--- a/src/Doc/Sugar/Sugar.thy Fri Nov 13 12:27:13 2015 +0000
+++ b/src/Doc/Sugar/Sugar.thy Fri Nov 13 12:43:54 2015 +0000
@@ -2,6 +2,8 @@
theory Sugar
imports "~~/src/HOL/Library/LaTeXsugar" "~~/src/HOL/Library/OptionalSugar"
begin
+no_translations
+ ("prop") "P \<and> Q \<Longrightarrow> R" <= ("prop") "P \<Longrightarrow> Q \<Longrightarrow> R"
(*>*)
text{*
\section{Introduction}
@@ -115,6 +117,10 @@
\section{Printing theorems}
+The @{prop "P \<Longrightarrow> Q \<Longrightarrow> R"} syntax is a bit idiosyncratic. If you would like
+to avoid it, you can easily print the premises as a conjunction:
+@{prop "P \<and> Q \<Longrightarrow> R"}. See \texttt{OptionalSugar} for the required ``code''.
+
\subsection{Question marks}
If you print anything, especially theorems, containing
--- a/src/HOL/Data_Structures/AVL_Map.thy Fri Nov 13 12:27:13 2015 +0000
+++ b/src/HOL/Data_Structures/AVL_Map.thy Fri Nov 13 12:43:54 2015 +0000
@@ -18,7 +18,7 @@
fun delete :: "'a::cmp \<Rightarrow> ('a*'b) avl_tree \<Rightarrow> ('a*'b) avl_tree" where
"delete _ Leaf = Leaf" |
"delete x (Node h l (a,b) r) = (case cmp x a of
- EQ \<Rightarrow> delete_root (Node h l (a,b) r) |
+ EQ \<Rightarrow> del_root (Node h l (a,b) r) |
LT \<Rightarrow> balR (delete x l) (a,b) r |
GT \<Rightarrow> balL l (a,b) (delete x r))"
@@ -34,7 +34,7 @@
"sorted1(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
by(induction t)
(auto simp: del_list_simps inorder_balL inorder_balR
- inorder_delete_root inorder_delete_maxD split: prod.splits)
+ inorder_del_root inorder_del_maxD split: prod.splits)
interpretation Map_by_Ordered
where empty = Leaf and lookup = lookup and update = update and delete = delete
--- a/src/HOL/Data_Structures/AVL_Set.thy Fri Nov 13 12:27:13 2015 +0000
+++ b/src/HOL/Data_Structures/AVL_Set.thy Fri Nov 13 12:43:54 2015 +0000
@@ -51,25 +51,23 @@
LT \<Rightarrow> balL (insert x l) a r |
GT \<Rightarrow> balR l a (insert x r))"
-fun delete_max :: "'a avl_tree \<Rightarrow> 'a avl_tree * 'a" where
-"delete_max (Node _ l a Leaf) = (l,a)" |
-"delete_max (Node _ l a r) =
- (let (r',a') = delete_max r in (balL l a r', a'))"
+fun del_max :: "'a avl_tree \<Rightarrow> 'a avl_tree * 'a" where
+"del_max (Node _ l a r) = (if r = Leaf then (l,a)
+ else let (r',a') = del_max r in (balL l a r', a'))"
-lemmas delete_max_induct = delete_max.induct[case_names Leaf Node]
+lemmas del_max_induct = del_max.induct[case_names Node Leaf]
-fun delete_root :: "'a avl_tree \<Rightarrow> 'a avl_tree" where
-"delete_root (Node h Leaf a r) = r" |
-"delete_root (Node h l a Leaf) = l" |
-"delete_root (Node h l a r) =
- (let (l', a') = delete_max l in balR l' a' r)"
+fun del_root :: "'a avl_tree \<Rightarrow> 'a avl_tree" where
+"del_root (Node h Leaf a r) = r" |
+"del_root (Node h l a Leaf) = l" |
+"del_root (Node h l a r) = (let (l', a') = del_max l in balR l' a' r)"
-lemmas delete_root_cases = delete_root.cases[case_names Leaf_t Node_Leaf Node_Node]
+lemmas del_root_cases = del_root.cases[case_names Leaf_t Node_Leaf Node_Node]
fun delete :: "'a::cmp \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
"delete _ Leaf = Leaf" |
"delete x (Node h l a r) = (case cmp x a of
- EQ \<Rightarrow> delete_root (Node h l a r) |
+ EQ \<Rightarrow> del_root (Node h l a r) |
LT \<Rightarrow> balR (delete x l) a r |
GT \<Rightarrow> balL l a (delete x r))"
@@ -97,22 +95,22 @@
subsubsection "Proofs for delete"
-lemma inorder_delete_maxD:
- "\<lbrakk> delete_max t = (t',a); t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
+lemma inorder_del_maxD:
+ "\<lbrakk> del_max t = (t',a); t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
inorder t' @ [a] = inorder t"
-by(induction t arbitrary: t' rule: delete_max.induct)
- (auto simp: inorder_balL split: prod.splits tree.split)
+by(induction t arbitrary: t' rule: del_max.induct)
+ (auto simp: inorder_balL split: if_splits prod.splits tree.split)
-lemma inorder_delete_root:
- "inorder (delete_root (Node h l a r)) = inorder l @ inorder r"
-by(induction "Node h l a r" arbitrary: l a r h rule: delete_root.induct)
- (auto simp: inorder_balR inorder_delete_maxD split: prod.splits)
+lemma inorder_del_root:
+ "inorder (del_root (Node h l a r)) = inorder l @ inorder r"
+by(induction "Node h l a r" arbitrary: l a r h rule: del_root.induct)
+ (auto simp: inorder_balL inorder_balR inorder_del_maxD split: if_splits prod.splits)
theorem inorder_delete:
"sorted(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
by(induction t)
(auto simp: del_list_simps inorder_balL inorder_balR
- inorder_delete_root inorder_delete_maxD split: prod.splits)
+ inorder_del_root inorder_del_maxD split: prod.splits)
subsubsection "Overall functional correctness"
@@ -295,65 +293,61 @@
subsubsection {* Deletion maintains AVL balance *}
-lemma avl_delete_max:
+lemma avl_del_max:
assumes "avl x" and "x \<noteq> Leaf"
- shows "avl (fst (delete_max x))" "height x = height(fst (delete_max x)) \<or>
- height x = height(fst (delete_max x)) + 1"
+ shows "avl (fst (del_max x))" "height x = height(fst (del_max x)) \<or>
+ height x = height(fst (del_max x)) + 1"
using assms
-proof (induct x rule: delete_max_induct)
- case (Node h l a rh rl b rr)
+proof (induct x rule: del_max_induct)
+ case (Node h l a r)
case 1
- with Node have "avl l" "avl (fst (delete_max (Node rh rl b rr)))" by auto
- with 1 Node have "avl (balL l a (fst (delete_max (Node rh rl b rr))))"
- by (intro avl_balL) fastforce+
- thus ?case
- by (auto simp: height_balL height_balL2
+ thus ?case using Node
+ by (auto simp: height_balL height_balL2 avl_balL
linorder_class.max.absorb1 linorder_class.max.absorb2
split:prod.split)
next
- case (Node h l a rh rl b rr)
+ case (Node h l a r)
case 2
- let ?r = "Node rh rl b rr"
- let ?r' = "fst (delete_max ?r)"
- from `avl x` Node 2 have "avl l" and "avl ?r" by simp_all
+ let ?r' = "fst (del_max r)"
+ from `avl x` Node 2 have "avl l" and "avl r" by simp_all
thus ?case using Node 2 height_balL[of l ?r' a] height_balL2[of l ?r' a]
apply (auto split:prod.splits simp del:avl.simps) by arith+
qed auto
-lemma avl_delete_root:
+lemma avl_del_root:
assumes "avl t" and "t \<noteq> Leaf"
- shows "avl(delete_root t)"
+ shows "avl(del_root t)"
using assms
-proof (cases t rule:delete_root_cases)
+proof (cases t rule:del_root_cases)
case (Node_Node h lh ll ln lr n rh rl rn rr)
let ?l = "Node lh ll ln lr"
let ?r = "Node rh rl rn rr"
- let ?l' = "fst (delete_max ?l)"
+ let ?l' = "fst (del_max ?l)"
from `avl t` and Node_Node have "avl ?r" by simp
from `avl t` and Node_Node have "avl ?l" by simp
hence "avl(?l')" "height ?l = height(?l') \<or>
- height ?l = height(?l') + 1" by (rule avl_delete_max,simp)+
+ height ?l = height(?l') + 1" by (rule avl_del_max,simp)+
with `avl t` Node_Node have "height ?l' = height ?r \<or> height ?l' = height ?r + 1
\<or> height ?r = height ?l' + 1 \<or> height ?r = height ?l' + 2" by fastforce
- with `avl ?l'` `avl ?r` have "avl(balR ?l' (snd(delete_max ?l)) ?r)"
+ with `avl ?l'` `avl ?r` have "avl(balR ?l' (snd(del_max ?l)) ?r)"
by (rule avl_balR)
with Node_Node show ?thesis by (auto split:prod.splits)
qed simp_all
-lemma height_delete_root:
+lemma height_del_root:
assumes "avl t" and "t \<noteq> Leaf"
- shows "height t = height(delete_root t) \<or> height t = height(delete_root t) + 1"
+ shows "height t = height(del_root t) \<or> height t = height(del_root t) + 1"
using assms
-proof (cases t rule: delete_root_cases)
+proof (cases t rule: del_root_cases)
case (Node_Node h lh ll ln lr n rh rl rn rr)
let ?l = "Node lh ll ln lr"
let ?r = "Node rh rl rn rr"
- let ?l' = "fst (delete_max ?l)"
- let ?t' = "balR ?l' (snd(delete_max ?l)) ?r"
+ let ?l' = "fst (del_max ?l)"
+ let ?t' = "balR ?l' (snd(del_max ?l)) ?r"
from `avl t` and Node_Node have "avl ?r" by simp
from `avl t` and Node_Node have "avl ?l" by simp
- hence "avl(?l')" by (rule avl_delete_max,simp)
- have l'_height: "height ?l = height ?l' \<or> height ?l = height ?l' + 1" using `avl ?l` by (intro avl_delete_max) auto
+ hence "avl(?l')" by (rule avl_del_max,simp)
+ have l'_height: "height ?l = height ?l' \<or> height ?l = height ?l' + 1" using `avl ?l` by (intro avl_del_max) auto
have t_height: "height t = 1 + max (height ?l) (height ?r)" using `avl t` Node_Node by simp
have "height t = height ?t' \<or> height t = height ?t' + 1" using `avl t` Node_Node
proof(cases "height ?r = height ?l' + 2")
@@ -362,7 +356,7 @@
next
case True
show ?thesis
- proof(cases rule: disjE[OF height_balR[OF True `avl ?l'` `avl ?r`, of "snd (delete_max ?l)"]])
+ proof(cases rule: disjE[OF height_balR[OF True `avl ?l'` `avl ?r`, of "snd (del_max ?l)"]])
case 1
thus ?thesis using l'_height t_height True by arith
next
@@ -385,7 +379,7 @@
with Node show ?case
proof(cases "x = n")
case True
- with Node 1 show ?thesis by (auto simp:avl_delete_root)
+ with Node 1 show ?thesis by (auto simp:avl_del_root)
next
case False
with Node 1 show ?thesis
@@ -401,9 +395,9 @@
with Node show ?case
proof(cases "x = n")
case True
- with 1 have "height (Node h l n r) = height(delete_root (Node h l n r))
- \<or> height (Node h l n r) = height(delete_root (Node h l n r)) + 1"
- by (subst height_delete_root,simp_all)
+ with 1 have "height (Node h l n r) = height(del_root (Node h l n r))
+ \<or> height (Node h l n r) = height(del_root (Node h l n r)) + 1"
+ by (subst height_del_root,simp_all)
with True show ?thesis by simp
next
case False
--- a/src/HOL/Data_Structures/Tree_Map.thy Fri Nov 13 12:27:13 2015 +0000
+++ b/src/HOL/Data_Structures/Tree_Map.thy Fri Nov 13 12:43:54 2015 +0000
@@ -44,7 +44,7 @@
"del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> sorted1(inorder t) \<Longrightarrow>
x # inorder t' = inorder t"
by(induction t arbitrary: t' rule: del_min.induct)
- (auto simp: del_list_simps split: prod.splits)
+ (auto simp: del_list_simps split: prod.splits if_splits)
lemma inorder_delete:
"sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
--- a/src/HOL/Data_Structures/Tree_Set.thy Fri Nov 13 12:27:13 2015 +0000
+++ b/src/HOL/Data_Structures/Tree_Set.thy Fri Nov 13 12:43:54 2015 +0000
@@ -24,8 +24,8 @@
GT \<Rightarrow> Node l a (insert x r))"
fun del_min :: "'a tree \<Rightarrow> 'a * 'a tree" where
-"del_min (Node Leaf a r) = (a, r)" |
-"del_min (Node l a r) = (let (x,l') = del_min l in (x, Node l' a r))"
+"del_min (Node l a r) = (if l = Leaf then (a,r)
+ else let (x,l') = del_min l in (x, Node l' a r))"
fun delete :: "'a::cmp \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
"delete x Leaf = Leaf" |
@@ -53,7 +53,7 @@
"del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> sorted(inorder t) \<Longrightarrow>
x # inorder t' = inorder t"
by(induction t arbitrary: t' rule: del_min.induct)
- (auto simp: sorted_lems split: prod.splits)
+ (auto simp: sorted_lems split: prod.splits if_splits)
lemma inorder_delete:
"sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
--- a/src/HOL/Library/OptionalSugar.thy Fri Nov 13 12:27:13 2015 +0000
+++ b/src/HOL/Library/OptionalSugar.thy Fri Nov 13 12:43:54 2015 +0000
@@ -7,6 +7,10 @@
imports Complex_Main LaTeXsugar
begin
+(* displaying theorems with conjunctions between premises: *)
+translations
+ ("prop") "P \<and> Q \<Longrightarrow> R" <= ("prop") "P \<Longrightarrow> Q \<Longrightarrow> R"
+
(* hiding set *)
translations
"xs" <= "CONST set xs"
--- a/src/HOL/Tools/Sledgehammer/sledgehammer_fact.ML Fri Nov 13 12:27:13 2015 +0000
+++ b/src/HOL/Tools/Sledgehammer/sledgehammer_fact.ML Fri Nov 13 12:43:54 2015 +0000
@@ -81,7 +81,7 @@
| explode_interval max (Facts.From i) = i upto i + max - 1
| explode_interval _ (Facts.Single i) = [i]
-val backquote = raw_explode #> map (fn "`" => "\\`" | s => s) #> implode #> enclose "`" "`"
+val backquote = enclose "\<open>" "\<close>"
(* unfolding these can yield really huge terms *)
val risky_defs = @{thms Bit0_def Bit1_def}