--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Predicate_Compile_Examples/Predicate_Compile_Quickcheck_Examples.thy Wed Mar 24 17:40:44 2010 +0100
@@ -0,0 +1,337 @@
+theory Predicate_Compile_Quickcheck_Examples
+imports Predicate_Compile_Quickcheck
+begin
+
+section {* Sets *}
+
+lemma "x \<in> {(1::nat)} ==> False"
+quickcheck[generator=predicate_compile_wo_ff, iterations=10]
+oops
+
+lemma "x \<in> {Suc 0, Suc (Suc 0)} ==> x \<noteq> Suc 0"
+quickcheck[generator=predicate_compile_wo_ff]
+oops
+
+lemma "x \<in> {Suc 0, Suc (Suc 0)} ==> x = Suc 0"
+quickcheck[generator=predicate_compile_wo_ff]
+oops
+
+lemma "x \<in> {Suc 0, Suc (Suc 0)} ==> x <= Suc 0"
+quickcheck[generator=predicate_compile_wo_ff]
+oops
+
+section {* Numerals *}
+
+lemma
+ "x \<in> {1, 2, (3::nat)} ==> x = 1 \<or> x = 2"
+quickcheck[generator=predicate_compile_wo_ff]
+oops
+
+lemma "x \<in> {1, 2, (3::nat)} ==> x < 3"
+quickcheck[generator=predicate_compile_wo_ff]
+oops
+
+lemma
+ "x \<in> {1, 2} \<union> {3, 4} ==> x = (1::nat) \<or> x = (2::nat)"
+quickcheck[generator=predicate_compile_wo_ff]
+oops
+
+section {* Context Free Grammar *}
+
+datatype alphabet = a | b
+
+inductive_set S\<^isub>1 and A\<^isub>1 and B\<^isub>1 where
+ "[] \<in> S\<^isub>1"
+| "w \<in> A\<^isub>1 \<Longrightarrow> b # w \<in> S\<^isub>1"
+| "w \<in> B\<^isub>1 \<Longrightarrow> a # w \<in> S\<^isub>1"
+| "w \<in> S\<^isub>1 \<Longrightarrow> a # w \<in> A\<^isub>1"
+| "w \<in> S\<^isub>1 \<Longrightarrow> b # w \<in> S\<^isub>1"
+| "\<lbrakk>v \<in> B\<^isub>1; v \<in> B\<^isub>1\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>1"
+
+lemma
+ "w \<in> S\<^isub>1 \<Longrightarrow> w = []"
+quickcheck[generator = predicate_compile_ff_nofs, iterations=1]
+oops
+
+theorem S\<^isub>1_sound:
+"w \<in> S\<^isub>1 \<Longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
+quickcheck[generator=predicate_compile_ff_nofs, size=15]
+oops
+
+
+inductive_set S\<^isub>2 and A\<^isub>2 and B\<^isub>2 where
+ "[] \<in> S\<^isub>2"
+| "w \<in> A\<^isub>2 \<Longrightarrow> b # w \<in> S\<^isub>2"
+| "w \<in> B\<^isub>2 \<Longrightarrow> a # w \<in> S\<^isub>2"
+| "w \<in> S\<^isub>2 \<Longrightarrow> a # w \<in> A\<^isub>2"
+| "w \<in> S\<^isub>2 \<Longrightarrow> b # w \<in> B\<^isub>2"
+| "\<lbrakk>v \<in> B\<^isub>2; v \<in> B\<^isub>2\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>2"
+(*
+code_pred [random_dseq inductify] S\<^isub>2 .
+thm S\<^isub>2.random_dseq_equation
+thm A\<^isub>2.random_dseq_equation
+thm B\<^isub>2.random_dseq_equation
+
+values [random_dseq 1, 2, 8] 10 "{x. S\<^isub>2 x}"
+
+lemma "w \<in> S\<^isub>2 ==> w \<noteq> [] ==> w \<noteq> [b, a] ==> w \<in> {}"
+quickcheck[generator=predicate_compile, size=8]
+oops
+
+lemma "[x <- w. x = a] = []"
+quickcheck[generator=predicate_compile]
+oops
+
+declare list.size(3,4)[code_pred_def]
+
+(*
+lemma "length ([x \<leftarrow> w. x = a]) = (0::nat)"
+quickcheck[generator=predicate_compile]
+oops
+*)
+
+lemma
+"w \<in> S\<^isub>2 ==> length [x \<leftarrow> w. x = a] <= Suc (Suc 0)"
+quickcheck[generator=predicate_compile, size = 10, iterations = 1]
+oops
+*)
+theorem S\<^isub>2_sound:
+"w \<in> S\<^isub>2 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
+quickcheck[generator=predicate_compile_ff_nofs, size=5, iterations=10]
+oops
+
+inductive_set S\<^isub>3 and A\<^isub>3 and B\<^isub>3 where
+ "[] \<in> S\<^isub>3"
+| "w \<in> A\<^isub>3 \<Longrightarrow> b # w \<in> S\<^isub>3"
+| "w \<in> B\<^isub>3 \<Longrightarrow> a # w \<in> S\<^isub>3"
+| "w \<in> S\<^isub>3 \<Longrightarrow> a # w \<in> A\<^isub>3"
+| "w \<in> S\<^isub>3 \<Longrightarrow> b # w \<in> B\<^isub>3"
+| "\<lbrakk>v \<in> B\<^isub>3; w \<in> B\<^isub>3\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>3"
+
+code_pred [inductify] S\<^isub>3 .
+thm S\<^isub>3.equation
+(*
+values 10 "{x. S\<^isub>3 x}"
+*)
+
+
+lemma S\<^isub>3_sound:
+"w \<in> S\<^isub>3 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
+quickcheck[generator=predicate_compile_ff_fs, size=10, iterations=10]
+oops
+
+lemma "\<not> (length w > 2) \<or> \<not> (length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b])"
+quickcheck[size=10, generator = predicate_compile_ff_fs]
+oops
+
+theorem S\<^isub>3_complete:
+"length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. b = x] \<longrightarrow> w \<in> S\<^isub>3"
+(*quickcheck[generator=SML]*)
+quickcheck[generator=predicate_compile_ff_fs, size=10, iterations=100]
+oops
+
+
+inductive_set S\<^isub>4 and A\<^isub>4 and B\<^isub>4 where
+ "[] \<in> S\<^isub>4"
+| "w \<in> A\<^isub>4 \<Longrightarrow> b # w \<in> S\<^isub>4"
+| "w \<in> B\<^isub>4 \<Longrightarrow> a # w \<in> S\<^isub>4"
+| "w \<in> S\<^isub>4 \<Longrightarrow> a # w \<in> A\<^isub>4"
+| "\<lbrakk>v \<in> A\<^isub>4; w \<in> A\<^isub>4\<rbrakk> \<Longrightarrow> b # v @ w \<in> A\<^isub>4"
+| "w \<in> S\<^isub>4 \<Longrightarrow> b # w \<in> B\<^isub>4"
+| "\<lbrakk>v \<in> B\<^isub>4; w \<in> B\<^isub>4\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>4"
+
+theorem S\<^isub>4_sound:
+"w \<in> S\<^isub>4 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
+quickcheck[generator = predicate_compile_ff_nofs, size=5, iterations=1]
+oops
+
+theorem S\<^isub>4_complete:
+"length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] \<longrightarrow> w \<in> S\<^isub>4"
+quickcheck[generator = predicate_compile_ff_nofs, size=5, iterations=1]
+oops
+
+hide const a b
+
+subsection {* Lexicographic order *}
+(* TODO *)
+(*
+lemma
+ "(u, v) : lexord r ==> (x @ u, y @ v) : lexord r"
+oops
+*)
+subsection {* IMP *}
+
+types
+ var = nat
+ state = "int list"
+
+datatype com =
+ Skip |
+ Ass var "int" |
+ Seq com com |
+ IF "state list" com com |
+ While "state list" com
+
+inductive exec :: "com => state => state => bool" where
+ "exec Skip s s" |
+ "exec (Ass x e) s (s[x := e])" |
+ "exec c1 s1 s2 ==> exec c2 s2 s3 ==> exec (Seq c1 c2) s1 s3" |
+ "s \<in> set b ==> exec c1 s t ==> exec (IF b c1 c2) s t" |
+ "s \<notin> set b ==> exec c2 s t ==> exec (IF b c1 c2) s t" |
+ "s \<notin> set b ==> exec (While b c) s s" |
+ "s1 \<in> set b ==> exec c s1 s2 ==> exec (While b c) s2 s3 ==> exec (While b c) s1 s3"
+
+code_pred [random_dseq] exec .
+
+values [random_dseq 1, 2, 3] 10 "{(c, s, s'). exec c s s'}"
+
+lemma
+ "exec c s s' ==> exec (Seq c c) s s'"
+(*quickcheck[generator = predicate_compile_wo_ff, size=2, iterations=10]*)
+oops
+
+subsection {* Lambda *}
+
+datatype type =
+ Atom nat
+ | Fun type type (infixr "\<Rightarrow>" 200)
+
+datatype dB =
+ Var nat
+ | App dB dB (infixl "\<degree>" 200)
+ | Abs type dB
+
+primrec
+ nth_el :: "'a list \<Rightarrow> nat \<Rightarrow> 'a option" ("_\<langle>_\<rangle>" [90, 0] 91)
+where
+ "[]\<langle>i\<rangle> = None"
+| "(x # xs)\<langle>i\<rangle> = (case i of 0 \<Rightarrow> Some x | Suc j \<Rightarrow> xs \<langle>j\<rangle>)"
+
+inductive nth_el' :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> bool"
+where
+ "nth_el' (x # xs) 0 x"
+| "nth_el' xs i y \<Longrightarrow> nth_el' (x # xs) (Suc i) y"
+
+inductive typing :: "type list \<Rightarrow> dB \<Rightarrow> type \<Rightarrow> bool" ("_ \<turnstile> _ : _" [50, 50, 50] 50)
+ where
+ Var [intro!]: "nth_el' env x T \<Longrightarrow> env \<turnstile> Var x : T"
+ | Abs [intro!]: "T # env \<turnstile> t : U \<Longrightarrow> env \<turnstile> Abs T t : (T \<Rightarrow> U)"
+ | App [intro!]: "env \<turnstile> s : U \<Rightarrow> T \<Longrightarrow> env \<turnstile> t : T \<Longrightarrow> env \<turnstile> (s \<degree> t) : U"
+
+primrec
+ lift :: "[dB, nat] => dB"
+where
+ "lift (Var i) k = (if i < k then Var i else Var (i + 1))"
+ | "lift (s \<degree> t) k = lift s k \<degree> lift t k"
+ | "lift (Abs T s) k = Abs T (lift s (k + 1))"
+
+primrec
+ subst :: "[dB, dB, nat] => dB" ("_[_'/_]" [300, 0, 0] 300)
+where
+ subst_Var: "(Var i)[s/k] =
+ (if k < i then Var (i - 1) else if i = k then s else Var i)"
+ | subst_App: "(t \<degree> u)[s/k] = t[s/k] \<degree> u[s/k]"
+ | subst_Abs: "(Abs T t)[s/k] = Abs T (t[lift s 0 / k+1])"
+
+inductive beta :: "[dB, dB] => bool" (infixl "\<rightarrow>\<^sub>\<beta>" 50)
+ where
+ beta [simp, intro!]: "Abs T s \<degree> t \<rightarrow>\<^sub>\<beta> s[t/0]"
+ | appL [simp, intro!]: "s \<rightarrow>\<^sub>\<beta> t ==> s \<degree> u \<rightarrow>\<^sub>\<beta> t \<degree> u"
+ | appR [simp, intro!]: "s \<rightarrow>\<^sub>\<beta> t ==> u \<degree> s \<rightarrow>\<^sub>\<beta> u \<degree> t"
+ | abs [simp, intro!]: "s \<rightarrow>\<^sub>\<beta> t ==> Abs T s \<rightarrow>\<^sub>\<beta> Abs T t"
+
+lemma
+ "\<Gamma> \<turnstile> t : U \<Longrightarrow> t \<rightarrow>\<^sub>\<beta> t' \<Longrightarrow> \<Gamma> \<turnstile> t' : U"
+quickcheck[generator = predicate_compile_ff_fs, size = 7, iterations = 10]
+oops
+
+subsection {* JAD *}
+
+definition matrix :: "('a :: semiring_0) list list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
+ "matrix M rs cs \<longleftrightarrow> (\<forall> row \<in> set M. length row = cs) \<and> length M = rs"
+(*
+code_pred [random_dseq inductify] matrix .
+thm matrix.random_dseq_equation
+
+thm matrix_aux.random_dseq_equation
+
+values [random_dseq 3, 2] 10 "{(M, rs, cs). matrix (M:: int list list) rs cs}"
+*)
+lemma [code_pred_intro]:
+ "matrix [] 0 m"
+ "matrix xss n m ==> length xs = m ==> matrix (xs # xss) (Suc n) m"
+proof -
+ show "matrix [] 0 m" unfolding matrix_def by auto
+next
+ show "matrix xss n m ==> length xs = m ==> matrix (xs # xss) (Suc n) m"
+ unfolding matrix_def by auto
+qed
+
+code_pred [random_dseq inductify] matrix
+ apply (cases x)
+ unfolding matrix_def apply fastsimp
+ apply fastsimp done
+
+
+values [random_dseq 2, 2, 15] 6 "{(M::int list list, n, m). matrix M n m}"
+
+definition "scalar_product v w = (\<Sum> (x, y)\<leftarrow>zip v w. x * y)"
+
+definition mv :: "('a \<Colon> semiring_0) list list \<Rightarrow> 'a list \<Rightarrow> 'a list"
+ where [simp]: "mv M v = map (scalar_product v) M"
+text {*
+ This defines the matrix vector multiplication. To work properly @{term
+"matrix M m n \<and> length v = n"} must hold.
+*}
+
+subsection "Compressed matrix"
+
+definition "sparsify xs = [i \<leftarrow> zip [0..<length xs] xs. snd i \<noteq> 0]"
+(*
+lemma sparsify_length: "(i, x) \<in> set (sparsify xs) \<Longrightarrow> i < length xs"
+ by (auto simp: sparsify_def set_zip)
+
+lemma listsum_sparsify[simp]:
+ fixes v :: "('a \<Colon> semiring_0) list"
+ assumes "length w = length v"
+ shows "(\<Sum>x\<leftarrow>sparsify w. (\<lambda>(i, x). v ! i) x * snd x) = scalar_product v w"
+ (is "(\<Sum>x\<leftarrow>_. ?f x) = _")
+ unfolding sparsify_def scalar_product_def
+ using assms listsum_map_filter[where f="?f" and P="\<lambda> i. snd i \<noteq> (0::'a)"]
+ by (simp add: listsum_setsum)
+*)
+definition [simp]: "unzip w = (map fst w, map snd w)"
+
+primrec insert :: "('a \<Rightarrow> 'b \<Colon> linorder) => 'a \<Rightarrow> 'a list => 'a list" where
+ "insert f x [] = [x]" |
+ "insert f x (y # ys) = (if f y < f x then y # insert f x ys else x # y # ys)"
+
+primrec sort :: "('a \<Rightarrow> 'b \<Colon> linorder) \<Rightarrow> 'a list => 'a list" where
+ "sort f [] = []" |
+ "sort f (x # xs) = insert f x (sort f xs)"
+
+definition
+ "length_permutate M = (unzip o sort (length o snd)) (zip [0 ..< length M] M)"
+(*
+definition
+ "transpose M = [map (\<lambda> xs. xs ! i) (takeWhile (\<lambda> xs. i < length xs) M). i \<leftarrow> [0 ..< length (M ! 0)]]"
+*)
+definition
+ "inflate upds = foldr (\<lambda> (i, x) upds. upds[i := x]) upds (replicate (length upds) 0)"
+
+definition
+ "jad = apsnd transpose o length_permutate o map sparsify"
+
+definition
+ "jad_mv v = inflate o split zip o apsnd (map listsum o transpose o map (map (\<lambda> (i, x). v ! i * x)))"
+
+lemma "matrix (M::int list list) rs cs \<Longrightarrow> False"
+quickcheck[generator = predicate_compile_ff_nofs, size = 6]
+oops
+
+lemma
+ "\<lbrakk> matrix M rs cs ; length v = cs \<rbrakk> \<Longrightarrow> jad_mv v (jad M) = mv M v"
+quickcheck[generator = predicate_compile_wo_ff]
+oops
+
+end