--- a/doc-src/HOL/HOL.tex Mon Mar 23 15:33:35 2009 +0100
+++ b/doc-src/HOL/HOL.tex Mon Mar 23 19:01:34 2009 +0100
@@ -1427,7 +1427,7 @@
provides a decision procedure for \emph{linear arithmetic}: formulae involving
addition and subtraction. The simplifier invokes a weak version of this
decision procedure automatically. If this is not sufficent, you can invoke the
-full procedure \ttindex{arith_tac} explicitly. It copes with arbitrary
+full procedure \ttindex{linear_arith_tac} explicitly. It copes with arbitrary
formulae involving {\tt=}, {\tt<}, {\tt<=}, {\tt+}, {\tt-}, {\tt Suc}, {\tt
min}, {\tt max} and numerical constants. Other subterms are treated as
atomic, while subformulae not involving numerical types are ignored. Quantified
@@ -1438,10 +1438,10 @@
If {\tt k} is a numeral, then {\tt div k}, {\tt mod k} and
{\tt k dvd} are also supported. The former two are eliminated
by case distinctions, again blowing up the running time.
-If the formula involves explicit quantifiers, \texttt{arith_tac} may take
+If the formula involves explicit quantifiers, \texttt{linear_arith_tac} may take
super-exponential time and space.
-If \texttt{arith_tac} fails, try to find relevant arithmetic results in
+If \texttt{linear_arith_tac} fails, try to find relevant arithmetic results in
the library. The theories \texttt{Nat} and \texttt{NatArith} contain
theorems about {\tt<}, {\tt<=}, \texttt{+}, \texttt{-} and \texttt{*}.
Theory \texttt{Divides} contains theorems about \texttt{div} and
--- a/src/HOL/Decision_Procs/Ferrack.thy Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Decision_Procs/Ferrack.thy Mon Mar 23 19:01:34 2009 +0100
@@ -1995,6 +1995,8 @@
"ferrack_test u = linrqe (A (A (Imp (Lt (Sub (Bound 1) (Bound 0)))
(E (Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))"
+code_reserved SML oo
+
ML {* @{code ferrack_test} () *}
oracle linr_oracle = {*
--- a/src/HOL/HoareParallel/OG_Examples.thy Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/HoareParallel/OG_Examples.thy Mon Mar 23 19:01:34 2009 +0100
@@ -443,7 +443,7 @@
--{* 32 subgoals left *}
apply(tactic {* ALLGOALS (clarify_tac @{claset}) *})
-apply(tactic {* TRYALL (simple_arith_tac @{context}) *})
+apply(tactic {* TRYALL (linear_arith_tac @{context}) *})
--{* 9 subgoals left *}
apply (force simp add:less_Suc_eq)
apply(drule sym)
--- a/src/HOL/Imperative_HOL/ROOT.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Imperative_HOL/ROOT.ML Mon Mar 23 19:01:34 2009 +0100
@@ -1,2 +1,2 @@
-use_thy "Imperative_HOL";
+use_thy "Imperative_HOL_ex";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Imperative_HOL/ex/Imperative_Quicksort.thy Mon Mar 23 19:01:34 2009 +0100
@@ -0,0 +1,639 @@
+(* Author: Lukas Bulwahn, TU Muenchen *)
+
+theory Imperative_Quicksort
+imports "~~/src/HOL/Imperative_HOL/Imperative_HOL" Subarray Multiset Efficient_Nat
+begin
+
+text {* We prove QuickSort correct in the Relational Calculus. *}
+
+definition swap :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap"
+where
+ "swap arr i j = (
+ do
+ x \<leftarrow> nth arr i;
+ y \<leftarrow> nth arr j;
+ upd i y arr;
+ upd j x arr;
+ return ()
+ done)"
+
+lemma swap_permutes:
+ assumes "crel (swap a i j) h h' rs"
+ shows "multiset_of (get_array a h')
+ = multiset_of (get_array a h)"
+ using assms
+ unfolding swap_def
+ by (auto simp add: Heap.length_def multiset_of_swap dest: sym [of _ "h'"] elim!: crelE crel_nth crel_return crel_upd)
+
+function part1 :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat Heap"
+where
+ "part1 a left right p = (
+ if (right \<le> left) then return right
+ else (do
+ v \<leftarrow> nth a left;
+ (if (v \<le> p) then (part1 a (left + 1) right p)
+ else (do swap a left right;
+ part1 a left (right - 1) p done))
+ done))"
+by pat_completeness auto
+
+termination
+by (relation "measure (\<lambda>(_,l,r,_). r - l )") auto
+
+declare part1.simps[simp del]
+
+lemma part_permutes:
+ assumes "crel (part1 a l r p) h h' rs"
+ shows "multiset_of (get_array a h')
+ = multiset_of (get_array a h)"
+ using assms
+proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
+ case (1 a l r p h h' rs)
+ thus ?case
+ unfolding part1.simps [of a l r p]
+ by (elim crelE crel_if crel_return crel_nth) (auto simp add: swap_permutes)
+qed
+
+lemma part_returns_index_in_bounds:
+ assumes "crel (part1 a l r p) h h' rs"
+ assumes "l \<le> r"
+ shows "l \<le> rs \<and> rs \<le> r"
+using assms
+proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
+ case (1 a l r p h h' rs)
+ note cr = `crel (part1 a l r p) h h' rs`
+ show ?case
+ proof (cases "r \<le> l")
+ case True (* Terminating case *)
+ with cr `l \<le> r` show ?thesis
+ unfolding part1.simps[of a l r p]
+ by (elim crelE crel_if crel_return crel_nth) auto
+ next
+ case False (* recursive case *)
+ note rec_condition = this
+ let ?v = "get_array a h ! l"
+ show ?thesis
+ proof (cases "?v \<le> p")
+ case True
+ with cr False
+ have rec1: "crel (part1 a (l + 1) r p) h h' rs"
+ unfolding part1.simps[of a l r p]
+ by (elim crelE crel_nth crel_if crel_return) auto
+ from rec_condition have "l + 1 \<le> r" by arith
+ from 1(1)[OF rec_condition True rec1 `l + 1 \<le> r`]
+ show ?thesis by simp
+ next
+ case False
+ with rec_condition cr
+ obtain h1 where swp: "crel (swap a l r) h h1 ()"
+ and rec2: "crel (part1 a l (r - 1) p) h1 h' rs"
+ unfolding part1.simps[of a l r p]
+ by (elim crelE crel_nth crel_if crel_return) auto
+ from rec_condition have "l \<le> r - 1" by arith
+ from 1(2) [OF rec_condition False rec2 `l \<le> r - 1`] show ?thesis by fastsimp
+ qed
+ qed
+qed
+
+lemma part_length_remains:
+ assumes "crel (part1 a l r p) h h' rs"
+ shows "Heap.length a h = Heap.length a h'"
+using assms
+proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
+ case (1 a l r p h h' rs)
+ note cr = `crel (part1 a l r p) h h' rs`
+
+ show ?case
+ proof (cases "r \<le> l")
+ case True (* Terminating case *)
+ with cr show ?thesis
+ unfolding part1.simps[of a l r p]
+ by (elim crelE crel_if crel_return crel_nth) auto
+ next
+ case False (* recursive case *)
+ with cr 1 show ?thesis
+ unfolding part1.simps [of a l r p] swap_def
+ by (auto elim!: crelE crel_if crel_nth crel_return crel_upd) fastsimp
+ qed
+qed
+
+lemma part_outer_remains:
+ assumes "crel (part1 a l r p) h h' rs"
+ shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"
+ using assms
+proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
+ case (1 a l r p h h' rs)
+ note cr = `crel (part1 a l r p) h h' rs`
+
+ show ?case
+ proof (cases "r \<le> l")
+ case True (* Terminating case *)
+ with cr show ?thesis
+ unfolding part1.simps[of a l r p]
+ by (elim crelE crel_if crel_return crel_nth) auto
+ next
+ case False (* recursive case *)
+ note rec_condition = this
+ let ?v = "get_array a h ! l"
+ show ?thesis
+ proof (cases "?v \<le> p")
+ case True
+ with cr False
+ have rec1: "crel (part1 a (l + 1) r p) h h' rs"
+ unfolding part1.simps[of a l r p]
+ by (elim crelE crel_nth crel_if crel_return) auto
+ from 1(1)[OF rec_condition True rec1]
+ show ?thesis by fastsimp
+ next
+ case False
+ with rec_condition cr
+ obtain h1 where swp: "crel (swap a l r) h h1 ()"
+ and rec2: "crel (part1 a l (r - 1) p) h1 h' rs"
+ unfolding part1.simps[of a l r p]
+ by (elim crelE crel_nth crel_if crel_return) auto
+ from swp rec_condition have
+ "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array a h ! i = get_array a h1 ! i"
+ unfolding swap_def
+ by (elim crelE crel_nth crel_upd crel_return) auto
+ with 1(2) [OF rec_condition False rec2] show ?thesis by fastsimp
+ qed
+ qed
+qed
+
+
+lemma part_partitions:
+ assumes "crel (part1 a l r p) h h' rs"
+ shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array (a::nat array) h' ! i \<le> p)
+ \<and> (\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array a h' ! i \<ge> p)"
+ using assms
+proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
+ case (1 a l r p h h' rs)
+ note cr = `crel (part1 a l r p) h h' rs`
+
+ show ?case
+ proof (cases "r \<le> l")
+ case True (* Terminating case *)
+ with cr have "rs = r"
+ unfolding part1.simps[of a l r p]
+ by (elim crelE crel_if crel_return crel_nth) auto
+ with True
+ show ?thesis by auto
+ next
+ case False (* recursive case *)
+ note lr = this
+ let ?v = "get_array a h ! l"
+ show ?thesis
+ proof (cases "?v \<le> p")
+ case True
+ with lr cr
+ have rec1: "crel (part1 a (l + 1) r p) h h' rs"
+ unfolding part1.simps[of a l r p]
+ by (elim crelE crel_nth crel_if crel_return) auto
+ from True part_outer_remains[OF rec1] have a_l: "get_array a h' ! l \<le> p"
+ by fastsimp
+ have "\<forall>i. (l \<le> i = (l = i \<or> Suc l \<le> i))" by arith
+ with 1(1)[OF False True rec1] a_l show ?thesis
+ by auto
+ next
+ case False
+ with lr cr
+ obtain h1 where swp: "crel (swap a l r) h h1 ()"
+ and rec2: "crel (part1 a l (r - 1) p) h1 h' rs"
+ unfolding part1.simps[of a l r p]
+ by (elim crelE crel_nth crel_if crel_return) auto
+ from swp False have "get_array a h1 ! r \<ge> p"
+ unfolding swap_def
+ by (auto simp add: Heap.length_def elim!: crelE crel_nth crel_upd crel_return)
+ with part_outer_remains [OF rec2] lr have a_r: "get_array a h' ! r \<ge> p"
+ by fastsimp
+ have "\<forall>i. (i \<le> r = (i = r \<or> i \<le> r - 1))" by arith
+ with 1(2)[OF lr False rec2] a_r show ?thesis
+ by auto
+ qed
+ qed
+qed
+
+
+fun partition :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat Heap"
+where
+ "partition a left right = (do
+ pivot \<leftarrow> nth a right;
+ middle \<leftarrow> part1 a left (right - 1) pivot;
+ v \<leftarrow> nth a middle;
+ m \<leftarrow> return (if (v \<le> pivot) then (middle + 1) else middle);
+ swap a m right;
+ return m
+ done)"
+
+declare partition.simps[simp del]
+
+lemma partition_permutes:
+ assumes "crel (partition a l r) h h' rs"
+ shows "multiset_of (get_array a h')
+ = multiset_of (get_array a h)"
+proof -
+ from assms part_permutes swap_permutes show ?thesis
+ unfolding partition.simps
+ by (elim crelE crel_return crel_nth crel_if crel_upd) auto
+qed
+
+lemma partition_length_remains:
+ assumes "crel (partition a l r) h h' rs"
+ shows "Heap.length a h = Heap.length a h'"
+proof -
+ from assms part_length_remains show ?thesis
+ unfolding partition.simps swap_def
+ by (elim crelE crel_return crel_nth crel_if crel_upd) auto
+qed
+
+lemma partition_outer_remains:
+ assumes "crel (partition a l r) h h' rs"
+ assumes "l < r"
+ shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"
+proof -
+ from assms part_outer_remains part_returns_index_in_bounds show ?thesis
+ unfolding partition.simps swap_def
+ by (elim crelE crel_return crel_nth crel_if crel_upd) fastsimp
+qed
+
+lemma partition_returns_index_in_bounds:
+ assumes crel: "crel (partition a l r) h h' rs"
+ assumes "l < r"
+ shows "l \<le> rs \<and> rs \<le> r"
+proof -
+ from crel obtain middle h'' p where part: "crel (part1 a l (r - 1) p) h h'' middle"
+ and rs_equals: "rs = (if get_array a h'' ! middle \<le> get_array a h ! r then middle + 1
+ else middle)"
+ unfolding partition.simps
+ by (elim crelE crel_return crel_nth crel_if crel_upd) simp
+ from `l < r` have "l \<le> r - 1" by arith
+ from part_returns_index_in_bounds[OF part this] rs_equals `l < r` show ?thesis by auto
+qed
+
+lemma partition_partitions:
+ assumes crel: "crel (partition a l r) h h' rs"
+ assumes "l < r"
+ shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array (a::nat array) h' ! i \<le> get_array a h' ! rs) \<and>
+ (\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array a h' ! rs \<le> get_array a h' ! i)"
+proof -
+ let ?pivot = "get_array a h ! r"
+ from crel obtain middle h1 where part: "crel (part1 a l (r - 1) ?pivot) h h1 middle"
+ and swap: "crel (swap a rs r) h1 h' ()"
+ and rs_equals: "rs = (if get_array a h1 ! middle \<le> ?pivot then middle + 1
+ else middle)"
+ unfolding partition.simps
+ by (elim crelE crel_return crel_nth crel_if crel_upd) simp
+ from swap have h'_def: "h' = Heap.upd a r (get_array a h1 ! rs)
+ (Heap.upd a rs (get_array a h1 ! r) h1)"
+ unfolding swap_def
+ by (elim crelE crel_return crel_nth crel_upd) simp
+ from swap have in_bounds: "r < Heap.length a h1 \<and> rs < Heap.length a h1"
+ unfolding swap_def
+ by (elim crelE crel_return crel_nth crel_upd) simp
+ from swap have swap_length_remains: "Heap.length a h1 = Heap.length a h'"
+ unfolding swap_def by (elim crelE crel_return crel_nth crel_upd) auto
+ from `l < r` have "l \<le> r - 1" by simp
+ note middle_in_bounds = part_returns_index_in_bounds[OF part this]
+ from part_outer_remains[OF part] `l < r`
+ have "get_array a h ! r = get_array a h1 ! r"
+ by fastsimp
+ with swap
+ have right_remains: "get_array a h ! r = get_array a h' ! rs"
+ unfolding swap_def
+ by (auto simp add: Heap.length_def elim!: crelE crel_return crel_nth crel_upd) (cases "r = rs", auto)
+ from part_partitions [OF part]
+ show ?thesis
+ proof (cases "get_array a h1 ! middle \<le> ?pivot")
+ case True
+ with rs_equals have rs_equals: "rs = middle + 1" by simp
+ {
+ fix i
+ assume i_is_left: "l \<le> i \<and> i < rs"
+ with swap_length_remains in_bounds middle_in_bounds rs_equals `l < r`
+ have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto
+ from i_is_left rs_equals have "l \<le> i \<and> i < middle \<or> i = middle" by arith
+ with part_partitions[OF part] right_remains True
+ have "get_array a h1 ! i \<le> get_array a h' ! rs" by fastsimp
+ with i_props h'_def in_bounds have "get_array a h' ! i \<le> get_array a h' ! rs"
+ unfolding Heap.upd_def Heap.length_def by simp
+ }
+ moreover
+ {
+ fix i
+ assume "rs < i \<and> i \<le> r"
+
+ hence "(rs < i \<and> i \<le> r - 1) \<or> (rs < i \<and> i = r)" by arith
+ hence "get_array a h' ! rs \<le> get_array a h' ! i"
+ proof
+ assume i_is: "rs < i \<and> i \<le> r - 1"
+ with swap_length_remains in_bounds middle_in_bounds rs_equals
+ have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto
+ from part_partitions[OF part] rs_equals right_remains i_is
+ have "get_array a h' ! rs \<le> get_array a h1 ! i"
+ by fastsimp
+ with i_props h'_def show ?thesis by fastsimp
+ next
+ assume i_is: "rs < i \<and> i = r"
+ with rs_equals have "Suc middle \<noteq> r" by arith
+ with middle_in_bounds `l < r` have "Suc middle \<le> r - 1" by arith
+ with part_partitions[OF part] right_remains
+ have "get_array a h' ! rs \<le> get_array a h1 ! (Suc middle)"
+ by fastsimp
+ with i_is True rs_equals right_remains h'_def
+ show ?thesis using in_bounds
+ unfolding Heap.upd_def Heap.length_def
+ by auto
+ qed
+ }
+ ultimately show ?thesis by auto
+ next
+ case False
+ with rs_equals have rs_equals: "middle = rs" by simp
+ {
+ fix i
+ assume i_is_left: "l \<le> i \<and> i < rs"
+ with swap_length_remains in_bounds middle_in_bounds rs_equals
+ have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto
+ from part_partitions[OF part] rs_equals right_remains i_is_left
+ have "get_array a h1 ! i \<le> get_array a h' ! rs" by fastsimp
+ with i_props h'_def have "get_array a h' ! i \<le> get_array a h' ! rs"
+ unfolding Heap.upd_def by simp
+ }
+ moreover
+ {
+ fix i
+ assume "rs < i \<and> i \<le> r"
+ hence "(rs < i \<and> i \<le> r - 1) \<or> i = r" by arith
+ hence "get_array a h' ! rs \<le> get_array a h' ! i"
+ proof
+ assume i_is: "rs < i \<and> i \<le> r - 1"
+ with swap_length_remains in_bounds middle_in_bounds rs_equals
+ have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto
+ from part_partitions[OF part] rs_equals right_remains i_is
+ have "get_array a h' ! rs \<le> get_array a h1 ! i"
+ by fastsimp
+ with i_props h'_def show ?thesis by fastsimp
+ next
+ assume i_is: "i = r"
+ from i_is False rs_equals right_remains h'_def
+ show ?thesis using in_bounds
+ unfolding Heap.upd_def Heap.length_def
+ by auto
+ qed
+ }
+ ultimately
+ show ?thesis by auto
+ qed
+qed
+
+
+function quicksort :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap"
+where
+ "quicksort arr left right =
+ (if (right > left) then
+ do
+ pivotNewIndex \<leftarrow> partition arr left right;
+ pivotNewIndex \<leftarrow> assert (\<lambda>x. left \<le> x \<and> x \<le> right) pivotNewIndex;
+ quicksort arr left (pivotNewIndex - 1);
+ quicksort arr (pivotNewIndex + 1) right
+ done
+ else return ())"
+by pat_completeness auto
+
+(* For termination, we must show that the pivotNewIndex is between left and right *)
+termination
+by (relation "measure (\<lambda>(a, l, r). (r - l))") auto
+
+declare quicksort.simps[simp del]
+
+
+lemma quicksort_permutes:
+ assumes "crel (quicksort a l r) h h' rs"
+ shows "multiset_of (get_array a h')
+ = multiset_of (get_array a h)"
+ using assms
+proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
+ case (1 a l r h h' rs)
+ with partition_permutes show ?case
+ unfolding quicksort.simps [of a l r]
+ by (elim crel_if crelE crel_assert crel_return) auto
+qed
+
+lemma length_remains:
+ assumes "crel (quicksort a l r) h h' rs"
+ shows "Heap.length a h = Heap.length a h'"
+using assms
+proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
+ case (1 a l r h h' rs)
+ with partition_length_remains show ?case
+ unfolding quicksort.simps [of a l r]
+ by (elim crel_if crelE crel_assert crel_return) auto
+qed
+
+lemma quicksort_outer_remains:
+ assumes "crel (quicksort a l r) h h' rs"
+ shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"
+ using assms
+proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
+ case (1 a l r h h' rs)
+ note cr = `crel (quicksort a l r) h h' rs`
+ thus ?case
+ proof (cases "r > l")
+ case False
+ with cr have "h' = h"
+ unfolding quicksort.simps [of a l r]
+ by (elim crel_if crel_return) auto
+ thus ?thesis by simp
+ next
+ case True
+ {
+ fix h1 h2 p ret1 ret2 i
+ assume part: "crel (partition a l r) h h1 p"
+ assume qs1: "crel (quicksort a l (p - 1)) h1 h2 ret1"
+ assume qs2: "crel (quicksort a (p + 1) r) h2 h' ret2"
+ assume pivot: "l \<le> p \<and> p \<le> r"
+ assume i_outer: "i < l \<or> r < i"
+ from partition_outer_remains [OF part True] i_outer
+ have "get_array a h !i = get_array a h1 ! i" by fastsimp
+ moreover
+ with 1(1) [OF True pivot qs1] pivot i_outer
+ have "get_array a h1 ! i = get_array a h2 ! i" by auto
+ moreover
+ with qs2 1(2) [of p h2 h' ret2] True pivot i_outer
+ have "get_array a h2 ! i = get_array a h' ! i" by auto
+ ultimately have "get_array a h ! i= get_array a h' ! i" by simp
+ }
+ with cr show ?thesis
+ unfolding quicksort.simps [of a l r]
+ by (elim crel_if crelE crel_assert crel_return) auto
+ qed
+qed
+
+lemma quicksort_is_skip:
+ assumes "crel (quicksort a l r) h h' rs"
+ shows "r \<le> l \<longrightarrow> h = h'"
+ using assms
+ unfolding quicksort.simps [of a l r]
+ by (elim crel_if crel_return) auto
+
+lemma quicksort_sorts:
+ assumes "crel (quicksort a l r) h h' rs"
+ assumes l_r_length: "l < Heap.length a h" "r < Heap.length a h"
+ shows "sorted (subarray l (r + 1) a h')"
+ using assms
+proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
+ case (1 a l r h h' rs)
+ note cr = `crel (quicksort a l r) h h' rs`
+ thus ?case
+ proof (cases "r > l")
+ case False
+ hence "l \<ge> r + 1 \<or> l = r" by arith
+ with length_remains[OF cr] 1(5) show ?thesis
+ by (auto simp add: subarray_Nil subarray_single)
+ next
+ case True
+ {
+ fix h1 h2 p
+ assume part: "crel (partition a l r) h h1 p"
+ assume qs1: "crel (quicksort a l (p - 1)) h1 h2 ()"
+ assume qs2: "crel (quicksort a (p + 1) r) h2 h' ()"
+ from partition_returns_index_in_bounds [OF part True]
+ have pivot: "l\<le> p \<and> p \<le> r" .
+ note length_remains = length_remains[OF qs2] length_remains[OF qs1] partition_length_remains[OF part]
+ from quicksort_outer_remains [OF qs2] quicksort_outer_remains [OF qs1] pivot quicksort_is_skip[OF qs1]
+ have pivot_unchanged: "get_array a h1 ! p = get_array a h' ! p" by (cases p, auto)
+ (*-- First of all, by induction hypothesis both sublists are sorted. *)
+ from 1(1)[OF True pivot qs1] length_remains pivot 1(5)
+ have IH1: "sorted (subarray l p a h2)" by (cases p, auto simp add: subarray_Nil)
+ from quicksort_outer_remains [OF qs2] length_remains
+ have left_subarray_remains: "subarray l p a h2 = subarray l p a h'"
+ by (simp add: subarray_eq_samelength_iff)
+ with IH1 have IH1': "sorted (subarray l p a h')" by simp
+ from 1(2)[OF True pivot qs2] pivot 1(5) length_remains
+ have IH2: "sorted (subarray (p + 1) (r + 1) a h')"
+ by (cases "Suc p \<le> r", auto simp add: subarray_Nil)
+ (* -- Secondly, both sublists remain partitioned. *)
+ from partition_partitions[OF part True]
+ have part_conds1: "\<forall>j. j \<in> set (subarray l p a h1) \<longrightarrow> j \<le> get_array a h1 ! p "
+ and part_conds2: "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h1) \<longrightarrow> get_array a h1 ! p \<le> j"
+ by (auto simp add: all_in_set_subarray_conv)
+ from quicksort_outer_remains [OF qs1] quicksort_permutes [OF qs1] True
+ length_remains 1(5) pivot multiset_of_sublist [of l p "get_array a h1" "get_array a h2"]
+ have multiset_partconds1: "multiset_of (subarray l p a h2) = multiset_of (subarray l p a h1)"
+ unfolding Heap.length_def subarray_def by (cases p, auto)
+ with left_subarray_remains part_conds1 pivot_unchanged
+ have part_conds2': "\<forall>j. j \<in> set (subarray l p a h') \<longrightarrow> j \<le> get_array a h' ! p"
+ by (simp, subst set_of_multiset_of[symmetric], simp)
+ (* -- These steps are the analogous for the right sublist \<dots> *)
+ from quicksort_outer_remains [OF qs1] length_remains
+ have right_subarray_remains: "subarray (p + 1) (r + 1) a h1 = subarray (p + 1) (r + 1) a h2"
+ by (auto simp add: subarray_eq_samelength_iff)
+ from quicksort_outer_remains [OF qs2] quicksort_permutes [OF qs2] True
+ length_remains 1(5) pivot multiset_of_sublist [of "p + 1" "r + 1" "get_array a h2" "get_array a h'"]
+ have multiset_partconds2: "multiset_of (subarray (p + 1) (r + 1) a h') = multiset_of (subarray (p + 1) (r + 1) a h2)"
+ unfolding Heap.length_def subarray_def by auto
+ with right_subarray_remains part_conds2 pivot_unchanged
+ have part_conds1': "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h') \<longrightarrow> get_array a h' ! p \<le> j"
+ by (simp, subst set_of_multiset_of[symmetric], simp)
+ (* -- Thirdly and finally, we show that the array is sorted
+ following from the facts above. *)
+ from True pivot 1(5) length_remains have "subarray l (r + 1) a h' = subarray l p a h' @ [get_array a h' ! p] @ subarray (p + 1) (r + 1) a h'"
+ by (simp add: subarray_nth_array_Cons, cases "l < p") (auto simp add: subarray_append subarray_Nil)
+ with IH1' IH2 part_conds1' part_conds2' pivot have ?thesis
+ unfolding subarray_def
+ apply (auto simp add: sorted_append sorted_Cons all_in_set_sublist'_conv)
+ by (auto simp add: set_sublist' dest: le_trans [of _ "get_array a h' ! p"])
+ }
+ with True cr show ?thesis
+ unfolding quicksort.simps [of a l r]
+ by (elim crel_if crel_return crelE crel_assert) auto
+ qed
+qed
+
+
+lemma quicksort_is_sort:
+ assumes crel: "crel (quicksort a 0 (Heap.length a h - 1)) h h' rs"
+ shows "get_array a h' = sort (get_array a h)"
+proof (cases "get_array a h = []")
+ case True
+ with quicksort_is_skip[OF crel] show ?thesis
+ unfolding Heap.length_def by simp
+next
+ case False
+ from quicksort_sorts [OF crel] False have "sorted (sublist' 0 (List.length (get_array a h)) (get_array a h'))"
+ unfolding Heap.length_def subarray_def by auto
+ with length_remains[OF crel] have "sorted (get_array a h')"
+ unfolding Heap.length_def by simp
+ with quicksort_permutes [OF crel] properties_for_sort show ?thesis by fastsimp
+qed
+
+subsection {* No Errors in quicksort *}
+text {* We have proved that quicksort sorts (if no exceptions occur).
+We will now show that exceptions do not occur. *}
+
+lemma noError_part1:
+ assumes "l < Heap.length a h" "r < Heap.length a h"
+ shows "noError (part1 a l r p) h"
+ using assms
+proof (induct a l r p arbitrary: h rule: part1.induct)
+ case (1 a l r p)
+ thus ?case
+ unfolding part1.simps [of a l r] swap_def
+ by (auto intro!: noError_if noErrorI noError_return noError_nth noError_upd elim!: crelE crel_upd crel_nth crel_return)
+qed
+
+lemma noError_partition:
+ assumes "l < r" "l < Heap.length a h" "r < Heap.length a h"
+ shows "noError (partition a l r) h"
+using assms
+unfolding partition.simps swap_def
+apply (auto intro!: noError_if noErrorI noError_return noError_nth noError_upd noError_part1 elim!: crelE crel_upd crel_nth crel_return)
+apply (frule part_length_remains)
+apply (frule part_returns_index_in_bounds)
+apply auto
+apply (frule part_length_remains)
+apply (frule part_returns_index_in_bounds)
+apply auto
+apply (frule part_length_remains)
+apply auto
+done
+
+lemma noError_quicksort:
+ assumes "l < Heap.length a h" "r < Heap.length a h"
+ shows "noError (quicksort a l r) h"
+using assms
+proof (induct a l r arbitrary: h rule: quicksort.induct)
+ case (1 a l ri h)
+ thus ?case
+ unfolding quicksort.simps [of a l ri]
+ apply (auto intro!: noError_if noErrorI noError_return noError_nth noError_upd noError_assert noError_partition)
+ apply (frule partition_returns_index_in_bounds)
+ apply auto
+ apply (frule partition_returns_index_in_bounds)
+ apply auto
+ apply (auto elim!: crel_assert dest!: partition_length_remains length_remains)
+ apply (subgoal_tac "Suc r \<le> ri \<or> r = ri")
+ apply (erule disjE)
+ apply auto
+ unfolding quicksort.simps [of a "Suc ri" ri]
+ apply (auto intro!: noError_if noError_return)
+ done
+qed
+
+
+subsection {* Example *}
+
+definition "qsort a = do
+ k \<leftarrow> length a;
+ quicksort a 0 (k - 1);
+ return a
+ done"
+
+ML {* @{code qsort} (Array.fromList [42, 2, 3, 5, 0, 1705, 8, 3, 15]) () *}
+
+export_code qsort in SML_imp module_name QSort
+export_code qsort in OCaml module_name QSort file -
+export_code qsort in OCaml_imp module_name QSort file -
+export_code qsort in Haskell module_name QSort file -
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Imperative_HOL/ex/Subarray.thy Mon Mar 23 19:01:34 2009 +0100
@@ -0,0 +1,66 @@
+theory Subarray
+imports Array Sublist
+begin
+
+definition subarray :: "nat \<Rightarrow> nat \<Rightarrow> ('a::heap) array \<Rightarrow> heap \<Rightarrow> 'a list"
+where
+ "subarray n m a h \<equiv> sublist' n m (get_array a h)"
+
+lemma subarray_upd: "i \<ge> m \<Longrightarrow> subarray n m a (Heap.upd a i v h) = subarray n m a h"
+apply (simp add: subarray_def Heap.upd_def)
+apply (simp add: sublist'_update1)
+done
+
+lemma subarray_upd2: " i < n \<Longrightarrow> subarray n m a (Heap.upd a i v h) = subarray n m a h"
+apply (simp add: subarray_def Heap.upd_def)
+apply (subst sublist'_update2)
+apply fastsimp
+apply simp
+done
+
+lemma subarray_upd3: "\<lbrakk> n \<le> i; i < m\<rbrakk> \<Longrightarrow> subarray n m a (Heap.upd a i v h) = subarray n m a h[i - n := v]"
+unfolding subarray_def Heap.upd_def
+by (simp add: sublist'_update3)
+
+lemma subarray_Nil: "n \<ge> m \<Longrightarrow> subarray n m a h = []"
+by (simp add: subarray_def sublist'_Nil')
+
+lemma subarray_single: "\<lbrakk> n < Heap.length a h \<rbrakk> \<Longrightarrow> subarray n (Suc n) a h = [get_array a h ! n]"
+by (simp add: subarray_def Heap.length_def sublist'_single)
+
+lemma length_subarray: "m \<le> Heap.length a h \<Longrightarrow> List.length (subarray n m a h) = m - n"
+by (simp add: subarray_def Heap.length_def length_sublist')
+
+lemma length_subarray_0: "m \<le> Heap.length a h \<Longrightarrow> List.length (subarray 0 m a h) = m"
+by (simp add: length_subarray)
+
+lemma subarray_nth_array_Cons: "\<lbrakk> i < Heap.length a h; i < j \<rbrakk> \<Longrightarrow> (get_array a h ! i) # subarray (Suc i) j a h = subarray i j a h"
+unfolding Heap.length_def subarray_def
+by (simp add: sublist'_front)
+
+lemma subarray_nth_array_back: "\<lbrakk> i < j; j \<le> Heap.length a h\<rbrakk> \<Longrightarrow> subarray i j a h = subarray i (j - 1) a h @ [get_array a h ! (j - 1)]"
+unfolding Heap.length_def subarray_def
+by (simp add: sublist'_back)
+
+lemma subarray_append: "\<lbrakk> i < j; j < k \<rbrakk> \<Longrightarrow> subarray i j a h @ subarray j k a h = subarray i k a h"
+unfolding subarray_def
+by (simp add: sublist'_append)
+
+lemma subarray_all: "subarray 0 (Heap.length a h) a h = get_array a h"
+unfolding Heap.length_def subarray_def
+by (simp add: sublist'_all)
+
+lemma nth_subarray: "\<lbrakk> k < j - i; j \<le> Heap.length a h \<rbrakk> \<Longrightarrow> subarray i j a h ! k = get_array a h ! (i + k)"
+unfolding Heap.length_def subarray_def
+by (simp add: nth_sublist')
+
+lemma subarray_eq_samelength_iff: "Heap.length a h = Heap.length a h' \<Longrightarrow> (subarray i j a h = subarray i j a h') = (\<forall>i'. i \<le> i' \<and> i' < j \<longrightarrow> get_array a h ! i' = get_array a h' ! i')"
+unfolding Heap.length_def subarray_def by (rule sublist'_eq_samelength_iff)
+
+lemma all_in_set_subarray_conv: "(\<forall>j. j \<in> set (subarray l r a h) \<longrightarrow> P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < Heap.length a h \<longrightarrow> P (get_array a h ! k))"
+unfolding subarray_def Heap.length_def by (rule all_in_set_sublist'_conv)
+
+lemma ball_in_set_subarray_conv: "(\<forall>j \<in> set (subarray l r a h). P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < Heap.length a h \<longrightarrow> P (get_array a h ! k))"
+unfolding subarray_def Heap.length_def by (rule ball_in_set_sublist'_conv)
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Imperative_HOL/ex/Sublist.thy Mon Mar 23 19:01:34 2009 +0100
@@ -0,0 +1,505 @@
+(* $Id$ *)
+
+header {* Slices of lists *}
+
+theory Sublist
+imports Multiset
+begin
+
+
+lemma sublist_split: "i \<le> j \<and> j \<le> k \<Longrightarrow> sublist xs {i..<j} @ sublist xs {j..<k} = sublist xs {i..<k}"
+apply (induct xs arbitrary: i j k)
+apply simp
+apply (simp only: sublist_Cons)
+apply simp
+apply safe
+apply simp
+apply (erule_tac x="0" in meta_allE)
+apply (erule_tac x="j - 1" in meta_allE)
+apply (erule_tac x="k - 1" in meta_allE)
+apply (subgoal_tac "0 \<le> j - 1 \<and> j - 1 \<le> k - 1")
+apply simp
+apply (subgoal_tac "{ja. Suc ja < j} = {0..<j - Suc 0}")
+apply (subgoal_tac "{ja. j \<le> Suc ja \<and> Suc ja < k} = {j - Suc 0..<k - Suc 0}")
+apply (subgoal_tac "{j. Suc j < k} = {0..<k - Suc 0}")
+apply simp
+apply fastsimp
+apply fastsimp
+apply fastsimp
+apply fastsimp
+apply (erule_tac x="i - 1" in meta_allE)
+apply (erule_tac x="j - 1" in meta_allE)
+apply (erule_tac x="k - 1" in meta_allE)
+apply (subgoal_tac " {ja. i \<le> Suc ja \<and> Suc ja < j} = {i - 1 ..<j - 1}")
+apply (subgoal_tac " {ja. j \<le> Suc ja \<and> Suc ja < k} = {j - 1..<k - 1}")
+apply (subgoal_tac "{j. i \<le> Suc j \<and> Suc j < k} = {i - 1..<k - 1}")
+apply (subgoal_tac " i - 1 \<le> j - 1 \<and> j - 1 \<le> k - 1")
+apply simp
+apply fastsimp
+apply fastsimp
+apply fastsimp
+apply fastsimp
+done
+
+lemma sublist_update1: "i \<notin> inds \<Longrightarrow> sublist (xs[i := v]) inds = sublist xs inds"
+apply (induct xs arbitrary: i inds)
+apply simp
+apply (case_tac i)
+apply (simp add: sublist_Cons)
+apply (simp add: sublist_Cons)
+done
+
+lemma sublist_update2: "i \<in> inds \<Longrightarrow> sublist (xs[i := v]) inds = (sublist xs inds)[(card {k \<in> inds. k < i}):= v]"
+proof (induct xs arbitrary: i inds)
+ case Nil thus ?case by simp
+next
+ case (Cons x xs)
+ thus ?case
+ proof (cases i)
+ case 0 with Cons show ?thesis by (simp add: sublist_Cons)
+ next
+ case (Suc i')
+ with Cons show ?thesis
+ apply simp
+ apply (simp add: sublist_Cons)
+ apply auto
+ apply (auto simp add: nat.split)
+ apply (simp add: card_less_Suc[symmetric])
+ apply (simp add: card_less_Suc2)
+ done
+ qed
+qed
+
+lemma sublist_update: "sublist (xs[i := v]) inds = (if i \<in> inds then (sublist xs inds)[(card {k \<in> inds. k < i}) := v] else sublist xs inds)"
+by (simp add: sublist_update1 sublist_update2)
+
+lemma sublist_take: "sublist xs {j. j < m} = take m xs"
+apply (induct xs arbitrary: m)
+apply simp
+apply (case_tac m)
+apply simp
+apply (simp add: sublist_Cons)
+done
+
+lemma sublist_take': "sublist xs {0..<m} = take m xs"
+apply (induct xs arbitrary: m)
+apply simp
+apply (case_tac m)
+apply simp
+apply (simp add: sublist_Cons sublist_take)
+done
+
+lemma sublist_all[simp]: "sublist xs {j. j < length xs} = xs"
+apply (induct xs)
+apply simp
+apply (simp add: sublist_Cons)
+done
+
+lemma sublist_all'[simp]: "sublist xs {0..<length xs} = xs"
+apply (induct xs)
+apply simp
+apply (simp add: sublist_Cons)
+done
+
+lemma sublist_single: "a < length xs \<Longrightarrow> sublist xs {a} = [xs ! a]"
+apply (induct xs arbitrary: a)
+apply simp
+apply(case_tac aa)
+apply simp
+apply (simp add: sublist_Cons)
+apply simp
+apply (simp add: sublist_Cons)
+done
+
+lemma sublist_is_Nil: "\<forall>i \<in> inds. i \<ge> length xs \<Longrightarrow> sublist xs inds = []"
+apply (induct xs arbitrary: inds)
+apply simp
+apply (simp add: sublist_Cons)
+apply auto
+apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
+apply auto
+done
+
+lemma sublist_Nil': "sublist xs inds = [] \<Longrightarrow> \<forall>i \<in> inds. i \<ge> length xs"
+apply (induct xs arbitrary: inds)
+apply simp
+apply (simp add: sublist_Cons)
+apply (auto split: if_splits)
+apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
+apply (case_tac x, auto)
+done
+
+lemma sublist_Nil[simp]: "(sublist xs inds = []) = (\<forall>i \<in> inds. i \<ge> length xs)"
+apply (induct xs arbitrary: inds)
+apply simp
+apply (simp add: sublist_Cons)
+apply auto
+apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
+apply (case_tac x, auto)
+done
+
+lemma sublist_eq_subseteq: " \<lbrakk> inds' \<subseteq> inds; sublist xs inds = sublist ys inds \<rbrakk> \<Longrightarrow> sublist xs inds' = sublist ys inds'"
+apply (induct xs arbitrary: ys inds inds')
+apply simp
+apply (drule sym, rule sym)
+apply (simp add: sublist_Nil, fastsimp)
+apply (case_tac ys)
+apply (simp add: sublist_Nil, fastsimp)
+apply (auto simp add: sublist_Cons)
+apply (erule_tac x="list" in meta_allE)
+apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
+apply (erule_tac x="{j. Suc j \<in> inds'}" in meta_allE)
+apply fastsimp
+apply (erule_tac x="list" in meta_allE)
+apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
+apply (erule_tac x="{j. Suc j \<in> inds'}" in meta_allE)
+apply fastsimp
+done
+
+lemma sublist_eq: "\<lbrakk> \<forall>i \<in> inds. ((i < length xs) \<and> (i < length ys)) \<or> ((i \<ge> length xs ) \<and> (i \<ge> length ys)); \<forall>i \<in> inds. xs ! i = ys ! i \<rbrakk> \<Longrightarrow> sublist xs inds = sublist ys inds"
+apply (induct xs arbitrary: ys inds)
+apply simp
+apply (rule sym, simp add: sublist_Nil)
+apply (case_tac ys)
+apply (simp add: sublist_Nil)
+apply (auto simp add: sublist_Cons)
+apply (erule_tac x="list" in meta_allE)
+apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
+apply fastsimp
+apply (erule_tac x="list" in meta_allE)
+apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
+apply fastsimp
+done
+
+lemma sublist_eq_samelength: "\<lbrakk> length xs = length ys; \<forall>i \<in> inds. xs ! i = ys ! i \<rbrakk> \<Longrightarrow> sublist xs inds = sublist ys inds"
+by (rule sublist_eq, auto)
+
+lemma sublist_eq_samelength_iff: "length xs = length ys \<Longrightarrow> (sublist xs inds = sublist ys inds) = (\<forall>i \<in> inds. xs ! i = ys ! i)"
+apply (induct xs arbitrary: ys inds)
+apply simp
+apply (rule sym, simp add: sublist_Nil)
+apply (case_tac ys)
+apply (simp add: sublist_Nil)
+apply (auto simp add: sublist_Cons)
+apply (case_tac i)
+apply auto
+apply (case_tac i)
+apply auto
+done
+
+section {* Another sublist function *}
+
+function sublist' :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+ "sublist' n m [] = []"
+| "sublist' n 0 xs = []"
+| "sublist' 0 (Suc m) (x#xs) = (x#sublist' 0 m xs)"
+| "sublist' (Suc n) (Suc m) (x#xs) = sublist' n m xs"
+by pat_completeness auto
+termination by lexicographic_order
+
+subsection {* Proving equivalence to the other sublist command *}
+
+lemma sublist'_sublist: "sublist' n m xs = sublist xs {j. n \<le> j \<and> j < m}"
+apply (induct xs arbitrary: n m)
+apply simp
+apply (case_tac n)
+apply (case_tac m)
+apply simp
+apply (simp add: sublist_Cons)
+apply (case_tac m)
+apply simp
+apply (simp add: sublist_Cons)
+done
+
+
+lemma "sublist' n m xs = sublist xs {n..<m}"
+apply (induct xs arbitrary: n m)
+apply simp
+apply (case_tac n, case_tac m)
+apply simp
+apply simp
+apply (simp add: sublist_take')
+apply (case_tac m)
+apply simp
+apply (simp add: sublist_Cons sublist'_sublist)
+done
+
+
+subsection {* Showing equivalence to use of drop and take for definition *}
+
+lemma "sublist' n m xs = take (m - n) (drop n xs)"
+apply (induct xs arbitrary: n m)
+apply simp
+apply (case_tac m)
+apply simp
+apply (case_tac n)
+apply simp
+apply simp
+done
+
+subsection {* General lemma about sublist *}
+
+lemma sublist'_Nil[simp]: "sublist' i j [] = []"
+by simp
+
+lemma sublist'_Cons[simp]: "sublist' i (Suc j) (x#xs) = (case i of 0 \<Rightarrow> (x # sublist' 0 j xs) | Suc i' \<Rightarrow> sublist' i' j xs)"
+by (cases i) auto
+
+lemma sublist'_Cons2[simp]: "sublist' i j (x#xs) = (if (j = 0) then [] else ((if (i = 0) then [x] else []) @ sublist' (i - 1) (j - 1) xs))"
+apply (cases j)
+apply auto
+apply (cases i)
+apply auto
+done
+
+lemma sublist_n_0: "sublist' n 0 xs = []"
+by (induct xs, auto)
+
+lemma sublist'_Nil': "n \<ge> m \<Longrightarrow> sublist' n m xs = []"
+apply (induct xs arbitrary: n m)
+apply simp
+apply (case_tac m)
+apply simp
+apply (case_tac n)
+apply simp
+apply simp
+done
+
+lemma sublist'_Nil2: "n \<ge> length xs \<Longrightarrow> sublist' n m xs = []"
+apply (induct xs arbitrary: n m)
+apply simp
+apply (case_tac m)
+apply simp
+apply (case_tac n)
+apply simp
+apply simp
+done
+
+lemma sublist'_Nil3: "(sublist' n m xs = []) = ((n \<ge> m) \<or> (n \<ge> length xs))"
+apply (induct xs arbitrary: n m)
+apply simp
+apply (case_tac m)
+apply simp
+apply (case_tac n)
+apply simp
+apply simp
+done
+
+lemma sublist'_notNil: "\<lbrakk> n < length xs; n < m \<rbrakk> \<Longrightarrow> sublist' n m xs \<noteq> []"
+apply (induct xs arbitrary: n m)
+apply simp
+apply (case_tac m)
+apply simp
+apply (case_tac n)
+apply simp
+apply simp
+done
+
+lemma sublist'_single: "n < length xs \<Longrightarrow> sublist' n (Suc n) xs = [xs ! n]"
+apply (induct xs arbitrary: n)
+apply simp
+apply simp
+apply (case_tac n)
+apply (simp add: sublist_n_0)
+apply simp
+done
+
+lemma sublist'_update1: "i \<ge> m \<Longrightarrow> sublist' n m (xs[i:=v]) = sublist' n m xs"
+apply (induct xs arbitrary: n m i)
+apply simp
+apply simp
+apply (case_tac i)
+apply simp
+apply simp
+done
+
+lemma sublist'_update2: "i < n \<Longrightarrow> sublist' n m (xs[i:=v]) = sublist' n m xs"
+apply (induct xs arbitrary: n m i)
+apply simp
+apply simp
+apply (case_tac i)
+apply simp
+apply simp
+done
+
+lemma sublist'_update3: "\<lbrakk>n \<le> i; i < m\<rbrakk> \<Longrightarrow> sublist' n m (xs[i := v]) = (sublist' n m xs)[i - n := v]"
+proof (induct xs arbitrary: n m i)
+ case Nil thus ?case by auto
+next
+ case (Cons x xs)
+ thus ?case
+ apply -
+ apply auto
+ apply (cases i)
+ apply auto
+ apply (cases i)
+ apply auto
+ done
+qed
+
+lemma "\<lbrakk> sublist' i j xs = sublist' i j ys; n \<ge> i; m \<le> j \<rbrakk> \<Longrightarrow> sublist' n m xs = sublist' n m ys"
+proof (induct xs arbitrary: i j ys n m)
+ case Nil
+ thus ?case
+ apply -
+ apply (rule sym, drule sym)
+ apply (simp add: sublist'_Nil)
+ apply (simp add: sublist'_Nil3)
+ apply arith
+ done
+next
+ case (Cons x xs i j ys n m)
+ note c = this
+ thus ?case
+ proof (cases m)
+ case 0 thus ?thesis by (simp add: sublist_n_0)
+ next
+ case (Suc m')
+ note a = this
+ thus ?thesis
+ proof (cases n)
+ case 0 note b = this
+ show ?thesis
+ proof (cases ys)
+ case Nil with a b Cons.prems show ?thesis by (simp add: sublist'_Nil3)
+ next
+ case (Cons y ys)
+ show ?thesis
+ proof (cases j)
+ case 0 with a b Cons.prems show ?thesis by simp
+ next
+ case (Suc j') with a b Cons.prems Cons show ?thesis
+ apply -
+ apply (simp, rule Cons.hyps [of "0" "j'" "ys" "0" "m'"], auto)
+ done
+ qed
+ qed
+ next
+ case (Suc n')
+ show ?thesis
+ proof (cases ys)
+ case Nil with Suc a Cons.prems show ?thesis by (auto simp add: sublist'_Nil3)
+ next
+ case (Cons y ys) with Suc a Cons.prems show ?thesis
+ apply -
+ apply simp
+ apply (cases j)
+ apply simp
+ apply (cases i)
+ apply simp
+ apply (rule_tac j="nat" in Cons.hyps [of "0" _ "ys" "n'" "m'"])
+ apply simp
+ apply simp
+ apply simp
+ apply simp
+ apply (rule_tac i="nata" and j="nat" in Cons.hyps [of _ _ "ys" "n'" "m'"])
+ apply simp
+ apply simp
+ apply simp
+ done
+ qed
+ qed
+ qed
+qed
+
+lemma length_sublist': "j \<le> length xs \<Longrightarrow> length (sublist' i j xs) = j - i"
+by (induct xs arbitrary: i j, auto)
+
+lemma sublist'_front: "\<lbrakk> i < j; i < length xs \<rbrakk> \<Longrightarrow> sublist' i j xs = xs ! i # sublist' (Suc i) j xs"
+apply (induct xs arbitrary: a i j)
+apply simp
+apply (case_tac j)
+apply simp
+apply (case_tac i)
+apply simp
+apply simp
+done
+
+lemma sublist'_back: "\<lbrakk> i < j; j \<le> length xs \<rbrakk> \<Longrightarrow> sublist' i j xs = sublist' i (j - 1) xs @ [xs ! (j - 1)]"
+apply (induct xs arbitrary: a i j)
+apply simp
+apply simp
+apply (case_tac j)
+apply simp
+apply auto
+apply (case_tac nat)
+apply auto
+done
+
+(* suffices that j \<le> length xs and length ys *)
+lemma sublist'_eq_samelength_iff: "length xs = length ys \<Longrightarrow> (sublist' i j xs = sublist' i j ys) = (\<forall>i'. i \<le> i' \<and> i' < j \<longrightarrow> xs ! i' = ys ! i')"
+proof (induct xs arbitrary: ys i j)
+ case Nil thus ?case by simp
+next
+ case (Cons x xs)
+ thus ?case
+ apply -
+ apply (cases ys)
+ apply simp
+ apply simp
+ apply auto
+ apply (case_tac i', auto)
+ apply (erule_tac x="Suc i'" in allE, auto)
+ apply (erule_tac x="i' - 1" in allE, auto)
+ apply (case_tac i', auto)
+ apply (erule_tac x="Suc i'" in allE, auto)
+ done
+qed
+
+lemma sublist'_all[simp]: "sublist' 0 (length xs) xs = xs"
+by (induct xs, auto)
+
+lemma sublist'_sublist': "sublist' n m (sublist' i j xs) = sublist' (i + n) (min (i + m) j) xs"
+by (induct xs arbitrary: i j n m) (auto simp add: min_diff)
+
+lemma sublist'_append: "\<lbrakk> i \<le> j; j \<le> k \<rbrakk> \<Longrightarrow>(sublist' i j xs) @ (sublist' j k xs) = sublist' i k xs"
+by (induct xs arbitrary: i j k) auto
+
+lemma nth_sublist': "\<lbrakk> k < j - i; j \<le> length xs \<rbrakk> \<Longrightarrow> (sublist' i j xs) ! k = xs ! (i + k)"
+apply (induct xs arbitrary: i j k)
+apply auto
+apply (case_tac k)
+apply auto
+apply (case_tac i)
+apply auto
+done
+
+lemma set_sublist': "set (sublist' i j xs) = {x. \<exists>k. i \<le> k \<and> k < j \<and> k < List.length xs \<and> x = xs ! k}"
+apply (simp add: sublist'_sublist)
+apply (simp add: set_sublist)
+apply auto
+done
+
+lemma all_in_set_sublist'_conv: "(\<forall>j. j \<in> set (sublist' l r xs) \<longrightarrow> P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < List.length xs \<longrightarrow> P (xs ! k))"
+unfolding set_sublist' by blast
+
+lemma ball_in_set_sublist'_conv: "(\<forall>j \<in> set (sublist' l r xs). P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < List.length xs \<longrightarrow> P (xs ! k))"
+unfolding set_sublist' by blast
+
+
+lemma multiset_of_sublist:
+assumes l_r: "l \<le> r \<and> r \<le> List.length xs"
+assumes left: "\<forall> i. i < l \<longrightarrow> (xs::'a list) ! i = ys ! i"
+assumes right: "\<forall> i. i \<ge> r \<longrightarrow> (xs::'a list) ! i = ys ! i"
+assumes multiset: "multiset_of xs = multiset_of ys"
+ shows "multiset_of (sublist' l r xs) = multiset_of (sublist' l r ys)"
+proof -
+ from l_r have xs_def: "xs = (sublist' 0 l xs) @ (sublist' l r xs) @ (sublist' r (List.length xs) xs)" (is "_ = ?xs_long")
+ by (simp add: sublist'_append)
+ from multiset have length_eq: "List.length xs = List.length ys" by (rule multiset_of_eq_length)
+ with l_r have ys_def: "ys = (sublist' 0 l ys) @ (sublist' l r ys) @ (sublist' r (List.length ys) ys)" (is "_ = ?ys_long")
+ by (simp add: sublist'_append)
+ from xs_def ys_def multiset have "multiset_of ?xs_long = multiset_of ?ys_long" by simp
+ moreover
+ from left l_r length_eq have "sublist' 0 l xs = sublist' 0 l ys"
+ by (auto simp add: length_sublist' nth_sublist' intro!: nth_equalityI)
+ moreover
+ from right l_r length_eq have "sublist' r (List.length xs) xs = sublist' r (List.length ys) ys"
+ by (auto simp add: length_sublist' nth_sublist' intro!: nth_equalityI)
+ moreover
+ ultimately show ?thesis by (simp add: multiset_of_append)
+qed
+
+
+end
--- a/src/HOL/IsaMakefile Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/IsaMakefile Mon Mar 23 19:01:34 2009 +0100
@@ -649,7 +649,11 @@
$(LOG)/HOL-Imperative_HOL.gz: $(OUT)/HOL Imperative_HOL/Heap.thy \
Imperative_HOL/Heap_Monad.thy Imperative_HOL/Array.thy \
Imperative_HOL/Relational.thy \
- Imperative_HOL/Ref.thy Imperative_HOL/Imperative_HOL.thy
+ Imperative_HOL/Ref.thy Imperative_HOL/Imperative_HOL.thy \
+ Imperative_HOL/Imperative_HOL_ex.thy \
+ Imperative_HOL/ex/Imperative_Quicksort.thy \
+ Imperative_HOL/ex/Subarray.thy \
+ Imperative_HOL/ex/Sublist.thy
@$(ISABELLE_TOOL) usedir $(OUT)/HOL Imperative_HOL
@@ -836,7 +840,7 @@
ex/Formal_Power_Series_Examples.thy ex/Fundefs.thy \
ex/Groebner_Examples.thy ex/Guess.thy ex/HarmonicSeries.thy \
ex/Hebrew.thy ex/Hex_Bin_Examples.thy ex/Higher_Order_Logic.thy \
- ex/Hilbert_Classical.thy ex/ImperativeQuicksort.thy \
+ ex/Hilbert_Classical.thy \
ex/Induction_Scheme.thy ex/InductiveInvariant.thy \
ex/InductiveInvariant_examples.thy ex/Intuitionistic.thy \
ex/Lagrange.thy ex/LocaleTest2.thy ex/MT.thy ex/MergeSort.thy \
@@ -845,8 +849,8 @@
ex/Quickcheck_Examples.thy ex/Quickcheck_Generators.thy ex/ROOT.ML \
ex/Recdefs.thy ex/Records.thy ex/ReflectionEx.thy \
ex/Refute_Examples.thy ex/SAT_Examples.thy ex/SVC_Oracle.thy \
- ex/Serbian.thy ex/Sqrt.thy ex/Sqrt_Script.thy ex/Subarray.thy \
- ex/Sublist.thy ex/Sudoku.thy ex/Tarski.thy ex/Term_Of_Syntax.thy \
+ ex/Serbian.thy ex/Sqrt.thy ex/Sqrt_Script.thy \
+ ex/Sudoku.thy ex/Tarski.thy ex/Term_Of_Syntax.thy \
ex/Termination.thy ex/Unification.thy ex/document/root.bib \
ex/document/root.tex ex/set.thy ex/svc_funcs.ML ex/svc_test.thy \
ex/Predicate_Compile.thy ex/predicate_compile.ML
--- a/src/HOL/NSA/hypreal_arith.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/NSA/hypreal_arith.ML Mon Mar 23 19:01:34 2009 +0100
@@ -30,10 +30,10 @@
Simplifier.simproc (the_context ())
"fast_hypreal_arith"
["(m::hypreal) < n", "(m::hypreal) <= n", "(m::hypreal) = n"]
- (K LinArith.lin_arith_simproc);
+ (K Lin_Arith.lin_arith_simproc);
val hypreal_arith_setup =
- LinArith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
+ Lin_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
{add_mono_thms = add_mono_thms,
mult_mono_thms = mult_mono_thms,
inj_thms = real_inj_thms @ inj_thms,
--- a/src/HOL/Nat.thy Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Nat.thy Mon Mar 23 19:01:34 2009 +0100
@@ -63,9 +63,8 @@
end
lemma Suc_not_Zero: "Suc m \<noteq> 0"
- apply (simp add: Zero_nat_def Suc_def Abs_Nat_inject [unfolded mem_def]
+ by (simp add: Zero_nat_def Suc_def Abs_Nat_inject [unfolded mem_def]
Rep_Nat [unfolded mem_def] Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep [unfolded mem_def])
- done
lemma Zero_not_Suc: "0 \<noteq> Suc m"
by (rule not_sym, rule Suc_not_Zero not_sym)
@@ -82,7 +81,7 @@
done
lemma nat_induct [case_names 0 Suc, induct type: nat]:
- -- {* for backward compatibility -- naming of variables differs *}
+ -- {* for backward compatibility -- names of variables differ *}
fixes n
assumes "P 0"
and "\<And>n. P n \<Longrightarrow> P (Suc n)"
@@ -1345,19 +1344,13 @@
shows "u = s"
using 2 1 by (rule trans)
+setup Arith_Data.setup
+
use "Tools/nat_arith.ML"
declaration {* K Nat_Arith.setup *}
-ML{*
-structure ArithFacts =
- NamedThmsFun(val name = "arith"
- val description = "arith facts - only ground formulas");
-*}
-
-setup ArithFacts.setup
-
use "Tools/lin_arith.ML"
-declaration {* K LinArith.setup *}
+declaration {* K Lin_Arith.setup *}
lemmas [arith_split] = nat_diff_split split_min split_max
--- a/src/HOL/NatBin.thy Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/NatBin.thy Mon Mar 23 19:01:34 2009 +0100
@@ -651,7 +651,7 @@
val numeral_ss = @{simpset} addsimps @{thms numerals};
val nat_bin_arith_setup =
- LinArith.map_data
+ Lin_Arith.map_data
(fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
{add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
inj_thms = inj_thms,
--- a/src/HOL/Presburger.thy Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Presburger.thy Mon Mar 23 19:01:34 2009 +0100
@@ -439,12 +439,7 @@
use "Tools/Qelim/presburger.ML"
-declaration {* fn _ =>
- arith_tactic_add
- (mk_arith_tactic "presburger" (fn ctxt => fn i => fn st =>
- (warning "Trying Presburger arithmetic ...";
- Presburger.cooper_tac true [] [] ctxt i st)))
-*}
+setup {* Arith_Data.add_tactic "Presburger arithmetic" (K (Presburger.cooper_tac true [] [])) *}
method_setup presburger = {*
let
--- a/src/HOL/Tools/Qelim/cooper.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Tools/Qelim/cooper.ML Mon Mar 23 19:01:34 2009 +0100
@@ -172,7 +172,7 @@
(* Canonical linear form for terms, formulae etc.. *)
fun provelin ctxt t = Goal.prove ctxt [] [] t
- (fn _ => EVERY [simp_tac lin_ss 1, TRY (simple_arith_tac ctxt 1)]);
+ (fn _ => EVERY [simp_tac lin_ss 1, TRY (linear_arith_tac ctxt 1)]);
fun linear_cmul 0 tm = zero
| linear_cmul n tm = case tm of
Const (@{const_name HOL.plus}, _) $ a $ b => addC $ linear_cmul n a $ linear_cmul n b
--- a/src/HOL/Tools/TFL/post.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Tools/TFL/post.ML Mon Mar 23 19:01:34 2009 +0100
@@ -55,7 +55,7 @@
Prim.postprocess strict
{wf_tac = REPEAT (ares_tac wfs 1),
terminator = asm_simp_tac ss 1
- THEN TRY (silent_arith_tac (Simplifier.the_context ss) 1 ORELSE
+ THEN TRY (Arith_Data.arith_tac (Simplifier.the_context ss) 1 ORELSE
fast_tac (cs addSDs [@{thm not0_implies_Suc}] addss ss) 1),
simplifier = Rules.simpl_conv ss []};
--- a/src/HOL/Tools/arith_data.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Tools/arith_data.ML Mon Mar 23 19:01:34 2009 +0100
@@ -6,6 +6,11 @@
signature ARITH_DATA =
sig
+ val arith_tac: Proof.context -> int -> tactic
+ val verbose_arith_tac: Proof.context -> int -> tactic
+ val add_tactic: string -> (bool -> Proof.context -> int -> tactic) -> theory -> theory
+ val get_arith_facts: Proof.context -> thm list
+
val prove_conv_nohyps: tactic list -> Proof.context -> term * term -> thm option
val prove_conv: tactic list -> Proof.context -> thm list -> term * term -> thm option
val prove_conv2: tactic -> (simpset -> tactic) -> simpset -> term * term -> thm
@@ -14,11 +19,54 @@
val trans_tac: thm option -> tactic
val prep_simproc: string * string list * (theory -> simpset -> term -> thm option)
-> simproc
+
+ val setup: theory -> theory
end;
structure Arith_Data: ARITH_DATA =
struct
+(* slots for pluging in arithmetic facts and tactics *)
+
+structure Arith_Facts = NamedThmsFun(
+ val name = "arith"
+ val description = "arith facts - only ground formulas"
+);
+
+val get_arith_facts = Arith_Facts.get;
+
+structure Arith_Tactics = TheoryDataFun
+(
+ type T = (serial * (string * (bool -> Proof.context -> int -> tactic))) list;
+ val empty = [];
+ val copy = I;
+ val extend = I;
+ fun merge _ = AList.merge (op =) (K true);
+);
+
+fun add_tactic name tac = Arith_Tactics.map (cons (serial (), (name, tac)));
+
+fun gen_arith_tac verbose ctxt =
+ let
+ val tactics = (Arith_Tactics.get o ProofContext.theory_of) ctxt
+ fun invoke (_, (name, tac)) k st = (if verbose
+ then warning ("Trying " ^ name ^ "...") else ();
+ tac verbose ctxt k st);
+ in FIRST' (map invoke (rev tactics)) end;
+
+val arith_tac = gen_arith_tac false;
+val verbose_arith_tac = gen_arith_tac true;
+
+val arith_method = Args.bang_facts >> (fn prems => fn ctxt =>
+ METHOD (fn facts => HEADGOAL (Method.insert_tac (prems @ get_arith_facts ctxt @ facts)
+ THEN' verbose_arith_tac ctxt)));
+
+val setup = Arith_Facts.setup
+ #> Method.setup @{binding arith} arith_method "various arithmetic decision procedures";
+
+
+(* various auxiliary and legacy *)
+
fun prove_conv_nohyps tacs ctxt (t, u) =
if t aconv u then NONE
else let val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u))
--- a/src/HOL/Tools/function_package/scnp_reconstruct.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Tools/function_package/scnp_reconstruct.ML Mon Mar 23 19:01:34 2009 +0100
@@ -197,7 +197,7 @@
else if b <= a then @{thm pair_leqI2} else @{thm pair_leqI1}
in
rtac rule 1 THEN PRIMITIVE (Thm.elim_implies stored_thm)
- THEN (if tag_flag then arith_tac ctxt 1 else all_tac)
+ THEN (if tag_flag then Arith_Data.verbose_arith_tac ctxt 1 else all_tac)
end
fun steps_tac MAX strict lq lp =
--- a/src/HOL/Tools/int_arith.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Tools/int_arith.ML Mon Mar 23 19:01:34 2009 +0100
@@ -530,7 +530,7 @@
:: Int_Numeral_Simprocs.cancel_numerals;
val setup =
- LinArith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
+ Lin_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
{add_mono_thms = add_mono_thms,
mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} :: mult_mono_thms,
inj_thms = nat_inj_thms @ inj_thms,
@@ -547,7 +547,7 @@
"fast_int_arith"
["(m::'a::{ordered_idom,number_ring}) < n",
"(m::'a::{ordered_idom,number_ring}) <= n",
- "(m::'a::{ordered_idom,number_ring}) = n"] (K LinArith.lin_arith_simproc);
+ "(m::'a::{ordered_idom,number_ring}) = n"] (K Lin_Arith.lin_arith_simproc);
end;
--- a/src/HOL/Tools/int_factor_simprocs.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Tools/int_factor_simprocs.ML Mon Mar 23 19:01:34 2009 +0100
@@ -232,7 +232,7 @@
val less = Const(@{const_name HOL.less}, [T,T] ---> HOLogic.boolT);
val pos = less $ zero $ t and neg = less $ t $ zero
fun prove p =
- Option.map Eq_True_elim (LinArith.lin_arith_simproc ss p)
+ Option.map Eq_True_elim (Lin_Arith.lin_arith_simproc ss p)
handle THM _ => NONE
in case prove pos of
SOME th => SOME(th RS pos_th)
--- a/src/HOL/Tools/lin_arith.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Tools/lin_arith.ML Mon Mar 23 19:01:34 2009 +0100
@@ -6,13 +6,9 @@
signature BASIC_LIN_ARITH =
sig
- type arith_tactic
- val mk_arith_tactic: string -> (Proof.context -> int -> tactic) -> arith_tactic
- val eq_arith_tactic: arith_tactic * arith_tactic -> bool
val arith_split_add: attribute
val arith_discrete: string -> Context.generic -> Context.generic
val arith_inj_const: string * typ -> Context.generic -> Context.generic
- val arith_tactic_add: arith_tactic -> Context.generic -> Context.generic
val fast_arith_split_limit: int Config.T
val fast_arith_neq_limit: int Config.T
val lin_arith_pre_tac: Proof.context -> int -> tactic
@@ -21,9 +17,7 @@
val trace_arith: bool ref
val lin_arith_simproc: simpset -> term -> thm option
val fast_nat_arith_simproc: simproc
- val simple_arith_tac: Proof.context -> int -> tactic
- val arith_tac: Proof.context -> int -> tactic
- val silent_arith_tac: Proof.context -> int -> tactic
+ val linear_arith_tac: Proof.context -> int -> tactic
end;
signature LIN_ARITH =
@@ -39,7 +33,7 @@
val setup: Context.generic -> Context.generic
end;
-structure LinArith: LIN_ARITH =
+structure Lin_Arith: LIN_ARITH =
struct
(* Parameters data for general linear arithmetic functor *)
@@ -72,7 +66,7 @@
let val _ $ t = Thm.prop_of thm
in t = Const("False",HOLogic.boolT) end;
-fun is_nat(t) = fastype_of1 t = HOLogic.natT;
+fun is_nat t = (fastype_of1 t = HOLogic.natT);
fun mk_nat_thm sg t =
let val ct = cterm_of sg t and cn = cterm_of sg (Var(("n",0),HOLogic.natT))
@@ -83,49 +77,35 @@
(* arith context data *)
-datatype arith_tactic =
- ArithTactic of {name: string, tactic: Proof.context -> int -> tactic, id: stamp};
-
-fun mk_arith_tactic name tactic = ArithTactic {name = name, tactic = tactic, id = stamp ()};
-
-fun eq_arith_tactic (ArithTactic {id = id1, ...}, ArithTactic {id = id2, ...}) = (id1 = id2);
-
structure ArithContextData = GenericDataFun
(
type T = {splits: thm list,
inj_consts: (string * typ) list,
- discrete: string list,
- tactics: arith_tactic list};
- val empty = {splits = [], inj_consts = [], discrete = [], tactics = []};
+ discrete: string list};
+ val empty = {splits = [], inj_consts = [], discrete = []};
val extend = I;
fun merge _
- ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1, tactics= tactics1},
- {splits= splits2, inj_consts= inj_consts2, discrete= discrete2, tactics= tactics2}) : T =
+ ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1},
+ {splits= splits2, inj_consts= inj_consts2, discrete= discrete2}) : T =
{splits = Library.merge Thm.eq_thm_prop (splits1, splits2),
inj_consts = Library.merge (op =) (inj_consts1, inj_consts2),
- discrete = Library.merge (op =) (discrete1, discrete2),
- tactics = Library.merge eq_arith_tactic (tactics1, tactics2)};
+ discrete = Library.merge (op =) (discrete1, discrete2)};
);
val get_arith_data = ArithContextData.get o Context.Proof;
val arith_split_add = Thm.declaration_attribute (fn thm =>
- ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
+ ArithContextData.map (fn {splits, inj_consts, discrete} =>
{splits = update Thm.eq_thm_prop thm splits,
- inj_consts = inj_consts, discrete = discrete, tactics = tactics}));
-
-fun arith_discrete d = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
- {splits = splits, inj_consts = inj_consts,
- discrete = update (op =) d discrete, tactics = tactics});
+ inj_consts = inj_consts, discrete = discrete}));
-fun arith_inj_const c = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
- {splits = splits, inj_consts = update (op =) c inj_consts,
- discrete = discrete, tactics= tactics});
+fun arith_discrete d = ArithContextData.map (fn {splits, inj_consts, discrete} =>
+ {splits = splits, inj_consts = inj_consts,
+ discrete = update (op =) d discrete});
-fun arith_tactic_add tac = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
- {splits = splits, inj_consts = inj_consts, discrete = discrete,
- tactics = update eq_arith_tactic tac tactics});
-
+fun arith_inj_const c = ArithContextData.map (fn {splits, inj_consts, discrete} =>
+ {splits = splits, inj_consts = update (op =) c inj_consts,
+ discrete = discrete});
val (fast_arith_split_limit, setup1) = Attrib.config_int "fast_arith_split_limit" 9;
val (fast_arith_neq_limit, setup2) = Attrib.config_int "fast_arith_neq_limit" 9;
@@ -794,7 +774,7 @@
Most of the work is done by the cancel tactics. *)
val init_arith_data =
- Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, ...} =>
+ map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, ...} =>
{add_mono_thms = add_mono_thms @
@{thms add_mono_thms_ordered_semiring} @ @{thms add_mono_thms_ordered_field},
mult_mono_thms = mult_mono_thms,
@@ -815,7 +795,7 @@
arith_discrete "nat";
fun add_arith_facts ss =
- add_prems (ArithFacts.get (MetaSimplifier.the_context ss)) ss;
+ add_prems (Arith_Data.get_arith_facts (MetaSimplifier.the_context ss)) ss;
val lin_arith_simproc = add_arith_facts #> Fast_Arith.lin_arith_simproc;
@@ -895,27 +875,16 @@
(REPEAT_DETERM o split_tac (#splits (get_arith_data ctxt)))
(fast_ex_arith_tac ctxt ex);
-fun more_arith_tacs ctxt =
- let val tactics = #tactics (get_arith_data ctxt)
- in FIRST' (map (fn ArithTactic {tactic, ...} => tactic ctxt) tactics) end;
-
in
-fun simple_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
- ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true];
-
-fun arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
- ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true,
- more_arith_tacs ctxt];
+fun gen_linear_arith_tac ex ctxt = FIRST' [fast_arith_tac ctxt,
+ ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt ex];
-fun silent_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
- ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt false,
- more_arith_tacs ctxt];
+val linear_arith_tac = gen_linear_arith_tac true;
-val arith_method = Args.bang_facts >>
- (fn prems => fn ctxt => METHOD (fn facts =>
- HEADGOAL (Method.insert_tac (prems @ ArithFacts.get ctxt @ facts)
- THEN' arith_tac ctxt)));
+val linarith_method = Args.bang_facts >> (fn prems => fn ctxt =>
+ METHOD (fn facts => HEADGOAL (Method.insert_tac (prems @ Arith_Data.get_arith_facts ctxt @ facts)
+ THEN' linear_arith_tac ctxt)));
end;
@@ -929,11 +898,12 @@
(add_arith_facts #> Fast_Arith.cut_lin_arith_tac))) #>
Context.mapping
(setup_options #>
- Method.setup @{binding arith} arith_method "decide linear arithmetic" #>
+ Arith_Data.add_tactic "linear arithmetic" gen_linear_arith_tac #>
+ Method.setup @{binding linarith} linarith_method "linear arithmetic" #>
Attrib.setup @{binding arith_split} (Scan.succeed arith_split_add)
"declaration of split rules for arithmetic procedure") I;
end;
-structure BasicLinArith: BASIC_LIN_ARITH = LinArith;
-open BasicLinArith;
+structure Basic_Lin_Arith: BASIC_LIN_ARITH = Lin_Arith;
+open Basic_Lin_Arith;
--- a/src/HOL/Tools/nat_simprocs.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Tools/nat_simprocs.ML Mon Mar 23 19:01:34 2009 +0100
@@ -565,7 +565,7 @@
in
val nat_simprocs_setup =
- LinArith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
+ Lin_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
{add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
inj_thms = inj_thms, lessD = lessD, neqE = neqE,
simpset = simpset addsimps add_rules
--- a/src/HOL/Tools/rat_arith.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Tools/rat_arith.ML Mon Mar 23 19:01:34 2009 +0100
@@ -35,7 +35,7 @@
in
val rat_arith_setup =
- LinArith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
+ Lin_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
{add_mono_thms = add_mono_thms,
mult_mono_thms = mult_mono_thms,
inj_thms = int_inj_thms @ nat_inj_thms @ inj_thms,
--- a/src/HOL/Tools/real_arith.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Tools/real_arith.ML Mon Mar 23 19:01:34 2009 +0100
@@ -29,7 +29,7 @@
in
val real_arith_setup =
- LinArith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
+ Lin_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
{add_mono_thms = add_mono_thms,
mult_mono_thms = mult_mono_thms,
inj_thms = int_inj_thms @ nat_inj_thms @ inj_thms,
--- a/src/HOL/Word/WordArith.thy Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/Word/WordArith.thy Mon Mar 23 19:01:34 2009 +0100
@@ -512,7 +512,7 @@
fun uint_arith_tacs ctxt =
let
- fun arith_tac' n t = arith_tac ctxt n t handle COOPER => Seq.empty;
+ fun arith_tac' n t = Arith_Data.verbose_arith_tac ctxt n t handle COOPER => Seq.empty;
val cs = local_claset_of ctxt;
val ss = local_simpset_of ctxt;
in
@@ -1075,7 +1075,7 @@
fun unat_arith_tacs ctxt =
let
- fun arith_tac' n t = arith_tac ctxt n t handle COOPER => Seq.empty;
+ fun arith_tac' n t = Arith_Data.verbose_arith_tac ctxt n t handle COOPER => Seq.empty;
val cs = local_claset_of ctxt;
val ss = local_simpset_of ctxt;
in
--- a/src/HOL/ex/Arith_Examples.thy Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/ex/Arith_Examples.thy Mon Mar 23 19:01:34 2009 +0100
@@ -1,5 +1,4 @@
(* Title: HOL/ex/Arith_Examples.thy
- ID: $Id$
Author: Tjark Weber
*)
@@ -14,13 +13,13 @@
@{ML fast_arith_tac} is a very basic version of the tactic. It performs no
meta-to-object-logic conversion, and only some splitting of operators.
- @{ML simple_arith_tac} performs meta-to-object-logic conversion, full
+ @{ML linear_arith_tac} performs meta-to-object-logic conversion, full
splitting of operators, and NNF normalization of the goal. The @{text arith}
method combines them both, and tries other methods (e.g.~@{text presburger})
as well. This is the one that you should use in your proofs!
An @{text arith}-based simproc is available as well (see @{ML
- LinArith.lin_arith_simproc}), which---for performance
+ Lin_Arith.lin_arith_simproc}), which---for performance
reasons---however does even less splitting than @{ML fast_arith_tac}
at the moment (namely inequalities only). (On the other hand, it
does take apart conjunctions, which @{ML fast_arith_tac} currently
@@ -83,7 +82,7 @@
by (tactic {* fast_arith_tac @{context} 1 *})
lemma "!!x. ((x::nat) <= y) = (x - y = 0)"
- by (tactic {* simple_arith_tac @{context} 1 *})
+ by (tactic {* linear_arith_tac @{context} 1 *})
lemma "[| (x::nat) < y; d < 1 |] ==> x - y = d"
by (tactic {* fast_arith_tac @{context} 1 *})
@@ -140,34 +139,34 @@
subsection {* Meta-Logic *}
lemma "x < Suc y == x <= y"
- by (tactic {* simple_arith_tac @{context} 1 *})
+ by (tactic {* linear_arith_tac @{context} 1 *})
lemma "((x::nat) == z ==> x ~= y) ==> x ~= y | z ~= y"
- by (tactic {* simple_arith_tac @{context} 1 *})
+ by (tactic {* linear_arith_tac @{context} 1 *})
subsection {* Various Other Examples *}
lemma "(x < Suc y) = (x <= y)"
- by (tactic {* simple_arith_tac @{context} 1 *})
+ by (tactic {* linear_arith_tac @{context} 1 *})
lemma "[| (x::nat) < y; y < z |] ==> x < z"
by (tactic {* fast_arith_tac @{context} 1 *})
lemma "(x::nat) < y & y < z ==> x < z"
- by (tactic {* simple_arith_tac @{context} 1 *})
+ by (tactic {* linear_arith_tac @{context} 1 *})
text {* This example involves no arithmetic at all, but is solved by
preprocessing (i.e. NNF normalization) alone. *}
lemma "(P::bool) = Q ==> Q = P"
- by (tactic {* simple_arith_tac @{context} 1 *})
+ by (tactic {* linear_arith_tac @{context} 1 *})
lemma "[| P = (x = 0); (~P) = (y = 0) |] ==> min (x::nat) y = 0"
- by (tactic {* simple_arith_tac @{context} 1 *})
+ by (tactic {* linear_arith_tac @{context} 1 *})
lemma "[| P = (x = 0); (~P) = (y = 0) |] ==> max (x::nat) y = x + y"
- by (tactic {* simple_arith_tac @{context} 1 *})
+ by (tactic {* linear_arith_tac @{context} 1 *})
lemma "[| (x::nat) ~= y; a + 2 = b; a < y; y < b; a < x; x < b |] ==> False"
by (tactic {* fast_arith_tac @{context} 1 *})
@@ -185,7 +184,7 @@
by (tactic {* fast_arith_tac @{context} 1 *})
lemma "[| (x::nat) < y; P (x - y) |] ==> P 0"
- by (tactic {* simple_arith_tac @{context} 1 *})
+ by (tactic {* linear_arith_tac @{context} 1 *})
lemma "(x - y) - (x::nat) = (x - x) - y"
by (tactic {* fast_arith_tac @{context} 1 *})
@@ -207,7 +206,7 @@
(* preprocessing negates the goal and tries to compute its negation *)
(* normal form, which creates lots of separate cases for this *)
(* disjunction of conjunctions *)
-(* by (tactic {* simple_arith_tac 1 *}) *)
+(* by (tactic {* linear_arith_tac 1 *}) *)
oops
lemma "2 * (x::nat) ~= 1"
--- a/src/HOL/ex/ImperativeQuicksort.thy Mon Mar 23 15:33:35 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,637 +0,0 @@
-theory ImperativeQuicksort
-imports "~~/src/HOL/Imperative_HOL/Imperative_HOL" Subarray Multiset Efficient_Nat
-begin
-
-text {* We prove QuickSort correct in the Relational Calculus. *}
-
-definition swap :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap"
-where
- "swap arr i j = (
- do
- x \<leftarrow> nth arr i;
- y \<leftarrow> nth arr j;
- upd i y arr;
- upd j x arr;
- return ()
- done)"
-
-lemma swap_permutes:
- assumes "crel (swap a i j) h h' rs"
- shows "multiset_of (get_array a h')
- = multiset_of (get_array a h)"
- using assms
- unfolding swap_def
- by (auto simp add: Heap.length_def multiset_of_swap dest: sym [of _ "h'"] elim!: crelE crel_nth crel_return crel_upd)
-
-function part1 :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat Heap"
-where
- "part1 a left right p = (
- if (right \<le> left) then return right
- else (do
- v \<leftarrow> nth a left;
- (if (v \<le> p) then (part1 a (left + 1) right p)
- else (do swap a left right;
- part1 a left (right - 1) p done))
- done))"
-by pat_completeness auto
-
-termination
-by (relation "measure (\<lambda>(_,l,r,_). r - l )") auto
-
-declare part1.simps[simp del]
-
-lemma part_permutes:
- assumes "crel (part1 a l r p) h h' rs"
- shows "multiset_of (get_array a h')
- = multiset_of (get_array a h)"
- using assms
-proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
- case (1 a l r p h h' rs)
- thus ?case
- unfolding part1.simps [of a l r p]
- by (elim crelE crel_if crel_return crel_nth) (auto simp add: swap_permutes)
-qed
-
-lemma part_returns_index_in_bounds:
- assumes "crel (part1 a l r p) h h' rs"
- assumes "l \<le> r"
- shows "l \<le> rs \<and> rs \<le> r"
-using assms
-proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
- case (1 a l r p h h' rs)
- note cr = `crel (part1 a l r p) h h' rs`
- show ?case
- proof (cases "r \<le> l")
- case True (* Terminating case *)
- with cr `l \<le> r` show ?thesis
- unfolding part1.simps[of a l r p]
- by (elim crelE crel_if crel_return crel_nth) auto
- next
- case False (* recursive case *)
- note rec_condition = this
- let ?v = "get_array a h ! l"
- show ?thesis
- proof (cases "?v \<le> p")
- case True
- with cr False
- have rec1: "crel (part1 a (l + 1) r p) h h' rs"
- unfolding part1.simps[of a l r p]
- by (elim crelE crel_nth crel_if crel_return) auto
- from rec_condition have "l + 1 \<le> r" by arith
- from 1(1)[OF rec_condition True rec1 `l + 1 \<le> r`]
- show ?thesis by simp
- next
- case False
- with rec_condition cr
- obtain h1 where swp: "crel (swap a l r) h h1 ()"
- and rec2: "crel (part1 a l (r - 1) p) h1 h' rs"
- unfolding part1.simps[of a l r p]
- by (elim crelE crel_nth crel_if crel_return) auto
- from rec_condition have "l \<le> r - 1" by arith
- from 1(2) [OF rec_condition False rec2 `l \<le> r - 1`] show ?thesis by fastsimp
- qed
- qed
-qed
-
-lemma part_length_remains:
- assumes "crel (part1 a l r p) h h' rs"
- shows "Heap.length a h = Heap.length a h'"
-using assms
-proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
- case (1 a l r p h h' rs)
- note cr = `crel (part1 a l r p) h h' rs`
-
- show ?case
- proof (cases "r \<le> l")
- case True (* Terminating case *)
- with cr show ?thesis
- unfolding part1.simps[of a l r p]
- by (elim crelE crel_if crel_return crel_nth) auto
- next
- case False (* recursive case *)
- with cr 1 show ?thesis
- unfolding part1.simps [of a l r p] swap_def
- by (auto elim!: crelE crel_if crel_nth crel_return crel_upd) fastsimp
- qed
-qed
-
-lemma part_outer_remains:
- assumes "crel (part1 a l r p) h h' rs"
- shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"
- using assms
-proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
- case (1 a l r p h h' rs)
- note cr = `crel (part1 a l r p) h h' rs`
-
- show ?case
- proof (cases "r \<le> l")
- case True (* Terminating case *)
- with cr show ?thesis
- unfolding part1.simps[of a l r p]
- by (elim crelE crel_if crel_return crel_nth) auto
- next
- case False (* recursive case *)
- note rec_condition = this
- let ?v = "get_array a h ! l"
- show ?thesis
- proof (cases "?v \<le> p")
- case True
- with cr False
- have rec1: "crel (part1 a (l + 1) r p) h h' rs"
- unfolding part1.simps[of a l r p]
- by (elim crelE crel_nth crel_if crel_return) auto
- from 1(1)[OF rec_condition True rec1]
- show ?thesis by fastsimp
- next
- case False
- with rec_condition cr
- obtain h1 where swp: "crel (swap a l r) h h1 ()"
- and rec2: "crel (part1 a l (r - 1) p) h1 h' rs"
- unfolding part1.simps[of a l r p]
- by (elim crelE crel_nth crel_if crel_return) auto
- from swp rec_condition have
- "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array a h ! i = get_array a h1 ! i"
- unfolding swap_def
- by (elim crelE crel_nth crel_upd crel_return) auto
- with 1(2) [OF rec_condition False rec2] show ?thesis by fastsimp
- qed
- qed
-qed
-
-
-lemma part_partitions:
- assumes "crel (part1 a l r p) h h' rs"
- shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array (a::nat array) h' ! i \<le> p)
- \<and> (\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array a h' ! i \<ge> p)"
- using assms
-proof (induct a l r p arbitrary: h h' rs rule:part1.induct)
- case (1 a l r p h h' rs)
- note cr = `crel (part1 a l r p) h h' rs`
-
- show ?case
- proof (cases "r \<le> l")
- case True (* Terminating case *)
- with cr have "rs = r"
- unfolding part1.simps[of a l r p]
- by (elim crelE crel_if crel_return crel_nth) auto
- with True
- show ?thesis by auto
- next
- case False (* recursive case *)
- note lr = this
- let ?v = "get_array a h ! l"
- show ?thesis
- proof (cases "?v \<le> p")
- case True
- with lr cr
- have rec1: "crel (part1 a (l + 1) r p) h h' rs"
- unfolding part1.simps[of a l r p]
- by (elim crelE crel_nth crel_if crel_return) auto
- from True part_outer_remains[OF rec1] have a_l: "get_array a h' ! l \<le> p"
- by fastsimp
- have "\<forall>i. (l \<le> i = (l = i \<or> Suc l \<le> i))" by arith
- with 1(1)[OF False True rec1] a_l show ?thesis
- by auto
- next
- case False
- with lr cr
- obtain h1 where swp: "crel (swap a l r) h h1 ()"
- and rec2: "crel (part1 a l (r - 1) p) h1 h' rs"
- unfolding part1.simps[of a l r p]
- by (elim crelE crel_nth crel_if crel_return) auto
- from swp False have "get_array a h1 ! r \<ge> p"
- unfolding swap_def
- by (auto simp add: Heap.length_def elim!: crelE crel_nth crel_upd crel_return)
- with part_outer_remains [OF rec2] lr have a_r: "get_array a h' ! r \<ge> p"
- by fastsimp
- have "\<forall>i. (i \<le> r = (i = r \<or> i \<le> r - 1))" by arith
- with 1(2)[OF lr False rec2] a_r show ?thesis
- by auto
- qed
- qed
-qed
-
-
-fun partition :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat Heap"
-where
- "partition a left right = (do
- pivot \<leftarrow> nth a right;
- middle \<leftarrow> part1 a left (right - 1) pivot;
- v \<leftarrow> nth a middle;
- m \<leftarrow> return (if (v \<le> pivot) then (middle + 1) else middle);
- swap a m right;
- return m
- done)"
-
-declare partition.simps[simp del]
-
-lemma partition_permutes:
- assumes "crel (partition a l r) h h' rs"
- shows "multiset_of (get_array a h')
- = multiset_of (get_array a h)"
-proof -
- from assms part_permutes swap_permutes show ?thesis
- unfolding partition.simps
- by (elim crelE crel_return crel_nth crel_if crel_upd) auto
-qed
-
-lemma partition_length_remains:
- assumes "crel (partition a l r) h h' rs"
- shows "Heap.length a h = Heap.length a h'"
-proof -
- from assms part_length_remains show ?thesis
- unfolding partition.simps swap_def
- by (elim crelE crel_return crel_nth crel_if crel_upd) auto
-qed
-
-lemma partition_outer_remains:
- assumes "crel (partition a l r) h h' rs"
- assumes "l < r"
- shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"
-proof -
- from assms part_outer_remains part_returns_index_in_bounds show ?thesis
- unfolding partition.simps swap_def
- by (elim crelE crel_return crel_nth crel_if crel_upd) fastsimp
-qed
-
-lemma partition_returns_index_in_bounds:
- assumes crel: "crel (partition a l r) h h' rs"
- assumes "l < r"
- shows "l \<le> rs \<and> rs \<le> r"
-proof -
- from crel obtain middle h'' p where part: "crel (part1 a l (r - 1) p) h h'' middle"
- and rs_equals: "rs = (if get_array a h'' ! middle \<le> get_array a h ! r then middle + 1
- else middle)"
- unfolding partition.simps
- by (elim crelE crel_return crel_nth crel_if crel_upd) simp
- from `l < r` have "l \<le> r - 1" by arith
- from part_returns_index_in_bounds[OF part this] rs_equals `l < r` show ?thesis by auto
-qed
-
-lemma partition_partitions:
- assumes crel: "crel (partition a l r) h h' rs"
- assumes "l < r"
- shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array (a::nat array) h' ! i \<le> get_array a h' ! rs) \<and>
- (\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array a h' ! rs \<le> get_array a h' ! i)"
-proof -
- let ?pivot = "get_array a h ! r"
- from crel obtain middle h1 where part: "crel (part1 a l (r - 1) ?pivot) h h1 middle"
- and swap: "crel (swap a rs r) h1 h' ()"
- and rs_equals: "rs = (if get_array a h1 ! middle \<le> ?pivot then middle + 1
- else middle)"
- unfolding partition.simps
- by (elim crelE crel_return crel_nth crel_if crel_upd) simp
- from swap have h'_def: "h' = Heap.upd a r (get_array a h1 ! rs)
- (Heap.upd a rs (get_array a h1 ! r) h1)"
- unfolding swap_def
- by (elim crelE crel_return crel_nth crel_upd) simp
- from swap have in_bounds: "r < Heap.length a h1 \<and> rs < Heap.length a h1"
- unfolding swap_def
- by (elim crelE crel_return crel_nth crel_upd) simp
- from swap have swap_length_remains: "Heap.length a h1 = Heap.length a h'"
- unfolding swap_def by (elim crelE crel_return crel_nth crel_upd) auto
- from `l < r` have "l \<le> r - 1" by simp
- note middle_in_bounds = part_returns_index_in_bounds[OF part this]
- from part_outer_remains[OF part] `l < r`
- have "get_array a h ! r = get_array a h1 ! r"
- by fastsimp
- with swap
- have right_remains: "get_array a h ! r = get_array a h' ! rs"
- unfolding swap_def
- by (auto simp add: Heap.length_def elim!: crelE crel_return crel_nth crel_upd) (cases "r = rs", auto)
- from part_partitions [OF part]
- show ?thesis
- proof (cases "get_array a h1 ! middle \<le> ?pivot")
- case True
- with rs_equals have rs_equals: "rs = middle + 1" by simp
- {
- fix i
- assume i_is_left: "l \<le> i \<and> i < rs"
- with swap_length_remains in_bounds middle_in_bounds rs_equals `l < r`
- have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto
- from i_is_left rs_equals have "l \<le> i \<and> i < middle \<or> i = middle" by arith
- with part_partitions[OF part] right_remains True
- have "get_array a h1 ! i \<le> get_array a h' ! rs" by fastsimp
- with i_props h'_def in_bounds have "get_array a h' ! i \<le> get_array a h' ! rs"
- unfolding Heap.upd_def Heap.length_def by simp
- }
- moreover
- {
- fix i
- assume "rs < i \<and> i \<le> r"
-
- hence "(rs < i \<and> i \<le> r - 1) \<or> (rs < i \<and> i = r)" by arith
- hence "get_array a h' ! rs \<le> get_array a h' ! i"
- proof
- assume i_is: "rs < i \<and> i \<le> r - 1"
- with swap_length_remains in_bounds middle_in_bounds rs_equals
- have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto
- from part_partitions[OF part] rs_equals right_remains i_is
- have "get_array a h' ! rs \<le> get_array a h1 ! i"
- by fastsimp
- with i_props h'_def show ?thesis by fastsimp
- next
- assume i_is: "rs < i \<and> i = r"
- with rs_equals have "Suc middle \<noteq> r" by arith
- with middle_in_bounds `l < r` have "Suc middle \<le> r - 1" by arith
- with part_partitions[OF part] right_remains
- have "get_array a h' ! rs \<le> get_array a h1 ! (Suc middle)"
- by fastsimp
- with i_is True rs_equals right_remains h'_def
- show ?thesis using in_bounds
- unfolding Heap.upd_def Heap.length_def
- by auto
- qed
- }
- ultimately show ?thesis by auto
- next
- case False
- with rs_equals have rs_equals: "middle = rs" by simp
- {
- fix i
- assume i_is_left: "l \<le> i \<and> i < rs"
- with swap_length_remains in_bounds middle_in_bounds rs_equals
- have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto
- from part_partitions[OF part] rs_equals right_remains i_is_left
- have "get_array a h1 ! i \<le> get_array a h' ! rs" by fastsimp
- with i_props h'_def have "get_array a h' ! i \<le> get_array a h' ! rs"
- unfolding Heap.upd_def by simp
- }
- moreover
- {
- fix i
- assume "rs < i \<and> i \<le> r"
- hence "(rs < i \<and> i \<le> r - 1) \<or> i = r" by arith
- hence "get_array a h' ! rs \<le> get_array a h' ! i"
- proof
- assume i_is: "rs < i \<and> i \<le> r - 1"
- with swap_length_remains in_bounds middle_in_bounds rs_equals
- have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto
- from part_partitions[OF part] rs_equals right_remains i_is
- have "get_array a h' ! rs \<le> get_array a h1 ! i"
- by fastsimp
- with i_props h'_def show ?thesis by fastsimp
- next
- assume i_is: "i = r"
- from i_is False rs_equals right_remains h'_def
- show ?thesis using in_bounds
- unfolding Heap.upd_def Heap.length_def
- by auto
- qed
- }
- ultimately
- show ?thesis by auto
- qed
-qed
-
-
-function quicksort :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap"
-where
- "quicksort arr left right =
- (if (right > left) then
- do
- pivotNewIndex \<leftarrow> partition arr left right;
- pivotNewIndex \<leftarrow> assert (\<lambda>x. left \<le> x \<and> x \<le> right) pivotNewIndex;
- quicksort arr left (pivotNewIndex - 1);
- quicksort arr (pivotNewIndex + 1) right
- done
- else return ())"
-by pat_completeness auto
-
-(* For termination, we must show that the pivotNewIndex is between left and right *)
-termination
-by (relation "measure (\<lambda>(a, l, r). (r - l))") auto
-
-declare quicksort.simps[simp del]
-
-
-lemma quicksort_permutes:
- assumes "crel (quicksort a l r) h h' rs"
- shows "multiset_of (get_array a h')
- = multiset_of (get_array a h)"
- using assms
-proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
- case (1 a l r h h' rs)
- with partition_permutes show ?case
- unfolding quicksort.simps [of a l r]
- by (elim crel_if crelE crel_assert crel_return) auto
-qed
-
-lemma length_remains:
- assumes "crel (quicksort a l r) h h' rs"
- shows "Heap.length a h = Heap.length a h'"
-using assms
-proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
- case (1 a l r h h' rs)
- with partition_length_remains show ?case
- unfolding quicksort.simps [of a l r]
- by (elim crel_if crelE crel_assert crel_return) auto
-qed
-
-lemma quicksort_outer_remains:
- assumes "crel (quicksort a l r) h h' rs"
- shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"
- using assms
-proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
- case (1 a l r h h' rs)
- note cr = `crel (quicksort a l r) h h' rs`
- thus ?case
- proof (cases "r > l")
- case False
- with cr have "h' = h"
- unfolding quicksort.simps [of a l r]
- by (elim crel_if crel_return) auto
- thus ?thesis by simp
- next
- case True
- {
- fix h1 h2 p ret1 ret2 i
- assume part: "crel (partition a l r) h h1 p"
- assume qs1: "crel (quicksort a l (p - 1)) h1 h2 ret1"
- assume qs2: "crel (quicksort a (p + 1) r) h2 h' ret2"
- assume pivot: "l \<le> p \<and> p \<le> r"
- assume i_outer: "i < l \<or> r < i"
- from partition_outer_remains [OF part True] i_outer
- have "get_array a h !i = get_array a h1 ! i" by fastsimp
- moreover
- with 1(1) [OF True pivot qs1] pivot i_outer
- have "get_array a h1 ! i = get_array a h2 ! i" by auto
- moreover
- with qs2 1(2) [of p h2 h' ret2] True pivot i_outer
- have "get_array a h2 ! i = get_array a h' ! i" by auto
- ultimately have "get_array a h ! i= get_array a h' ! i" by simp
- }
- with cr show ?thesis
- unfolding quicksort.simps [of a l r]
- by (elim crel_if crelE crel_assert crel_return) auto
- qed
-qed
-
-lemma quicksort_is_skip:
- assumes "crel (quicksort a l r) h h' rs"
- shows "r \<le> l \<longrightarrow> h = h'"
- using assms
- unfolding quicksort.simps [of a l r]
- by (elim crel_if crel_return) auto
-
-lemma quicksort_sorts:
- assumes "crel (quicksort a l r) h h' rs"
- assumes l_r_length: "l < Heap.length a h" "r < Heap.length a h"
- shows "sorted (subarray l (r + 1) a h')"
- using assms
-proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
- case (1 a l r h h' rs)
- note cr = `crel (quicksort a l r) h h' rs`
- thus ?case
- proof (cases "r > l")
- case False
- hence "l \<ge> r + 1 \<or> l = r" by arith
- with length_remains[OF cr] 1(5) show ?thesis
- by (auto simp add: subarray_Nil subarray_single)
- next
- case True
- {
- fix h1 h2 p
- assume part: "crel (partition a l r) h h1 p"
- assume qs1: "crel (quicksort a l (p - 1)) h1 h2 ()"
- assume qs2: "crel (quicksort a (p + 1) r) h2 h' ()"
- from partition_returns_index_in_bounds [OF part True]
- have pivot: "l\<le> p \<and> p \<le> r" .
- note length_remains = length_remains[OF qs2] length_remains[OF qs1] partition_length_remains[OF part]
- from quicksort_outer_remains [OF qs2] quicksort_outer_remains [OF qs1] pivot quicksort_is_skip[OF qs1]
- have pivot_unchanged: "get_array a h1 ! p = get_array a h' ! p" by (cases p, auto)
- (*-- First of all, by induction hypothesis both sublists are sorted. *)
- from 1(1)[OF True pivot qs1] length_remains pivot 1(5)
- have IH1: "sorted (subarray l p a h2)" by (cases p, auto simp add: subarray_Nil)
- from quicksort_outer_remains [OF qs2] length_remains
- have left_subarray_remains: "subarray l p a h2 = subarray l p a h'"
- by (simp add: subarray_eq_samelength_iff)
- with IH1 have IH1': "sorted (subarray l p a h')" by simp
- from 1(2)[OF True pivot qs2] pivot 1(5) length_remains
- have IH2: "sorted (subarray (p + 1) (r + 1) a h')"
- by (cases "Suc p \<le> r", auto simp add: subarray_Nil)
- (* -- Secondly, both sublists remain partitioned. *)
- from partition_partitions[OF part True]
- have part_conds1: "\<forall>j. j \<in> set (subarray l p a h1) \<longrightarrow> j \<le> get_array a h1 ! p "
- and part_conds2: "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h1) \<longrightarrow> get_array a h1 ! p \<le> j"
- by (auto simp add: all_in_set_subarray_conv)
- from quicksort_outer_remains [OF qs1] quicksort_permutes [OF qs1] True
- length_remains 1(5) pivot multiset_of_sublist [of l p "get_array a h1" "get_array a h2"]
- have multiset_partconds1: "multiset_of (subarray l p a h2) = multiset_of (subarray l p a h1)"
- unfolding Heap.length_def subarray_def by (cases p, auto)
- with left_subarray_remains part_conds1 pivot_unchanged
- have part_conds2': "\<forall>j. j \<in> set (subarray l p a h') \<longrightarrow> j \<le> get_array a h' ! p"
- by (simp, subst set_of_multiset_of[symmetric], simp)
- (* -- These steps are the analogous for the right sublist \<dots> *)
- from quicksort_outer_remains [OF qs1] length_remains
- have right_subarray_remains: "subarray (p + 1) (r + 1) a h1 = subarray (p + 1) (r + 1) a h2"
- by (auto simp add: subarray_eq_samelength_iff)
- from quicksort_outer_remains [OF qs2] quicksort_permutes [OF qs2] True
- length_remains 1(5) pivot multiset_of_sublist [of "p + 1" "r + 1" "get_array a h2" "get_array a h'"]
- have multiset_partconds2: "multiset_of (subarray (p + 1) (r + 1) a h') = multiset_of (subarray (p + 1) (r + 1) a h2)"
- unfolding Heap.length_def subarray_def by auto
- with right_subarray_remains part_conds2 pivot_unchanged
- have part_conds1': "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h') \<longrightarrow> get_array a h' ! p \<le> j"
- by (simp, subst set_of_multiset_of[symmetric], simp)
- (* -- Thirdly and finally, we show that the array is sorted
- following from the facts above. *)
- from True pivot 1(5) length_remains have "subarray l (r + 1) a h' = subarray l p a h' @ [get_array a h' ! p] @ subarray (p + 1) (r + 1) a h'"
- by (simp add: subarray_nth_array_Cons, cases "l < p") (auto simp add: subarray_append subarray_Nil)
- with IH1' IH2 part_conds1' part_conds2' pivot have ?thesis
- unfolding subarray_def
- apply (auto simp add: sorted_append sorted_Cons all_in_set_sublist'_conv)
- by (auto simp add: set_sublist' dest: le_trans [of _ "get_array a h' ! p"])
- }
- with True cr show ?thesis
- unfolding quicksort.simps [of a l r]
- by (elim crel_if crel_return crelE crel_assert) auto
- qed
-qed
-
-
-lemma quicksort_is_sort:
- assumes crel: "crel (quicksort a 0 (Heap.length a h - 1)) h h' rs"
- shows "get_array a h' = sort (get_array a h)"
-proof (cases "get_array a h = []")
- case True
- with quicksort_is_skip[OF crel] show ?thesis
- unfolding Heap.length_def by simp
-next
- case False
- from quicksort_sorts [OF crel] False have "sorted (sublist' 0 (List.length (get_array a h)) (get_array a h'))"
- unfolding Heap.length_def subarray_def by auto
- with length_remains[OF crel] have "sorted (get_array a h')"
- unfolding Heap.length_def by simp
- with quicksort_permutes [OF crel] properties_for_sort show ?thesis by fastsimp
-qed
-
-subsection {* No Errors in quicksort *}
-text {* We have proved that quicksort sorts (if no exceptions occur).
-We will now show that exceptions do not occur. *}
-
-lemma noError_part1:
- assumes "l < Heap.length a h" "r < Heap.length a h"
- shows "noError (part1 a l r p) h"
- using assms
-proof (induct a l r p arbitrary: h rule: part1.induct)
- case (1 a l r p)
- thus ?case
- unfolding part1.simps [of a l r] swap_def
- by (auto intro!: noError_if noErrorI noError_return noError_nth noError_upd elim!: crelE crel_upd crel_nth crel_return)
-qed
-
-lemma noError_partition:
- assumes "l < r" "l < Heap.length a h" "r < Heap.length a h"
- shows "noError (partition a l r) h"
-using assms
-unfolding partition.simps swap_def
-apply (auto intro!: noError_if noErrorI noError_return noError_nth noError_upd noError_part1 elim!: crelE crel_upd crel_nth crel_return)
-apply (frule part_length_remains)
-apply (frule part_returns_index_in_bounds)
-apply auto
-apply (frule part_length_remains)
-apply (frule part_returns_index_in_bounds)
-apply auto
-apply (frule part_length_remains)
-apply auto
-done
-
-lemma noError_quicksort:
- assumes "l < Heap.length a h" "r < Heap.length a h"
- shows "noError (quicksort a l r) h"
-using assms
-proof (induct a l r arbitrary: h rule: quicksort.induct)
- case (1 a l ri h)
- thus ?case
- unfolding quicksort.simps [of a l ri]
- apply (auto intro!: noError_if noErrorI noError_return noError_nth noError_upd noError_assert noError_partition)
- apply (frule partition_returns_index_in_bounds)
- apply auto
- apply (frule partition_returns_index_in_bounds)
- apply auto
- apply (auto elim!: crel_assert dest!: partition_length_remains length_remains)
- apply (subgoal_tac "Suc r \<le> ri \<or> r = ri")
- apply (erule disjE)
- apply auto
- unfolding quicksort.simps [of a "Suc ri" ri]
- apply (auto intro!: noError_if noError_return)
- done
-qed
-
-
-subsection {* Example *}
-
-definition "qsort a = do
- k \<leftarrow> length a;
- quicksort a 0 (k - 1);
- return a
- done"
-
-ML {* @{code qsort} (Array.fromList [42, 2, 3, 5, 0, 1705, 8, 3, 15]) () *}
-
-export_code qsort in SML_imp module_name QSort
-export_code qsort in OCaml module_name QSort file -
-export_code qsort in OCaml_imp module_name QSort file -
-export_code qsort in Haskell module_name QSort file -
-
-end
\ No newline at end of file
--- a/src/HOL/ex/ROOT.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/HOL/ex/ROOT.ML Mon Mar 23 19:01:34 2009 +0100
@@ -21,7 +21,6 @@
use_thys [
"Numeral",
- "ImperativeQuicksort",
"Higher_Order_Logic",
"Abstract_NAT",
"Guess",
--- a/src/HOL/ex/Subarray.thy Mon Mar 23 15:33:35 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,66 +0,0 @@
-theory Subarray
-imports Array Sublist
-begin
-
-definition subarray :: "nat \<Rightarrow> nat \<Rightarrow> ('a::heap) array \<Rightarrow> heap \<Rightarrow> 'a list"
-where
- "subarray n m a h \<equiv> sublist' n m (get_array a h)"
-
-lemma subarray_upd: "i \<ge> m \<Longrightarrow> subarray n m a (Heap.upd a i v h) = subarray n m a h"
-apply (simp add: subarray_def Heap.upd_def)
-apply (simp add: sublist'_update1)
-done
-
-lemma subarray_upd2: " i < n \<Longrightarrow> subarray n m a (Heap.upd a i v h) = subarray n m a h"
-apply (simp add: subarray_def Heap.upd_def)
-apply (subst sublist'_update2)
-apply fastsimp
-apply simp
-done
-
-lemma subarray_upd3: "\<lbrakk> n \<le> i; i < m\<rbrakk> \<Longrightarrow> subarray n m a (Heap.upd a i v h) = subarray n m a h[i - n := v]"
-unfolding subarray_def Heap.upd_def
-by (simp add: sublist'_update3)
-
-lemma subarray_Nil: "n \<ge> m \<Longrightarrow> subarray n m a h = []"
-by (simp add: subarray_def sublist'_Nil')
-
-lemma subarray_single: "\<lbrakk> n < Heap.length a h \<rbrakk> \<Longrightarrow> subarray n (Suc n) a h = [get_array a h ! n]"
-by (simp add: subarray_def Heap.length_def sublist'_single)
-
-lemma length_subarray: "m \<le> Heap.length a h \<Longrightarrow> List.length (subarray n m a h) = m - n"
-by (simp add: subarray_def Heap.length_def length_sublist')
-
-lemma length_subarray_0: "m \<le> Heap.length a h \<Longrightarrow> List.length (subarray 0 m a h) = m"
-by (simp add: length_subarray)
-
-lemma subarray_nth_array_Cons: "\<lbrakk> i < Heap.length a h; i < j \<rbrakk> \<Longrightarrow> (get_array a h ! i) # subarray (Suc i) j a h = subarray i j a h"
-unfolding Heap.length_def subarray_def
-by (simp add: sublist'_front)
-
-lemma subarray_nth_array_back: "\<lbrakk> i < j; j \<le> Heap.length a h\<rbrakk> \<Longrightarrow> subarray i j a h = subarray i (j - 1) a h @ [get_array a h ! (j - 1)]"
-unfolding Heap.length_def subarray_def
-by (simp add: sublist'_back)
-
-lemma subarray_append: "\<lbrakk> i < j; j < k \<rbrakk> \<Longrightarrow> subarray i j a h @ subarray j k a h = subarray i k a h"
-unfolding subarray_def
-by (simp add: sublist'_append)
-
-lemma subarray_all: "subarray 0 (Heap.length a h) a h = get_array a h"
-unfolding Heap.length_def subarray_def
-by (simp add: sublist'_all)
-
-lemma nth_subarray: "\<lbrakk> k < j - i; j \<le> Heap.length a h \<rbrakk> \<Longrightarrow> subarray i j a h ! k = get_array a h ! (i + k)"
-unfolding Heap.length_def subarray_def
-by (simp add: nth_sublist')
-
-lemma subarray_eq_samelength_iff: "Heap.length a h = Heap.length a h' \<Longrightarrow> (subarray i j a h = subarray i j a h') = (\<forall>i'. i \<le> i' \<and> i' < j \<longrightarrow> get_array a h ! i' = get_array a h' ! i')"
-unfolding Heap.length_def subarray_def by (rule sublist'_eq_samelength_iff)
-
-lemma all_in_set_subarray_conv: "(\<forall>j. j \<in> set (subarray l r a h) \<longrightarrow> P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < Heap.length a h \<longrightarrow> P (get_array a h ! k))"
-unfolding subarray_def Heap.length_def by (rule all_in_set_sublist'_conv)
-
-lemma ball_in_set_subarray_conv: "(\<forall>j \<in> set (subarray l r a h). P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < Heap.length a h \<longrightarrow> P (get_array a h ! k))"
-unfolding subarray_def Heap.length_def by (rule ball_in_set_sublist'_conv)
-
-end
\ No newline at end of file
--- a/src/HOL/ex/Sublist.thy Mon Mar 23 15:33:35 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,505 +0,0 @@
-(* $Id$ *)
-
-header {* Slices of lists *}
-
-theory Sublist
-imports Multiset
-begin
-
-
-lemma sublist_split: "i \<le> j \<and> j \<le> k \<Longrightarrow> sublist xs {i..<j} @ sublist xs {j..<k} = sublist xs {i..<k}"
-apply (induct xs arbitrary: i j k)
-apply simp
-apply (simp only: sublist_Cons)
-apply simp
-apply safe
-apply simp
-apply (erule_tac x="0" in meta_allE)
-apply (erule_tac x="j - 1" in meta_allE)
-apply (erule_tac x="k - 1" in meta_allE)
-apply (subgoal_tac "0 \<le> j - 1 \<and> j - 1 \<le> k - 1")
-apply simp
-apply (subgoal_tac "{ja. Suc ja < j} = {0..<j - Suc 0}")
-apply (subgoal_tac "{ja. j \<le> Suc ja \<and> Suc ja < k} = {j - Suc 0..<k - Suc 0}")
-apply (subgoal_tac "{j. Suc j < k} = {0..<k - Suc 0}")
-apply simp
-apply fastsimp
-apply fastsimp
-apply fastsimp
-apply fastsimp
-apply (erule_tac x="i - 1" in meta_allE)
-apply (erule_tac x="j - 1" in meta_allE)
-apply (erule_tac x="k - 1" in meta_allE)
-apply (subgoal_tac " {ja. i \<le> Suc ja \<and> Suc ja < j} = {i - 1 ..<j - 1}")
-apply (subgoal_tac " {ja. j \<le> Suc ja \<and> Suc ja < k} = {j - 1..<k - 1}")
-apply (subgoal_tac "{j. i \<le> Suc j \<and> Suc j < k} = {i - 1..<k - 1}")
-apply (subgoal_tac " i - 1 \<le> j - 1 \<and> j - 1 \<le> k - 1")
-apply simp
-apply fastsimp
-apply fastsimp
-apply fastsimp
-apply fastsimp
-done
-
-lemma sublist_update1: "i \<notin> inds \<Longrightarrow> sublist (xs[i := v]) inds = sublist xs inds"
-apply (induct xs arbitrary: i inds)
-apply simp
-apply (case_tac i)
-apply (simp add: sublist_Cons)
-apply (simp add: sublist_Cons)
-done
-
-lemma sublist_update2: "i \<in> inds \<Longrightarrow> sublist (xs[i := v]) inds = (sublist xs inds)[(card {k \<in> inds. k < i}):= v]"
-proof (induct xs arbitrary: i inds)
- case Nil thus ?case by simp
-next
- case (Cons x xs)
- thus ?case
- proof (cases i)
- case 0 with Cons show ?thesis by (simp add: sublist_Cons)
- next
- case (Suc i')
- with Cons show ?thesis
- apply simp
- apply (simp add: sublist_Cons)
- apply auto
- apply (auto simp add: nat.split)
- apply (simp add: card_less_Suc[symmetric])
- apply (simp add: card_less_Suc2)
- done
- qed
-qed
-
-lemma sublist_update: "sublist (xs[i := v]) inds = (if i \<in> inds then (sublist xs inds)[(card {k \<in> inds. k < i}) := v] else sublist xs inds)"
-by (simp add: sublist_update1 sublist_update2)
-
-lemma sublist_take: "sublist xs {j. j < m} = take m xs"
-apply (induct xs arbitrary: m)
-apply simp
-apply (case_tac m)
-apply simp
-apply (simp add: sublist_Cons)
-done
-
-lemma sublist_take': "sublist xs {0..<m} = take m xs"
-apply (induct xs arbitrary: m)
-apply simp
-apply (case_tac m)
-apply simp
-apply (simp add: sublist_Cons sublist_take)
-done
-
-lemma sublist_all[simp]: "sublist xs {j. j < length xs} = xs"
-apply (induct xs)
-apply simp
-apply (simp add: sublist_Cons)
-done
-
-lemma sublist_all'[simp]: "sublist xs {0..<length xs} = xs"
-apply (induct xs)
-apply simp
-apply (simp add: sublist_Cons)
-done
-
-lemma sublist_single: "a < length xs \<Longrightarrow> sublist xs {a} = [xs ! a]"
-apply (induct xs arbitrary: a)
-apply simp
-apply(case_tac aa)
-apply simp
-apply (simp add: sublist_Cons)
-apply simp
-apply (simp add: sublist_Cons)
-done
-
-lemma sublist_is_Nil: "\<forall>i \<in> inds. i \<ge> length xs \<Longrightarrow> sublist xs inds = []"
-apply (induct xs arbitrary: inds)
-apply simp
-apply (simp add: sublist_Cons)
-apply auto
-apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
-apply auto
-done
-
-lemma sublist_Nil': "sublist xs inds = [] \<Longrightarrow> \<forall>i \<in> inds. i \<ge> length xs"
-apply (induct xs arbitrary: inds)
-apply simp
-apply (simp add: sublist_Cons)
-apply (auto split: if_splits)
-apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
-apply (case_tac x, auto)
-done
-
-lemma sublist_Nil[simp]: "(sublist xs inds = []) = (\<forall>i \<in> inds. i \<ge> length xs)"
-apply (induct xs arbitrary: inds)
-apply simp
-apply (simp add: sublist_Cons)
-apply auto
-apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
-apply (case_tac x, auto)
-done
-
-lemma sublist_eq_subseteq: " \<lbrakk> inds' \<subseteq> inds; sublist xs inds = sublist ys inds \<rbrakk> \<Longrightarrow> sublist xs inds' = sublist ys inds'"
-apply (induct xs arbitrary: ys inds inds')
-apply simp
-apply (drule sym, rule sym)
-apply (simp add: sublist_Nil, fastsimp)
-apply (case_tac ys)
-apply (simp add: sublist_Nil, fastsimp)
-apply (auto simp add: sublist_Cons)
-apply (erule_tac x="list" in meta_allE)
-apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
-apply (erule_tac x="{j. Suc j \<in> inds'}" in meta_allE)
-apply fastsimp
-apply (erule_tac x="list" in meta_allE)
-apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
-apply (erule_tac x="{j. Suc j \<in> inds'}" in meta_allE)
-apply fastsimp
-done
-
-lemma sublist_eq: "\<lbrakk> \<forall>i \<in> inds. ((i < length xs) \<and> (i < length ys)) \<or> ((i \<ge> length xs ) \<and> (i \<ge> length ys)); \<forall>i \<in> inds. xs ! i = ys ! i \<rbrakk> \<Longrightarrow> sublist xs inds = sublist ys inds"
-apply (induct xs arbitrary: ys inds)
-apply simp
-apply (rule sym, simp add: sublist_Nil)
-apply (case_tac ys)
-apply (simp add: sublist_Nil)
-apply (auto simp add: sublist_Cons)
-apply (erule_tac x="list" in meta_allE)
-apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
-apply fastsimp
-apply (erule_tac x="list" in meta_allE)
-apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
-apply fastsimp
-done
-
-lemma sublist_eq_samelength: "\<lbrakk> length xs = length ys; \<forall>i \<in> inds. xs ! i = ys ! i \<rbrakk> \<Longrightarrow> sublist xs inds = sublist ys inds"
-by (rule sublist_eq, auto)
-
-lemma sublist_eq_samelength_iff: "length xs = length ys \<Longrightarrow> (sublist xs inds = sublist ys inds) = (\<forall>i \<in> inds. xs ! i = ys ! i)"
-apply (induct xs arbitrary: ys inds)
-apply simp
-apply (rule sym, simp add: sublist_Nil)
-apply (case_tac ys)
-apply (simp add: sublist_Nil)
-apply (auto simp add: sublist_Cons)
-apply (case_tac i)
-apply auto
-apply (case_tac i)
-apply auto
-done
-
-section {* Another sublist function *}
-
-function sublist' :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where
- "sublist' n m [] = []"
-| "sublist' n 0 xs = []"
-| "sublist' 0 (Suc m) (x#xs) = (x#sublist' 0 m xs)"
-| "sublist' (Suc n) (Suc m) (x#xs) = sublist' n m xs"
-by pat_completeness auto
-termination by lexicographic_order
-
-subsection {* Proving equivalence to the other sublist command *}
-
-lemma sublist'_sublist: "sublist' n m xs = sublist xs {j. n \<le> j \<and> j < m}"
-apply (induct xs arbitrary: n m)
-apply simp
-apply (case_tac n)
-apply (case_tac m)
-apply simp
-apply (simp add: sublist_Cons)
-apply (case_tac m)
-apply simp
-apply (simp add: sublist_Cons)
-done
-
-
-lemma "sublist' n m xs = sublist xs {n..<m}"
-apply (induct xs arbitrary: n m)
-apply simp
-apply (case_tac n, case_tac m)
-apply simp
-apply simp
-apply (simp add: sublist_take')
-apply (case_tac m)
-apply simp
-apply (simp add: sublist_Cons sublist'_sublist)
-done
-
-
-subsection {* Showing equivalence to use of drop and take for definition *}
-
-lemma "sublist' n m xs = take (m - n) (drop n xs)"
-apply (induct xs arbitrary: n m)
-apply simp
-apply (case_tac m)
-apply simp
-apply (case_tac n)
-apply simp
-apply simp
-done
-
-subsection {* General lemma about sublist *}
-
-lemma sublist'_Nil[simp]: "sublist' i j [] = []"
-by simp
-
-lemma sublist'_Cons[simp]: "sublist' i (Suc j) (x#xs) = (case i of 0 \<Rightarrow> (x # sublist' 0 j xs) | Suc i' \<Rightarrow> sublist' i' j xs)"
-by (cases i) auto
-
-lemma sublist'_Cons2[simp]: "sublist' i j (x#xs) = (if (j = 0) then [] else ((if (i = 0) then [x] else []) @ sublist' (i - 1) (j - 1) xs))"
-apply (cases j)
-apply auto
-apply (cases i)
-apply auto
-done
-
-lemma sublist_n_0: "sublist' n 0 xs = []"
-by (induct xs, auto)
-
-lemma sublist'_Nil': "n \<ge> m \<Longrightarrow> sublist' n m xs = []"
-apply (induct xs arbitrary: n m)
-apply simp
-apply (case_tac m)
-apply simp
-apply (case_tac n)
-apply simp
-apply simp
-done
-
-lemma sublist'_Nil2: "n \<ge> length xs \<Longrightarrow> sublist' n m xs = []"
-apply (induct xs arbitrary: n m)
-apply simp
-apply (case_tac m)
-apply simp
-apply (case_tac n)
-apply simp
-apply simp
-done
-
-lemma sublist'_Nil3: "(sublist' n m xs = []) = ((n \<ge> m) \<or> (n \<ge> length xs))"
-apply (induct xs arbitrary: n m)
-apply simp
-apply (case_tac m)
-apply simp
-apply (case_tac n)
-apply simp
-apply simp
-done
-
-lemma sublist'_notNil: "\<lbrakk> n < length xs; n < m \<rbrakk> \<Longrightarrow> sublist' n m xs \<noteq> []"
-apply (induct xs arbitrary: n m)
-apply simp
-apply (case_tac m)
-apply simp
-apply (case_tac n)
-apply simp
-apply simp
-done
-
-lemma sublist'_single: "n < length xs \<Longrightarrow> sublist' n (Suc n) xs = [xs ! n]"
-apply (induct xs arbitrary: n)
-apply simp
-apply simp
-apply (case_tac n)
-apply (simp add: sublist_n_0)
-apply simp
-done
-
-lemma sublist'_update1: "i \<ge> m \<Longrightarrow> sublist' n m (xs[i:=v]) = sublist' n m xs"
-apply (induct xs arbitrary: n m i)
-apply simp
-apply simp
-apply (case_tac i)
-apply simp
-apply simp
-done
-
-lemma sublist'_update2: "i < n \<Longrightarrow> sublist' n m (xs[i:=v]) = sublist' n m xs"
-apply (induct xs arbitrary: n m i)
-apply simp
-apply simp
-apply (case_tac i)
-apply simp
-apply simp
-done
-
-lemma sublist'_update3: "\<lbrakk>n \<le> i; i < m\<rbrakk> \<Longrightarrow> sublist' n m (xs[i := v]) = (sublist' n m xs)[i - n := v]"
-proof (induct xs arbitrary: n m i)
- case Nil thus ?case by auto
-next
- case (Cons x xs)
- thus ?case
- apply -
- apply auto
- apply (cases i)
- apply auto
- apply (cases i)
- apply auto
- done
-qed
-
-lemma "\<lbrakk> sublist' i j xs = sublist' i j ys; n \<ge> i; m \<le> j \<rbrakk> \<Longrightarrow> sublist' n m xs = sublist' n m ys"
-proof (induct xs arbitrary: i j ys n m)
- case Nil
- thus ?case
- apply -
- apply (rule sym, drule sym)
- apply (simp add: sublist'_Nil)
- apply (simp add: sublist'_Nil3)
- apply arith
- done
-next
- case (Cons x xs i j ys n m)
- note c = this
- thus ?case
- proof (cases m)
- case 0 thus ?thesis by (simp add: sublist_n_0)
- next
- case (Suc m')
- note a = this
- thus ?thesis
- proof (cases n)
- case 0 note b = this
- show ?thesis
- proof (cases ys)
- case Nil with a b Cons.prems show ?thesis by (simp add: sublist'_Nil3)
- next
- case (Cons y ys)
- show ?thesis
- proof (cases j)
- case 0 with a b Cons.prems show ?thesis by simp
- next
- case (Suc j') with a b Cons.prems Cons show ?thesis
- apply -
- apply (simp, rule Cons.hyps [of "0" "j'" "ys" "0" "m'"], auto)
- done
- qed
- qed
- next
- case (Suc n')
- show ?thesis
- proof (cases ys)
- case Nil with Suc a Cons.prems show ?thesis by (auto simp add: sublist'_Nil3)
- next
- case (Cons y ys) with Suc a Cons.prems show ?thesis
- apply -
- apply simp
- apply (cases j)
- apply simp
- apply (cases i)
- apply simp
- apply (rule_tac j="nat" in Cons.hyps [of "0" _ "ys" "n'" "m'"])
- apply simp
- apply simp
- apply simp
- apply simp
- apply (rule_tac i="nata" and j="nat" in Cons.hyps [of _ _ "ys" "n'" "m'"])
- apply simp
- apply simp
- apply simp
- done
- qed
- qed
- qed
-qed
-
-lemma length_sublist': "j \<le> length xs \<Longrightarrow> length (sublist' i j xs) = j - i"
-by (induct xs arbitrary: i j, auto)
-
-lemma sublist'_front: "\<lbrakk> i < j; i < length xs \<rbrakk> \<Longrightarrow> sublist' i j xs = xs ! i # sublist' (Suc i) j xs"
-apply (induct xs arbitrary: a i j)
-apply simp
-apply (case_tac j)
-apply simp
-apply (case_tac i)
-apply simp
-apply simp
-done
-
-lemma sublist'_back: "\<lbrakk> i < j; j \<le> length xs \<rbrakk> \<Longrightarrow> sublist' i j xs = sublist' i (j - 1) xs @ [xs ! (j - 1)]"
-apply (induct xs arbitrary: a i j)
-apply simp
-apply simp
-apply (case_tac j)
-apply simp
-apply auto
-apply (case_tac nat)
-apply auto
-done
-
-(* suffices that j \<le> length xs and length ys *)
-lemma sublist'_eq_samelength_iff: "length xs = length ys \<Longrightarrow> (sublist' i j xs = sublist' i j ys) = (\<forall>i'. i \<le> i' \<and> i' < j \<longrightarrow> xs ! i' = ys ! i')"
-proof (induct xs arbitrary: ys i j)
- case Nil thus ?case by simp
-next
- case (Cons x xs)
- thus ?case
- apply -
- apply (cases ys)
- apply simp
- apply simp
- apply auto
- apply (case_tac i', auto)
- apply (erule_tac x="Suc i'" in allE, auto)
- apply (erule_tac x="i' - 1" in allE, auto)
- apply (case_tac i', auto)
- apply (erule_tac x="Suc i'" in allE, auto)
- done
-qed
-
-lemma sublist'_all[simp]: "sublist' 0 (length xs) xs = xs"
-by (induct xs, auto)
-
-lemma sublist'_sublist': "sublist' n m (sublist' i j xs) = sublist' (i + n) (min (i + m) j) xs"
-by (induct xs arbitrary: i j n m) (auto simp add: min_diff)
-
-lemma sublist'_append: "\<lbrakk> i \<le> j; j \<le> k \<rbrakk> \<Longrightarrow>(sublist' i j xs) @ (sublist' j k xs) = sublist' i k xs"
-by (induct xs arbitrary: i j k) auto
-
-lemma nth_sublist': "\<lbrakk> k < j - i; j \<le> length xs \<rbrakk> \<Longrightarrow> (sublist' i j xs) ! k = xs ! (i + k)"
-apply (induct xs arbitrary: i j k)
-apply auto
-apply (case_tac k)
-apply auto
-apply (case_tac i)
-apply auto
-done
-
-lemma set_sublist': "set (sublist' i j xs) = {x. \<exists>k. i \<le> k \<and> k < j \<and> k < List.length xs \<and> x = xs ! k}"
-apply (simp add: sublist'_sublist)
-apply (simp add: set_sublist)
-apply auto
-done
-
-lemma all_in_set_sublist'_conv: "(\<forall>j. j \<in> set (sublist' l r xs) \<longrightarrow> P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < List.length xs \<longrightarrow> P (xs ! k))"
-unfolding set_sublist' by blast
-
-lemma ball_in_set_sublist'_conv: "(\<forall>j \<in> set (sublist' l r xs). P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < List.length xs \<longrightarrow> P (xs ! k))"
-unfolding set_sublist' by blast
-
-
-lemma multiset_of_sublist:
-assumes l_r: "l \<le> r \<and> r \<le> List.length xs"
-assumes left: "\<forall> i. i < l \<longrightarrow> (xs::'a list) ! i = ys ! i"
-assumes right: "\<forall> i. i \<ge> r \<longrightarrow> (xs::'a list) ! i = ys ! i"
-assumes multiset: "multiset_of xs = multiset_of ys"
- shows "multiset_of (sublist' l r xs) = multiset_of (sublist' l r ys)"
-proof -
- from l_r have xs_def: "xs = (sublist' 0 l xs) @ (sublist' l r xs) @ (sublist' r (List.length xs) xs)" (is "_ = ?xs_long")
- by (simp add: sublist'_append)
- from multiset have length_eq: "List.length xs = List.length ys" by (rule multiset_of_eq_length)
- with l_r have ys_def: "ys = (sublist' 0 l ys) @ (sublist' l r ys) @ (sublist' r (List.length ys) ys)" (is "_ = ?ys_long")
- by (simp add: sublist'_append)
- from xs_def ys_def multiset have "multiset_of ?xs_long = multiset_of ?ys_long" by simp
- moreover
- from left l_r length_eq have "sublist' 0 l xs = sublist' 0 l ys"
- by (auto simp add: length_sublist' nth_sublist' intro!: nth_equalityI)
- moreover
- from right l_r length_eq have "sublist' r (List.length xs) xs = sublist' r (List.length ys) ys"
- by (auto simp add: length_sublist' nth_sublist' intro!: nth_equalityI)
- moreover
- ultimately show ?thesis by (simp add: multiset_of_append)
-qed
-
-
-end
--- a/src/Provers/Arith/fast_lin_arith.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/Provers/Arith/fast_lin_arith.ML Mon Mar 23 19:01:34 2009 +0100
@@ -466,7 +466,7 @@
NONE => ( the (try_add ([thm2] RL inj_thms) thm1)
handle Option =>
(trace_thm "" thm1; trace_thm "" thm2;
- sys_error "Lin.arith. failed to add thms")
+ sys_error "Linear arithmetic: failed to add thms")
)
| SOME thm => thm)
| SOME thm => thm;
@@ -588,8 +588,8 @@
handle NoEx => NONE
in
case ex of
- SOME s => (warning "arith failed - see trace for a counterexample"; tracing s)
- | NONE => warning "arith failed"
+ SOME s => (warning "Linear arithmetic failed - see trace for a counterexample."; tracing s)
+ | NONE => warning "Linear arithmetic failed"
end;
(* ------------------------------------------------------------------------- *)
--- a/src/Pure/Isar/code_unit.ML Mon Mar 23 15:33:35 2009 +0100
+++ b/src/Pure/Isar/code_unit.ML Mon Mar 23 19:01:34 2009 +0100
@@ -218,7 +218,7 @@
|> burrow_thms (canonical_tvars thy purify_tvar)
|> map (canonical_vars thy purify_var)
|> map (canonical_absvars purify_var)
- |> map Drule.zero_var_indexes
+ |> Drule.zero_var_indexes_list
end;