "bst t \ bst (ins x t)" +apply(induction t) +apply (auto simp: set_tree_ins) +done + + +section "BST Deletion" + +fun split_min :: "'a tree \ 'a * 'a tree" where +"split_min (Node l a r) = + (if l = Leaf then (a,r) + else let (x,l') = split_min l + in (x, Node l' a r))" + +fun delete :: "'a::linorder \ 'a tree \ 'a tree" where +"delete x Leaf = Leaf" | +"delete x (Node l a r) = + (if x < a then Node (delete x l) a r else + if x > a then Node l a (delete x r) + else if r = Leaf then l else let (a',r') = split_min r in Node l a' r')" + +(* A proof attempt *) + +lemma "split_min t = (x,t') \ set_tree t' = set_tree t - {x}" +oops + +(* The final proof (needs more than auto!): *) + +lemma "\ split_min t = (x,t'); bst t; t \ Leaf \ \ + set_tree t' = set_tree t - {x} \ x \ set_tree t" +apply(induction t arbitrary: x t') + apply simp +apply (force split: if_split_asm prod.splits) +done + +end diff --git a/04/Homework04.thy b/04/Homework04.thy new file mode 100644 index 0000000..ed1f7df --- /dev/null +++ b/04/Homework04.thy @@ -0,0 +1,71 @@ +theory Homework04 + imports "BST_Demo" +begin + + +datatype 'a rtree = Leaf|Node "'a rtree" nat 'a "'a rtree" + +fun num_nodes :: "'a rtree \ nat" where + "num_nodes Leaf = 0"| + "num_nodes (Node l a b r) = num_nodes l + num_nodes r + 1" + + +fun rbst :: "'a::linorder rtree \ bool" where + "rbst Leaf = True"| + "rbst (Node l a b r) = + ((rbst l)\ + (\x\set_rtree l. x + (a = num_nodes l)\ + (\x\set_rtree r. b + (rbst r))" + + +value "rbst (Node (Node Leaf (0::nat) (1::nat) Leaf) (1::nat) 2 (Node Leaf (0::nat) 3 Leaf))" +value "set_rtree(Node (Node Leaf (0::nat) (1::nat) Leaf) (1::nat) 2 (Node Leaf (0::nat) 3 Leaf))" + +fun rins :: "'a::linorder \ 'a rtree \ 'a rtree" where + "rins x Leaf = (Node Leaf 0 x Leaf)"| + "rins x (Node l a b r) = (if xset_rtree t) \num_nodes (rins x t) = Suc(num_nodes t)" + apply(induction t arbitrary:x) + apply auto + done + +lemma aux[simp]: "rbst(Node l a b r) \ (num_nodes l = a) " + apply auto + done + + +lemma "x\set_rtree t \ rbst t \ rbst (rins x t)" + apply(induction t arbitrary: x rule:rbst.induct) + apply auto + done + +fun risin:: "'a::linorder \ 'a rtree \ bool" where + "risin _ Leaf = False"| + "risin x (Node l a b r) = ((x=b)\(risin x l)\(risin x r))" + +lemma "rbst t \ risin x t \ x\set_rtree t" + apply(induction t) + apply auto + done + +fun inorder :: "'a rtree \ 'a list" where + "inorder Leaf = []"| + "inorder (Node l a b r) = inorder l @ b # inorder r" + +fun rank:: "'a::linorder \ _" where + + + + + +end diff --git a/04/ex04.pdf b/04/ex04.pdf new file mode 100644 index 0000000000000000000000000000000000000000..969ac94754c1d04e6e81ae27cb7243ed21d2281f GIT binary patch literal 121511 zcma&MLy#~`(52h<+qP}nwr$(CZQHhO+qP}nJ>P%h&T3{cxyr1ls9I&kQ|Ba!ys#(@ z11$>_$^7!jIur{&J-(fxB@{O|6rHq*t(mhqJ|iO&BmVyvP;{ae*3Kr5_;jMy2F@nJ 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list" where + "in_range Leaf x y = []" +| "in_range (Node l x r) u v = + (if x > u then in_range l u v else []) +@ (if u \ x \ x \ v then [x] else []) +@ (if v > x then in_range r u v else [])" + +lemma "bst t \ set (in_range t u v) = {x\set_tree t. u\x \ x\v}" + apply (induction t) + apply auto + done + + +text \Show that your list is actually in-order\ +lemma "bst t \ in_range t u v = filter (\x. u\x \ x\v) (inorder t)" + apply (induction t) + apply simp + apply (clarsimp) + apply safe + + +end diff --git a/04/hw04.thy b/04/hw04.thy new file mode 100644 index 0000000..ebe9a68 --- /dev/null +++ b/04/hw04.thy @@ -0,0 +1,101 @@ +theory hw04 + imports Main "~~/src/HOL/Library/Tree" +begin + +declare Let_def [simp] + +datatype 'a rtree = Leaf | Node "'a rtree" nat 'a "'a rtree" + +fun num_nodes:: "'a rtree \ nat" where + "num_nodes Leaf = 0" +| "num_nodes (Node l n b r) = 1 + num_nodes l + num_nodes r" + +fun rbst:: "'a::linorder rtree \ bool" where + "rbst Leaf = True" +| "rbst (Node l a b r) = (rbst l \ (\x\set_rtree l. (b > x)) \ (\x\set_rtree r. (b < x)) \ (a = num_nodes l) \ rbst r)" + + +value "rbst (Node (Node Leaf (0::nat) (1::nat) Leaf) (1::nat) 2 (Node Leaf (0::nat) 3 Leaf))" + +fun rins:: "'a::linorder \ 'a rtree \ 'a rtree" where + "rins x Leaf = (Node Leaf 0 x Leaf)" +| "rins x (Node l n b r) = + (if (x < b) then (Node (rins x l) (Suc n) b r) else (Node l n b (rins x r)))" + +value "rins (4::nat) (Node (Node Leaf (0::nat) (1::nat) Leaf) (1::nat) 2 (Node Leaf (0::nat) 3 Leaf))" + +lemma rins_set[simp]: "set_rtree (rins x t) = insert x (set_rtree t)" + apply(induction t arbitrary:x) + apply(auto) + done + + +lemma aux1[simp]: "(x\set_rtree t) \ num_nodes (rins x t) = Suc(num_nodes t)" + apply(induction t arbitrary: x) + apply(auto) + done + +lemma aux2[simp]: "rbst(Node l a b r) \ (num_nodes l = a) " + apply(auto) + done + +value "rbst (rtree.Node rtree.Leaf 0 (1::nat) (rtree.Node rtree.Leaf 0 (1::nat) rtree.Leaf))" + + + +lemma "x\set_rtree t \ rbst t \ rbst (rins x t)" + apply(induction t arbitrary: x rule:rbst.induct) + apply(auto) + done + +fun risin :: "'a::linorder \ 'a rtree \ bool" where + "risin x Leaf = False" +| "risin x (Node l a b r) = + ( + if (b < x) then + (risin x r) + else if (b > x) then + (risin x l) + else + True + )" + +value "risin (3::nat) (Node (Node Leaf (0::nat) (1::nat) Leaf) (1::nat) 2 (Node Leaf (0::nat) 3 Leaf))" + +lemma "rbst t \ risin x t \ x\set_rtree t" + apply(induction t) + apply(auto) + done + +fun inorder:: "'a rtree \ 'a list" where + "inorder Leaf = []" +| "inorder (Node l a b r) = inorder l @ [b] @ inorder r" + +fun rank::"'a::linorder \ _" where + "rank x Leaf = undefined" +| "rank x (Node l a b r) = (if (x = b) then a else if (x < b) then (rank x l) else (a + 1 + rank x r))" + +definition "at_index i l x \ i l!i=x" + +lemma aux3[simp]:"rbst t \ num_nodes t = length (inorder t)" + apply(induction t) + apply(auto) + done + + +lemma "rbst t \ x\set_rtree t \ at_index (rank x t) (inorder t) x" + unfolding at_index_def + apply(induction t rule:inorder.induct) + apply(auto) + sorry + +fun select :: "nat \ 'a::linorder rtree \ 'a" where + "select n Leaf = undefined" +| "select n (Node l a b r) = (if (n = a) then b else if n < a then select n l else select (n-a-1) r)" + +lemma select_correct: "rbst t \ i select i t = inorder t ! i" + apply(induction i t rule:select.induct) + apply(auto) + sorry + +end \ No newline at end of file diff --git a/04/hw04.thy~ b/04/hw04.thy~ new file mode 100644 index 0000000..714958e --- /dev/null +++ b/04/hw04.thy~ @@ -0,0 +1,101 @@ +theory hw04 + imports Main "~~/src/HOL/Library/Tree" +begin + +declare Let_def [simp] + +datatype 'a rtree = Leaf | Node "'a rtree" nat 'a "'a rtree" + +fun num_nodes:: "'a rtree \ nat" where + "num_nodes Leaf = 0" +| "num_nodes (Node l n b r) = 1 + num_nodes l + num_nodes r" + +fun rbst:: "'a::linorder rtree \ bool" where + "rbst Leaf = True" +| "rbst (Node l a b r) = (rbst l \ (\x\set_rtree l. (b > x)) \ (\x\set_rtree r. (b < x)) \ (a = num_nodes l) \ rbst r)" + + +value "rbst (Node (Node Leaf (0::nat) (1::nat) Leaf) (1::nat) 2 (Node Leaf (0::nat) 3 Leaf))" + +fun rins:: "'a::linorder \ 'a rtree \ 'a rtree" where + "rins x Leaf = (Node Leaf 0 x Leaf)" +| "rins x (Node l n b r) = + (if (x < b) then (Node (rins x l) (Suc n) b r) else (Node l n b (rins x r)))" + +value "rins (4::nat) (Node (Node Leaf (0::nat) (1::nat) Leaf) (1::nat) 2 (Node Leaf (0::nat) 3 Leaf))" + +lemma rins_set[simp]: "set_rtree (rins x t) = insert x (set_rtree t)" + apply(induction t arbitrary:x) + apply(auto) + done + + +lemma aux1[simp]: "(x\set_rtree t) \ num_nodes (rins x t) = Suc(num_nodes t)" + apply(induction t arbitrary: x) + apply(auto) + done + +lemma aux2[simp]: "rbst(Node l a b r) \ (num_nodes l = a) " + apply(auto) + done + +value "rbst (rtree.Node rtree.Leaf 0 (1::nat) (rtree.Node rtree.Leaf 0 (1::nat) rtree.Leaf))" + + + +lemma "x\set_rtree t \ rbst t \ rbst (rins x t)" + apply(induction t arbitrary: x rule:rbst.induct) + apply(auto) + done + +fun risin :: "'a::linorder \ 'a rtree \ bool" where + "risin x Leaf = False" +| "risin x (Node l a b r) = + ( + if (b < x) then + (risin x r) + else if (b > x) then + (risin x l) + else + True + )" + +value "risin (3::nat) (Node (Node Leaf (0::nat) (1::nat) Leaf) (1::nat) 2 (Node Leaf (0::nat) 3 Leaf))" + +lemma "rbst t \ risin x t \ x\set_rtree t" + apply(induction t) + apply(auto) + done + +fun inorder:: "'a rtree \ 'a list" where + "inorder Leaf = []" +| "inorder (Node l a b r) = inorder l @ [b] @ inorder r" + +fun rank::"'a::linorder \ _" where + "rank x Leaf = undefined" +| "rank x (Node l a b r) = (if (x = b) then a else if (x < b) then (rank x l) else (a + 1 + rank x r))" + +definition "at_index i l x \ i l!i=x" + +lemma aux3[simp]:"rbst t \ num_nodes t = length (inorder t)" + apply(induction t) + apply(auto) + done + + +lemma "rbst t \ x\set_rtree t \ at_index (rank x t) (inorder t) x" + unfolding at_index_def + apply(induction t rule:inorder.induct) + apply(auto) + sorry + +fun select :: "nat \ 'a::linorder rtree \ 'a" where + "select n Leaf = undefined" +| "select n (Node l a b r) = (if (n = a) then b else if n < a then select n l else select (n-a-1) r)" + +lemma select_correct: "rbst t \ i select i t = inorder t ! i" + apply(induction t rule:inorder.induct) + apply(auto) + + +end \ No newline at end of file diff --git a/05/ex05.pdf b/05/ex05.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7b9ba4485fe2ce128767ab9016d1725f56d69b04 GIT binary patch literal 169299 zcma%?Q?MXGx22D5+qP}nwr$(CZQJT&^|5W+Hm2{)+^2sc=H|nST~YN`nHk?&xsy~u zM2wb^junb@Zt3qD6e|G(fxVFx6b}y+y^N`yxr+q>6AKGF!T%mm^kSAaE~ZWd^kO!K zE~X-;#`Y$rP<(t)&Mr=-hPF^18^@Zq_FE%}KXv+o$I_k^c0<00Np?vQy z1rf#pormWT+llB&=#m0~KhHB)SNF2i<8s^ZwtW)bpG@4`+>cYtlN23rMSJw(iJ!mM z$Mfm+o;Y_ArHT{5ld1_NG6It%VV0Iblcsmh9-PvL4Ej?g5UZy13dH%-rbUDJlLzcy zxvR4CGf8*(%(&SM){ZJ+r+4xK`Z$>m`kkT2HZ!r2sveGlOd`X{>-6cnpg?S>WEsxm 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z|66h~Xh)L!vlLbdOm?Of*;;=I-X+ZL0tG4|Mvi0-YEJKXfbkG-9jYIfh zPCET=PO8ChTUjYy2$Q#ex+KTnRuDkYL!?aR-8u^-h?PLCuhUcVCi#AKU%u1Daz|5l r>2UdXd$Qvposu!{`QMl9<7?yX8*l-j=z;Qz3Q9nFLBWTH+VuYf$I?TB literal 0 HcmV?d00001 diff --git a/05/hw05.thy b/05/hw05.thy new file mode 100644 index 0000000..f1d08e1 --- /dev/null +++ b/05/hw05.thy @@ -0,0 +1,49 @@ +theory hw05 + imports + Complex_Main + "HOL-Library.Tree" +begin + +value "(0::nat) div 0" + +lemma + assumes "n\0" + shows "\ys zs. length ys = length xs div n \ xs=ys@zs" +proof (intro exI) + let ?n = "length xs div n" + let ?ys = "take ?n xs" + let ?zs = "drop ?n xs" + show "length ?ys = length xs div n \ xs = ?ys @ ?zs" by(simp add:min.absorb2) +qed + + +fun a :: "nat \ int" where + "a 0 = 0" +| "a (Suc n) = a n ^ 2 + 1" + +thm power_mono[where n = 2] + +find_theorems "(_-_)^2" +find_theorems "(_ + _) > _" + +lemma "a n \ 2 ^ (2 ^ n) - 1" +proof(induction n) + case 0 thus ?case by simp +next + case (Suc n) + assume IH: "a n \ 2 ^ 2 ^ n - 1" + -- \Refer to the induction hypothesis by name \IH\ or \Suc.IH\\ + show "a (Suc n) \ 2 ^ 2 ^ Suc n - 1" + proof - + from IH have azer:"0 \ a n" by (smt a.elims power2_less_eq_zero_iff zero_eq_power2) + + from IH have "a (Suc n) = (a n)^2 +1" by simp + also have "(a n)^2 \ (2 ^ 2 ^ n - 1) ^ 2" using power_mono[where n = 2] IH azer by blast + also have "... \ (2 ^ 2 ^ (n + 1)) + 1 - 2*2^(2^(n))" by (simp add: power2_diff power_even_eq) + also have "... \ 2 ^ 2 ^ Suc n - 2 " by (smt Suc.IH Suc_eq_plus1 a.elims power.simps(1) power2_less_eq_zero_iff power_one_right zero_eq_power2) + finally show ?thesis by auto + + qed +qed + +end \ No newline at end of file diff --git a/05/hw05.thy~ b/05/hw05.thy~ new file mode 100644 index 0000000..f85a649 --- /dev/null +++ b/05/hw05.thy~ @@ -0,0 +1,44 @@ +theory hw05 + imports + Complex_Main + "HOL-Library.Tree" +begin + +lemma + assumes "n\0" + shows "\ys zs. length ys = length xs div n \ xs=ys@zs" + apply(simp) + apply(induction xs) + apply(auto) + oops + +fun a :: "nat \ int" where + "a 0 = 0" +| "a (Suc n) = a n ^ 2 + 1" + +thm power_mono[where n = 2] + +find_theorems "(_-_)^2" +find_theorems "(_ + _) > _" + +lemma "a n \ 2 ^ (2 ^ n) - 1" +proof(induction n) + case 0 thus ?case by simp +next + case (Suc n) + assume IH: "a n \ 2 ^ 2 ^ n - 1" + -- \Refer to the induction hypothesis by name \IH\ or \Suc.IH\\ + show "a (Suc n) \ 2 ^ 2 ^ Suc n - 1" + proof - + from IH have azer:"0 \ a n" by (smt a.elims power2_less_eq_zero_iff zero_eq_power2) + + from IH have "a (Suc n) = (a n)^2 +1" by simp + also have "(a n)^2 \ (2 ^ 2 ^ n - 1) ^ 2" using power_mono[where n = 2] IH azer by blast + also have "... \ (2 ^ 2 ^ (n + 1)) + 1 - 2*2^(2^(n))" by (simp add: power2_diff power_even_eq) + also have "... \ 2 ^ 2 ^ Suc n - 2 " by (smt Suc.IH Suc_eq_plus1 a.elims power.simps(1) power2_less_eq_zero_iff power_one_right zero_eq_power2) + finally show ?thesis by auto + + qed +qed + +end \ No newline at end of file diff --git a/05/tmpl05.thy b/05/tmpl05.thy new file mode 100644 index 0000000..4393d77 --- /dev/null +++ b/05/tmpl05.thy @@ -0,0 +1,183 @@ +(*<*) +theory tmpl05 + imports + Complex_Main + "HOL-Library.Tree" +begin +(*>*) + + +text {* \ExerciseSheet{5}{11.~5.~2017} *} + +text \ + \<^item> Import \Complex_Main\ and \HOL-Library.Tree\ + \<^item> For this exercise sheet (and Homework 1), you are not allowed to use sledgehammer! + Proofs using the \smt, metis, meson, or moura\ methods are forbidden! +\ + +text \ + \Exercise{Bounding power-of-two by factorial} + Prove that, for all natural numbers $n>3$, we have $2^n < n!$. + We have already prepared the proof skeleton for you. +\ +lemma exp_fact_estimate: "n>3 \ (2::nat)^n < fact n" +proof (induction n) + case 0 then show ?case by auto +next + case (Suc n) + assume IH: "3 < n \ (2::nat) ^ n < fact n" + assume PREM: "3 < Suc n" + show "(2::nat) ^ Suc n < fact (Suc n)" + text \Fill in a proof here. Hint: Start with a case distinction + whether \n>3\ or \n=3\. \ + sorry +qed + +text \ + \vspace{1em} + {\bfseries Warning!} + Make sure that your numerals have the right type, otherwise + proofs will not work! To check the type of a numeral, hover the mouse over + it with pressed CTRL (Mac: CMD) key. Example: +\ +lemma "2^n \ 2^Suc n" + apply auto oops -- \Leaves the subgoal \2 ^ n \ 2 * 2 ^ n\\ + text \You will find out that the numeral \2\ has type @{typ 'a}, + for which you do not have any ordering laws. So you have to + manually restrict the numeral's type to, e.g., @{typ nat}.\ +lemma "(2::nat)^n \ 2^Suc n" by simp -- \Note: Type inference will + infer \nat\ for the unannotated numeral, too. Use CTRL+hover to double check!\ + +text \ + \vspace{1em} +\ + +text \\Exercise{Sum Squared is Sum of Cubes} + \<^item> Define a recursive function $sumto~f~n = \sum_{i=0\ldots n} f(i)$. + \<^item> Show that $\left(\sum_{i=0\ldots n}i\right)^2 = \sum_{i=0\ldots n} i^3$. +\ + + +fun sumto :: "(nat \ nat) \ nat \ nat" +where +"sumto f 0 = 0" + +text \You may need the following lemma:\ +lemma sum_of_naturals: "2 * sumto (\x. x) n = n * Suc n" + (*by (induction n) auto*) oops + +lemma "sumto (\x. x) n ^ 2 = sumto (\x. x^3) n" +proof (induct n) + case 0 show ?case by simp +next + case (Suc n) + assume IH: "(sumto (\x. x) n)\<^sup>2 = sumto (\x. x ^ 3) n" + note [simp] = algebra_simps -- \Extend the simpset only in this block\ + show "(sumto (\x. x) (Suc n))\<^sup>2 = sumto (\x. x ^ 3) (Suc n)" + text \Insert a proof here\ + sorry +qed + +text \ + \Exercise{Paths in Graphs} + A graph is described by its adjacency matrix, i.e., \G :: 'a \ 'a \ bool\. + + Define a predicate \path G u p v\ that is true if \p\ is a path from + \u\ to \v\, i.e., \p\ is a list of nodes, not including \u\, such that + the nodes on the path are connected with edges. + In other words, \path G u (p\<^sub>1\p\<^sub>n) v\, iff \G u p\<^sub>1\, \G p\<^sub>i p\<^sub>i\<^sub>+\<^sub>1\, + and \p\<^sub>n = v\. For the empty path (\n=0\), we have \u=v\. +\ + +fun path :: "('a \ 'a \ bool) \ 'a \ 'a list \ 'a \ bool" + where + "path _ _ _ _ \ False" + +text \Test cases\ +definition "nat_graph x y \ y=Suc x" +value \path nat_graph 2 [] 2\ +value \path nat_graph 2 [3,4,5] 5\ +value \\ path nat_graph 3 [3,4,5] 6\ +value \\ path nat_graph 2 [3,4,5] 6\ + +text \Show the following lemma, that decomposes paths. Register it as simp-lemma.\ +lemma path_append[simp]: "path G u (p1@p2) v \ (\w. path G u p1 w \ path G w p2 v)" + oops + +text \ + Show that, for a non-distinct path from \u\ to \v\, + we find a longer non-distinct path from \u\ to \v\. + Note: This can be seen as a simple pumping-lemma, + allowing to pump the length of the path. + + Hint: Theorem @{thm [source] not_distinct_decomp}. +\ +lemma pump_nondistinct_path: + assumes P: "path G u p v" + assumes ND: "\distinct p" + shows "\p'. length p' > length p \ \distinct p' \ path G u p' v" + oops + + +text \ + \NumHomework{Split Lists}{May 18} + Recall: Use Isar where appropriate, proofs using + \metis, smt, meson, or moura\ (as generated by sledgehammer) are forbidden! + + Show that every list can be split into a prefix and a suffix, + such that the length of the prefix is \1/n\ of the original lists's length. + +\ + +lemma + assumes "n\0" -- \Note: This assumption is actually not needed, + as @{lemma "n div 0 = 0" by auto}, so don't be puzzled if you do + not use it at all in your proof.\ + shows "\ys zs. length ys = length xs div n \ xs=ys@zs" + oops + +text \ + \NumHomework{Estimate Recursion Equation}{May 18} + + (Sledgehammer allowed again) + + Show that the function defined by \a 0 = 0\ and \a (n+1) = (a n)\<^sup>2 + 1\ + is bounded by the double-exponential function \2^(2^n)\ +\ + +fun a :: "nat \ int" where +"a 0 = 0" | +"a (Suc n) = a n ^ 2 + 1" + +text \ + We have given you a proof skeleton, setting up the induction. + To complete your proof, you should come up with a chain of inequations. + You may try to solve the intermediate steps with sledgehammer. + + Hint: It is a bit tricky to get the approximation right. + We strongly recommend to sketch the inequations on paper first. + + Hint: Have a look at the lemma @{thm [source] power_mono}, in particular its + instance for squares: +\ + +thm power_mono[where n=2] + +lemma "a n \ 2 ^ (2 ^ n) - 1" +proof(induction n) + case 0 thus ?case by simp +next + case (Suc n) + assume IH: "a n \ 2 ^ 2 ^ n - 1" + -- \Refer to the induction hypothesis by name \IH\ or \Suc.IH\\ + show "a (Suc n) \ 2 ^ 2 ^ Suc n - 1" + proof - + text \Insert your proof here\ + show ?thesis sorry + qed +qed + + +(*<*) +end +(*>*) diff --git a/06/ex06.pdf b/06/ex06.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7ea780e42886de7fba7b98ed5f7af18339f726c1 GIT binary patch literal 134272 zcma&MQ;aZN5T@C-ZQHhO+qTWGZQHhO+qP}n-Sf|8XK!Ybol2^5PU^n$)cY2xf`}L` zBONOg>HPA@Iut7b1A)Df6%-E-6upe8ow(TO z(n8Y1Bm%(svE+Z9-f%i9%E^saDtZvp`DI;YcXp*sb4@O;CmPHZAUTz(kx}<5&Rf#^FXEd5=qR4uWAA#5j-&HhrJ6v^Hg*jQrYaJ?F z6IyR6a9KW^Q6_>WFX#BuN#^sXbgn&AUz))~4rbbeS!n!-dzz(VzDtVFBkXLUbx8`H6 zC;x6|hfc#-J%NjnkkZY9agBVGf%zmY9|xvNXNar(=32Li!UuA0P#1Xne5V#MSArEO zv)PJ?Qa3k*6pd7df86Xskj=DJOj`lFwaDxn*`V#wZF}@vzqssi0)%falN~NRSc4%^ z?|uF9)}{(4hIZ&yJ$J#(R)R$hV&*7F)uc`pXv+_SfvmGJ!oC4eF#?EK?)#75^U)l5 zK{J=DrEN5ZFd395s#s|;8(I8sJZ0s z8m1_KMmGd?#1~}+_INxrTQ+>x?O*enxVi(4jZta{10x~;1mVEQXJ88&+uEwG?Dw6T z-QlFGjV80;%KAwmU<;8OcsAv_c3*K|>Z85G!|p= zT~FiogC*E89(V&qk%qi1e*p&!AmqJkZ?Qo_NE06s@tc#d3N0WI5)C8aX8jA$;}zO( zW_{3I0|?PcSj%oF!Q%J~SL@_XkTlxcNeLMNwOF$1mo|5$Bz4%P|@|{@PzCFA($wycn^}2JWl-B!rKJ`vK>K=(#y$=90vXtNd5Q= zJ|PaJz=N5BkQ9FLx1SUESWD;BH8w-6A|;&T9b!o*pB*Ewb(g0u!tU3N6EGjY5Iz>3 zIULuupMPykW{Gl?A+q-j8ZbX0SBo}_V&%gI0jBu+Z{sWN7j$Hmg`?@q^)jpfq$_w6 z+D6dNY<~y4V3Wi34nN(#UAvK@6*;sMq(jFMYH=keB zV(xm%*PeH!(jDKJrknl61jkwG6#G&*RIn_GMY(^p0Q7sw2Ev;fQ*c_xJO9++<2wc6 zq7cM)1~Pq?&NdKlI2Mh-vD&|r`}mH{8GZ=H&vpQoiu(-Uc?Lc{JX{~4orG)YIoTtm z#R*^I+Zd>DT;(rh`&<8T(tA-E^`R1NMkM!1hGPp}?1*yF@Ypp1zYnkCWvsjw(mGrSK)Y*f(H_$Q<(+{l^n znTtHxClJiej~r09@f;*`8|kL|Aqi_^ zPsI;Y={DoJUVixma7r#@2fUfemFqH;QmN-!uepwA5o$GRpQ^e}1stGyOkbTk<$I%? 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where + "quickselect (x#xs) k = (let + xs1 = [y\xs. yxs. \(yx\set xs. x < p) \ (\y\set ys. \ y < p) \ (a < p) \ insort a (xs@ys) = (insort a xs) @ ys" + apply(induction xs) + apply(induction ys) + by auto + +lemma aux2[simp]: "(\x\set xs. x < p) \ (\y\set ys. \ y < p) \ \(a < p) \ insort a (xs@ys) = (xs) @ insort a ys" + apply(induction xs) + apply(induction ys) + by auto + + +lemma partition_correct[simp]: "sort xs = sort [x\xs. xxs. \(x quickselect xs k = sort xs ! k" +proof (induction xs k rule: quickselect.induct) + print_cases + case (1 x xs k) + note IH = "1.IH"[OF refl refl] + + let ?xs1 = "[y\xs. y(k int" where + "funky k = (k - 3)^2" + +value "sort_key funky [(2::int), 1, 3, 4]" + +find_theorems "insort" + +lemma "[x\sort_key k xs. k x = a] = [x\xs. k x = a]" + apply(induction xs) + apply(auto simp: sort_key_def filter_sort filter_insort_triv) + sorry + +fun quickselect :: "'a::linorder list \ nat \ 'a" where + "quickselect (x#xs) k = (let + xs1 = [y\xs. yxs. \(y*) + + +text {* \ExerciseSheet{6}{18.~5.~2018} *} + +text \\Exercise{Complexity of Naive Reverse} + Show that the naive reverse function needs quadratically many + \Cons\ operations in the length of the input list. + (Note that \[x]\ is syntax sugar for \Cons x []\!) +\ + +thm append.simps + +fun reverse where + "reverse [] = []" +| "reverse (x#xs) = reverse xs @ [x]" + +(** Define cost functions and prove that they are equal to quadratic function *) + +text \ + \Exercise{Simple Paths} + Recall the definition of paths from last exercise sheet: +\ +fun path :: "('a \ 'a \ bool) \ 'a \ 'a list \ 'a \ bool" + where + "path G u [] v \ u=v" +| "path G u (x#xs) v \ G u x \ path G x xs v" + +lemma path_append[simp]: "path G u (p1@p2) v \ (\w. path G u p1 w \ path G w p2 v)" + by (induction p1 arbitrary: u) auto + + +text \ + A simple path is a path without loops, or, in other words, a path + where no node occurs twice. (Note that the first node of the path is + not included, such that there may be a simple path from \u\ to \u\.) + + Show that for every path, there is a corresponding simple path. + + Hint: Induction on the length of the path +\ +thm measure_induct_rule[where f=length, case_names shorter] + +thm not_distinct_decomp + +lemma exists_simple_path: + assumes "path G u p v" + shows "\p'. path G u p' v \ distinct p'" + oops + + +text \\NumHomework{Stability of Insertion Sort}{May 25} + Have a look at Isabelle's standard implementation of sorting: @{const sort_key}. + (Use Ctrl-Click to jump to the definition in @{file "~~/src/HOL/List.thy"}) + Show that this function is a stable sorting algorithm, i.e., the order of elements + with the same key is not changed during sorting! +\ + +lemma "[x\sort_key k xs. k x = a] = [x\xs. k x = a]" + oops + +term "[x\xs. P x]" +text \ + Note: @{term [source] \[x\xs. P x] \} is syntax sugar for @{term [source] \filter P xs\}, + where the filter function returns only the elements of list \xs\ for which \P xs = True\. + + Hint: You do not necessarily need Isar, and the auxiliary lemmas + you need are already in Isabelle's library. @{command find_theorems} is your friend! +\ + + +text \\NumHomework{Quickselect}{May 25} + +From \<^url>\https://en.wikipedia.org/wiki/Quickselect\: + +Quickselect is a selection algorithm to find the kth smallest element in an unordered list. +It is related to the quicksort sorting algorithm. +Like quicksort, it was developed by Tony Hoare, and thus is also known as Hoare's selection +algorithm. Like quicksort, it is efficient in practice and has good average-case performance, +but has poor worst-case performance. Quickselect and its variants are the selection algorithms +most often used in efficient real-world implementations. + +Quickselect uses the same overall approach as quicksort, choosing one element as a pivot and +partitioning the data in two based on the pivot, accordingly as less than or greater than the +pivot. However, instead of recursing into both sides, as in quicksort, quickselect only +recurses into one side --- the side with the element it is searching for. + + +Your task is to prove correct the quickselect algorithm, which can be + implemented in Isabelle as follows: +\ + +fun quickselect :: "'a::linorder list \ nat \ 'a" where + "quickselect (x#xs) k = (let + xs1 = [y\xs. yxs. \(yYour first task is to prove the crucial idea of quicksort, i.e., that + partitioning wrt.\ a pivot element $p$ is correct. +\ + +lemma partition_correct: "sort xs = sort [x\xs. xxs. \(x + Hint: Induction, and auxiliary lemmas to transform a term of the + form @{term \insort x (xs@ys)\} when you know that \x\ is greater than + all elements in \xs\ / less than or equal all elements in \ys\. +\ + + + +text \Next, show that quickselect is correct\ +lemma "k quickselect xs k = sort xs ! k" + text \Proceed by computation induction, and a case distinction according to the + cases in the body of the quickselect function\ +proof (induction xs k rule: quickselect.induct) + case (1 x xs k) + + text \Note: To make the induction hypothesis more readable, + you can collapse the first two premises of the form \?x=\\ + by reflexivity:\ + note IH = "1.IH"[OF refl refl] + + text \Insert your proof here!\ + + show ?case sorry +next + case 2 then show ?case by simp +qed + + + + + +(*<*) +end +(*>*) diff --git a/07/ex07.pdf b/07/ex07.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ce05bf01bb138d6588b11bbd780046c45f48a7fc GIT binary patch literal 118615 zcma&NLy#~$*rnaJZQHhO+qP}nw%z@-ZQHhOoAdsE&9|CaOsbNcY_hv6*EvZ@6-2~n z8R=M|NavSF)}dGl7zpf*te|*!py*{x?aW;)2$8`JEiREu6Z+Sn&>J43uBqjt+7Dw4@8=yDp`8aS@<}A3JAAQqG!n(hCK=64Me<#9q zlm(eb$rDRsp0>3?q*)##{QZ zWxdv3-x(zjTkJPizZVJHLNUgR2RdCcR!f0xMQHB>=gsT~5B3IDmR=5Y4HDP%`qVVM zgW685wX7R11}%OQFY(xr6nad<+{j<-T2I29m&^znhGG`sFrSQ>RjRo<_ih^#W=3`P z1M`6a`{IPP;hrGu56(I5o-i~jXGS!pTEOUtF!kuOe|Uop&$SNbiq#f;?@(`#k3R5| 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zoaxfMvU-YaYXk@NxM0Ne#P5RywUj(#yy`Ze7mfB-czY9SnI$~c>_9C&MaT@j?k zLNn2ULp)>FYJ-Nzk?(OF8f>@Syf3Gm4d0~8%*X=%o^o(ENrHP7A5N+>F4UF7jKEc6 zNrW<(lJUUT@TJ&W2EHBch9xoEZ?w1+(3qc41pa*-N-NmsjXS~~#TRzJ!MQ<9M zwy&tu9^#1O#)MMot*v2(oWq{-GY1xzEq}9g{kuwsZ=uvoMscRJvDxWI;%;|qygbEg zS_vks8Pj*cy3Yi!d%PLPkZQ~Z(RFdgXYrJu1XA0Rw?EM-wy@PzfuKJ9+faCWF-lr` zd(tZJO@>PW=SJtTHB&m^kI`=}iJ{1`xWA=3-W`RLpq!QvL;F&;0jl%vgxeKj1E|fy2JODT+k!)-E>?)q^1rwi8y=0ZErD zXb34OD{0-;TZ8EMo?E(m)%WcpP^EXIxcn66XU<2!Sm`CcZ4YP`~R zI!oCkctdf&B70Ihu4xtj;cPuNb%4jxuxpijeK257WTd%dF=wGxcc-mk@A`OS*5si# zW!K%RfwO6>7`hpp+0?0HO*-aUC25?8Z${r-nZZDHf`px;}c;3>N zvRggbup963mpsltfj#JRMVuojm;LyN(CS#X=?{T`Q(%v=3AY8(!(ZU+--8nijxQ~E zyCAGyGFds|NnPDa)Wzc5?AKHjN!R&C7*!gUa-}A9rtYyGomcF(>#5bv4Z#2p&^rbAwzC*B&`0AkIc{IhaV(*3aiMHotP9iMI zhZr5Sf;LHo1N^UuuYEevnNtTtiRshmH_NsU3;hpe=a35%<1Dc_%pQF?=qnR6UY@; zXifOTREp4c8IYKy38$8E`)~ivAA&mCwfpcF!w(o%o#EGF9B?xl?|(P?`WMnZYxj2W&s-H@JGSDj*KG?e9}D1?^}i-4M**=Xt{g*% zFyZgtynTgFH;m{nWv&7tc<&*uAs8wVp`1J+-fWEWTcf_ezyDM|{ZgD-(c_rzd?Z~u z$ypw3>$5p`ahdnaSNto=kL_bGYK_ukKaFM!GoBfFt|f38eG%nUDRbI8@fef6KG-&T zHQIHs@CsqCWUq3~lce{&FTReL$Y$Zyoqo%v!1 z#rd1PEuM+bX>=XmVhIZF(qQm;>{)q5Px!TwApkCzWdm#KMS>Ssk;Q|z_4|WD%zm;( z(z^=UVf?G(LRVLI;@3(E{O&HNAq=?;-Mu3bm5H&i{|8s&{tud=s}8b42Ff{Eg1i6* ztN1qP|G{lY zrj6M9p>ZOvAKoB9&-YxQ9+96}j=E7&ZE7EICff(Sm5Pjx&^x>`Z< zVkzYUqtizI_~3%FTGC?Kg6VlfEV3eS+`gifPlvMPi4j_D!9mDT2pDG*Ze3Y1P^Lqq zbi?z~17m(r>qZFTd_`icM>tgtL8BX>(fo?cwbVE^!28{C-LuG}I*) + + +text {* \ExerciseSheet{9}{8.~6.~2018} *} + +text \\Exercise{Indicate Unchanged by Option} + + Write an insert function for red-black trees that either inserts the element + and returns a new tree, or returns None if the element was already in the tree +\ + +fun ins' :: "'a::linorder \ 'a rbt \ 'a rbt option" +(*<*) +where +"ins' x Leaf = Some (R Leaf x Leaf)" | +"ins' x (B l a r) = + (case cmp x a of + LT \ (case ins' x l of None \ None | Some l \ Some (baliL l a r)) | + GT \ (case ins' x r of None \ None | Some r \ Some (baliR l a r)) | + EQ \ None)" | +"ins' x (R l a r) = + (case cmp x a of + LT \ (case ins' x l of None \ None | Some l \ Some (R l a r)) | + GT \ (case ins' x r of None \ None | Some r \ Some (R l a r)) | + EQ \ None)" + +lemma baliR_id: "\invc l; invc r\ \ baliR l a r = B l a r" + by (cases "(l,a,r)" rule: baliR.cases; auto) + +lemma baliL_id: "\invc l; invc r\ \ baliL l a r = B l a r" + by (cases "(l,a,r)" rule: baliL.cases; auto) +(*>*) + +lemma "invc t \ case ins' x t of None \ ins x t = t | Some t' \ ins x t = t'" +(*<*) +proof (induction x t rule: ins.induct) + case (1 x) + then show ?case by simp +next + case (2 x l a r) + + then show ?case + apply (auto split: option.splits simp: baliL_id baliR_id) + done + +next + case (3 x l a r) + then show ?case by (auto split: option.splits) +qed +(*>*) + +(*<*) +end +(*>*) diff --git a/09/ex09_2.thy b/09/ex09_2.thy new file mode 100644 index 0000000..8a083d9 --- /dev/null +++ b/09/ex09_2.thy @@ -0,0 +1,115 @@ +(*<*) +theory ex09_2 +imports + "~~/src/HOL/Data_Structures/Tree23_Set" +begin +(*>*) + +text \\Exercise{Joining 2-3-Trees} + + Write a join function for 2-3-trees: The function shall take two + 2-3-trees \l\ and \r\ and an element \x\, and return a new 2-3-tree with + the inorder-traversal \l x r\ . + + Write two functions, one for the height of \l\ being greater, the + other for the height of \r\ being greater. +\ + + + +text \\height r\ greater\ +fun joinL :: "'a tree23 \ 'a \ 'a tree23 \ 'a up\<^sub>i" +(*<*) +where +"joinL l x r = + (if height l = height r then Up\<^sub>i l x r + else case r of + Node2 r1 a r2 \ + (case joinL l x r1 of + T\<^sub>i t \ T\<^sub>i (Node2 t a r2) | + Up\<^sub>i t1 b t2 \ T\<^sub>i (Node3 t1 b t2 a r2)) | + Node3 r1 a r2 b r3 \ (case joinL l x r1 of + T\<^sub>i t \ T\<^sub>i (Node3 t a r2 b r3) | + Up\<^sub>i t1 y t2 \ Up\<^sub>i (Node2 t1 y t2) a (Node2 r2 b r3)))" +(*>*) + +lemma bal_joinL: "\ bal l; bal r; height l \ height r \ \ + bal (tree\<^sub>i (joinL l x r)) \ height(joinL l x r) = height r" +(*<*) +apply(induction r) + apply simp + apply (fastforce simp: le_less split: up\<^sub>i.split) +apply (fastforce simp: le_less split: up\<^sub>i.split) +done +(*>*) + +lemma inorder_joinL: "\ bal l; bal r; height l \ height r \ \ inorder (tree\<^sub>i (joinL l x r)) = inorder l @x # inorder r" +(*<*) + apply(induction r) + apply (auto split: up\<^sub>i.splits) + done +(*>*) + +text \\height l\ greater\ +fun joinR :: "'a tree23 \ 'a \ 'a tree23 \ 'a up\<^sub>i" +(*<*) +where +"joinR l x r = + (if height l = height r then Up\<^sub>i l x r + else case l of + Node2 l1 a l2 \ + (case joinR l2 x r of + T\<^sub>i t \ T\<^sub>i (Node2 l1 a t) | + Up\<^sub>i t1 b t2 \ T\<^sub>i (Node3 l1 a t1 b t2)) | + Node3 l1 a l2 b l3 \ (case joinR l3 x r of + T\<^sub>i t \ T\<^sub>i (Node3 l1 a l2 b t) | + Up\<^sub>i t1 y t2 \ Up\<^sub>i (Node2 l1 a l2) b (Node2 t1 y t2)))" +(*>*) + +lemma bal_joinR: "\ bal l; bal r; height l \ height r \ \ + bal (tree\<^sub>i (joinR l x r)) \ height(joinR l x r) = height l" + text \Note the generalization: We augmented the lemma with a statement about the height of the result.\ +(*<*) +apply(induction l) + apply simp + apply (auto simp: le_less split!: up\<^sub>i.splits tree23.split)[] +apply (fastforce simp: le_less split: up\<^sub>i.split) +done +(*>*) + +lemma inorder_joinR: "\ bal l; bal r; height l \ height r \ \ inorder (tree\<^sub>i (joinR l x r)) = inorder l @x # inorder r" +(*<*) + apply(induction l) + apply simp + apply (auto split!: up\<^sub>i.splits tree23.split) + done +(*>*) + + +text \Combine both functions\ +fun join :: "'a tree23 \ 'a \ 'a tree23 \ 'a tree23" +(*<*) +where +"join l x r = + (if height l > height r + then tree\<^sub>i(joinR l x r) + else if height l < height r + then tree\<^sub>i(joinL l x r) + else Node2 l x r)" +(*>*) + +lemma "\ bal l; bal r \ \ bal (join l x r)" +(*<*) + by(auto simp: bal_joinL bal_joinR simp del: joinL.simps joinR.simps) +(*>*) + +lemma "\ bal l; bal r \ \ inorder (join l x r) = inorder l @x # inorder r" +(*<*) + by(auto simp: inorder_joinL inorder_joinR simp del: joinL.simps joinR.simps) +(*>*) + + + +(*<*) +end +(*>*) diff --git a/09/ex09_2tmpl.thy b/09/ex09_2tmpl.thy new file mode 100644 index 0000000..93e6248 --- /dev/null +++ b/09/ex09_2tmpl.thy @@ -0,0 +1,78 @@ +(*<*) +theory ex09_2tmpl +imports + "~~/src/HOL/Data_Structures/Tree23_Set" +begin +(*>*) + +text \\Exercise{Joining 2-3-Trees} + + Write a join function for 2-3-trees: The function shall take two + 2-3-trees \l\ and \r\ and an element \x\, and return a new 2-3-tree with + the inorder-traversal \l x r\ . + + Write two functions, one for the height of \l\ being greater, the + other for the height of \r\ being greater. +\ + +text \\height l\ greater\ +fun joinR :: "'a tree23 \ 'a \ 'a tree23 \ 'a up\<^sub>i" +where + "joinR l x r = ( + if height l = height r then Up\<^sub>i l x r + else case l of + Node2 l1 a l2 \ (case joinR l2 x r of + T\<^sub>i t \ T\<^sub>i (Node2 l1 a t) + | Up\<^sub>i l2' x' r' \ T\<^sub>i (Node3 l1 a l2' x' r') + ) + | Node3 l1 a l2 b l3 \ (case joinR l3 x r of + T\<^sub>i t \ T\<^sub>i (Node3 l1 a l2 b t) + | Up\<^sub>i l2' x' r' \ Up\<^sub>i (Node2 l1 a l2) b (Node2 l2' x' r') + ) + )" + +declare joinR.simps[simp del] + + +lemma inorder_joinR: "\ bal l; bal r; height l \ height r \ \ inorder (tree\<^sub>i (joinR l x r)) = inorder l @x # inorder r" + apply (induction l x r rule: joinR.induct) + apply (subst joinR.simps) + apply (fastforce split!: tree23.split up\<^sub>i.split) + done + +lemma bal_joinR: "\ bal l; bal r; height l \ height r \ \ + bal (tree\<^sub>i (joinR l x r)) \ height(joinR l x r) = height l" + sorry + + + +text \\height r\ greater\ +fun joinL :: "'a tree23 \ 'a \ 'a tree23 \ 'a up\<^sub>i" +where + "joinL l x r = undefined" + +lemma bal_joinL: "\ bal l; bal r; height l \ height r \ \ + bal (tree\<^sub>i (joinL l x r)) \ height(joinL l x r) = height r" + text \Note the generalization: We augmented the lemma with a statement about the height of the result.\ + sorry + +lemma inorder_joinL: "\ bal l; bal r; height l \ height r \ \ inorder (tree\<^sub>i (joinL l x r)) = inorder l @x # inorder r" + sorry + + +text \Combine both functions\ +fun join :: "'a tree23 \ 'a \ 'a tree23 \ 'a tree23" +where +"join l x r = undefined" + + +lemma "\ bal l; bal r \ \ bal (join l x r)" + sorry + +lemma "\ bal l; bal r \ \ inorder (join l x r) = inorder l @x # inorder r" + sorry + + +(*<*) +end +(*>*) diff --git a/09/ex09tmpl.thy b/09/ex09tmpl.thy new file mode 100644 index 0000000..932ffa7 --- /dev/null +++ b/09/ex09tmpl.thy @@ -0,0 +1,59 @@ +(*<*) +theory ex09tmpl + imports Main "HOL-Data_Structures.RBT_Set" +begin +(*>*) + + +text {* \ExerciseSheet{9}{8.~6.~2018} *} + +text \\Exercise{Indicate Unchanged by Option} + + Write an insert function for red-black trees that either inserts the element + and returns a new tree, or returns None if the element was already in the tree +\ + +term map_option + +fun ins' :: "'a::linorder \ 'a rbt \ 'a rbt option" +where +"ins' x Leaf = Some (R Leaf x Leaf)" | +"ins' x (B l a r) = + (case cmp x a of + LT \ (case ins' x l of None \ None | Some l' \ Some (baliL l' a r)) | + GT \ (case ins' x r of None \ None | Some r' \ Some (baliR l a r')) | + EQ \ None)" | +"ins' x (R l a r) = + (case cmp x a of + LT \ (case ins' x l of None \ None | Some l' \ Some (R l' a r)) | + GT \ (case ins' x r of None \ None | Some r' \ Some (R l a r')) | + EQ \ None)" + +lemma [simp]: "\invc r\ \ baliR l a r = B l a r" + by (cases "(l,a,r)" rule: baliR.cases; simp) + +lemma [simp]: "\invc l\ \ baliL l a r = B l a r" + by (cases "(l,a,r)" rule: baliL.cases; simp) + +lemma X1: "invc t \ case ins' x t of None \ ins x t = t | Some t' \ ins x t = t'" + by (induction x t rule: ins'.induct) (auto split: option.split) + +lemma [simp]: "size (baliR l x r) = size l + size r + 1" + apply (cases "(l,x,r)" rule: baliR.cases) by auto + +lemma [simp]: "size (baliL l x r) = size l + size r + 1" + apply (cases "(l,x,r)" rule: baliL.cases) by auto + +lemma X2: "invc t \ ins' x t = Some t' \ size t < size t'" + apply (induction x t arbitrary: t' rule: ins'.induct) + apply (auto split: option.splits if_splits) + done + + +lemma "invc t \ case ins' x t of None \ ins x t = t | Some t' \ ins x t = t' \ t\t'" + using X1[of t x] X2[of t x] by (auto split: option.split) + + +(*<*) +end +(*>*) diff --git a/09/ex09tmpl.thy~ b/09/ex09tmpl.thy~ new file mode 100644 index 0000000..f7ec48a --- /dev/null +++ b/09/ex09tmpl.thy~ @@ -0,0 +1,24 @@ +(*<*) +theory ex09tmpl + imports Main "HOL-Data_Structures.RBT_Set" +begin +(*>*) + + +text {* \ExerciseSheet{9}{8.~6.~2018} *} + +text \\Exercise{Indicate Unchanged by Option} + + Write an insert function for red-black trees that either inserts the element + and returns a new tree, or returns None if the element was already in the tree +\ + +fun ins' :: "'a::linorder \ 'a rbt \ 'a rbt option" +where +"ins' _ _ = undefined" + +lemma "invc t \ case ins' x t of None \ ins x t = t | Some t' \ ins x t = t'" + sorry +(*<*) +end +(*>*) diff --git a/09/hw09.thy b/09/hw09.thy new file mode 100644 index 0000000..2ee9ddd --- /dev/null +++ b/09/hw09.thy @@ -0,0 +1,158 @@ +theory hw09 + imports Main "HOL-Data_Structures.RBT_Set" +begin + + + +fun min_height :: "('a,'b) tree \ nat" where +"min_height Leaf = 0" | +"min_height (Node _ l _ r) = min (min_height l) (min_height r) + 1" + +definition "balanced t \ height t - min_height t \ 1" + + + +fun mk_rbt :: "('a,unit) tree \ 'a rbt" where + "mk_rbt Leaf = Leaf" +| "mk_rbt (Node _ l a r) = (let + l'=mk_rbt l; + r'=mk_rbt r + in + if min_height l > min_height r then + B (paint Red l') a r' + else if min_height l < min_height r then + B l' a (paint Red r') + else + B l' a r' + )" + + +text \ + \subsection*{Warmup} + +\ + +lemma balanced_subt: "balanced (Node c l a r) \ balanced l \ balanced r" + unfolding balanced_def by auto + + +lemma aux1:"height t \ min_height t" + apply(induction t) + by auto + +lemma balanced_alt: + "balanced t \ height t = min_height t \ height t = min_height t + 1" + unfolding balanced_def + by (metis One_nat_def add_diff_cancel_left' aux1 cancel_comm_monoid_add_class.diff_cancel le_Suc_eq le_add_diff_inverse le_zero_eq order_refl) + +text \ + \subsection*{The Easy Parts} +\ + + +lemma mk_rbt_inorder: "inorder (mk_rbt t) = inorder t" + apply(induction t) + apply(auto) + by (smt inorder.simps(2) inorder_paint) + + +lemma mk_rbt_color: "color (mk_rbt t) = Black" + apply(induction t) + apply(auto) + by (smt RBT_Set.color.simps(2)) + +text \ + \subsection*{Medium Complex Parts} + +\ + + + +lemma mk_rbt_bheight: "balanced t \ bheight (mk_rbt t) = min_height t" +proof(induction t) + case Leaf + then show ?case by auto +next + case (Node x1 t1 x3 t2) + note IH = Node.IH + have "min_height t1 = min_height t2 \ min_height t1 < min_height t2 \ min_height t1 > min_height t2" (is "?C1 \ ?C2 \ ?C3") + by auto + moreover + { + assume ?C1 + have ?case using IH Node.prems \?C1\ + apply(cases "min_height t1") + apply(auto) + apply (meson balanced_subt) + by (meson balanced_subt) + } + moreover + { + assume ?C2 + have ?case using IH Node.prems \?C2\ + apply(cases "min_height t1") + apply(auto) + apply (meson balanced_subt) + by (metis balanced_subt min.strict_order_iff) + } + moreover + { + assume ?C3 + then have h0:"min_height (Node x1 t1 x3 t2) = min_height t2 + 1" using IH Node.prems by force + then have h1:"min_height t2 = 0 \ bheight (paint Red (mk_rbt t1)) = 0" using IH Node.prems + by (metis (no_types, lifting) One_nat_def Suc_eq_plus1 Suc_leI \min_height t2 < min_height t1\ add_diff_cancel_right' aux1 balanced_def bheight_paint_Red height.simps(2) le_trans max.commute max_def mk_rbt_color) + then have h2: "min_height t1 > min_height t2 \ min_height t1 = Suc (min_height t2)" using IH Node.prems + by (smt Suc_eq_plus1 Suc_leI Suc_mono balanced_alt balanced_subt h0 height.simps(2) less_Suc_eq max.commute max_def not_le) + then have h3: " min_height t1 > min_height t2 \ bheight (paint Red (mk_rbt t1)) = min_height t2" using IH Node.prems + apply(cases "min_height t1") + apply(auto) + by (metis Suc_eq_plus1 add_diff_cancel_right' balanced_subt bheight_paint_Red mk_rbt_color) + + then have ?case using IH Node.prems \?C3\ + apply(cases "min_height t2") + by (auto) + + } + ultimately show ?case by blast +qed + + +lemma mk_rbt_invh: "balanced t \ invh (mk_rbt t)" + apply(induction t rule:mk_rbt.induct) + apply(simp add: balanced_alt mk_rbt_color mk_rbt_inorder mk_rbt_bheight) + apply(simp split:if_splits) + by (smt Suc_eq_plus1 Suc_le_mono add_diff_cancel_right' antisym_conv balanced_alt balanced_subt bheight_paint_Red height.simps(2) invh.simps(2) le_Suc_eq le_simps(3) max_def min_def invh_paint min_height.simps(2) mk_rbt_bheight mk_rbt_color) + + + + + + +fun mk_rbt' :: "('a,unit) tree \ 'a rbt \ nat" +where + "mk_rbt' Leaf = (Leaf, 0)" +| "mk_rbt' (Node _ l a r) = (let + (l',lx)=mk_rbt' l; + (r',rx)=mk_rbt' r + in + if lx > rx then + (B (paint Red l') a r', Suc(rx)) + else if lx < rx then + (B l' a (paint Red r'), Suc(lx)) + else + (B l' a r', Suc(lx)) + )" + +lemma mk_rbt'_refine_aux: "mk_rbt' t = (mk_rbt t, min_height t)" + apply(induction t) + apply(auto) + done + + +lemma mk_rbt'_refine: "fst (mk_rbt' t) = mk_rbt t" + apply(induction t) + by(auto simp:mk_rbt'_refine_aux) + + + +end diff --git a/09/hw09tmpl.thy b/09/hw09tmpl.thy new file mode 100644 index 0000000..c339d28 --- /dev/null +++ b/09/hw09tmpl.thy @@ -0,0 +1,239 @@ +(*<*) +theory hw09tmpl + imports Main "HOL-Data_Structures.RBT_Set" +begin +(*>*) + +text \\NumHomework{Balanced Tree to RBT}{15.~6.~2018} + + A tree is balanced, if its minimum height and its height differ by at most 1. +\ + +fun min_height :: "('a,'b) tree \ nat" where +"min_height Leaf = 0" | +"min_height (Node _ l _ r) = min (min_height l) (min_height r) + 1" + +definition "balanced t \ height t - min_height t \ 1" + +text \The following function paints a balanced tree to form a valid red-black tree + with the same structure. The task of this homework is to prove this! +\ + +fun mk_rbt :: "('a,unit) tree \ 'a rbt" where + "mk_rbt Leaf = Leaf" +| "mk_rbt (Node _ l a r) = (let + l'=mk_rbt l; + r'=mk_rbt r + in + if min_height l > min_height r then + B (paint Red l') a r' + else if min_height l < min_height r then + B l' a (paint Red r') + else + B l' a r' + )" + + +text \ + \subsection*{Warmup} + + Show that the left and right subtree of a balanced tree are, again, balanced +\ + +lemma balanced_subt: "balanced (Node c l a r) \ balanced l \ balanced r" + unfolding balanced_def by auto + + +text \Show the following alternative characterization of balanced:\ + +lemma aux1:"height t \ min_height t" + apply(induction t) + by auto + +lemma balanced_alt: + "balanced t \ height t = min_height t \ height t = min_height t + 1" + text \Hint: Auxiliary lemma relating @{term \height t\} and @{term \min_height t\}\ + unfolding balanced_def + by (metis One_nat_def add_diff_cancel_left' aux1 cancel_comm_monoid_add_class.diff_cancel le_Suc_eq le_add_diff_inverse le_zero_eq order_refl) + +text \ + \subsection*{The Easy Parts} + + Show that \mk_rbt\ does not change the inorder-traversal +\ +lemma mk_rbt_inorder: "inorder (mk_rbt t) = inorder t" + apply(induction t) + apply(auto) + by (smt inorder.simps(2) inorder_paint) + +text \Show that the color of the root node is always black\ +lemma mk_rbt_color: "color (mk_rbt t) = Black" + apply(induction t) + apply(auto) + by (smt RBT_Set.color.simps(2)) + +text \ + \subsection*{Medium Complex Parts} + + Show that the black-height of the returned tree is the minimum height of the argument tree +\ + + + +lemma mk_rbt_bheight: "balanced t \ bheight (mk_rbt t) = min_height t" +text \Hint: Use Isar to have better control when to unfold with @{thm [source] balanced_alt}, + and when to use @{thm [source] balanced_subt} (e.g. to discharge the premises of the IH) +\ +proof(induction t) + case Leaf + then show ?case by auto +next + case (Node x1 t1 x3 t2) + note IH = Node.IH + have "min_height t1 = min_height t2 \ min_height t1 < min_height t2 \ min_height t1 > min_height t2" (is "?C1 \ ?C2 \ ?C3") + by auto + moreover + { + assume ?C1 + have ?case using IH Node.prems \?C1\ + apply(cases "min_height t1") + apply(auto) + apply (meson balanced_subt) + by (meson balanced_subt) + } + moreover + { + assume ?C2 + have ?case using IH Node.prems \?C2\ + apply(cases "min_height t1") + apply(auto) + apply (meson balanced_subt) + by (metis balanced_subt min.strict_order_iff) + } + moreover + { + assume ?C3 + then have h0:"min_height (Node x1 t1 x3 t2) = min_height t2 + 1" using IH Node.prems by force + then have h1:"min_height t2 = 0 \ bheight (paint Red (mk_rbt t1)) = 0" using IH Node.prems + by (metis (no_types, lifting) One_nat_def Suc_eq_plus1 Suc_leI \min_height t2 < min_height t1\ add_diff_cancel_right' aux1 balanced_def bheight_paint_Red height.simps(2) le_trans max.commute max_def mk_rbt_color) + then have h2: "min_height t1 > min_height t2 \ min_height t1 = Suc (min_height t2)" using IH Node.prems + by (smt Suc_eq_plus1 Suc_leI Suc_mono balanced_alt balanced_subt h0 height.simps(2) less_Suc_eq max.commute max_def not_le) + then have h3: " min_height t1 > min_height t2 \ bheight (paint Red (mk_rbt t1)) = min_height t2" using IH Node.prems + apply(cases "min_height t1") + apply(auto) + by (metis Suc_eq_plus1 add_diff_cancel_right' balanced_subt bheight_paint_Red mk_rbt_color) + + then have ?case using IH Node.prems \?C3\ + apply(cases "min_height t2") + by (auto) + + } + ultimately show ?case by blast +qed + +text \ + Show that the returned tree satisfies the height invariant. +\ + +lemma mk_rbt_invh: "balanced t \ invh (mk_rbt t)" + apply(induction t rule:mk_rbt.induct) + apply(simp add: balanced_alt mk_rbt_color mk_rbt_inorder mk_rbt_bheight) + apply(simp split:if_splits) + by (smt Suc_eq_plus1 Suc_le_mono add_diff_cancel_right' antisym_conv balanced_alt balanced_subt bheight_paint_Red height.simps(2) invh.simps(2) le_Suc_eq le_simps(3) max_def min_def invh_paint min_height.simps(2) mk_rbt_bheight mk_rbt_color) + + +text \ + \subsection*{The Hard Part (3 Bonus Points)} + + For {\bf three bonus points}, show that the returned tree satisfies the color invariant. + + Warning: This requires careful case splitting, via a clever combination of + automation and manual proof (Isar, aux-lemmas), in order to deal with the + multiple cases without a combinatorial explosion of the proofs. +\ + +lemma mk_rbt_invc: "balanced t \ invc (mk_rbt t)" +proof(induction t) +case Leaf +then show ?case by auto +next + case (Node x1 t1 x3 t2) + note IH = Node.IH + have "min_height t1 = min_height t2 \ min_height t1 < min_height t2 \ min_height t1 > min_height t2" (is "?C1 \ ?C2 \ ?C3") + by auto + moreover + { + assume ?C1 + have ?case using IH Node.prems \?C1\ + apply(cases "min_height t1") + apply(auto) + apply (meson balanced_subt) + apply (meson balanced_subt) + apply (meson balanced_subt) + by (meson balanced_subt) + } + moreover + { + assume ?C2 + then have h0:"min_height (Node x1 t1 x3 t2) = min_height t1 + 1" using IH Node.prems by force + then have h1: "min_height t1 < min_height t2 \ min_height t2 = Suc (min_height t1)" using IH Node.prems + by (smt Suc_eq_plus1 Suc_leI Suc_mono balanced_alt balanced_subt h0 height.simps(2) less_Suc_eq max.commute max_def not_le) + then have h2:"invc (paint Red (mk_rbt t2))" using IH Node.prems sorry + then have ?case using IH Node.prems \?C2\ + apply(cases "min_height t1") + apply(auto) + apply (smt balanced_subt) + apply (simp add: h2) + by (meson balanced_subt) + } + +qed + + +(* Now you can combine everything, to show that you are, indeed, generating an RBT *) +theorem mk_rbt_is_rbt: "balanced t \ rbt (mk_rbt t)" + using mk_rbt_invh mk_rbt_invc mk_rbt_color unfolding rbt_def by auto + + + +text \\NumHomework{Linear-Time Repainting}{15.~6.~2018} + + Write a linear-time version of \mk_rbt\, and show that it behaves like + \mk_rbt\. + + Idea: Compute the min-height during the same recursion as you build + the tree. + + Note: No formal complexity proof required. +\ + +fun mk_rbt' :: "('a,unit) tree \ 'a rbt \ nat" -- \Returns the RBT and the min-height of the argument\ +where + "mk_rbt' Leaf = (Leaf, 0)" +| "mk_rbt' (Node _ l a r) = (let + (l',lx)=mk_rbt' l; + (r',rx)=mk_rbt' r + in + if lx > rx then + (B (paint Red l') a r', Suc(rx)) + else if lx < rx then + (B l' a (paint Red r'), Suc(lx)) + else + (B l' a r', Suc(lx)) + )" + +lemma mk_rbt'_refine_aux: "mk_rbt' t = (mk_rbt t, min_height t)" + apply(induction t) + apply(auto) + done + + +lemma mk_rbt'_refine: "fst (mk_rbt' t) = mk_rbt t" + apply(induction t) + by(auto simp:mk_rbt'_refine_aux) + + + +(*<*) +end +(*>*) diff --git a/09/hw09tmpl.thy~ b/09/hw09tmpl.thy~ new file mode 100644 index 0000000..de121b0 --- /dev/null +++ b/09/hw09tmpl.thy~ @@ -0,0 +1,192 @@ +(*<*) +theory hw09tmpl + imports Main "HOL-Data_Structures.RBT_Set" +begin +(*>*) + +text \\NumHomework{Balanced Tree to RBT}{15.~6.~2018} + + A tree is balanced, if its minimum height and its height differ by at most 1. +\ + +fun min_height :: "('a,'b) tree \ nat" where +"min_height Leaf = 0" | +"min_height (Node _ l _ r) = min (min_height l) (min_height r) + 1" + +definition "balanced t \ height t - min_height t \ 1" + +text \The following function paints a balanced tree to form a valid red-black tree + with the same structure. The task of this homework is to prove this! +\ + +fun mk_rbt :: "('a,unit) tree \ 'a rbt" where + "mk_rbt Leaf = Leaf" +| "mk_rbt (Node _ l a r) = (let + l'=mk_rbt l; + r'=mk_rbt r + in + if min_height l > min_height r then + B (paint Red l') a r' + else if min_height l < min_height r then + B l' a (paint Red r') + else + B l' a r' + )" + + +text \ + \subsection*{Warmup} + + Show that the left and right subtree of a balanced tree are, again, balanced +\ + +lemma balanced_subt: "balanced (Node c l a r) \ balanced l \ balanced r" + unfolding balanced_def by auto + + +text \Show the following alternative characterization of balanced:\ + +lemma aux1:"height t \ min_height t" + apply(induction t) + by auto + +lemma balanced_alt: + "balanced t \ height t = min_height t \ height t = min_height t + 1" + text \Hint: Auxiliary lemma relating @{term \height t\} and @{term \min_height t\}\ + unfolding balanced_def + by (metis One_nat_def add_diff_cancel_left' aux1 cancel_comm_monoid_add_class.diff_cancel le_Suc_eq le_add_diff_inverse le_zero_eq order_refl) + +text \ + \subsection*{The Easy Parts} + + Show that \mk_rbt\ does not change the inorder-traversal +\ +lemma mk_rbt_inorder: "inorder (mk_rbt t) = inorder t" + apply(induction t) + apply(auto) + by (smt inorder.simps(2) inorder_paint) + +text \Show that the color of the root node is always black\ +lemma mk_rbt_color: "color (mk_rbt t) = Black" + apply(induction t) + apply(auto) + by (smt RBT_Set.color.simps(2)) + +text \ + \subsection*{Medium Complex Parts} + + Show that the black-height of the returned tree is the minimum height of the argument tree +\ + + + +lemma mk_rbt_bheight: "balanced t \ bheight (mk_rbt t) = min_height t" +text \Hint: Use Isar to have better control when to unfold with @{thm [source] balanced_alt}, + and when to use @{thm [source] balanced_subt} (e.g. to discharge the premises of the IH) +\ +proof(induction t) + case Leaf + then show ?case by auto +next + case (Node x1 t1 x3 t2) + note IH = Node.IH + have "min_height t1 = min_height t2 \ min_height t1 < min_height t2 \ min_height t1 > min_height t2" (is "?C1 \ ?C2 \ ?C3") + by auto + moreover + { + assume ?C1 + have ?case using IH Node.prems \?C1\ + apply(cases "min_height t1") + apply(auto) + apply (meson balanced_subt) + by (meson balanced_subt) + } + moreover + { + assume ?C2 + have ?case using IH Node.prems \?C2\ + apply(cases "min_height t1") + apply(auto) + apply (meson balanced_subt) + by (metis balanced_subt min.strict_order_iff) + } + moreover + { + assume ?C3 + have ?case using IH Node.prems \?C3\ + apply(cases "min_height t2") + apply(auto) + + then show ?case +qed + +text \ + Show that the returned tree satisfies the height invariant. +\ + +lemma mk_rbt_invh: "balanced t \ invh (mk_rbt t)" + unfolding balanced_def + apply(induction t) sorry + + +text \ + \subsection*{The Hard Part (3 Bonus Points)} + + For {\bf three bonus points}, show that the returned tree satisfies the color invariant. + + Warning: This requires careful case splitting, via a clever combination of + automation and manual proof (Isar, aux-lemmas), in order to deal with the + multiple cases without a combinatorial explosion of the proofs. +\ + +lemma mk_rbt_invc: "balanced t \ invc (mk_rbt t)" + sorry + + +(* Now you can combine everything, to show that you are, indeed, generating an RBT *) +theorem mk_rbt_is_rbt: "balanced t \ rbt (mk_rbt t)" + using mk_rbt_invh mk_rbt_invc mk_rbt_color unfolding rbt_def by auto + + + +text \\NumHomework{Linear-Time Repainting}{15.~6.~2018} + + Write a linear-time version of \mk_rbt\, and show that it behaves like + \mk_rbt\. + + Idea: Compute the min-height during the same recursion as you build + the tree. + + Note: No formal complexity proof required. +\ + +fun mk_rbt' :: "('a,unit) tree \ 'a rbt \ nat" -- \Returns the RBT and the min-height of the argument\ +where + "mk_rbt' Leaf = (Leaf, 0)" +| "mk_rbt' (Node _ l a r) = (let + (l',lx)=mk_rbt' l; + (r',rx)=mk_rbt' r + in + if lx > rx then + (B (paint Red l') a r', Suc(rx)) + else if lx < rx then + (B l' a (paint Red r'), Suc(lx)) + else + (B l' a r', Suc(lx)) + )" + +lemma mk_rbt'_refine_aux: "mk_rbt' t = (mk_rbt t, min_height t)" + apply(induction t) + apply(auto) + done + + +lemma mk_rbt'_refine: "fst (mk_rbt' t) = mk_rbt t" + apply(induction t) + by(auto simp:mk_rbt'_refine_aux) + + + +(*<*) +end +(*>*) diff --git a/10/Trie1.thy b/10/Trie1.thy new file mode 100644 index 0000000..c179f23 --- /dev/null +++ b/10/Trie1.thy @@ -0,0 +1,168 @@ +(* Author: Tobias Nipkow *) + +section "Trie1" + +theory Trie1 +imports Main +begin + +lemma [simp]: "(\x. x \ y) = False" +by blast + +hide_const (open) insert + +declare Let_def[simp] + + +subsection "Trie" + +datatype trie = Leaf | Node bool "trie * trie" + +fun isin :: "trie \ bool list \ bool" where +"isin Leaf ks = False" | +"isin (Node b (l,r)) ks = + (case ks of + [] \ b | + k#ks \ isin (if k then r else l) ks)" + +fun insert :: "bool list \ trie \ trie" where +"insert [] Leaf = Node True (Leaf,Leaf)" | +"insert [] (Node b lr) = Node True lr" | +"insert (k#ks) Leaf = + Node False (if k then (Leaf, insert ks Leaf) + else (insert ks Leaf, Leaf))" | +"insert (k#ks) (Node b (l,r)) = + Node b (if k then (l, insert ks r) + else (insert ks l, r))" + +lemma isin_insert: "isin (insert as t) bs = (as = bs \ isin t bs)" +apply(induction as t arbitrary: bs rule: insert.induct) +apply (auto split: list.splits) +done + +text \A simple implementation of delete; does not shrink the trie!\ + +fun delete :: "bool list \ trie \ trie" where +"delete ks Leaf = Leaf" | +"delete ks (Node b (l,r)) = + (case ks of + [] \ Node False (l,r) | + k#ks' \ Node b (if k then (l, delete ks' r) else (delete ks' l, r)))" + +lemma "isin (delete as t) bs = (as \ bs \ isin t bs)" +apply(induction as t arbitrary: bs rule: delete.induct) +apply (auto split: list.splits) +done + +fun node :: "bool \ trie * trie \ trie" where +"node b lr = (if \ b \ lr = (Leaf,Leaf) then Leaf else Node b lr)" + +fun delete2 :: "bool list \ trie \ trie" where +"delete2 ks Leaf = Leaf" | +"delete2 ks (Node b (l,r)) = + (case ks of + [] \ node False (l,r) | + k#ks' \ node b (if k then (l, delete2 ks' r) else (delete2 ks' l, r)))" + +lemma "isin (delete2 as t) bs = isin (delete as t) bs" +apply(induction as t arbitrary: bs rule: delete2.induct) + apply simp +apply (force split: list.splits) +done + + +subsection "Patricia Trie" + +datatype ptrie = LeafP | NodeP "bool list" bool "ptrie * ptrie" + +fun isinP :: "ptrie \ bool list \ bool" where +"isinP LeafP ks = False" | +"isinP (NodeP ps b (l,r)) ks = + (let n = length ps in + if ps = take n ks + then case drop n ks of [] \ b | k#ks' \ isinP (if k then r else l) ks' + else False)" + +fun split where +"split [] ys = ([],[],ys)" | +"split xs [] = ([],xs,[])" | +"split (x#xs) (y#ys) = + (if x\y then ([],x#xs,y#ys) + else let (ps,xs',ys') = split xs ys in (x#ps,xs',ys'))" + +fun insertP :: "bool list \ ptrie \ ptrie" where +"insertP ks LeafP = NodeP ks True (LeafP,LeafP)" | +"insertP ks (NodeP ps b (l,r)) = + (case split ks ps of + (qs,k#ks',p#ps') \ + let tp = NodeP ps' b (l,r); tk = NodeP ks' True (LeafP,LeafP) in + NodeP qs False (if k then (tp,tk) else (tk,tp)) | + (qs,k#ks',[]) \ + NodeP ps b (if k then (l, insertP ks' r) else (insertP ks' l, r)) | + (qs,[],p#ps') \ + let t = NodeP ps' b (l,r) in + NodeP qs True (if p then (LeafP,t) else (t,LeafP)) | + (qs,[],[]) \ NodeP ps True (l,r))" + +fun prefix_trie :: "bool list \ trie \ trie" where +"prefix_trie [] t = t" | +"prefix_trie (k#ks) t = + (let t' = prefix_trie ks t in Node False (if k then (Leaf,t') else (t',Leaf)))" + +fun abs_ptrie :: "ptrie \ trie" where +"abs_ptrie LeafP = Leaf" | +"abs_ptrie (NodeP ps b (l,r)) = prefix_trie ps (Node b (abs_ptrie l, abs_ptrie r))" + +lemma isin_prefix_trie: "isin (prefix_trie ps t) ks = + (length ks \ length ps \ + (let n = length ps in ps = take n ks \ isin t (drop n ks)))" +apply(induction ps arbitrary: ks) +apply(auto split: list.split) +done + +lemma isinP: "isinP t ks = isin (abs_ptrie t) ks" +apply(induction t arbitrary: ks rule: abs_ptrie.induct) + apply(auto simp: isin_prefix_trie split: list.split) + using nat_le_linear apply force +using nat_le_linear apply force +done + +lemma prefix_trie_Leafs: "prefix_trie ks (Node True (Leaf,Leaf)) = insert ks Leaf" +apply(induction ks) +apply auto +done + +lemma insert_prefix_trie_same: + "insert ps (prefix_trie ps (Node b lr)) = prefix_trie ps (Node True lr)" +apply(induction ps) +apply auto +done + +lemma insert_append: "insert (ks @ k # ks') (prefix_trie ks (Node b (t1,t2))) = + prefix_trie ks (Node b (if k then (t1, insert ks' t2) else (insert ks' t1, t2)))" +apply(induction ks) +apply auto +done + +lemma prefix_trie_append: "prefix_trie (ps @ qs) t = prefix_trie ps (prefix_trie qs t)" +apply(induction ps) +apply auto +done + +lemma split_if: "split ks ps = (qs, ks', ps') \ + ks = qs @ ks' \ ps = qs @ ps' \ (ks' \ [] \ ps' \ [] \ hd ks' \ hd ps')" +apply(induction ks ps arbitrary: qs ks' ps' rule: split.induct) +apply(auto split: prod.splits if_splits) +done + +lemma abs_ptrie_insertP: + "abs_ptrie (insertP ks t) = insert ks (abs_ptrie t)" +apply(induction t arbitrary: ks) +apply(auto simp: prefix_trie_Leafs insert_prefix_trie_same insert_append prefix_trie_append + dest!: split_if split: list.split prod.split) +done + +corollary isinP_insertP: "isinP (insertP ks t) ks' = (ks=ks' \ isinP t ks')" +by (simp add: isin_insert isinP abs_ptrie_insertP) + +end diff --git a/10/ex10.pdf b/10/ex10.pdf new file mode 100644 index 0000000000000000000000000000000000000000..aa869af8320e20225e2e7528bd6bd20b4c813459 GIT binary patch literal 125177 zcma&sQ*b6+&?xE{Z!)oM+qP}n&cwED+qP{x6PpuEY`^C_*jN9qI=ioW)w=Ggr$MeH zCPB~4zz!f^Tp8N{uoE#6Ihxo2`1k;fa^?<}u2w`WtSp>F|4#rIC9LgS&7FxDCG3n{ z&Be@39nH)E0s;URS7&o$dw}QGwa%`~b|-SctKm>rhDsicq%U~Zg)C*p{H~tka@1$4 zEN?1=WG(qJIrZnmN4{WWz=D2JvKiIaea8eLy^d(CIK 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Node "trie * trie" + +fun is_trie :: "nat \ trie \ bool" + where + "is_trie 0 (Node(x,y)) \ False" +| "is_trie 0 _ \ True" +| "is_trie (Suc n) LeafF \ True" +| "is_trie (Suc n) LeafT \ False" +| "is_trie (Suc n) (Node (x,y)) \ is_trie n x \ is_trie n y" + +text \Hint: The following should evaluate to true!\ +value "is_trie 42 LeafF" +value "is_trie 2 (Node (LeafF,Node (LeafT,LeafF)))" +text \Whereas these should be false\ +value "is_trie 42 LeafT" -- \Wrong key length\ +value "is_trie 2 (Node (LeafT,Node (LeafT,LeafF)))" -- \Wrong key length\ +value "is_trie 1 (Node (LeafT,Node (LeafF,LeafF)))" -- \Superfluous node\ + +fun isin :: "trie \ bool list \ bool" + where + "isin (Node (x,y)) [] = False" +| "isin LeafT [] = True" +| "isin LeafF _ = False" +| "isin (Node (x,y)) (b#bx) = (if b then isin y bx else isin x bx)" +| "isin LeafT (b#bx) = False" + +value "isin (Node (LeafF,Node (LeafT,LeafF))) [True, False]" +value "isin LeafT []" +value "isin LeafF []" + +fun ins :: "bool list \ trie \ trie" + where + "ins [] _ = LeafT" +| "ins (b#bx) LeafF = (if b then (Node (LeafF, ins bx LeafF)) else (Node ((ins bx LeafF), LeafF)))" +| "ins (b#bx) (Node (x,y)) = (if b then (Node (x, ins bx y)) else (Node ((ins bx x),y)))" +| "ins _ x = x" + +value "ins [] LeafF" +value "ins [] LeafT" +value "ins [] (Node(LeafF,LeafT))" +value "ins [True] (Node(LeafF,LeafT))" +value "ins [False] (Node(LeafF,LeafT))" +value "ins [False, False] (Node (LeafF,Node (LeafF,LeafT)))" +value "ins [False] LeafF" +value "ins [True,True,True] (Node (LeafF,Node (LeafT,LeafF)))" + + +lemma aux1: "\b bx x y bs. + (\bs n. is_trie n y \ length bx = n \ isin (ins bx y) bs = (bx = bs \ isin y bs)) \ + (\bs n. False \ is_trie n x \ length bx = n \ isin (ins bx x) bs = (bx = bs \ isin x bs)) \ + b \ isin (Node (x, ins bx y)) bs \ + \ isin (Node (x, y)) bs \ is_trie (length bx) x \ is_trie (length bx) y \ True # bx = bs" + proof - +fix b :: bool and bx :: "bool list" and x :: trie and y :: trie and bsa :: "bool list" +assume a1: "isin (Node (x, ins bx y)) bsa" +assume a2: "\ isin (Node (x, y)) bsa" +assume a3: "\bs n. \is_trie n y; length bx = n\ \ isin (ins bx y) bs = (bx = bs \ isin y bs)" +assume a4: "is_trie (length bx) y" +have "bsa \ []" +using a1 by force +then show "True # bx = bsa" + using a4 a3 a2 a1 by (metis (full_types) isin.simps(4) list.exhaust) +qed + + + +lemma isin_ins1: + assumes "is_trie n t" and "length as = n" + shows "isin (ins as t) bs = (as = bs \ isin t bs)" +proof - + show ?thesis using assms + apply (induction as t arbitrary: bs n rule: ins.induct) + apply (auto split:if_splits) + using isin.elims(2) apply blast + using is_trie.simps(1) isin.elims(2) apply blast + apply (metis (full_types) is_trie.simps(2) is_trie.simps(4) isin.simps(1) isin.simps(3) isin.simps(4) length_0_conv length_Cons list.exhaust) + using is_trie.elims(3) apply blast + apply (smt Suc_length_conv ins.simps(2) is_trie.simps(2) is_trie.simps(4) isin.simps(1) isin.simps(3) isin.simps(4) length_0_conv length_Cons list.exhaust list.inject neq_Nil_conv) + using is_trie.elims(3) apply blast + + using aux1 apply metis + apply (metis isin.elims(2) isin.simps(4) trie.distinct(5)) + apply (smt isin.elims(2) isin.simps(4) trie.distinct(5)) + by (metis isin.elims(2) isin.simps(4) trie.distinct(5)) +qed + +lemma isin_ins2: + assumes "is_trie n t" and "length as = n" + shows "is_trie n (ins as t)" +proof - + show ?thesis using assms + apply (induction as t arbitrary: n rule: ins.induct) + apply (auto split:if_splits) + using is_trie.elims(3) apply blast + using is_trie.elims(3) apply blast + using is_trie.elims(3) apply blast + using is_trie.elims(3) by blast +qed + +lemma isin_ins: + assumes "is_trie n t" and "length as = n" + shows "isin (ins as t) bs = (as = bs \ isin t bs)" + and "is_trie n (ins as t)" + using assms(1) assms(2) isin_ins1 apply auto[1] + by (simp add: assms(1) assms(2) isin_ins2) + +fun node :: "trie \ trie \ trie" where + "node (x,y) = (if (x = LeafF \ y = LeafF) then LeafF else (Node(x,y)))" + + +fun delete2 :: "bool list \ trie \ trie" where + "delete2 [] _ = LeafF" +| "delete2 (b#bx) LeafF = LeafF" +| "delete2 (b#bx) (Node (x,y)) = (if b then (node (x, delete2 bx y)) else (node ((delete2 bx x),y)))" +| "delete2 _ x = x" + +lemma delaux1: + assumes "is_trie n t" and "length as = n" + shows "isin (delete2 as t) bs = (as\bs \ isin t bs)" +proof - + show ?thesis using assms + apply (induction as t arbitrary: bs n rule: delete2.induct) + apply (auto split!:if_splits) + apply (metis gen_length_code(1) ins.simps(1) is_trie.simps(2) isin.simps(3) isin_ins1 length_code) + apply (metis isin.elims(2) isin.simps(4) trie.distinct(5)) + apply (metis isin.elims(2) isin.simps(4) trie.distinct(5)) + apply (metis (full_types) isin.simps(1) isin.simps(3) isin.simps(4) list.exhaust) + apply (smt isin.elims(2) isin.simps(4) trie.simps(7)) + apply (smt isin.elims(2) isin.simps(4) trie.simps(7)) + apply (metis isin.elims(2) isin.simps(4) trie.distinct(5)) + apply (metis isin.elims(2) isin.simps(4) trie.distinct(5)) + apply (metis (full_types) isin.simps(1) isin.simps(3) isin.simps(4) list.exhaust) + apply (smt isin.elims(2) isin.simps(4) trie.simps(7)) + by (smt isin.elims(2) isin.simps(4) trie.simps(7)) +qed + +lemma delaux2: + assumes "is_trie n t" + shows"(is_trie n (delete2 as t))" +proof - + show ?thesis using assms + apply (induction as t arbitrary: n rule: delete2.induct) + apply (auto split!:if_splits) + using is_trie.elims(3) apply blast + using is_trie.elims(3) apply blast + apply (metis is_trie.simps(1) is_trie.simps(6) old.nat.exhaust) + apply (metis is_trie.simps(1) is_trie.simps(6) not0_implies_Suc) + apply (metis is_trie.elims(2) is_trie.simps(4) trie.simps(7)) + apply (metis is_trie.simps(1) is_trie.simps(6) old.nat.exhaust) + by (metis is_trie.simps(1) is_trie.simps(6) not0_implies_Suc) +qed + +lemma + assumes "is_trie n t" and "length as = n" + shows "isin (delete2 as t) bs = (as\bs \ isin t bs)" + and "(is_trie n (delete2 as t))" + using assms(1) assms(2) delaux1 apply blast + by (simp add: assms(1) delaux2) + +end \ No newline at end of file diff --git a/10/hw10_1_tmpl.thy b/10/hw10_1_tmpl.thy new file mode 100644 index 0000000..bf61fcd --- /dev/null +++ b/10/hw10_1_tmpl.thy @@ -0,0 +1,134 @@ +(*<*) +theory hw10_1_tmpl +imports Main +begin +(*>*) + + +text \\NumHomework{Tries with Same-Length Keys}{22.~6.~2018} + + Consider the following trie datatype: +\ + +datatype trie = LeafF | LeafT | Node "trie * trie" + +text \It is meant to store keys of the same length only. + Thus, the @{const Node} constructor stores inner nodes, and there are two + types of leaves, @{const LeafF} if this path is not in the set, + and @{const LeafT} if it is in the set. + + Define an invariant \is_trie N t\ that states that all keys in \t\ + have length \N\, and that there are no superfluous nodes, i.e., + no nodes of the form @{term \Node (LeafF, LeafF)\}. +\ + +fun is_trie :: "nat \ trie \ bool" +where + "is_trie 0 (Node (x,y)) \ False" +| "is_trie 0 _ \ True" +| "is_trie _ (Node (LeafF, LeafF)) \ False" +| "is_trie (Suc n) LeafT \ False" +| "is_trie (Suc n) LeafF \ True" +(*| "is_trie (Suc 0) (Node (x,y)) \ ((x = LeafF) \ (x = LeafT)) \ ((y = LeafF) \ (y = LeafT))"*) +| "is_trie (Suc n) (Node (x,y)) \ is_trie n x \ is_trie n y" + +text \Hint: The following should evaluate to true!\ +value "is_trie 42 LeafF" +value "is_trie 2 (Node (LeafF,Node (LeafT,LeafF)))" +text \Whereas these should be false\ +value "is_trie 42 LeafT" -- \Wrong key length\ +value "is_trie 2 (Node (LeafT,Node (LeafT,LeafF)))" -- \Wrong key length\ +value "is_trie 1 (Node (LeafT,Node (LeafF,LeafF)))" -- \Superfluous node\ + +value "is_trie 1 (Node (LeafT,LeafF))" + + +text \Define membership, insert, and delete functions, and prove them correct! \ + +fun isin :: "trie \ bool list \ bool" + where + "isin (Node (x, y)) (b#[]) = (if b then (x = LeafT) else (y = LeafT))" +| "isin (Node (x, y)) (b#bs) = (if b then (isin x bs) else (isin y bs))" +| "isin LeafT [] = True" +| "isin _ _ = False" + +value "isin (Node (LeafF,Node (LeafT,LeafF))) [False, True]" + +fun ins :: "bool list \ trie \ trie" + where + "ins (b#[]) LeafF = (if b then (Node (LeafT, LeafF)) else (Node(LeafF, LeafT)))" +| "ins (b#[]) LeafT = (if b then (Node (LeafT, LeafF)) else (Node(LeafF, LeafT)))" +| "ins (b#[]) (Node(x,y)) = (if b then (Node (LeafT, y)) else (Node(x, LeafT)))" +| "ins [] x = x" +| "ins (b#bs) (Node(x,y)) = (if b then (Node ((ins bs x),y)) else (Node (x, (ins bs y))))" +| "ins (b#bs) LeafF = (if b then (Node ((ins bs LeafF),LeafF)) else (Node (LeafF, (ins bs LeafF))))" +| "ins (b#bs) LeafT = (if b then (Node ((ins bs LeafT),LeafF)) else (Node (LeafF, (ins bs LeafT))))" + +(*fun ins :: "bool list \ trie \ trie" + where + "ins [] x = LeafT" +| "ins (b#bs) (Node(x,y)) = (if b then (Node ((ins bs x),y)) else (Node (x, (ins bs y))))" +| "ins (b#bs) x = (if b then (Node ((ins bs x),LeafF)) else (Node (LeafF, (ins bs x))))"*) + +value "ins [True, False] (Node (LeafF,Node (LeafT,LeafF)))" +value "ins [False] LeafF" +value "ins [True,True,True] (Node (LeafF,Node (LeafT,LeafF)))" + + + +lemma isin_ins1: + assumes "is_trie n t" and "length as = n" + shows "isin (ins as t) bs = (as = bs \ isin t bs)" + apply (induction as t arbitrary: bs n rule: ins.induct) + + + + + + + +lemma isin_ins2: + assumes "is_trie n t" and "length as = n" + shows "is_trie n (ins as t)" +proof(induction as) + case Nil +then show ?case + by (simp add: assms(1)) +next +case (Cons a as) + then show ?case + by (metis One_nat_def ins.simps(2) ins.simps(3) is_trie.simps(2) isin.simps(4) isin_ins1 le_numeral_extra(4) length_Cons list.size(3) not_less_eq_eq not_one_le_zero order_antisym_conv zero_induct) +qed + + + + +lemma isin_ins: + assumes "is_trie n t" and "length as = n" + shows "isin (ins as t) bs = (as = bs \ isin t bs)" + and "is_trie n (ins as t)" + using assms(1) assms(2) isin_ins1 apply auto[1] + + +fun delete2 :: "bool list \ trie \ trie" where + "delete2 _ _ = undefined" + +lemma + assumes "is_trie n t" + shows "isin (delete2 as t) bs = (as\bs \ isin t bs)" + and "(is_trie n (delete2 as t))" + oops + +text \Hints: + \<^item> Like in the \delete2\ function for standard tries, you may want to define + a "smart-constructor" \node :: trie \ trie \ trie\ for nodes, + that constructs a node and handles the case that both successors are \LeafF\. + \<^item> Consider proving auxiliary lemmas about the smart-constructor, instead of + always unfolding it with the simplifier. +\ + + + +(*<*) +end +(*>*) diff --git a/10/hw10_1_tmpl.thy~ b/10/hw10_1_tmpl.thy~ new file mode 100644 index 0000000..d21ae20 --- /dev/null +++ b/10/hw10_1_tmpl.thy~ @@ -0,0 +1,144 @@ +(*<*) +theory hw10_1_tmpl +imports Main +begin +(*>*) + + +text \\NumHomework{Tries with Same-Length Keys}{22.~6.~2018} + + Consider the following trie datatype: +\ + +datatype trie = LeafF | LeafT | Node "trie * trie" + +text \It is meant to store keys of the same length only. + Thus, the @{const Node} constructor stores inner nodes, and there are two + types of leaves, @{const LeafF} if this path is not in the set, + and @{const LeafT} if it is in the set. + + Define an invariant \is_trie N t\ that states that all keys in \t\ + have length \N\, and that there are no superfluous nodes, i.e., + no nodes of the form @{term \Node (LeafF, LeafF)\}. +\ + +fun is_trie :: "nat \ trie \ bool" +where + "is_trie 0 LeafT \ True" +| "is_trie _ LeafF \ True" +| "is_trie _ (Node (LeafF, LeafF)) \ False" +| "is_trie (Suc n) LeafT \ False" +| "is_trie 0 (Node (x,y)) \ False" +| "is_trie (Suc n) (Node (x,y)) \ is_trie n x \ is_trie n y" + +text \Hint: The following should evaluate to true!\ +value "is_trie 42 LeafF" +value "is_trie 2 (Node (LeafF,Node (LeafT,LeafF)))" +text \Whereas these should be false\ +value "is_trie 42 LeafT" -- \Wrong key length\ +value "is_trie 2 (Node (LeafT,Node (LeafT,LeafF)))" -- \Wrong key length\ +value "is_trie 1 (Node (LeafT,Node (LeafF,LeafF)))" -- \Superfluous node\ + +value "is_trie 1 (Node (LeafT,LeafF))" + + +text \Define membership, insert, and delete functions, and prove them correct! \ + +fun isin :: "trie \ bool list \ bool" + where + "isin (Node (x, y)) (b#[]) = (if b then (x = LeafT) else (y = LeafT))" +| "isin (Node (x, y)) (b#bs) = (if b then (isin x bs) else (isin y bs))" +| "isin _ _ = False" + +value "isin (Node (LeafF,Node (LeafT,LeafF))) [False, True]" + +fun ins' :: "bool list \ trie \ trie" + where + "ins' (b#[]) LeafF = (if b then (Node (LeafT, LeafF)) else (Node(LeafF, LeafT)))" +| "ins' (b#[]) LeafT = (if b then (Node (LeafT, LeafF)) else (Node(LeafF, LeafT)))" +| "ins' (b#[]) (Node(x,y)) = (if b then (Node (LeafT, y)) else (Node(x, LeafT)))" +| "ins' [] x = x" +| "ins' (b#bs) (Node(x,y)) = (if b then (Node ((ins' bs x),y)) else (Node (x, (ins' bs y))))" +| "ins' (b#bs) LeafF = (if b then (Node ((ins' bs LeafF),LeafF)) else (Node (LeafF, (ins' bs LeafF))))" +| "ins' (b#bs) LeafT = (if b then (Node ((ins' bs LeafT),LeafF)) else (Node (LeafF, (ins' bs LeafT))))" + +fun ins :: "bool list \ trie \ trie" + where + "ins [] x = LeafT" +| "ins (b#bs) (Node(x,y)) = (if b then (Node ((ins bs x),y)) else (Node (x, (ins bs y))))" +| "ins (b#bs) x = (if b then (Node ((ins bs x),LeafF)) else (Node (LeafF, (ins bs x))))" + +value "ins [True, False] (Node (LeafF,Node (LeafT,LeafF)))" +value "ins [False] LeafF" +value "ins [True,True,True] (Node (LeafF,Node (LeafT,LeafF)))" + +lemma isin_ins1: + assumes "is_trie n t" and "length as = n" + shows "isin (ins as t) bs = (as = bs \ isin t bs)" +proof (induction as t arbitrary: bs n rule: ins.induct) +case (1 x) + then show ?case + apply(auto) + +next +case (2 b bs x y) +then show ?case sorry +next +case ("3_1" b bs) + then show ?case sorry +next + case ("3_2" b bs) + then show ?case sorry +qed + + + + + + + +lemma isin_ins2: + assumes "is_trie n t" and "length as = n" + shows "is_trie n (ins as t)" +proof(induction as) + case Nil +then show ?case + by (simp add: assms(1)) +next +case (Cons a as) + then show ?case + by (metis One_nat_def ins.simps(2) ins.simps(3) is_trie.simps(2) isin.simps(4) isin_ins1 le_numeral_extra(4) length_Cons list.size(3) not_less_eq_eq not_one_le_zero order_antisym_conv zero_induct) +qed + + + + +lemma isin_ins: + assumes "is_trie n t" and "length as = n" + shows "isin (ins as t) bs = (as = bs \ isin t bs)" + and "is_trie n (ins as t)" + using assms(1) assms(2) isin_ins1 apply auto[1] + + +fun delete2 :: "bool list \ trie \ trie" where + "delete2 _ _ = undefined" + +lemma + assumes "is_trie n t" + shows "isin (delete2 as t) bs = (as\bs \ isin t bs)" + and "(is_trie n (delete2 as t))" + oops + +text \Hints: + \<^item> Like in the \delete2\ function for standard tries, you may want to define + a "smart-constructor" \node :: trie \ trie \ trie\ for nodes, + that constructs a node and handles the case that both successors are \LeafF\. + \<^item> Consider proving auxiliary lemmas about the smart-constructor, instead of + always unfolding it with the simplifier. +\ + + + +(*<*) +end +(*>*) diff --git a/10/hw10_2_tmpl.thy b/10/hw10_2_tmpl.thy new file mode 100644 index 0000000..c077b4d --- /dev/null +++ b/10/hw10_2_tmpl.thy @@ -0,0 +1,54 @@ +(*<*) +theory hw10_2_tmpl +imports Trie1 "HOL-Library.List_lexord" +begin +(*>*) +(* Incomplete *) + +fun path_trav :: "trie \ bool list \ bool list list" where + "path_trav Leaf bs = []" +| "path_trav (Node b (x,y)) bs = (if b then ([bs] @ (path_trav x (bs@[True])) @ (path_trav y (bs@[False]))) + else ( (path_trav x (bs@[True])) @ (path_trav y (bs@[False]))))" + +fun enum :: "trie \ bool list list" + where + "enum Leaf = []" + | "enum t = path_trav t []" + +value "enum (Node False ((Node True (Leaf, Leaf)),Leaf))" +value "path_trav (Node False ((Node True (Leaf, Leaf)),Leaf)) []" + +lemma enum_correct1: "set (enum t) = { xs. isin t xs }" +proof(induction rule: isin.induct) + case (1 ks) +then show ?case by auto +next +case (2 b l r ks) + then show ?case +qed + + oops + +lemma enum_correct2: "sorted_wrt op< (enum t)" + + oops + +lemma enum_correct: + "set (enum t) = { xs. isin t xs }" and "sorted_wrt op< (enum t)" + oops + + + +text \ + Note that Booleans are ordered by \False < True\, and + that we imported @{theory "List_lexord"}, which defines + a lexicographic ordering on lists, if the elements are ordered. +\ + +value "[True,True,False] < [True,True,True,True]" + + + +(*<*) +end +(*>*) diff --git a/10/hw10_2_tmpl.thy~ b/10/hw10_2_tmpl.thy~ new file mode 100644 index 0000000..f80abbf --- /dev/null +++ b/10/hw10_2_tmpl.thy~ @@ -0,0 +1,35 @@ +(*<*) +theory hw10_2_tmpl +imports Trie1 "HOL-Library.List_lexord" +begin +(*>*) + +text \\NumHomework{Enumeration of Keys in Tries}{22.~6.~2018} + + Write a function that enumerates all keys in a trie, in lexicographic order! + Prove it correct. +\ + + + fun enum :: "trie \ bool list list" + where + "enum _ = undefined" + + + lemma enum_correct: + "set (enum t) = { xs. isin t xs }" and "sorted_wrt op< (enum t)" + oops + +text \ + Note that Booleans are ordered by \False < True\, and + that we imported @{theory "List_lexord"}, which defines + a lexicographic ordering on lists, if the elements are ordered. +\ + +value "[True,True,False] < [True,True,True,True]" + + + +(*<*) +end +(*>*) diff --git a/11/ex11_tmpl.thy b/11/ex11_tmpl.thy new file mode 100644 index 0000000..ec762b0 --- /dev/null +++ b/11/ex11_tmpl.thy @@ -0,0 +1,197 @@ +(*<*) +theory ex11_tmpl +imports + "HOL-Data_Structures.Leftist_Heap" +begin +(*>*) +text {* \ExerciseSheet{11}{22.~6.~2018} *} + +text \ + \Exercise{Insert for Leftist Heap} + + \<^item> Define a function to directly insert an element into a leftist heap. + Do not construct an intermediate heap like insert via merge does! + + \<^item> Show that your function is correct + + \<^item> Define a timing function for your insert function, and show that it is + linearly bounded by the rank of the tree. +\ + +fun lh_insert :: "'a::ord \ 'a lheap \ 'a lheap" + where + "lh_insert _ _ = undefined" + +lemma mset_lh_insert: "mset_tree (lh_insert x t) = mset_tree t + {# x #}" + sorry + +lemma "heap t \ heap (lh_insert x t)" + sorry + +lemma "ltree t \ ltree (lh_insert x t)" + sorry + + +fun t_lh_insert :: "'a::ord \ 'a lheap \ nat" + where + "t_lh_insert _ _ = undefined" + +lemma "t_lh_insert x t \ rank t + 1" + sorry + +text \ + \Exercise{Bootstrapping a Priority Queue} + + Given a generic priority queue implementation with + \O(1)\ \empty\, \is_empty\ operations, \O(f\<^sub>1 n)\ insert, + and \O(f\<^sub>2 n)\ \get_min\ and \del_min\ operations. + + Derive an implementation with \O(1)\ \get_min\, and the + asymptotic complexities of the other operations unchanged! + + Hint: Store the current minimal element! As you know nothing + about \f\<^sub>1\ and \f\<^sub>2\, you must not use \get_min/del_min\ + in your new \insert\ operation, and vice versa! +\ + +text \For technical reasons, you have to define the new implementations type + outside the locale!\ +datatype ('a,'s) bs_pq = PUT_CONSTRUCTORS_HERE + +locale Bs_Priority_Queue = + orig: Priority_Queue where + empty = orig_empty and + is_empty = orig_is_empty and + insert = orig_insert and + get_min = orig_get_min and + del_min = orig_del_min and + invar = orig_invar and + mset = orig_mset + for orig_empty orig_is_empty orig_insert orig_get_min orig_del_min orig_invar + and orig_mset :: "'s \ 'a::linorder multiset" +begin + text \In here, the original implementation is available with the prefix \orig\, e.g. \ + term orig_empty term orig_invar + thm orig.invar_empty + + definition empty :: "('a,'s) bs_pq" + where "empty = undefined" + + fun is_empty :: "('a,'s) bs_pq \ bool" + where + "is_empty _ \ undefined" + + fun insert :: "'a \ ('a,'s) bs_pq \ ('a,'s) bs_pq" + where + "insert _ _ = undefined" + + fun get_min :: "('a,'s) bs_pq \ 'a" + where + "get_min _ = undefined" + + fun del_min :: "('a,'s) bs_pq \ ('a,'s) bs_pq" + where + "del_min _ = undefined" + + fun invar :: "('a,'s) bs_pq \ bool" + where + "invar _ = undefined" + + fun mset :: "('a,'s) bs_pq \ 'a multiset" + where + "mset _ = undefined" + + lemmas [simp] = orig.is_empty orig.mset_get_min orig.mset_del_min + orig.mset_insert orig.mset_empty + orig.invar_empty orig.invar_insert orig.invar_del_min + + text \Show that your new implementation satisfies the priority queue interface!\ + sublocale Priority_Queue empty is_empty insert get_min del_min invar mset + apply unfold_locales + proof goal_cases + case 1 + then show ?case sorry + next + case (2 q) -- \and so on\ + oops + + +end + +text\ +\Homework{Heap}{29.~6.~2018} + +A binary tree can be encoded as an array \[a\<^sub>1,...,a\<^sub>n]\, such that +the parent of node \a\<^sub>i\ is node \a\<^sub>(\<^sub>i \<^sub>d\<^sub>i\<^sub>v \<^sub>2\<^sub>)\. + +Thus, for a heap, each node is greater than or equal to its parent: +\ + +definition parent :: "nat \ nat" where "parent i \ (i+1) div 2 - 1" +definition is_heap :: "'a::linorder list \ bool" + where "is_heap h \ \i h!parent i" + + +text \A heap with a single defect at index \j\ is characterized as follows: + The heap property holds for all elements except \j\, + and the children of \j\ must also be greater than their grand-parent. +\ +definition is_heap_except :: "nat \ 'a::linorder list \ bool" where + "is_heap_except j h \ + (\ij \ h!i \ h!(parent i)) + \ (\i h!i \ h!(parent j))" + + + +text \ + The function \sift_up\ corrects a single defect in a heap by + iterated swapping of the defect with its parent, until the heap property + is restored. +\ + +definition "swap i j h \ h[i:=h!j, j:=h!i]" + +(* Required for termination proof of sift_up *) +lemma parent_less[simp]: "i\0 \ parent i < i" + by (auto simp: parent_def) + +fun sift_up where + "sift_up h 0 = h" +| "sift_up h i = ( + if h!i \ h!parent i then h + else sift_up (swap i (parent i) h) (parent i))" + + +text \Show that @{const sift_up} restores the heap + and preserves the multiset of elements in the heap\ + +lemma sift_up_restore_heap: + "is_heap_except j h \ j is_heap (sift_up h j)" + sorry + + +lemma sift_up_mset: "j mset (sift_up h j) = mset h" + sorry + +text \For \textbf{3 bonus points}, add an empty, insert, and get-min + function to the heap implementation, and prove their essential properties. +\ + +definition emp :: "'a::linorder list" where "emp \ undefined" +definition get_min :: "'a::linorder list \ 'a" where "get_min h \ undefined" +definition ins :: "'a::linorder \ 'a list \ 'a list" where "ins x h = undefined" + +lemma invar_empty: "is_heap emp" sorry + +lemma mset_empty: "mset emp = {#}" sorry + +lemma mset_get_min: "is_heap h \ mset h \ {#} \ get_min h = Min_mset (mset h)" sorry + +lemma invar_ins: "is_heap h \ is_heap (ins x h)" sorry + +lemma mset_ins: "mset (ins x h) = mset h + {#x#}" sorry + + +(*<*) +end +(*>*) diff --git a/12/ex12.pdf b/12/ex12.pdf new file mode 100644 index 0000000000000000000000000000000000000000..54c49a6aa4a314048ab0eb5429ff16188a92e7d6 GIT binary patch literal 126412 zcma&qL$EMP&?e|(+qP}nwr$(CZQHhO+tzn%8-06X{@L^_x>i|L5xL0KlP{@)h!`y+ z9V-;+!pi6d6e|G(fxVFx6b}y+y^N`yxr+q>6C)!V!T)oh=*28;Tuhw^=*4UdT}(wx zjqOcLq4@ZqoL!tu4Q-)3wobI9owvo1`mWRqUP07mCZBu8TRTl~?MW(tobDp9mlohSMipl{TXRgm`&>zE81XEM5tVH~H%=di=bdUtX*gFcpj{ zL$6Qyn9~?~mc@`KUEQ41duQ#%)J{%JMl|_qtJ!u^iW%97{z~ikI`>VN)pjS2|1c#> zpNgFSPCnv24BurGbLp=*8t`HTtw|rRt_p$`8H(T7|E4hHrd-&-Y`XJ)Ys?AD8`br`s<_R!%e)Tez^7f}!Gx~|iZ?>13tvk$ttr7GJls0n8% z2=2T0r_PpGO&@H(w9s@L&@=?bT42uuM5HEh>KZ4cX@urci?y)Gw;19A5F=b0H1V$~ z6fF{Ajp@)wAufCV6K72I*IqpwJiGrYKe={eF&jhOOp_$|x{WQ~V>}&!!h^PG^Pn)- zaIF~d9P{AQ7jX@RHp0e}ivlAT>DLe0SLKa#7xhKAzdf4mfm?S|6sgwqi&g`V_Piz6 zp36a=bGo6?r>4d`#Xp^XC7g15@)%8XdJzdPeO^(UdkU}meJ9#cYh9laWkJTDoeK9j zyBVS0V#`q%+DECgvq7J 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+imports + Complex_Main + "HOL-Library.Multiset" +begin +(*>*) +text {* \ExerciseSheet{12}{29.~6.~2018} *} + +text \ + \Exercise{Sparse Binary Numbers} + + Implement operations carry, inc, and add on sparse binary numbers, + analogously to the operations link, ins, and meld on binomial heaps. + + Show that the operations have logarithmic worst-case complexity. +\ + +type_synonym rank = nat +type_synonym snat = "rank list" + +abbreviation invar :: "snat \ bool" where "invar s \ sorted_wrt op< s" +definition \ :: "snat \ nat" where "\ s = (\i\s. 2^i)" + +lemmas [simp] = sorted_wrt_Cons sorted_wrt_append + + +(*<*) +lemma \_Nil[simp]: "\ [] = 0" + unfolding \_def by auto + +lemma \_Cons[simp]: "\ (r#rs) = 2^r + \ rs" + unfolding \_def by auto + +lemma \_append[simp]: "\ (rs\<^sub>1@rs\<^sub>2) = \ rs\<^sub>1 + \ rs\<^sub>2" + unfolding \_def by auto + +(*>*) + +fun carry :: "rank \ snat \ snat" +(*<*) + where + "carry r [] = [r]" +| "carry r (r'#rest) = (if r*) + +lemma carry_invar[simp]: +(*<*) + assumes "invar rs" + shows "invar (carry r rs)" + using assms + apply (induction r rs rule: carry.induct) + apply (auto) + done +(*>*) + +lemma carry_\: +(*<*) + assumes "invar rs" + assumes "\r'\set rs. r\r'" + shows "\ (carry r rs) = 2^r + \ rs" + using assms + by (induction r rs rule: carry.induct) force+ +(*>*) + +(*<*) +lemma carry_rank_bound: + assumes "r' \ set (carry r rs)" + assumes "\r'\set rs. r\<^sub>0 < r'" + assumes "r\<^sub>0 < r" + shows "r\<^sub>0 < r'" + using assms + by (induction r rs rule: carry.induct) (auto split: if_splits) +(*>*) + + +definition inc :: "snat \ snat" +(*<*) + where "inc rs = carry 0 rs" +(*>*) + +lemma inc_invar[simp]: "invar rs \ invar (inc rs)" +(*<*) + unfolding inc_def by auto +(*>*) + +lemma inc_\[simp]: "invar rs \ \ (inc rs) = Suc (\ rs)" +(*<*) + unfolding inc_def by (auto simp: carry_\) +(*>*) + +fun add :: "snat \ snat \ snat" +(*<*) + where + "add rs [] = rs" +| "add [] rs = rs" +| "add (r\<^sub>1#rs\<^sub>1) (r\<^sub>2#rs\<^sub>2) = ( + if r\<^sub>1 < r\<^sub>2 then r\<^sub>1#add rs\<^sub>1 (r\<^sub>2#rs\<^sub>2) + else if r\<^sub>21 then r\<^sub>2#add (r\<^sub>1#rs\<^sub>1) rs\<^sub>2 + else carry (r\<^sub>1 + 1) (add rs\<^sub>1 rs\<^sub>2) + )" +(*>*) + +(*<*) +lemma add_rank_bound: + assumes "r' \ set (add rs\<^sub>1 rs\<^sub>2)" + assumes "\r'\set rs\<^sub>1. r < r'" + assumes "\r'\set rs\<^sub>2. r < r'" + shows "r < r'" + using assms + apply (induction rs\<^sub>1 rs\<^sub>2 arbitrary: r' rule: add.induct) + apply (auto split: if_splits simp: carry_rank_bound) + done +(*>*) + +lemma add_invar[simp]: + assumes "invar rs\<^sub>1" + assumes "invar rs\<^sub>2" + shows "invar (add rs\<^sub>1 rs\<^sub>2)" +(*<*) + using assms +proof (induction rs\<^sub>1 rs\<^sub>2 rule: add.induct) + case (3 r\<^sub>1 rs\<^sub>1 r\<^sub>2 rs\<^sub>2) + + consider (LT) "r\<^sub>1 < r\<^sub>2" + | (GT) "r\<^sub>1 > r\<^sub>2" + | (EQ) "r\<^sub>1 = r\<^sub>2" + using antisym_conv3 by blast + then show ?case proof cases + case LT + then show ?thesis using 3 + by (force elim!: add_rank_bound) + next + case GT + then show ?thesis using 3 + by (force elim!: add_rank_bound) + next + case [simp]: EQ + + from "3.IH"(3) "3.prems" have [simp]: "invar (add rs\<^sub>1 rs\<^sub>2)" + by auto + + have "r\<^sub>2 < r'" if "r' \ set (add rs\<^sub>1 rs\<^sub>2)" for r' + using that + apply (rule add_rank_bound) + using "3.prems" by auto + with EQ show ?thesis by auto + qed +qed simp_all +(*>*) + +lemma add_\[simp]: + assumes "invar rs\<^sub>1" + assumes "invar rs\<^sub>2" + shows "\ (add rs\<^sub>1 rs\<^sub>2) = \ rs\<^sub>1 + \ rs\<^sub>2" +(*<*) + using assms + apply (induction rs\<^sub>1 rs\<^sub>2 rule: add.induct) + apply (auto simp: carry_\ Suc_leI add_rank_bound) + done +(*>*) + + +lemma size_snat: + assumes "invar rs" + shows "2^length rs \ \ rs + 1" +(*<*) +proof - + have "(2::nat)^length rs = (\i\{0.. \ \ rs + 1" + using assms unfolding \_def + by (simp add: sorted_wrt_less_sum_mono_lowerbound) + finally show ?thesis . +qed +(*>*) + +fun t_carry :: "rank \ snat \ nat" +(*<*) + where + "t_carry r [] = 1" +| "t_carry r (r'#rest) = (if r*) + +definition t_inc :: "snat \ nat" +(*<*) + where "t_inc rs = t_carry 0 rs" +(*>*) + +lemma t_inc_bound: + assumes "invar rs" + shows "t_inc rs \ log 2 (\ rs + 1) + 1" +(*<*) +proof - + have "t_carry r rs \ length rs + 1" for r + by (induction r rs rule: t_carry.induct) auto + hence "t_inc rs \ length rs + 1" unfolding t_inc_def by auto + hence "(2::nat)^t_inc rs \ 2^(length rs + 1)" + by (rule power_increasing) auto + also have "\ \ 2*(\ rs + 1)" using size_snat[OF \invar rs\] by auto + finally have "2 ^ t_inc rs \ 2 * (\ rs + 1)" . + from le_log2_of_power[OF this] show ?thesis + by (simp only: log_mult of_nat_mult) auto +qed +(*>*) + +fun t_add :: "snat \ snat \ nat" +(*<*) + where + "t_add rs [] = 1" +| "t_add [] rs = 1" +| "t_add (r\<^sub>1#rs\<^sub>1) (r\<^sub>2#rs\<^sub>2) = 1 + ( + if r\<^sub>1 < r\<^sub>2 then t_add rs\<^sub>1 (r\<^sub>2#rs\<^sub>2) + else if r\<^sub>21 then t_add (r\<^sub>1#rs\<^sub>1) rs\<^sub>2 + else t_carry (r\<^sub>1 + 1) (add rs\<^sub>1 rs\<^sub>2) + t_add rs\<^sub>1 rs\<^sub>2 + )" +(*>*) + +(*<*) +lemma l_carry_estimate: + "t_carry r rs + length (carry r rs) = 2 + length rs" + by (induction r rs rule: carry.induct) auto + +lemma l_add_estimate: + "length (add rs\<^sub>1 rs\<^sub>2) + t_add rs\<^sub>1 rs\<^sub>2 \ 2 * (length rs\<^sub>1 + length rs\<^sub>2) + 1" + apply (induction rs\<^sub>1 rs\<^sub>2 rule: t_add.induct) + apply (auto simp: l_carry_estimate algebra_simps) + done +(*>*) + +lemma t_add_bound: + fixes rs\<^sub>1 rs\<^sub>2 + defines "n\<^sub>1 \ \ rs\<^sub>1" + defines "n\<^sub>2 \ \ rs\<^sub>2" + assumes INVARS: "invar rs\<^sub>1" "invar rs\<^sub>2" + shows "t_add rs\<^sub>1 rs\<^sub>2 \ 4*log 2 (n\<^sub>1 + n\<^sub>2 + 1) + 2" +(*<*) +proof - + define n where "n = n\<^sub>1 + n\<^sub>2" + + from l_add_estimate[of rs\<^sub>1 rs\<^sub>2] + have "t_add rs\<^sub>1 rs\<^sub>2 \ 2 * (length rs\<^sub>1 + length rs\<^sub>2) + 1" by auto + hence "(2::nat)^t_add rs\<^sub>1 rs\<^sub>2 \ 2^(2 * (length rs\<^sub>1 + length rs\<^sub>2) + 1)" + by (rule power_increasing) auto + also have "\ = 2*(2^length rs\<^sub>1)\<^sup>2*(2^length rs\<^sub>2)\<^sup>2" + by (auto simp: algebra_simps power_add power_mult) + also note INVARS(1)[THEN size_snat] + also note INVARS(2)[THEN size_snat] + finally have "2 ^ t_add rs\<^sub>1 rs\<^sub>2 \ 2 * (n\<^sub>1 + 1)\<^sup>2 * (n\<^sub>2 + 1)\<^sup>2" + by (auto simp: power2_nat_le_eq_le n\<^sub>1_def n\<^sub>2_def) + from le_log2_of_power[OF this] have "t_add rs\<^sub>1 rs\<^sub>2 \ log 2 \" + by simp + also have "\ = log 2 2 + 2*log 2 (n\<^sub>1 + 1) + 2*log 2 (n\<^sub>2 + 1)" + by (simp add: log_mult log_nat_power) + also have "n\<^sub>2 \ n" by (auto simp: n_def) + finally have "t_add rs\<^sub>1 rs\<^sub>2 \ log 2 2 + 2*log 2 (n\<^sub>1 + 1) + 2*log 2 (n + 1)" + by auto + also have "n\<^sub>1 \ n" by (auto simp: n_def) + finally have "t_add rs\<^sub>1 rs\<^sub>2 \ log 2 2 + 4*log 2 (n + 1)" + by auto + also have "log 2 2 \ 2" by auto + finally have "t_add rs\<^sub>1 rs\<^sub>2 \ 4*log 2 (n + 1) + 2" by auto + thus ?thesis unfolding n_def by (auto simp: algebra_simps) +qed +(*>*) + + + +(*<*) +end +(*>*) diff --git a/12/ex12_tmpl.thy b/12/ex12_tmpl.thy new file mode 100644 index 0000000..0dbd897 --- /dev/null +++ b/12/ex12_tmpl.thy @@ -0,0 +1,98 @@ +(*<*) +theory ex12_tmpl +imports + Complex_Main + "HOL-Library.Multiset" +begin +(*>*) +text {* \ExerciseSheet{12}{29.~6.~2018} *} + +text \ + \Exercise{Sparse Binary Numbers} + + Implement operations carry, inc, and add on sparse binary numbers, + analogously to the operations link, ins, and meld on binomial heaps. + + Show that the operations have logarithmic worst-case complexity. +\ + +type_synonym rank = nat +type_synonym snat = "rank list" + +abbreviation invar :: "snat \ bool" where "invar s \ sorted_wrt op< s" +definition \ :: "snat \ nat" where "\ s = (\i\s. 2^i)" + +lemmas [simp] = sorted_wrt_Cons sorted_wrt_append + + +fun carry :: "rank \ snat \ snat" + where + "carry _ _ = undefined" + +lemma carry_invar[simp]: specify_something_meaningful oops + +lemma carry_\: specify_something_meaningful oops + + +definition inc :: "snat \ snat" where "inc \ undefined" + +lemma inc_invar[simp]: "invar rs \ invar (inc rs)" sorry + +lemma inc_\[simp]: "invar rs \ \ (inc rs) = Suc (\ rs)" sorry + +fun add :: "snat \ snat \ snat" + where + "add _ _ = undefined" + + +lemma add_invar[simp]: + assumes "invar rs\<^sub>1" + assumes "invar rs\<^sub>2" + shows "invar (add rs\<^sub>1 rs\<^sub>2)" + sorry + +lemma add_\[simp]: + assumes "invar rs\<^sub>1" + assumes "invar rs\<^sub>2" + shows "\ (add rs\<^sub>1 rs\<^sub>2) = \ rs\<^sub>1 + \ rs\<^sub>2" + sorry + + +(** Size in relation to represented number *) +lemma size_snat: + assumes "invar rs" + shows "2^length rs \ \ rs + 1" + sorry + + +(** Timing *) + +fun t_carry :: "rank \ snat \ nat" + where + "t_carry _ _ = undefined" + +definition t_inc :: "snat \ nat" + where "t_inc rs = undefined" + +lemma t_inc_bound: + assumes "invar rs" + shows "t_inc rs \ log 2 (\ rs + 1) + 1" + sorry + +fun t_add :: "snat \ snat \ nat" + where + "t_add _ _ = undefined" + + +lemma t_add_bound: + fixes rs\<^sub>1 rs\<^sub>2 + defines "n\<^sub>1 \ \ rs\<^sub>1" + defines "n\<^sub>2 \ \ rs\<^sub>2" + assumes INVARS: "invar rs\<^sub>1" "invar rs\<^sub>2" + shows "t_add rs\<^sub>1 rs\<^sub>2 \ 4*log 2 (n\<^sub>1 + n\<^sub>2 + 1) + 2" + sorry + + +(*<*) +end +(*>*) diff --git a/12/hw12_tmpl.thy b/12/hw12_tmpl.thy new file mode 100644 index 0000000..31c6937 --- /dev/null +++ b/12/hw12_tmpl.thy @@ -0,0 +1,240 @@ +theory hw12_tmpl +imports + "HOL-Data_Structures.Base_FDS" + "HOL-Data_Structures.Tree2" + "HOL-Library.Multiset" + Complex_Main +begin + + + +text \ + \Homework{Explicit Priorities}{6.7.2018} + + Modify the priority queue interface to handle multisets of pairs of + data and priority, i.e., the new \mset\ function has the signature + @{text \mset::'q \ ('d\'a::linorder) multiset\}. + + Next, implement the new interface using leftist heaps. + No complexity proofs are required. + + Hints: + \<^item> Start with the existing theories, and modify them! + \<^item> Be careful to design a good specification for \get_min\! + +\ + +(** This template provides a copy of the relevant theories, ready for modification. *) + + +text \Priority queue interface:\ + +locale Priority_Queue = +fixes empty :: "'q" +and is_empty :: "'q \ bool" +and insert :: "('d\'a::linorder) \ 'q \ 'q" +and get_min :: "'q \ ('d\'a::linorder)" +and del_min :: "'q \ 'q" +and invar :: "'q \ bool" +and mset :: "'q \ ('d\'a::linorder) multiset" +assumes mset_empty: "mset empty = {#}" +and is_empty: "invar q \ is_empty q = (mset q = {#})" +and mset_insert: "invar q \ mset (insert x q) = mset q + {#x#}" +and mset_del_min: "invar q \ mset q \ {#} \ + mset (del_min q) = mset q - {# get_min q #}" +and mset_get_min: "invar q \ mset q \ {#} \ get_min q \ set_mset(mset q) \ \p \ set_mset(mset q). snd (get_min q) \ snd p" +and invar_empty: "invar empty" +and invar_insert: "invar q \ invar (insert x q)" +and invar_del_min: "invar q \ mset q \ {#} \ invar (del_min q)" + +text \Extend locale with \merge\. Need to enforce that \'q\ is the same in both locales.\ + +locale Priority_Queue_Merge = Priority_Queue where empty = empty for empty :: 'q + +fixes merge :: "'q \ 'q \ 'q" +assumes mset_merge: "\ invar q1; invar q2 \ \ mset (merge q1 q2) = mset q1 + mset q2" +and invar_merge: "\ invar q1; invar q2 \ \ invar (merge q1 q2)" + + + + + + + + + + + + +fun mset_tree :: "(('d\'a),'b) tree \ ('d\'a) multiset" where +"mset_tree Leaf = {#}" | +"mset_tree (Node _ l a r) = {#a#} + mset_tree l + mset_tree r" + +type_synonym ('d, 'a) lheap = "(('d\'a),nat)tree" + +fun rank :: "('d, 'a) lheap \ nat" where +"rank Leaf = 0" | +"rank (Node _ _ _ r) = rank r + 1" + +fun rk :: "('d, 'a) lheap \ nat" where +"rk Leaf = 0" | +"rk (Node n _ _ _) = n" + +text{* The invariants: *} + +fun (in linorder) heap :: "(('d\'a),'b) tree \ bool" where +"heap Leaf = True" | +"heap (Node _ l (d,m) r) = + (heap l \ heap r \ (\(x,y) \ set_mset(mset_tree l + mset_tree r). m \ y))" + +fun ltree :: "('d, 'a) lheap \ bool" where +"ltree Leaf = True" | +"ltree (Node n l a r) = + (n = rank r + 1 \ rank l \ rank r \ ltree l & ltree r)" + +definition node :: "('d, 'a) lheap \ ('d\'a) \ ('d, 'a) lheap \ ('d, 'a) lheap" where +"node l a r = + (let rl = rk l; rr = rk r + in if rl \ rr then Node (rr+1) l a r else Node (rl+1) r a l)" + +fun get_min :: "('d, 'a) lheap \ ('d\'a::linorder)" where +"get_min(Node n l a r) = a" + +text \For function \merge\:\ +unbundle pattern_aliases +declare size_prod_measure[measure_function] + +fun merge :: "('d, 'a::ord) lheap \ ('d, 'a) lheap \ ('d, 'a) lheap" where +"merge Leaf t2 = t2" | +"merge t1 Leaf = t1" | +"merge (Node n1 l1 (d1,a1) r1 =: t1) (Node n2 l2 (d2,a2) r2 =: t2) = + (if a1 \ a2 then node l1 (d1,a1) (merge r1 t2) + else node l2 (d2,a2) (merge r2 t1))" + +lemma merge_code: "merge t1 t2 = (case (t1,t2) of + (Leaf, _) \ t2 | + (_, Leaf) \ t1 | + (Node n1 l1 a1 r1, Node n2 l2 a2 r2) \ + if snd a1 \ snd a2 then node l1 a1 (merge r1 t2) else node l2 a2 (merge r2 t1))" +by(induction t1 t2 rule: merge.induct) (simp_all split: tree.split) + +hide_const (open) insert + +definition insert :: "('d\'a::ord) \ ('d, 'a) lheap \ ('d, 'a) lheap" where +"insert x t = merge (Node 1 Leaf x Leaf) t" + +fun del_min :: "('d, 'a::ord) lheap \ ('d, 'a) lheap" where +"del_min Leaf = Leaf" | +"del_min (Node n l x r) = merge l r" + + +subsection "Lemmas" + +lemma mset_tree_empty: "mset_tree t = {#} \ t = Leaf" +by(cases t) auto + +lemma rk_eq_rank[simp]: "ltree t \ rk t = rank t" +by(cases t) auto + +lemma ltree_node: "ltree (node l a r) \ ltree l \ ltree r" +by(auto simp add: node_def) + +lemma heap_node: "heap (node l (d,a) r) \ + heap l \ heap r \ (\(x,y) \ set_mset(mset_tree l + mset_tree r). a \ y)" +by(auto simp add: node_def) + + +subsection "Functional Correctness" + +lemma mset_merge: "mset_tree (merge h1 h2) = mset_tree h1 + mset_tree h2" +by (induction h1 h2 rule: merge.induct) (auto simp add: node_def ac_simps) + +lemma mset_insert: "mset_tree (insert x t) = mset_tree t + {#x#}" +by (auto simp add: insert_def mset_merge) + +lemma get_min: "\ heap h; h \ Leaf \ \ get_min h \ set_mset(mset_tree h) \ \(x,y) \ set_mset(mset_tree h). snd (get_min h) \ y" + apply(induction h) + apply auto + apply blast + by blast + +lemma mset_del_min: "mset_tree (del_min h) = mset_tree h - {# get_min h #}" +by (cases h) (auto simp: mset_merge) + +lemma ltree_merge: "\ ltree l; ltree r \ \ ltree (merge l r)" +proof(induction l r rule: merge.induct) + case (3 n1 l1 d1 a1 r1 n2 l2 d2 a2 r2) + show ?case (is "ltree(merge ?t1 ?t2)") + proof cases + assume "a1 \ a2" + hence "ltree (merge ?t1 ?t2) = ltree (node l1 (d1,a1) (merge r1 ?t2))" by simp + also have "\ = (ltree l1 \ ltree(merge r1 ?t2))" + by(simp add: ltree_node) + also have "..." using "3.prems" "3.IH"(1)[OF `a1 \ a2`] by (simp) + finally show ?thesis . + next (* analogous but automatic *) + assume "\ a1 \ a2" + thus ?thesis using 3 by(simp)(auto simp: ltree_node) + qed +qed simp_all + +lemma heap_merge: "\ heap l; heap r \ \ heap (merge l r)" +proof(induction l r rule: merge.induct) + case 3 thus ?case by(auto simp: heap_node mset_merge ball_Un) +qed simp_all + +lemma ltree_insert: "ltree t \ ltree(insert x t)" +by(simp add: insert_def ltree_merge del: merge.simps split: tree.split) + +lemma heap_insert_aux1_aux1[simp]: "heap \a, \\, x, \\\" + by(induction x)(auto) + +lemma heap_insert_aux1[simp]: "heap t \ heap (merge \a, \\, x, \\\ t)" + apply(induction t) + by(auto simp add:heap_merge) + +lemma heap_insert: "heap t \ heap(insert x t)" +by(simp add: insert_def heap_merge del: merge.simps split: tree.split) + +lemma ltree_del_min: "ltree t \ ltree(del_min t)" +by(cases t)(auto simp add: ltree_merge simp del: merge.simps) + +lemma heap_del_min: "heap t \ heap(del_min t)" +by(cases t)(auto simp add: heap_merge simp del: merge.simps) + +text \Last step of functional correctness proof: combine all the above lemmas +to show that leftist heaps satisfy the specification of priority queues with merge.\ + +interpretation lheap: Priority_Queue_Merge +where empty = Leaf and is_empty = "\h. h = Leaf" +and insert = insert and del_min = del_min +and get_min = get_min and merge = merge +and invar = "\h. heap h \ ltree h" and mset = mset_tree +proof(standard, goal_cases) + case 1 show ?case by simp +next + case (2 q) show ?case by (cases q) auto +next + case 3 show ?case by(rule mset_insert) +next + case 4 show ?case by(rule mset_del_min) +next + case 5 thus ?case + by (metis (full_types) case_prodE get_min mset_tree.simps(1) prod.sel(2)) +next + case 6 thus ?case by(simp) +next + case 7 thus ?case by(simp add: heap_insert ltree_insert) +next + case 8 thus ?case by(simp add: heap_del_min ltree_del_min) +next + case 9 thus ?case by (simp add: mset_merge) +next + case 10 thus ?case by (simp add: heap_merge ltree_merge) +qed + + + + + + +end diff --git a/12/hw12_tmpl.thy~ b/12/hw12_tmpl.thy~ new file mode 100644 index 0000000..2a87cb8 --- /dev/null +++ b/12/hw12_tmpl.thy~ @@ -0,0 +1,240 @@ +theory hw12_tmpl +imports + "HOL-Data_Structures.Base_FDS" + "HOL-Data_Structures.Tree2" + "HOL-Library.Multiset" + Complex_Main +begin + + + +text \ + \Homework{Explicit Priorities}{6.7.2018} + + Modify the priority queue interface to handle multisets of pairs of + data and priority, i.e., the new \mset\ function has the signature + @{text \mset::'q \ ('d\'a::linorder) multiset\}. + + Next, implement the new interface using leftist heaps. + No complexity proofs are required. + + Hints: + \<^item> Start with the existing theories, and modify them! + \<^item> Be careful to design a good specification for \get_min\! + +\ + +(** This template provides a copy of the relevant theories, ready for modification. *) + + +text \Priority queue interface:\ + +locale Priority_Queue = +fixes empty :: "'q" +and is_empty :: "'q \ bool" +and insert :: "('d\'a::linorder) \ 'q \ 'q" +and get_min :: "'q \ ('d\'a::linorder)" +and del_min :: "'q \ 'q" +and invar :: "'q \ bool" +and mset :: "'q \ ('d\'a::linorder) multiset" +assumes mset_empty: "mset empty = {#}" +and is_empty: "invar q \ is_empty q = (mset q = {#})" +and mset_insert: "invar q \ mset (insert x q) = mset q + {#x#}" +and mset_del_min: "invar q \ mset q \ {#} \ + mset (del_min q) = mset q - {# get_min q #}" +and mset_get_min: "invar q \ mset q \ {#} \ get_min q \ set_mset(mset q) \ \p \ set_mset(mset q). snd (get_min q) \ snd p" +and invar_empty: "invar empty" +and invar_insert: "invar q \ invar (insert x q)" +and invar_del_min: "invar q \ mset q \ {#} \ invar (del_min q)" + +text \Extend locale with \merge\. Need to enforce that \'q\ is the same in both locales.\ + +locale Priority_Queue_Merge = Priority_Queue where empty = empty for empty :: 'q + +fixes merge :: "'q \ 'q \ 'q" +assumes mset_merge: "\ invar q1; invar q2 \ \ mset (merge q1 q2) = mset q1 + mset q2" +and invar_merge: "\ invar q1; invar q2 \ \ invar (merge q1 q2)" + + + + + + + + + + + + +fun mset_tree :: "(('d\'a),'b) tree \ ('d\'a) multiset" where +"mset_tree Leaf = {#}" | +"mset_tree (Node _ l a r) = {#a#} + mset_tree l + mset_tree r" + +type_synonym ('d, 'a) lheap = "(('d\'a),nat)tree" + +fun rank :: "('d, 'a) lheap \ nat" where +"rank Leaf = 0" | +"rank (Node _ _ _ r) = rank r + 1" + +fun rk :: "('d, 'a) lheap \ nat" where +"rk Leaf = 0" | +"rk (Node n _ _ _) = n" + +text{* The invariants: *} + +fun (in linorder) heap :: "(('d\'a),'b) tree \ bool" where +"heap Leaf = True" | +"heap (Node _ l (d,m) r) = + (heap l \ heap r \ (\(x,y) \ set_mset(mset_tree l + mset_tree r). m \ y))" + +fun ltree :: "('d, 'a) lheap \ bool" where +"ltree Leaf = True" | +"ltree (Node n l a r) = + (n = rank r + 1 \ rank l \ rank r \ ltree l & ltree r)" + +definition node :: "('d, 'a) lheap \ ('d\'a) \ ('d, 'a) lheap \ ('d, 'a) lheap" where +"node l a r = + (let rl = rk l; rr = rk r + in if rl \ rr then Node (rr+1) l a r else Node (rl+1) r a l)" + +fun get_min :: "('d, 'a) lheap \ ('d\'a::linorder)" where +"get_min(Node n l a r) = a" + +text \For function \merge\:\ +unbundle pattern_aliases +declare size_prod_measure[measure_function] + +fun merge :: "('d, 'a::ord) lheap \ ('d, 'a) lheap \ ('d, 'a) lheap" where +"merge Leaf t2 = t2" | +"merge t1 Leaf = t1" | +"merge (Node n1 l1 (d1,a1) r1 =: t1) (Node n2 l2 (d2,a2) r2 =: t2) = + (if a1 \ a2 then node l1 (d1,a1) (merge r1 t2) + else node l2 (d2,a2) (merge r2 t1))" + +lemma merge_code: "merge t1 t2 = (case (t1,t2) of + (Leaf, _) \ t2 | + (_, Leaf) \ t1 | + (Node n1 l1 a1 r1, Node n2 l2 a2 r2) \ + if snd a1 \ snd a2 then node l1 a1 (merge r1 t2) else node l2 a2 (merge r2 t1))" +by(induction t1 t2 rule: merge.induct) (simp_all split: tree.split) + +hide_const (open) insert + +definition insert :: "('d\'a::ord) \ ('d, 'a) lheap \ ('d, 'a) lheap" where +"insert x t = merge (Node 1 Leaf x Leaf) t" + +fun del_min :: "('d, 'a::ord) lheap \ ('d, 'a) lheap" where +"del_min Leaf = Leaf" | +"del_min (Node n l x r) = merge l r" + + +subsection "Lemmas" + +lemma mset_tree_empty: "mset_tree t = {#} \ t = Leaf" +by(cases t) auto + +lemma rk_eq_rank[simp]: "ltree t \ rk t = rank t" +by(cases t) auto + +lemma ltree_node: "ltree (node l a r) \ ltree l \ ltree r" +by(auto simp add: node_def) + +lemma heap_node: "heap (node l (d,a) r) \ + heap l \ heap r \ (\(x,y) \ set_mset(mset_tree l + mset_tree r). a \ y)" +by(auto simp add: node_def) + + +subsection "Functional Correctness" + +lemma mset_merge: "mset_tree (merge h1 h2) = mset_tree h1 + mset_tree h2" +by (induction h1 h2 rule: merge.induct) (auto simp add: node_def ac_simps) + +lemma mset_insert: "mset_tree (insert x t) = mset_tree t + {#x#}" +by (auto simp add: insert_def mset_merge) + +lemma get_min: "\ heap h; h \ Leaf \ \ get_min h \ set_mset(mset_tree h) \ \(x,y) \ set_mset(mset_tree h). snd (get_min h) \ y" + apply(induction h) + apply auto + apply blast + by blast + +lemma mset_del_min: "mset_tree (del_min h) = mset_tree h - {# get_min h #}" +by (cases h) (auto simp: mset_merge) + +lemma ltree_merge: "\ ltree l; ltree r \ \ ltree (merge l r)" +proof(induction l r rule: merge.induct) + case (3 n1 l1 d1 a1 r1 n2 l2 d2 a2 r2) + show ?case (is "ltree(merge ?t1 ?t2)") + proof cases + assume "a1 \ a2" + hence "ltree (merge ?t1 ?t2) = ltree (node l1 (d1,a1) (merge r1 ?t2))" by simp + also have "\ = (ltree l1 \ ltree(merge r1 ?t2))" + by(simp add: ltree_node) + also have "..." using "3.prems" "3.IH"(1)[OF `a1 \ a2`] by (simp) + finally show ?thesis . + next (* analogous but automatic *) + assume "\ a1 \ a2" + thus ?thesis using 3 by(simp)(auto simp: ltree_node) + qed +qed simp_all + +lemma heap_merge: "\ heap l; heap r \ \ heap (merge l r)" +proof(induction l r rule: merge.induct) + case 3 thus ?case by(auto simp: heap_node mset_merge ball_Un) +qed simp_all + +lemma ltree_insert: "ltree t \ ltree(insert x t)" +by(simp add: insert_def ltree_merge del: merge.simps split: tree.split) + +lemma heap_insert_aux1_aux1[simp]: "heap \a, \\, x, \\\" + by(induction x)(auto) + +lemma heap_insert_aux1[simp]: "heap t \ heap (merge \Suc 0, \\, x, \\\ t)" + apply(induction t) + by(auto simp add:heap_merge) + +lemma heap_insert: "heap t \ heap(insert x t)" +by(simp add: insert_def heap_merge del: merge.simps split: tree.split) + +lemma ltree_del_min: "ltree t \ ltree(del_min t)" +by(cases t)(auto simp add: ltree_merge simp del: merge.simps) + +lemma heap_del_min: "heap t \ heap(del_min t)" +by(cases t)(auto simp add: heap_merge simp del: merge.simps) + +text \Last step of functional correctness proof: combine all the above lemmas +to show that leftist heaps satisfy the specification of priority queues with merge.\ + +interpretation lheap: Priority_Queue_Merge +where empty = Leaf and is_empty = "\h. h = Leaf" +and insert = insert and del_min = del_min +and get_min = get_min and merge = merge +and invar = "\h. heap h \ ltree h" and mset = mset_tree +proof(standard, goal_cases) + case 1 show ?case by simp +next + case (2 q) show ?case by (cases q) auto +next + case 3 show ?case by(rule mset_insert) +next + case 4 show ?case by(rule mset_del_min) +next + case 5 thus ?case + by (metis (full_types) case_prodE get_min mset_tree.simps(1) prod.sel(2)) +next + case 6 thus ?case by(simp) +next + case 7 thus ?case by(simp add: heap_insert ltree_insert) +next + case 8 thus ?case by(simp add: heap_del_min ltree_del_min) +next + case 9 thus ?case by (simp add: mset_merge) +next + case 10 thus ?case by (simp add: heap_merge ltree_merge) +qed + + + + + + +end diff --git a/13/ex13.pdf b/13/ex13.pdf new file mode 100644 index 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\You may assume\ + +definition t_push :: "'a \ 'a list \ nat" where +"t_push x xs = 1" + +definition t_pop :: "nat \ 'a list \ nat" where +"t_pop n xs = min n (length xs)" + +definition \ :: "'a list \ int" where +"\ bs = int(length bs)" + +lemma \_non_neg: "\ bs \ 0" +by(simp add: \_def) + +definition "init = []" + +lemma \_init: "\ init = 0" +by(simp add: \_def init_def) + +lemma push_const: "int(t_push x xs) + \(push x xs) - \ xs = 2" +apply(induction xs rule: push.induct) + apply (auto simp add: \_def t_push_def) + done + +lemma pop_const: "int(t_pop n xs) + \(pop n xs) - \ xs = 0" + apply(induction xs rule: pop.induct) + apply (auto simp add: \_def t_pop_def) + done + +end + + diff --git a/13/hw13tmpl.thy~ b/13/hw13tmpl.thy~ new file mode 100644 index 0000000..ac403d2 --- /dev/null +++ b/13/hw13tmpl.thy~ @@ -0,0 +1,45 @@ +(*<*) +theory hw13tmpl +imports Main +begin +(*>*) + +text \\Homework{Amortized Complexity}{13.07.2018} + + This is an old exam question, which we have converted to a homework to + be done with Isabelle! +\ + +text \A ``stack with multipop'' is a list with the following two interface functions:\ + +fun push :: "'a \ 'a list \ 'a list" where +"push x xs = x # xs" + +fun pop :: "nat \ 'a list \ 'a list" where +"pop n xs = drop n xs" + +text \You may assume\ + +definition t_push :: "'a \ 'a list \ nat" where +"t_push x xs = 1" + +definition t_pop :: "nat \ 'a list \ nat" where +"t_pop n xs = min n (length xs)" + +text \Use the potential method to show that +the amortized complexity of \push\ and \pop\ is constant. + +Provide complete proofs in Isabelle! + + + +Original text here was: +\emph{ +If you need any properties of the auxiliary functions \length\, \drop\ and \min\, +you should state them but you do not need to prove them.}\ + +(*<*) +end +(*>*) + + diff --git a/BeCreative/UFDS.thy b/BeCreative/UFDS.thy new file mode 100644 index 0000000..a78f0c0 --- /dev/null +++ b/BeCreative/UFDS.thy @@ -0,0 +1,169 @@ +theory UFDS + imports Main +begin +fun fpow where + "fpow f 0 = id" +| "fpow f (Suc n) = f o fpow f n" + +lemma fpow_is_compow[simp]: "fpow = op ^^" +proof (intro ext) + fix f n x + show "fpow f n x = (f^^n) x" + apply (induction n arbitrary: x) + apply auto + done +qed + + + +(*location of array contains parent*) + +partial_function (tailrec) find' where + "find' ls i = (if ls!i = i then i else find' ls (ls!i))" + +thm find'.simps + + +definition "init n = [0.. pred ls i = i" +(*definition find where "find ls i == fpow (pred ls) (length ls) i"*) +definition "invar ls \ + (set ls \ {0.. (\in\{0.. :: "nat list \ eqcs" where "\ ls i = + (if i find' ls j = find' ls i} else {i})" + +definition union_eqc :: "nat \ nat \ eqcs \ eqcs" + where + "union_eqc x y eqc i = (if i \ eqc x \ eqc y then eqc x \ eqc y else eqc i)" + +value "([5,1,2]::nat list)[1:=2]" + +lemma + assumes "invar ls" "x (union x y ls) = union_eqc x y (\ ls)" + + +lemma "invar [0.. n \ {0.. ?C2") + + qed + + thm find'.simps + +lemma "\ [0.._def) + apply (auto simp add: \_def find'.simps split del: if_split) + + apply (subst (asm) find'.simps) + apply (subst (asm) (2) find'.simps) + apply auto + done + +lemma "find " + + +fun natbelow :: "nat \ nat list" where + "natbelow 0 = [0]" +| "natbelow (Suc n) = (Suc n) # natbelow n" + +value "rev [1::nat,2,3]" + +fun init :: "nat \ nat list" where + "init n = rev (natbelow n)" + +value "init 10" +value "[0..<10]" + +term upto +term upt + +fun findParentAux :: "nat \ nat \ nat list \ nat" where + "findParentAux n 0 xs = (xs!n)" +| "findParentAux n (Suc b) xs = (if (xs!n = n) then n else findParentAux (xs!n) b xs)" + +fun findParent :: "nat \ nat list \ nat" where + "findParent n xs = findParentAux n (size xs) xs" + + +fun isSameSet :: "nat \ nat \ nat list \ bool" where + "isSameSet a b l = (findParent a l = findParent b l)" + +value "init 10" +value "findParent 5 [0, 1, 2, 3, 4, 5, 6]" +value "isSameSet 2 3 [0, 1, 4, 5, 5, 1]" + +fun replace :: "nat \ nat \ nat list \ nat list" where + "replace _ _ [] = []" +| "replace a b (x#xs) = (if (a = x) then b#xs else x#(replace a b xs))" + +value "let ls = [0::nat,1,2,3,4,5,6] in ls[4:=9]" + + +fun union :: "nat \ nat \ nat list \ nat list" where + "union a b ls = (if (isSameSet a b ls) then ls else (replace (findParent a ls) b ls))" + +value "isSameSet 2 3 (union 5 3 [0, 1, 2, 3, 4, 2, 6])" +value "union 5 0 [0, 1, 2, 3, 4, 2, 6]" +value "union 1 0 [1, 0]" + +fun valid :: "nat list \ bool" where + "valid [] = False" +| "valid t = (\x \ set t. t!x = x)" + +value "valid [1, 0]" +value "init 10" +value "valid (init 10)" + +fun sizeufd :: "nat list \ nat" where + "sizeufd fn = size fn + 1" + +(* here be dragons *) + +lemma isSameSetWorks:"\ isSameSet a b t \ findParent a t \ findParent b t" + by simp + +lemma unionWorks:"valid t \ n \ max a b \ isSameSet a b (union a b (init n))" + apply(auto) + +end \ No newline at end of file diff --git a/BeCreative/UFDS.thy~ b/BeCreative/UFDS.thy~ new file mode 100644 index 0000000..a78f0c0 --- /dev/null +++ b/BeCreative/UFDS.thy~ @@ -0,0 +1,169 @@ +theory UFDS + imports Main +begin +fun fpow where + "fpow f 0 = id" +| "fpow f (Suc n) = f o fpow f n" + +lemma fpow_is_compow[simp]: "fpow = op ^^" +proof (intro ext) + fix f n x + show "fpow f n x = (f^^n) x" + apply (induction n arbitrary: x) + apply auto + done +qed + + + +(*location of array contains parent*) + +partial_function (tailrec) find' where + "find' ls i = (if ls!i = i then i else find' ls (ls!i))" + +thm find'.simps + + +definition "init n = [0.. pred ls i = i" +(*definition find where "find ls i == fpow (pred ls) (length ls) i"*) +definition "invar ls \ + (set ls \ {0.. (\in\{0.. :: "nat list \ eqcs" where "\ ls i = + (if i find' ls j = find' ls i} else {i})" + +definition union_eqc :: "nat \ nat \ eqcs \ eqcs" + where + "union_eqc x y eqc i = (if i \ eqc x \ eqc y then eqc x \ eqc y else eqc i)" + +value "([5,1,2]::nat list)[1:=2]" + +lemma + assumes "invar ls" "x (union x y ls) = union_eqc x y (\ ls)" + + +lemma "invar [0.. n \ {0.. ?C2") + + qed + + thm find'.simps + +lemma "\ [0.._def) + apply (auto simp add: \_def find'.simps split del: if_split) + + apply (subst (asm) find'.simps) + apply (subst (asm) (2) find'.simps) + apply auto + done + +lemma "find " + + +fun natbelow :: "nat \ nat list" where + "natbelow 0 = [0]" +| "natbelow (Suc n) = (Suc n) # natbelow n" + +value "rev [1::nat,2,3]" + +fun init :: "nat \ nat list" where + "init n = rev (natbelow n)" + +value "init 10" +value "[0..<10]" + +term upto +term upt + +fun findParentAux :: "nat \ nat \ nat list \ nat" where + "findParentAux n 0 xs = (xs!n)" +| "findParentAux n (Suc b) xs = (if (xs!n = n) then n else findParentAux (xs!n) b xs)" + +fun findParent :: "nat \ nat list \ nat" where + "findParent n xs = findParentAux n (size xs) xs" + + +fun isSameSet :: "nat \ nat \ nat list \ bool" where + "isSameSet a b l = (findParent a l = findParent b l)" + +value "init 10" +value "findParent 5 [0, 1, 2, 3, 4, 5, 6]" +value "isSameSet 2 3 [0, 1, 4, 5, 5, 1]" + +fun replace :: "nat \ nat \ nat list \ nat list" where + "replace _ _ [] = []" +| "replace a b (x#xs) = (if (a = x) then b#xs else x#(replace a b xs))" + +value "let ls = [0::nat,1,2,3,4,5,6] in ls[4:=9]" + + +fun union :: "nat \ nat \ nat list \ nat list" where + "union a b ls = (if (isSameSet a b ls) then ls else (replace (findParent a ls) b ls))" + +value "isSameSet 2 3 (union 5 3 [0, 1, 2, 3, 4, 2, 6])" +value "union 5 0 [0, 1, 2, 3, 4, 2, 6]" +value "union 1 0 [1, 0]" + +fun valid :: "nat list \ bool" where + "valid [] = False" +| "valid t = (\x \ set t. t!x = x)" + +value "valid [1, 0]" +value "init 10" +value "valid (init 10)" + +fun sizeufd :: "nat list \ nat" where + "sizeufd fn = size fn + 1" + +(* here be dragons *) + +lemma isSameSetWorks:"\ isSameSet a b t \ findParent a t \ findParent b t" + by simp + +lemma unionWorks:"valid t \ n \ max a b \ isSameSet a b (union a b (init n))" + apply(auto) + +end \ No newline at end of file diff --git a/BeCreative/UnionFind.thy b/BeCreative/UnionFind.thy new file mode 100644 index 0000000..f0fa84b --- /dev/null +++ b/BeCreative/UnionFind.thy @@ -0,0 +1,118 @@ +theory UnionFind + imports Main +begin + +(*Initial definitions*) + +definition "init n = [0.. pred ls i = i" (*if own parent, then root*) + +function (domintros) root_of where (*function with custom termination*) + "root_of ls i = (if is_root ls i then i else root_of ls (ls!i))" + apply auto[1] + by blast + +thm "root_of.psimps" +thm "root_of.pinduct" + + +(*A list is a valid UFDS if all its indices are less than the length of the list, and the indices terminate*) +definition "invar ls \ \i root_of_dom (ls,i)" + +(*Lemmas*) + +(*If an element has itself as a root, it is a root*) +lemma is_root_corr : "ls!i = i \ is_root ls i" + apply(subst is_root_def) + apply(subst pred_def) + by simp + +(*The init function sets all the members to themselves as roots*) +lemma init_corr : "\i\ set(init n). (is_root (init n) i)" + apply(subst is_root_def) + apply(auto) + by (simp add: pred_def init_def) + +(*If a member is a root, it is its own root*) +lemma root_of_base: "is_root ls i \ root_of ls i = i" + apply(subst root_of.psimps) + apply(auto) + using root_of.domintros by blast + +(*The root of a non-root member is the same as the root of its parent's member*) +lemma root_of_step: " \ invar ls; i < length ls; ls!i \ i \ \ root_of ls i = root_of ls (ls!i)" + apply(subst root_of.psimps) + apply (simp add: invar_def) + by (simp add: pred_def is_root_def) + +(*The root of a parent is the root of its child*) +lemma root_of_back: " \ invar ls; i < length ls; ls!i \ i \ \ root_of ls (ls!i) = root_of ls i" + by (simp add: root_of_step) + +(*The init function returns a valid UFDS*) +lemma invar_init: "invar (init n)" + apply(subst init_def) + apply(subst invar_def) + apply(auto) + using is_root_corr root_of.domintros by fastforce + +(*union of two sets*) +definition union where "union i j ls = ls[root_of ls i := root_of ls j]" + +type_synonym eqcs = "nat \ nat set" + +(*return disjoint set the element belongs to*) +definition \ :: "nat list \ eqcs" where "\ ls i = + (if i root_of ls j = root_of ls i} else {i})" + +(*return if an element is part of a disjoint set after uniting two other elements*) +definition union_eqc :: "nat \ nat \ eqcs \ eqcs" + where + "union_eqc x y eqc i = (if i \ eqc x \ eqc y then eqc x \ eqc y else eqc i)" + +(*Lemmas*) + +(*Every disjoint set from init list has only itself as an element*) +lemma "\ (init n) i = {i}" + apply (auto simp add: init_def \_def) + using is_root_corr root_of_base by auto + +(*Union does not change length of list*) +lemma length_same : "length ls = length (union a b ls)" + by (simp add: union_def) + + + + + + +(*Lemmas I couldn't prove*) +lemma length_fixed : "\invar ls ; x < length ls ; y < length ls ; i < length ls \ \ union x y ls ! i < length (union x y ls)" + apply(simp add: union_def) + apply(subst root_of.psimps) + apply(auto) + using invar_def apply auto[1] + + oops + +lemma + assumes "invar ls" "x (union x y ls) = union_eqc x y (\ ls)" + oops + + +lemma + assumes "invar ls" "x i < length ls \ root_of ls i < length ls" + oops + + +end \ No newline at end of file diff --git a/BeCreative/UnionFind.thy~ b/BeCreative/UnionFind.thy~ new file mode 100644 index 0000000..efd15e6 --- /dev/null +++ b/BeCreative/UnionFind.thy~ @@ -0,0 +1,96 @@ +theory UnionFind + imports Main +begin + +definition "init n = [0.. pred ls i = i" + +function (domintros) root_of where + "root_of ls i = (if is_root ls i then i else root_of ls (ls!i))" + apply auto[1] + by blast + +thm "root_of.psimps" +thm "root_of.pinduct" + +definition "invar ls \ \i root_of_dom (ls,i)" + +lemma is_root_corr : "ls!i = i \ is_root ls i" + apply(subst is_root_def) + apply(subst pred_def) + by simp + +lemma init_corr : "\i\ set(init n). (is_root (init n) i)" + apply(subst is_root_def) + apply(auto) + by (simp add: pred_def init_def) + + +lemma root_of_base: "is_root ls i \ root_of ls i = i" + apply(subst root_of.psimps) + apply(auto) + using root_of.domintros by blast + + +lemma root_of_step: " \ invar ls; i < length ls; ls!i \ i \ \ root_of ls i = root_of ls (ls!i)" + apply(subst root_of.psimps) + apply (simp add: invar_def) + by (simp add: pred_def is_root_def) + + +lemma invar_init: "invar (init n)" + apply(subst init_def) + apply(subst invar_def) + apply(auto) + using is_root_corr root_of.domintros by fastforce + +lemma all_have_root: "invar ls \ i < length ls \ root_of ls i < length ls" + oops + +definition union where "union i j ls = ls[root_of ls i := root_of ls j]" + +type_synonym eqcs = "nat \ nat set" + +definition \ :: "nat list \ eqcs" where "\ ls i = + (if i root_of ls j = root_of ls i} else {i})" + +definition union_eqc :: "nat \ nat \ eqcs \ eqcs" + where + "union_eqc x y eqc i = (if i \ eqc x \ eqc y then eqc x \ eqc y else eqc i)" + +lemma "\ (init n) i = {i}" + apply (auto simp add: init_def \_def) + using is_root_corr root_of_base by auto + +lemma length_same : "length ls = length (union a b ls)" + by (simp add: union_def) + + +lemma length_fixed : "\invar ls ; x < length ls ; y < length ls ; i < length ls \ \ union x y ls ! i < length (union x y ls)" + apply(simp add: union_def) + apply(subst root_of.psimps) + apply(auto) + using invar_def apply auto[1] + + oops + +lemma + assumes "invar ls" "x (union x y ls) = union_eqc x y (\ ls)" + oops + + +lemma + assumes "invar ls" "x

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inv' 0" + + +fun set_of :: "intervals \ nat set" +where + "set_of [] = {}"| + "set_of ((a,b)#ivs) = {a..b} \ set_of ivs" + + +fun del :: "nat \ intervals \ intervals" + where + "del x [] = []" +| "del x ((a,b)#lis) = ( + if (x < a) then (a,b)#lis + else if (x = a \ x = b) then lis + else if (x = a) then ((a+1), b)#lis + else if (x < b) then (a,x-1)#(x+1,b)#lis + else if (x = b) then (a,b-1)#lis + else (a,b)#(del x lis))" + +value "del (12::nat) [(2,5),(7,7),(9,11)]" +value "set_of [] - {1::nat}" + +lemma del_pres_inv: "n\x \ inv' n itl \ inv' n (del x itl)" +proof (induction itl arbitrary:n) + case Nil + then show ?case by auto +next + case (Cons a itl) + then show ?case + apply (cases a) + apply (cases itl) + apply auto + done +qed + +lemma set_of_del: + assumes "n\x" + assumes "inv' n itl" + shows "set_of (del x itl) = set_of itl - {x}" + using assms +proof (induction itl arbitrary:n) +case Nil + then show ?case by auto +next + case (Cons a itl) + then show ?case + apply (cases a) + apply (cases itl) + apply (auto split: if_splits) + apply (auto) + done +qed + + +lemma del_correct: + assumes "inv itl" + shows "inv (del x itl)" "set_of (del x itl) = (set_of itl) - {x}" + using assms del_pres_inv hw07.inv_def apply fastforce + using assms hw07.inv_def set_of_del by fastforce + + + + + + + + + +fun addi :: "nat \ nat \ intervals \ intervals" +where + "addi i j [] = [(i,j)]"| + "addi i j ((a,b)#lis) = ( + if j + 1 < a then ((i,j)#(a,b)#lis) + else if j+1 = a then ((i,b)#lis) + else if j \ b then( + if i < a then ((i,b)#lis) + else ((a,b)#lis)) + else ( + if i < a then addi i j lis + else if i \ b+1 then addi a j lis + else ((a,b)#(addi i j lis))) + )" + +value "addi 5 99 [(2,5),(7,7),(9,11)]" + +lemma addi_pres_inv: "n\i \ i\j \ inv' n itl \ inv' n (addi i j itl)" +proof (induction i j itl arbitrary:n rule:addi.induct) + case (1 i j) + then show ?case by auto +next + case (2 i j a b itl) + then show ?case + apply(cases itl) + apply(auto) + done +qed + +lemma set_of_addi: + assumes "n\i" "i\j" + assumes "inv' n itl" + shows "set_of (addi i j itl) = {i..j} \ set_of itl" + using assms +proof (induction i j itl arbitrary:n rule:addi.induct) + case (1 i j) + then show ?case by auto +next + case (2 i j a b lis) + then show ?case + apply(cases lis) + apply(cases a) + apply(cases b) + apply (auto split:if_splits) + apply fastforce + apply fastforce + apply fastforce + apply fastforce + apply fastforce + apply fastforce + apply fastforce + done +qed + +lemma addi_correct: + assumes "inv is" "i\j" + shows "inv (addi i j is)" "set_of (addi i j is) = {i..j} \ (set_of is)" + using addi_pres_inv assms hw07.inv_def zero_order(3) apply force + by (metis assms hw07.inv_def le0 set_of_addi) + + +end diff --git a/07/hw07.thy~ b/07/hw07.thy~ new file mode 100644 index 0000000..455d335 --- /dev/null +++ b/07/hw07.thy~ @@ -0,0 +1,129 @@ +(*<*) +theory hw07 + imports Main +begin +(*>*) + +hide_const (open) inv (*inv is used as a constant in isabelle, renaming it pretty prints the const +with the long name if the const is not hidden*) + +type_synonym intervals = "(nat*nat) list" + +fun inv' :: "nat \ intervals \ bool" where + "inv' n [] \ True"| + "inv' n ((a,b)#ivs) \ n\a \ a\b \ inv' (b+2) ivs" + +definition inv where "inv = inv' 0" + + +fun set_of :: "intervals => nat set" +where + "set_of [] = {}"| + "set_of ((a,b)#ivs) = {a..b} \ set_of ivs" + + +fun del :: "nat \ intervals \ intervals" + where + "del x [] = []" +| "del x ((a,b)#lis) = ( + if (x < a) then (a,b)#lis + else if (x = a \ x = b) then lis + else if (x = a) then ((a+1), b)#lis + else if (x < b) then (a,x-1)#(x+1,b)#lis + else if (x = b) then (a,b-1)#lis + else (a,b)#(del x lis))" + +value "del (12::nat) [(2,5),(7,7),(9,11)]" +value "set_of [] - {1::nat}" + +lemma del_pres_inv: "n\x \ inv' n itl \ inv' n (del x itl)" +proof (induction itl arbitrary:n) + case Nil + then show ?case by auto +next + case (Cons a itl) + then show ?case + apply (cases a) + apply (cases itl) + apply auto + done +qed + +lemma set_of_del: + assumes "n\x" + assumes "inv' n itl" + shows "set_of (del x itl) = set_of itl - {x}" + using assms +proof (induction itl arbitrary:n) +case Nil + then show ?case by auto +next + case (Cons a itl) + then show ?case + apply (cases a) + apply (cases itl) + apply (auto split: if_splits) + apply (auto) + done +qed + + +lemma del_correct: + assumes "inv itl" + shows "inv (del x itl)" "set_of (del x itl) = (set_of itl) - {x}" + using assms del_pres_inv hw07.inv_def apply fastforce + using assms hw07.inv_def set_of_del by fastforce + + + +text \ \NumHomework{Addition of Interval to Interval List}{June 1} + For 3 \<^bold>\bonus points\, implement and prove correct a function + to add a whole interval to an interval list. The runtime must + not depend on the size of the interval, e.g., iterating over the + interval and adding the elements separately is not allowed! +\ +fun merge_aux where + "merge_aux a j [] = [(a,j)]"| + "merge_aux a j ((c,d)#ivs) = ( + if (j+1 \ c \ j+1 \ d) then (a,d)#ivs + else if (j+1 < c) then ((a,j)#(c,d)#ivs) + else merge_aux a j ivs)" + +fun addi :: "nat \ nat \ intervals \ intervals" +where + "addi i j [] = [(i,j)]" +| "addi i j ((a,b)#lis) = ( + if (j+1 < a) then (i,j)#(a,b)#lis + else if (i < a \ j+1 \ a \ j+1 < b) then (i,b)#lis + else if (i \ a \ j \ b) then (a,b)#lis + else if (j+1 \ b \ i \ b+1) then merge_aux (min a i) j lis + else (a,b)#(addi i j lis))" + +value "addi 5 99 [(2,5),(7,7),(9,11)]" + +lemma addi_pres_inv: "n\i \ i\j \ inv' n itl \ inv' n (addi i j itl)" +proof (induction itl arbitrary:n) + case Nil + then show ?case by auto +next + case (Cons a itl) + then show ?case + apply (cases a) + apply (cases itl) + apply auto + apply linarith + apply linarith + apply (smt Suc_leD min.left_commute min_def) + using Cons.prems(2) min_le_iff_disj apply blast + apply linarith +qed + +lemma addi_correct: + assumes "inv is" "i\j" + shows "inv (addi i j is)" "set_of (addi i j is) = {i..j} \ (set_of is)" + sorry + + +(*<*) +end +(*>*) diff --git a/07/tmpl07.thy b/07/tmpl07.thy new file mode 100644 index 0000000..e373aa1 --- /dev/null +++ b/07/tmpl07.thy @@ -0,0 +1,135 @@ +(*<*) +theory tmpl07 + imports Main "HOL-Data_Structures.Sorting" +begin +(*>*) + + +text {* \ExerciseSheet{7}{25.~5.~2018} *} + + +text {* \Exercise{Interval Lists} + + Sets of natural numbers can be implemented as lists of intervals, where +an interval is simply a pair of numbers. For example the set @{term "{2, 3, 5, +7, 8, 9::nat}"} can be represented by the list @{term "[(2, 3), (5, 5), +(7::nat, 9::nat)]"}. A typical application is the list of free blocks of +dynamically allocated memory. *} + +text {* We introduce the type *} + +type_synonym intervals = "(nat*nat) list" + +text {* Next, define an \emph{invariant} +that characterizes valid interval lists: +For efficiency reasons intervals should be sorted in ascending order, the lower +bound of each interval should be less than or equal to the upper bound, and the +intervals should be chosen as large as possible, i.e.\ no two adjacent +intervals should overlap or even touch each other. It turns out to be +convenient to define @{term inv} in terms of a more general function +such that the additional argument is a lower bound for the intervals in +the list:*} + +fun inv' :: "nat \ intervals \ bool" where + "inv' _ _ \ undefined" + +definition inv where "inv = inv' 0" + + + +text {* To relate intervals back to sets define an \emph{abstraction function}*} + +fun set_of :: "intervals => nat set" +where + "set_of _ = undefined" + +text \Define a function to add a single element to the interval list, + and show its correctness\ + + +fun add :: "nat \ intervals \ intervals" + where + "add _ _ = undefined" + +lemma add_correct: + assumes "inv is" + shows "inv (add x is)" "set_of (add x is) = insert x (set_of is)" + oops + +text \Hints: + \<^item> Sketch the different cases (position of element relative to the first interval of the list) + on paper first + \<^item> In one case, you will also need information about the second interval of the list. + Do this case split via an auxiliary function! Otherwise, you may end up with a recursion equation of the form + \f (x#xs) = \ case xs of x'#xs' \ \ f (x'#xs') \\ + combined with \split: list.splits\ this will make the simplifier loop! + +\ + + +text \\Exercise{Optimized Mergesort} + + Import @{theory "Sorting"} for this exercise. + The @{const msort} function recomputes the length of the list in each iteration. + Implement an optimized version that has an additional parameter keeping track + of the length, and show that it is equal to the original @{const msort}. +\ + +(* Optimized mergesort *) + +fun msort2 :: "nat \ 'a::linorder list \ 'a list" + where "msort2 _ _ = undefined" + +lemma "n = length xs \ msort2 n xs = msort xs" + oops + +text \Hint: + Use @{thm [source] msort.simps} only when instantiated to a particular \xs\ + (@{thm [source] msort.simps[of xs]}), + otherwise the simplifier will loop! +\ + + + +text \ \NumHomework{Deletion from Interval Lists}{June 1} + + Implement and prove correct a delete function. + + Hints: + \<^item> The correctness lemma is analogous to the one for add. + \<^item> A monotonicity property on \inv'\ may be useful, i.e., + @{prop \inv' m is \ inv' m' is\} if @{prop \m'\m\} + \<^item> A bounding lemma, relating \m\ and the elements of @{term \set_of is\} + if @{prop \inv' m is\}, may be useful. +\ + + + +fun del :: "nat \ intervals \ intervals" +where + "del _ _ = undefined" + +lemma del_correct: "Come up with a meaningful spec yourself" oops + + + +text \ \NumHomework{Addition of Interval to Interval List}{June 1} + For 3 \<^bold>\bonus points\, implement and prove correct a function + to add a whole interval to an interval list. The runtime must + not depend on the size of the interval, e.g., iterating over the + interval and adding the elements separately is not allowed! +\ + +fun addi :: "nat \ nat \ intervals \ intervals" +where + "addi i j is = undefined" + +lemma addi_correct: + assumes "inv is" "i\j" + shows "inv (addi i j is)" "set_of (addi i j is) = {i..j} \ (set_of is)" + sorry + + +(*<*) +end +(*>*) diff --git a/07/tmpl0arun7.thy b/07/tmpl0arun7.thy new file mode 100644 index 0000000..a4e25f8 --- /dev/null +++ b/07/tmpl0arun7.thy @@ -0,0 +1,222 @@ +(*<*) +theory tmpl0arun7 + imports Main "HOL-Data_Structures.Sorting" +begin +(*>*) + +hide_const (open) inv (*inv is used as a constant in isabelle, renaming it pretty prints the const +with the long name if the const is not hidden*) + +text {* \ExerciseSheet{7}{25.~5.~2018} *} + + +text {* \Exercise{Interval Lists} + + Sets of natural numbers can be implemented as lists of intervals, where +an interval is simply a pair of numbers. For example the set @{term "{2, 3, 5, +7, 8, 9::nat}"} can be represented by the list @{term "[(2, 3), (5, 5), +(7::nat, 9::nat)]"}. A typical application is the list of free blocks of +dynamically allocated memory. *} + +text {* We introduce the type *} + +type_synonym intervals = "(nat*nat) list" + +text {* Next, define an \emph{invariant} +that characterizes valid interval lists: +For efficiency reasons intervals should be sorted in ascending order, the lower +bound of each interval should be less than or equal to the upper bound, and the +intervals should be chosen as large as possible, i.e.\ no two adjacent +intervals should overlap or even touch each other. It turns out to be +convenient to define @{term inv} in terms of a more general function +such that the additional argument is a lower bound for the intervals in +the list:*} + +fun inv' :: "nat \ intervals \ bool" where + "inv' n [] \ True"| + "inv' n ((a,b)#ivs) \ n\a \ a\b \ inv' (b+2) ivs" + +definition inv where "inv = inv' 0" + + + +text {* To relate intervals back to sets define an \emph{abstraction function}*} + +fun set_of :: "intervals => nat set" +where + "set_of [] = {}"| + "set_of ((a,b)#ivs) = {a..b} \ set_of ivs" +(*{a..Define a function to add a single element to the interval list, + and show its correctness\ + +fun merge_aux where + "merge_aux a b [] = [(a,b)]"| + "merge_aux a b ((c,d)#ivs) = (if b+1 = c then (a,d)#ivs else (a,b)#(c,d)#ivs)" + +fun add :: "nat \ intervals \ intervals" + where + "add i [] = [(i,i)]"| +(*i can be less than a, a-1, between a and b, b+1, greater than b. if b+1 might need to merge with next element too*) + "add i ((a,b)#ivs) = ( + if i+1 < a then (i,i)#(a,b)#ivs + else if i+1 = a then (i,b)#ivs + else if i\b then (a,b)#ivs + else if i = b+1 then merge_aux a i ivs + else (a,b)#add i ivs +)" + +(* the (a,b)#add i ivs uses ivs which might cause infinite case splits on ivs when originally +the case split on ivs is done in the add function itself, therefore we write a different function for +the case splitting + else if i = b+1 then case ivs of + [] \ [(a,i)]| + (c,d)#ivs' \ if i+1 = c then (a,d)#ivs' else (a,i)#ivs*) + +lemma add_pres_inv': "n\x \ inv' n itl \ inv' n (add x itl)" + apply(induction itl arbitrary: n) (*arbitrary n as the value of n changes in the inv' function definition*) + apply simp (*simp here keeps induction step as (add x (a #itl)) instead of (a,b) *) + apply auto (*after this sledgehammer can solve this, auto case splits on products not on anything else, + hence we need to case split on merge_aux*) + apply (case_tac itl) (*case split by apply (case_tac itl) but it is unstable so don't use except to verify*) + apply auto + done + +lemma add_pres_inv: "n\x \ inv' n itl \ inv' n (add x itl)" +proof (induction itl arbitrary:n) + case Nil + then show ?case by auto +next + case (Cons a itl) + then show ?case + apply (cases a) + apply (cases itl) + apply auto + done +qed + +lemma set_of_add: + assumes "n\x" + assumes "inv' n itl" + shows "set_of (add x itl) = {x} \ set_of itl" + using assms +proof (induction itl arbitrary:n) +case Nil + then show ?case by auto +next + case (Cons a itl) + then show ?case + apply (cases a) + apply (cases itl) + apply (auto split: if_splits) (*the apply auto without splitting on ifs shows 10 subcases where it shows all + the if cases in each subcase split: if_splits/if_split_asm where asm is + assumptions*) + done +qed +(*not in ex but done in class*) +consts + a :: "nat" + b :: "nat set" + c :: "nat set" + +lemma A: "{a} \ b = c" sorry + +lemma B: "{a} \ b \ d = c \ d" + using A apply simp (*doing apply (simp add: A) will simplify it to insert a (b \ d) = c \ d *) + done +(*not in ex*) +lemma add_correct: + assumes "inv itl" + shows "inv (add x itl)" "set_of (add x itl) = insert x (set_of itl)" + using add_pres_inv' assms tmpl0arun7.inv_def apply fastforce + using assms set_of_add tmpl0arun7.inv_def by fastforce + +text \Hints: + \<^item> Sketch the different cases (position of element relative to the first interval of the list) + on paper first + \<^item> In one case, you will also need information about the second interval of the list. + Do this case split via an auxiliary function! Otherwise, you may end up with a recursion equation of the form + \f (x#xs) = \ case xs of x'#xs' \ \ f (x'#xs') \\ + combined with \split: list.splits\ this will make the simplifier loop! + +\ + + +text \\Exercise{Optimized Mergesort} + + Import @{theory "Sorting"} for this exercise. + The @{const msort} function recomputes the length of the list in each iteration. + Implement an optimized version that has an additional parameter keeping track + of the length, and show that it is equal to the original @{const msort}. +\ + +(* Optimized mergesort *) + +fun msort2 :: "nat \ 'a::linorder list \ 'a list" where + "msort2 n xs = ( + if n \ 1 then xs + else merge (msort2 (n div 2) (take (n div 2) xs)) (msort2 (n - n div 2) (drop (n div 2) xs)))" + +declare msort2.simps [simp del] + +lemma "n = length xs \ msort2 n xs = msort xs" +proof (induction n xs rule:msort2.induct) + case (1 n xs) + then show ?case + apply (auto simp: msort.simps[of xs] msort2.simps[of _ xs]) (*simplifier works innermost to out and using + substitutes instead*) + done +qed + + +text \Hint: + Use @{thm [source] msort.simps} only when instantiated to a particular \xs\ + (@{thm [source] msort.simps[of xs]}), + otherwise the simplifier will loop! +\ + + + +text \ \NumHomework{Deletion from Interval Lists}{June 1} + + Implement and prove correct a delete function. + + Hints: + \<^item> The correctness lemma is analogous to the one for add. + \<^item> A monotonicity property on \inv'\ may be useful, i.e., + @{prop \inv' m is \ inv' m' is\} if @{prop \m'\m\} + \<^item> A bounding lemma, relating \m\ and the elements of @{term \set_of is\} + if @{prop \inv' m is\}, may be useful. +\ + + + +fun del :: "nat \ intervals \ intervals" +where + "del _ _ = undefined" + +lemma del_correct: "Come up with a meaningful spec yourself" oops + + + +text \ \NumHomework{Addition of Interval to Interval List}{June 1} + For 3 \<^bold>\bonus points\, implement and prove correct a function + to add a whole interval to an interval list. The runtime must + not depend on the size of the interval, e.g., iterating over the + interval and adding the elements separately is not allowed! +\ + +fun addi :: "nat \ nat \ intervals \ intervals" +where + "addi i j is = undefined" + +lemma addi_correct: + assumes "inv is" "i\j" + shows "inv (addi i j is)" "set_of (addi i j is) = {i..j} \ (set_of is)" + sorry + + +(*<*) +end +(*>*) diff --git a/07/tmpl0arun7.thy~ b/07/tmpl0arun7.thy~ new file mode 100644 index 0000000..91dd884 --- /dev/null +++ b/07/tmpl0arun7.thy~ @@ -0,0 +1,222 @@ +(*<*) +theory tmpl07 + imports Main "HOL-Data_Structures.Sorting" +begin +(*>*) + +hide_const (open) inv (*inv is used as a constant in isabelle, renaming it pretty prints the const +with the long name if the const is not hidden*) + +text {* \ExerciseSheet{7}{25.~5.~2018} *} + + +text {* \Exercise{Interval Lists} + + Sets of natural numbers can be implemented as lists of intervals, where +an interval is simply a pair of numbers. For example the set @{term "{2, 3, 5, +7, 8, 9::nat}"} can be represented by the list @{term "[(2, 3), (5, 5), +(7::nat, 9::nat)]"}. A typical application is the list of free blocks of +dynamically allocated memory. *} + +text {* We introduce the type *} + +type_synonym intervals = "(nat*nat) list" + +text {* Next, define an \emph{invariant} +that characterizes valid interval lists: +For efficiency reasons intervals should be sorted in ascending order, the lower +bound of each interval should be less than or equal to the upper bound, and the +intervals should be chosen as large as possible, i.e.\ no two adjacent +intervals should overlap or even touch each other. It turns out to be +convenient to define @{term inv} in terms of a more general function +such that the additional argument is a lower bound for the intervals in +the list:*} + +fun inv' :: "nat \ intervals \ bool" where + "inv' n [] \ True"| + "inv' n ((a,b)#ivs) \ n\a \ a\b \ inv' (b+2) ivs" + +definition inv where "inv = inv' 0" + + + +text {* To relate intervals back to sets define an \emph{abstraction function}*} + +fun set_of :: "intervals => nat set" +where + "set_of [] = {}"| + "set_of ((a,b)#ivs) = {a..b} \ set_of ivs" +(*{a..Define a function to add a single element to the interval list, + and show its correctness\ + +fun merge_aux where + "merge_aux a b [] = [(a,b)]"| + "merge_aux a b ((c,d)#ivs) = (if b+1 = c then (a,d)#ivs else (a,b)#(c,d)#ivs)" + +fun add :: "nat \ intervals \ intervals" + where + "add i [] = [(i,i)]"| +(*i can be less than a, a-1, between a and b, b+1, greater than b. if b+1 might need to merge with next element too*) + "add i ((a,b)#ivs) = ( + if i+1 < a then (i,i)#(a,b)#ivs + else if i+1 = a then (i,b)#ivs + else if i\b then (a,b)#ivs + else if i = b+1 then merge_aux a i ivs + else (a,b)#add i ivs +)" + +(* the (a,b)#add i ivs uses ivs which might cause infinite case splits on ivs when originally +the case split on ivs is done in the add function itself, therefore we write a different function for +the case splitting + else if i = b+1 then case ivs of + [] \ [(a,i)]| + (c,d)#ivs' \ if i+1 = c then (a,d)#ivs' else (a,i)#ivs*) + +lemma add_pres_inv': "n\x \ inv' n itl \ inv' n (add x itl)" + apply(induction itl arbitrary: n) (*arbitrary n as the value of n changes in the inv' function definition*) + apply simp (*simp here keeps induction step as (add x (a #itl)) instead of (a,b) *) + apply auto (*after this sledgehammer can solve this, auto case splits on products not on anything else, + hence we need to case split on merge_aux*) + apply (case_tac itl) (*case split by apply (case_tac itl) but it is unstable so don't use except to verify*) + apply auto + done + +lemma add_pres_inv: "n\x \ inv' n itl \ inv' n (add x itl)" +proof (induction itl arbitrary:n) + case Nil + then show ?case by auto +next + case (Cons a itl) + then show ?case + apply (cases a) + apply (cases itl) + apply auto + done +qed + +lemma set_of_add: + assumes "n\x" + assumes "inv' n itl" + shows "set_of (add x itl) = {x} \ set_of itl" + using assms +proof (induction itl arbitrary:n) +case Nil + then show ?case by auto +next + case (Cons a itl) + then show ?case + apply (cases a) + apply (cases itl) + apply (auto split: if_splits) (*the apply auto without splitting on ifs shows 10 subcases where it shows all + the if cases in each subcase split: if_splits/if_split_asm where asm is + assumptions*) + done +qed +(*not in ex but done in class*) +consts + a :: "nat" + b :: "nat set" + c :: "nat set" + +lemma A: "{a} \ b = c" sorry + +lemma B: "{a} \ b \ d = c \ d" + using A apply simp (*doing apply (simp add: A) will simplify it to insert a (b \ d) = c \ d *) + done +(*not in ex*) +lemma add_correct: + assumes "inv itl" + shows "inv (add x itl)" "set_of (add x itl) = insert x (set_of itl)" + using add_pres_inv' assms tmpl07.inv_def apply fastforce + using assms set_of_add tmpl07.inv_def by fastforce + +text \Hints: + \<^item> Sketch the different cases (position of element relative to the first interval of the list) + on paper first + \<^item> In one case, you will also need information about the second interval of the list. + Do this case split via an auxiliary function! Otherwise, you may end up with a recursion equation of the form + \f (x#xs) = \ case xs of x'#xs' \ \ f (x'#xs') \\ + combined with \split: list.splits\ this will make the simplifier loop! + +\ + + +text \\Exercise{Optimized Mergesort} + + Import @{theory "Sorting"} for this exercise. + The @{const msort} function recomputes the length of the list in each iteration. + Implement an optimized version that has an additional parameter keeping track + of the length, and show that it is equal to the original @{const msort}. +\ + +(* Optimized mergesort *) + +fun msort2 :: "nat \ 'a::linorder list \ 'a list" where + "msort2 n xs = ( + if n \ 1 then xs + else merge (msort2 (n div 2) (take (n div 2) xs)) (msort2 (n - n div 2) (drop (n div 2) xs)))" + +declare msort2.simps [simp del] + +lemma "n = length xs \ msort2 n xs = msort xs" +proof (induction n xs rule:msort2.induct) + case (1 n xs) + then show ?case + apply (auto simp: msort.simps[of xs] msort2.simps[of _ xs]) (*simplifier works innermost to out and using + substitutes instead*) + done +qed + + +text \Hint: + Use @{thm [source] msort.simps} only when instantiated to a particular \xs\ + (@{thm [source] msort.simps[of xs]}), + otherwise the simplifier will loop! +\ + + + +text \ \NumHomework{Deletion from Interval Lists}{June 1} + + Implement and prove correct a delete function. + + Hints: + \<^item> The correctness lemma is analogous to the one for add. + \<^item> A monotonicity property on \inv'\ may be useful, i.e., + @{prop \inv' m is \ inv' m' is\} if @{prop \m'\m\} + \<^item> A bounding lemma, relating \m\ and the elements of @{term \set_of is\} + if @{prop \inv' m is\}, may be useful. +\ + + + +fun del :: "nat \ intervals \ intervals" +where + "del _ _ = undefined" + +lemma del_correct: "Come up with a meaningful spec yourself" oops + + + +text \ \NumHomework{Addition of Interval to Interval List}{June 1} + For 3 \<^bold>\bonus points\, implement and prove correct a function + to add a whole interval to an interval list. The runtime must + not depend on the size of the interval, e.g., iterating over the + interval and adding the elements separately is not allowed! +\ + +fun addi :: "nat \ nat \ intervals \ intervals" +where + "addi i j is = undefined" + +lemma addi_correct: + assumes "inv is" "i\j" + shows "inv (addi i j is)" "set_of (addi i j is) = {i..j} \ (set_of is)" + sorry + + +(*<*) +end +(*>*) diff --git a/07/tut07.thy b/07/tut07.thy new file mode 100644 index 0000000..b334158 --- /dev/null +++ b/07/tut07.thy @@ -0,0 +1,197 @@ +(*<*) +theory tut07 + imports Main "HOL-Data_Structures.Sorting" +begin +(*>*) + +hide_const (open) inv + + +text {* \ExerciseSheet{7}{25.~5.~2018} *} + + +text {* \Exercise{Interval Lists} + + Sets of natural numbers can be implemented as lists of intervals, where +an interval is simply a pair of numbers. For example the set @{term "{2, 3, 5, +7, 8, 9::nat}"} can be represented by the list @{term "[(2, 3), (5, 5), +(7::nat, 9::nat)]"}. A typical application is the list of free blocks of +dynamically allocated memory. *} + +text {* We introduce the type *} + +type_synonym intervals = "(nat*nat) list" + +text {* Next, define an \emph{invariant} +that characterizes valid interval lists: +For efficiency reasons intervals should be sorted in ascending order, the lower +bound of each interval should be less than or equal to the upper bound, and the +intervals should be chosen as large as possible, i.e.\ no two adjacent +intervals should overlap or even touch each other. It turns out to be +convenient to define @{term inv} in terms of a more general function +such that the additional argument is a lower bound for the intervals in +the list:*} + +fun inv' :: "nat \ intervals \ bool" where + "inv' n [] \ True" +| "inv' n ((a,b)#ivs) \ n\a \ a\b \ inv' (b+2) ivs" + +definition inv where "inv = inv' 0" + + + +text {* To relate intervals back to sets define an \emph{abstraction function}*} + +fun set_of :: "intervals => nat set" +where + "set_of [] = {}" +| "set_of ((a,b)#ivs) = {a..b} \ set_of ivs" + +text \Define a function to add a single element to the interval list, + and show its correctness\ + + +fun merge_aux where + "merge_aux a b [] = [(a,b)]" +| "merge_aux a b ((c,d)#ivs) = (if b+1=c then (a,d)#ivs else (a,b)#(c,d)#ivs)" + +fun add :: "nat \ intervals \ intervals" + where + "add i [] = [(i,i)]" +| "add i ((a,b)#ivs) = ( + if i+1 < a then (i,i)#(a,b)#ivs + else if i+1 = a then (i,b)#ivs + else if i\b then (a,b)#ivs + else if i=b+1 then merge_aux a i ivs + else (a,b)#add i ivs + )" + + +lemma add_pres_inv: + "n\x \ inv' n ivs \ inv' n (add x ivs)" +proof (induction ivs arbitrary: n) + case Nil + then show ?case by auto +next + case (Cons a ivs) + then show ?case + apply (cases a) + apply (cases ivs) + apply auto + done +qed + +lemma set_of_add: + assumes "n\x" + assumes "inv' n ivs" + shows "set_of (add x ivs) = {x} \ set_of ivs" + using assms +proof (induction ivs arbitrary: n) + case Nil + then show ?case by auto +next + case (Cons a ivs) + then show ?case + apply (cases a) + apply (cases ivs) + apply (auto split: if_splits) + done +qed + + +lemma add_correct: + assumes "inv is" + shows "inv (add x is)" "set_of (add x is) = {x} \ set_of is" + using add_pres_inv assms tut07.inv_def apply fastforce + using assms set_of_add tut07.inv_def by fastforce + + +text \Hints: + \<^item> Sketch the different cases (position of element relative to the first interval of the list) + on paper first + \<^item> In one case, you will also need information about the second interval of the list. + Do this case split via an auxiliary function! Otherwise, you may end up with a recursion equation of the form + \f (x#xs) = \ case xs of x'#xs' \ \ f (x'#xs') \\ + combined with \split: list.splits\ this will make the simplifier loop! + +\ + + +text \\Exercise{Optimized Mergesort} + + Import @{theory "Sorting"} for this exercise. + The @{const msort} function recomputes the length of the list in each iteration. + Implement an optimized version that has an additional parameter keeping track + of the length, and show that it is equal to the original @{const msort}. +\ + +(* Optimized mergesort *) + +fun msort2 :: "nat \ 'a::linorder list \ 'a list" where +"msort2 n xs = ( + if n \ 1 then xs + else merge (msort2 (n div 2) (take (n div 2) xs)) (msort2 (n - n div 2) (drop (n div 2) xs)))" + +declare msort2.simps[simp del] + +lemma "n = length xs \ msort2 n xs = msort xs" +proof (induction n xs rule: msort2.induct) + case (1 n xs) + then show ?case + using msort.simps[of xs] msort2.simps[of n xs] + apply (auto simp: ) + done + +qed + + +text \Hint: + Use @{thm [source] msort.simps} only when instantiated to a particular \xs\ + (@{thm [source] msort.simps[of xs]}), + otherwise the simplifier will loop! +\ + + + +text \ \NumHomework{Deletion from Interval Lists}{June 1} + + Implement and prove correct a delete function. + + Hints: + \<^item> The correctness lemma is analogous to the one for add. + \<^item> A monotonicity property on \inv'\ may be useful, i.e., + @{prop \inv' m is \ inv' m' is\} if @{prop \m'\m\} + \<^item> A bounding lemma, relating \m\ and the elements of @{term \set_of is\} + if @{prop \inv' m is\}, may be useful. +\ + + + +fun del :: "nat \ intervals \ intervals" +where + "del _ _ = undefined" + +lemma del_correct: "Come up with a meaningful spec yourself" oops + + + +text \ \NumHomework{Addition of Interval to Interval List}{June 1} + For 3 \<^bold>\bonus points\, implement and prove correct a function + to add a whole interval to an interval list. The runtime must + not depend on the size of the interval, e.g., iterating over the + interval and adding the elements separately is not allowed! +\ + +fun addi :: "nat \ nat \ intervals \ intervals" +where + "addi i j is = undefined" + +lemma addi_correct: + assumes "inv is" "i\j" + shows "inv (addi i j is)" "set_of (addi i j is) = {i..j} \ (set_of is)" + sorry + + +(*<*) +end +(*>*) diff --git a/08/.hw08.thy.marks b/08/.hw08.thy.marks new file mode 100644 index 0000000000000000000000000000000000000000..eb6908e12e0da0aeaefe86c69805dbf6cc538ae4 GIT binary patch literal 13 QcmY#nur@L>FaQ%=01oE@DgXcg literal 0 HcmV?d00001 diff --git a/08/Untitled-1 b/08/Untitled-1 new file mode 100644 index 0000000..52f9218 --- /dev/null +++ b/08/Untitled-1 @@ -0,0 +1,26 @@ +lemma min_corr: "avl t \ height t = h \ avl_minnodes_height h \ size1 t" +proof(induction t arbitrary:h) + case Leaf + then show ?case by auto +next + case (Node t1 x2 t2) + + note IH = Node.IH[OF _ refl] + then have ?case using IH + using Node.prems(1) Suc_eq_plus1 avl_size by force + have "height t1 = height t2 \ height t1 < height t2 \ height t1 > height t2" (is "?C1 \ ?C2 \ ?C3") + by auto + moreover { + assume ?C1 + (* have "h = Suc (height t1)" using Node.prems(2) \height t1 = height t2\ by force*) + then have ?case using IH + using Node.prems(1) Suc_eq_plus1 avl_size by force + } moreover { + assume ?C2 have ?case sorry + } moreover { + assume ?C3 have ?case sorry + } ultimately show ?case by blast + + + then show ?case +qed \ No newline at end of file diff --git a/08/ex08.pdf b/08/ex08.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1c164e63929b69ec60a5e3ed32103d3c6a28a26a GIT binary patch literal 159937 zcma&NLy#^^u&&$oYTLGL+qPEQwr$(CZQI6b+qU~V|GjbdXy3u9K}Oc7B63jqM85fw zDT<2IG10TZkS(l?{)1s7WF)jVvV!5|g<+62wKI3IAY|g=VkP{44-A92rHzZJ6Cs1R z&0iN&QBz}k6H^#|ei&yLC)2;SFdmyXT08bz?1=vJ21M6VV@C4dPEF~DN;_$ucm;{TS@6Oo3V?{qJf59O;oYnv%A{~=d^87t;b<4^t?T~xjs)} 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zvp$6EY6FXobdJ85tnTV7*^PK007$8PX`-0Er>f{cK_yn>KH#mVe8gjx)rU_1o&pG# zxAk#=W$REz*SsDfyIMNCD!u%xE!~TFG zW)J*-@|&r!gp zcIKaQo0XmWzXT)jf94w!&hl{f8)Apvy!!fyk&^>>YHLZ6y=QC<8;^}d(mDEMlL5ui(zp)@r=uRtC%%TT_pdTTFmR nat" + where + fib0: "fib 0 = 0" + | fib1: "fib (Suc 0) = 1" + | fib2: "fib (Suc (Suc n)) = fib (Suc n) + fib n" + +lemma f_alt_induct [consumes 1, case_names 1 2 rec]: + assumes "n > 0" + and "P (Suc 0)" "P 2" "\n. n > 0 \ P n \ P (Suc n) \ P (Suc (Suc n))" + shows "P n" + using assms(1) +proof (induction n rule: fib.induct) + case (3 n) + thus ?case using assms by (cases n) (auto simp: eval_nat_numeral) +qed (auto simp: \P (Suc 0)\ \P 2\) + +lemma fib_lowerbound: "n > 0 \ real (fib n) \ 1.5 ^ n / 3" +proof (induction n rule: f_alt_induct) +case 1 +then show ?case by auto +next + case 2 + then show ?case by (auto simp:eval_nat_numeral) +next + case (rec n) + then show ?case by auto +qed + + + + +fun avl :: "'a tree \ bool" +where + "avl (Node l _ r) = ((((height r::int) -(height l::int))\ {-1..1})\ avl l \ avl r)" +| "avl (Leaf) = True" + +value "avl \\\,a,\\\,a,\\\,a,\\\\\" + +fun avl_minnodes_height::"nat \ nat" + where + "avl_minnodes_height 0 = 1" +| "avl_minnodes_height (Suc 0) = 2" +| "avl_minnodes_height (Suc (Suc n)) = avl_minnodes_height n + avl_minnodes_height (Suc n)" + + +value "height \\\,a,\\\,a,\\\\" +value "size1 \\\,a,\\\,a,\\\\" + +value "height \\\,a,\\\" +value "size1 \\\,a,\\\" + +value "height \\" +value "size1 \\" + +value "height \\\\,a,\\\,a,\\\,a,\\\\" +value "size1 \\\\,a,\\\,a,\\\,a,\\\\" +value "avl \\\\,a,\\\,a,\\\,a,\\\\" + +lemma avl_size: " size1 (Node t1 x t2) = size1 t1 + size1 t2" + by simp + +lemma avl_min_mono: "avl_minnodes_height n \ avl_minnodes_height (Suc n)" + apply(induction n rule: avl_minnodes_height.induct) + apply(auto) + done + + +lemma min_corr: "avl t \ height t = h \ avl_minnodes_height h \ size1 t" +proof(induction t arbitrary:h) + case Leaf + then show ?case by auto +next + case (Node t1 x2 t2) + note IH = Node.IH[OF _ refl] + have "height t1 = height t2 \ height t1 < height t2 \ height t1 > height t2" (is "?C1 \ ?C2 \ ?C3") + by auto + moreover { + assume ?C1 + have ?case using IH Node.prems \?C1\ + apply (cases "height t2") + apply (auto) + by (meson add_mono_thms_linordered_semiring(1) avl_min_mono le_trans) + + } moreover { + assume ?C2 + have "h = Suc (height t2)" using IH Node.prems using \?C2\ by force + then have ?case using IH Node.prems \?C2\ + apply (cases "height t2") + apply (auto) + by (simp add: le_less_Suc_eq) + + } moreover { + assume ?C3 + have "h = Suc (height t1)" using IH Node.prems using \?C3\ by force + then have ?case using IH Node.prems \?C3\ + apply (cases "height t1") + apply (auto) + by (simp add: le_less_Suc_eq) + } ultimately show ?case by blast + +qed + + + + +lemma fib_min: "fib (h+2) = avl_minnodes_height h" + apply(induction h rule:avl_minnodes_height.induct) + apply(auto) + done + +lemma avl_fib_bound: "avl t \ height t = h \ fib (h+2) \ size1 t" + using fib_min le_trans min_corr by metis + + + +lemma avl_lowerbound: + assumes "avl t" + shows "1.5 ^ (height t + 2) / 3 \ real (size1 t)" +proof - + show ?thesis using assms fib_min le_trans min_corr fib_lowerbound + by (smt add_gr_0 avl_fib_bound of_nat_le_iff pos2) +qed + + + + + + + +end \ No newline at end of file diff --git a/08/hw08.thy~ b/08/hw08.thy~ new file mode 100644 index 0000000..3f3326e --- /dev/null +++ b/08/hw08.thy~ @@ -0,0 +1,142 @@ +theory hw08 + imports Complex_Main "HOL-Library.Tree" +begin + +fun fib :: "nat \ nat" + where + fib0: "fib 0 = 0" + | fib1: "fib (Suc 0) = 1" + | fib2: "fib (Suc (Suc n)) = fib (Suc n) + fib n" + +lemma f_alt_induct [consumes 1, case_names 1 2 rec]: + assumes "n > 0" + and "P (Suc 0)" "P 2" "\n. n > 0 \ P n \ P (Suc n) \ P (Suc (Suc n))" + shows "P n" + using assms(1) +proof (induction n rule: fib.induct) + case (3 n) + thus ?case using assms by (cases n) (auto simp: eval_nat_numeral) +qed (auto simp: \P (Suc 0)\ \P 2\) + +lemma fib_lowerbound: "n > 0 \ real (fib n) \ 1.5 ^ n / 3" +proof (induction n rule: f_alt_induct) +case 1 +then show ?case by auto +next + case 2 + then show ?case by (auto simp:eval_nat_numeral) +next + case (rec n) + then show ?case by auto +qed + + + + +fun avl :: "'a tree \ bool" +where + "avl (Node l _ r) = ((((height r::int) -(height l::int))\ {-1..1})\ avl l \ avl r)" +| "avl (Leaf) = True" + +value "avl \\\,a,\\\,a,\\\,a,\\\\\" + +fun avl_minnodes_height::"nat \ nat" + where + "avl_minnodes_height 0 = 1" +| "avl_minnodes_height (Suc 0) = 2" +| "avl_minnodes_height (Suc (Suc n)) = avl_minnodes_height n + avl_minnodes_height (Suc n)" + +lemma "avl_minnodes_height h = fib (h+2)" + apply(induction h rule:avl_minnodes_height.induct) + apply(auto) + done + + + +value "height \\\,a,\\\,a,\\\\" +value "size1 \\\,a,\\\,a,\\\\" + +value "height \\\,a,\\\" +value "size1 \\\,a,\\\" + +value "height \\" +value "size1 \\" + +value "height \\\\,a,\\\,a,\\\,a,\\\\" +value "size1 \\\\,a,\\\,a,\\\,a,\\\\" +value "avl \\\\,a,\\\,a,\\\,a,\\\\" + +lemma avl_size: " size1 (Node t1 x t2) = size1 t1 + size1 t2" + by simp + +lemma avl_min_mono: "avl_minnodes_height n \ avl_minnodes_height (Suc n)" + apply(induction n rule: avl_minnodes_height.induct) + apply(auto) + done + + +lemma min_corr: "avl t \ height t = h \ avl_minnodes_height h \ size1 t" +proof(induction t arbitrary:h) + case Leaf + then show ?case by auto +next + case (Node t1 x2 t2) + note IH = Node.IH[OF _ refl] + have "height t1 = height t2 \ height t1 < height t2 \ height t1 > height t2" (is "?C1 \ ?C2 \ ?C3") + by auto + moreover { + assume ?C1 + have ?case using IH Node.prems \?C1\ + apply (cases "height t2") + apply (auto) + by (meson add_mono_thms_linordered_semiring(1) avl_min_mono le_trans) + + } moreover { + assume ?C2 + have "h = Suc (height t2)" using IH Node.prems by (simp add: \height t1 < height t2\ add.right_neutral less_imp_le_nat max_def) + then have ?case using IH Node.prems \?C2\ + apply (cases "height t2") + apply (auto) + by (simp add: le_less_Suc_eq) + + } moreover { + assume ?C3 + have "h = Suc (height t1)" using IH Node.prems using \?C3\ by force + then have ?case using IH Node.prems \?C3\ + apply (cases "height t1") + apply (auto) + by (simp add: le_less_Suc_eq) + } ultimately show ?case by blast + +qed + + + + +lemma fib_min: "fib (h+2) \ avl_minnodes_height h" + apply(induction h rule:avl_minnodes_height.induct) + apply(auto) + done + +lemma avl_fib_bound: "avl t \ height t = h \ fib (h+2) \ size1 t" + using fib_min le_trans min_corr by blast + + + + + +lemma avl_lowerbound: + assumes "avl t" + shows "1.5 ^ (height t + 2) / 3 \ real (size1 t)" +proof - + show ?thesis using assms fib_min le_trans min_corr fib_lowerbound + by (smt add_gr_0 avl_fib_bound of_nat_le_iff pos2) +qed + + + + + + + +end \ No newline at end of file diff --git a/08/hw08bonustmpl.thy b/08/hw08bonustmpl.thy new file mode 100644 index 0000000..6504fe4 --- /dev/null +++ b/08/hw08bonustmpl.thy @@ -0,0 +1,127 @@ +(*<*) +theory hw08bonustmpl + imports + Main + "HOL-Data_Structures.Tree23_Set" +begin +(*>*) + +fun join :: "'a tree23 \ 'a tree23 \ 'a up\<^sub>i" +where + "join Leaf Leaf = T\<^sub>i Leaf" +| "join (Node2 t1 a t2) (Node2 t3 b t4) = ( + case (join t2 t3) of + T\<^sub>i t23 \ T\<^sub>i (Node3 t1 a t23 b t4) + | Up\<^sub>i t2' x t3' \ Up\<^sub>i (Node2 t1 a t2') x (Node2 t3' b t4) + )" +| "join (Node2 t1 a t2) (Node3 t3 b t4 c t5) = ( + case (join t2 t3) of + T\<^sub>i t23 \ Up\<^sub>i (Node2 t1 a t23) b (Node2 t4 c t5) + | Up\<^sub>i t2' x t3' \ Up\<^sub>i (Node2 t1 a t2') x (Node3 t3' b t4 c t5) +)" +| "join (Node3 t1 a t2 b t3) (Node2 t4 c t5) = ( + case (join t3 t4) of + T\<^sub>i t34 \ Up\<^sub>i (Node2 t1 a t2) b (Node2 t34 c t5) + | Up\<^sub>i t3' x t4' \ Up\<^sub>i (Node2 t1 a t2) b (Node3 t3' x t4' c t5) +)" +| "join (Node3 t1 a t2 b t3) (Node3 t4 c t5 d t6) = ( + case (join t3 t4) of + T\<^sub>i t34 \ Up\<^sub>i (Node2 t1 a t2) b (Node3 t34 c t5 d t6) + | Up\<^sub>i t3' x t4' \ Up\<^sub>i (Node3 t1 a t2 b t3') x (Node3 t4' c t5 d t6) +)" +| "join _ _ = T\<^sub>i Leaf" + + +fun del' :: "'a::linorder \ 'a tree23 \ 'a up\<^sub>d" where + "del' x Leaf = T\<^sub>d Leaf" +| "del' x (Node2 Leaf a Leaf) = (if (x = a) then (Up\<^sub>d Leaf) else T\<^sub>d(Node2 Leaf a Leaf))" +| "del' x (Node3 Leaf a Leaf b Leaf) = T\<^sub>d(if x = a then Node2 Leaf b Leaf else + if x = b then Node2 Leaf a Leaf + else Node3 Leaf a Leaf b Leaf)" +| "del' x (Node2 l a r) = (case cmp x a of + LT \ node21 (del' x l) a r | + GT \ node22 l a (del' x r) | + EQ \ T\<^sub>d(tree\<^sub>i(join l r)))" +|"del' x (Node3 l a m b r) = + (case cmp x a of + LT \ node31 (del' x l) a m b r | + EQ \ T\<^sub>d(Node2 tree\<^sub>i((join (l m)::'a tree23) d r)) | + GT \ + (case cmp x b of + LT \ node32 l a (del x m) b r | + EQ \ let (b',r') = del_min r in node33 l a m b' r' | + GT \ node33 l a m b (del x r)))" + + +fun del :: "'a::linorder \ 'a tree23 \ 'a up\<^sub>d" where +"del x Leaf = T\<^sub>d Leaf" | +"del x (Node2 Leaf a Leaf) = + (if x = a then Up\<^sub>d Leaf else T\<^sub>d(Node2 Leaf a Leaf))" | +"del x (Node3 Leaf a Leaf b Leaf) = + T\<^sub>d(if x = a then Node2 Leaf b Leaf else + if x = b then Node2 Leaf a Leaf + else Node3 Leaf a Leaf b Leaf)" | +"del x (Node2 l a r) = + (case cmp x a of + LT \ node21 (del x l) a r | + GT \ node22 l a (del x r) | + EQ \ let (a',t) = del_min r in node22 l a' t)" | +"del x (Node3 l a m b r) = + (case cmp x a of + LT \ node31 (del x l) a m b r | + EQ \ let (a',m') = del_min m in node32 l a' m' b r | + GT \ + (case cmp x b of + LT \ node32 l a (del x m) b r | + EQ \ let (b',r') = del_min r in node33 l a m b' r' | + GT \ node33 l a m b (del x r)))" + +(* These are the two essential lemmas needed to instantiate the + set_by_ordered infrastructure with the new del function: + + (You're not required to repeat the instantiation proof, just these two lemmas are enough. ) +*) +lemma inorder_del': "\ bal t ; sorted(inorder t) \ \ + inorder(tree\<^sub>d (del' x t)) = del_list x (inorder t)" + sorry + +lemma bal_tree\<^sub>d_del: "bal t \ bal(tree\<^sub>d(del' x t))" + sorry + +text \A few hints: + \<^item> Prove auxiliary lemmas on \join\ that are suitable to discharge your proof obligations, and + disable simplification of join for your main proof (\declare join.simps[simp del]\). + This will make proof obligations more readable. + \<^item> Case splitting by \simp\ or \auto\ may take quite a long time. + Use \split!:\ instead of \split:\ to make it a bit faster. + +\ + + +(* In case you are interested how to instantiate the infrastructure with + the new delete function: *) + +definition delete' :: "'a::linorder \ 'a tree23 \ 'a tree23" where + "delete' x t = tree\<^sub>d(del' x t)" + + +interpretation Set_by_Ordered +where empty = Leaf and isin = isin and insert = insert and delete = delete' +and inorder = inorder and inv = bal +proof (standard, goal_cases) + case 2 thus ?case by(simp add: isin_set) +next + case 3 thus ?case by(simp add: inorder_insert) +next + case 4 thus ?case by(simp add: delete'_def inorder_del') +next + case 6 thus ?case by(simp add: bal_insert) +next + case 7 thus ?case by(simp add: delete'_def bal_tree\<^sub>d_del) +qed simp+ + + + +(*<*) +end +(*>*) diff --git a/08/hw08bonustmpl.thy~ b/08/hw08bonustmpl.thy~ new file mode 100644 index 0000000..b5b5fa1 --- /dev/null +++ b/08/hw08bonustmpl.thy~ @@ -0,0 +1,87 @@ +(*<*) +theory hw08bonustmpl + imports + Main + "HOL-Data_Structures.Tree23_Set" +begin +(*>*) + +fun join :: "'a tree23 \ 'a tree23 \ 'a up\<^sub>i" +where + "join Leaf Leaf = T\<^sub>i Leaf" +| "join (Node2 t1 a t2) (Node2 t3 b t4) = ( + case (join t2 t3) of + T\<^sub>i t23 \ T\<^sub>i (Node3 t1 a t23 b t4) + | Up\<^sub>i t2' x t3' \ Up\<^sub>i (Node2 t1 a t2') x (Node2 t3' b t4) + )" +| "join (Node2 t1 a t2) (Node3 t3 b t4 c t5) = ( + case (join t2 t3) of + T\<^sub>i t23 \ Up\<^sub>i (Node2 t1 a t23) b (Node2 t4 c t5) + | Up\<^sub>i t2' x t3' \ Up\<^sub>i (Node2 t1 a t2') x (Node3 t3' b t4 c t5) +)" +| "join (Node3 t1 a t2 b t3) (Node2 t4 c t5) = ( + case (join t3 t4) of + T\<^sub>i t34 \ Up\<^sub>i (Node2 t1 a t2) b (Node2 t34 c t5) + | Up\<^sub>i t3' x t4' \ Up\<^sub>i (Node2 t1 a t2) b (Node3 t3' x t4' c t5) +)" +| "join (Node3 t1 a t2 b t3) (Node3 t4 c t5 d t6) = ( + case (join t3 t4) of + T\<^sub>i t34 \ Up\<^sub>i (Node2 t1 a t2) b (Node3 t34 c t5 d t6) + | Up\<^sub>i t3' x t4' \ Up\<^sub>i (Node3 t1 a t2 b t3') x (Node3 t4' c t5 d t6) +)" +| "join _ _ = T\<^sub>i Leaf" + + +fun del' :: "'a::linorder \ 'a tree23 \ 'a up\<^sub>d" where + "del' x Leaf = T\<^sub>d Leaf" + + +(* These are the two essential lemmas needed to instantiate the + set_by_ordered infrastructure with the new del function: + + (You're not required to repeat the instantiation proof, just these two lemmas are enough. ) +*) +lemma inorder_del': "\ bal t ; sorted(inorder t) \ \ + inorder(tree\<^sub>d (del' x t)) = del_list x (inorder t)" + sorry + +lemma bal_tree\<^sub>d_del: "bal t \ bal(tree\<^sub>d(del' x t))" + sorry + +text \A few hints: + \<^item> Prove auxiliary lemmas on \join\ that are suitable to discharge your proof obligations, and + disable simplification of join for your main proof (\declare join.simps[simp del]\). + This will make proof obligations more readable. + \<^item> Case splitting by \simp\ or \auto\ may take quite a long time. + Use \split!:\ instead of \split:\ to make it a bit faster. + +\ + + +(* In case you are interested how to instantiate the infrastructure with + the new delete function: *) + +definition delete' :: "'a::linorder \ 'a tree23 \ 'a tree23" where + "delete' x t = tree\<^sub>d(del' x t)" + + +interpretation Set_by_Ordered +where empty = Leaf and isin = isin and insert = insert and delete = delete' +and inorder = inorder and inv = bal +proof (standard, goal_cases) + case 2 thus ?case by(simp add: isin_set) +next + case 3 thus ?case by(simp add: inorder_insert) +next + case 4 thus ?case by(simp add: delete'_def inorder_del') +next + case 6 thus ?case by(simp add: bal_insert) +next + case 7 thus ?case by(simp add: delete'_def bal_tree\<^sub>d_del) +qed simp+ + + + +(*<*) +end +(*>*) diff --git a/08/hw08tmpl.thy b/08/hw08tmpl.thy new file mode 100644 index 0000000..5210424 --- /dev/null +++ b/08/hw08tmpl.thy @@ -0,0 +1,70 @@ +(*<*) +theory hw08tmpl + imports Complex_Main "HOL-Library.Tree" +begin +(*>*) + +text {* \NumHomework{Bounding Fibonacci}{June 8} + + We start by defining the Fibonacci sequence, and an alternative + induction scheme for indexes greater 0: +*} + +fun fib :: "nat \ nat" + where + fib0: "fib 0 = 0" + | fib1: "fib (Suc 0) = 1" + | fib2: "fib (Suc (Suc n)) = fib (Suc n) + fib n" + +lemma f_alt_induct [consumes 1, case_names 1 2 rec]: + assumes "n > 0" + and "P (Suc 0)" "P 2" "\n. n > 0 \ P n \ P (Suc n) \ P (Suc (Suc n))" + shows "P n" + using assms(1) +proof (induction n rule: fib.induct) + case (3 n) + thus ?case using assms by (cases n) (auto simp: eval_nat_numeral) +qed (auto simp: \P (Suc 0)\ \P 2\) + +text \Show that the Fibonacci numbers grow exponentially, i.e., that they are + bounded from below by \1.5\<^sup>n/3\. + + Use the alternative induction scheme defined above. +\ +lemma fib_lowerbound: "n > 0 \ real (fib n) \ 1.5 ^ n / 3" +proof (induction n rule: f_alt_induct) +oops + +text \ + \NumHomework{AVL Trees}{June 8} + + AVL trees are binary search trees where, for each node, the heights of + its subtrees differ by at most one. In this homework, you are to bound + the minimal number of nodes in an AVL tree of a given height. + + First, define the AVL invariant on binary trees. + Note: In practice, one additionally stores the heights or height difference + in the nodes, but this is not required for this exercise. +\ + + +fun avl :: "'a tree \ bool" +where +"avl _ = undefined" + + +text \Show that an AVL tree of height \h\ has at least \fib (h+2)\ nodes:\ +lemma avl_fib_bound: "avl t \ height t = h \ fib (h+2) \ size1 t" + oops + +text \Combine your results to get an exponential lower bound on the number + of nodes in an AVL tree.\ +lemma avl_lowerbound: + assumes "avl t" + shows "1.5 ^ (height t + 2) / 3 \ real (size1 t)" + oops + + +(*<*) +end +(*>*) diff --git a/09/ex09.pdf b/09/ex09.pdf new file mode 100644 index 0000000000000000000000000000000000000000..55dda14975a62d6bee0781bee8b367cb0a3ca3f4 GIT binary patch literal 125362 zcma&sL$EL~vnJqU+qP|6-?44mwr$(CZQHhO8}rZ9+?vIm#m%Zy)ypLH)Y~DI7ZIan 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