(*  Title:      HOL/Nitpick_Examples/Manual_Nits.thy

    Author:     Jasmin Blanchette, TU Muenchen

    Copyright   2009

Examples from the Nitpick manual.

*)

header {* Examples from the Nitpick Manual *}

theory Manual_Nits

imports Main Coinductive_List RealDef

begin

chapter {* 3. First Steps *}

nitpick_params [sat_solver = MiniSatJNI, max_threads = 1]

subsection {* 3.1. Propositional Logic *}

lemma "P \<longleftrightarrow> Q"

nitpick

apply auto

nitpick 1

nitpick 2

oops

subsection {* 3.2. Type Variables *}

lemma "P x \<Longrightarrow> P (THE y. P y)"

nitpick [verbose]

oops

subsection {* 3.3. Constants *}

lemma "P x \<Longrightarrow> P (THE y. P y)"

nitpick [show_consts]

nitpick [full_descrs, show_consts]

nitpick [dont_specialize, full_descrs, show_consts]

oops

lemma "\<exists>!x. P x \<Longrightarrow> P (THE y. P y)"

nitpick

nitpick [card 'a = 1-50]

(* sledgehammer *)

apply (metis the_equality)

done

subsection {* 3.4. Skolemization *}

lemma "\<exists>g. \<forall>x. g (f x) = x \<Longrightarrow> \<forall>y. \<exists>x. y = f x"

nitpick

oops

lemma "\<exists>x. \<forall>f. f x = x"

nitpick

oops

lemma "refl r \<Longrightarrow> sym r"

nitpick

oops

subsection {* 3.5. Natural Numbers and Integers *}

lemma "\<lbrakk>i \<le> j; n \<le> (m\<Colon>int)\<rbrakk> \<Longrightarrow> i * n + j * m \<le> i * m + j * n"

nitpick

oops

lemma "\<forall>n. Suc n \<noteq> n \<Longrightarrow> P"

nitpick [card nat = 100, check_potential]

oops

lemma "P Suc"

nitpick [card = 1-6]

oops

lemma "P (op +\<Colon>nat\<Rightarrow>nat\<Rightarrow>nat)"

nitpick [card nat = 1]

nitpick [card nat = 2]

oops

subsection {* 3.6. Inductive Datatypes *}

lemma "hd (xs @ [y, y]) = hd xs"

nitpick

nitpick [show_consts, show_datatypes]

oops

lemma "\<lbrakk>length xs = 1; length ys = 1\<rbrakk> \<Longrightarrow> xs = ys"

nitpick [show_datatypes]

oops

subsection {* 3.7. Typedefs, Records, Rationals, and Reals *}

typedef three = "{0\<Colon>nat, 1, 2}"

by blast

definition A :: three where "A \<equiv> Abs_three 0"

definition B :: three where "B \<equiv> Abs_three 1"

definition C :: three where "C \<equiv> Abs_three 2"

lemma "\<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P x"

nitpick [show_datatypes]

oops

record point =

  Xcoord :: int

  Ycoord :: int

lemma "Xcoord (p\<Colon>point) = Xcoord (q\<Colon>point)"

nitpick [show_datatypes]

oops

lemma "4 * x + 3 * (y\<Colon>real) \<noteq> 1 / 2"

nitpick [show_datatypes]

oops

subsection {* 3.8. Inductive and Coinductive Predicates *}

inductive even where

"even 0" |

"even n \<Longrightarrow> even (Suc (Suc n))"

lemma "\<exists>n. even n \<and> even (Suc n)"

nitpick [card nat = 100, unary_ints, verbose]

oops

lemma "\<exists>n \<le> 99. even n \<and> even (Suc n)"

nitpick [card nat = 100, unary_ints, verbose]

oops

inductive even' where

"even' (0\<Colon>nat)" |

"even' 2" |

"\<lbrakk>even' m; even' n\<rbrakk> \<Longrightarrow> even' (m + n)"

lemma "\<exists>n \<in> {0, 2, 4, 6, 8}. \<not> even' n"

nitpick [card nat = 10, unary_ints, verbose, show_consts] (* FIXME: should be genuine *)

oops

lemma "even' (n - 2) \<Longrightarrow> even' n"

nitpick [card nat = 10, show_consts]

oops

coinductive nats where

"nats (x\<Colon>nat) \<Longrightarrow> nats x"

lemma "nats = {0, 1, 2, 3, 4}"

nitpick [card nat = 10, show_consts]

oops

inductive odd where

"odd 1" |

"\<lbrakk>odd m; even n\<rbrakk> \<Longrightarrow> odd (m + n)"

lemma "odd n \<Longrightarrow> odd (n - 2)"

nitpick [card nat = 10, show_consts]

oops

subsection {* 3.9. Coinductive Datatypes *}

lemma "xs \<noteq> LCons a xs"

nitpick

oops

lemma "\<lbrakk>xs = LCons a xs; ys = iterates (\<lambda>b. a) b\<rbrakk> \<Longrightarrow> xs = ys"

nitpick [verbose]

nitpick [bisim_depth = -1, verbose]

oops

lemma "\<lbrakk>xs = LCons a xs; ys = LCons a ys\<rbrakk> \<Longrightarrow> xs = ys"

nitpick [bisim_depth = -1, show_datatypes]

nitpick

sorry

subsection {* 3.10. Boxing *}

datatype tm = Var nat | Lam tm | App tm tm

primrec lift where

"lift (Var j) k = Var (if j < k then j else j + 1)" |

"lift (Lam t) k = Lam (lift t (k + 1))" |

"lift (App t u) k = App (lift t k) (lift u k)"

primrec loose where

"loose (Var j) k = (j \<ge> k)" |

"loose (Lam t) k = loose t (Suc k)" |

"loose (App t u) k = (loose t k \<or> loose u k)"

primrec subst\<^isub>1 where

"subst\<^isub>1 \<sigma> (Var j) = \<sigma> j" |

"subst\<^isub>1 \<sigma> (Lam t) =

 Lam (subst\<^isub>1 (\<lambda>n. case n of 0 \<Rightarrow> Var 0 | Suc m \<Rightarrow> lift (\<sigma> m) 1) t)" |

"subst\<^isub>1 \<sigma> (App t u) = App (subst\<^isub>1 \<sigma> t) (subst\<^isub>1 \<sigma> u)"

lemma "\<not> loose t 0 \<Longrightarrow> subst\<^isub>1 \<sigma> t = t"

nitpick [verbose]

nitpick [eval = "subst\<^isub>1 \<sigma> t"]

nitpick [dont_box]

oops

primrec subst\<^isub>2 where

"subst\<^isub>2 \<sigma> (Var j) = \<sigma> j" |

"subst\<^isub>2 \<sigma> (Lam t) =

 Lam (subst\<^isub>2 (\<lambda>n. case n of 0 \<Rightarrow> Var 0 | Suc m \<Rightarrow> lift (\<sigma> m) 0) t)" |

"subst\<^isub>2 \<sigma> (App t u) = App (subst\<^isub>2 \<sigma> t) (subst\<^isub>2 \<sigma> u)"

lemma "\<not> loose t 0 \<Longrightarrow> subst\<^isub>2 \<sigma> t = t"

nitpick

sorry

subsection {* 3.11. Scope Monotonicity *}

lemma "length xs = length ys \<Longrightarrow> rev (zip xs ys) = zip xs (rev ys)"

nitpick [verbose]

nitpick [card = 8, verbose]

oops

lemma "\<exists>g. \<forall>x\<Colon>'b. g (f x) = x \<Longrightarrow> \<forall>y\<Colon>'a. \<exists>x. y = f x"

nitpick [mono]

nitpick

oops

subsection {* 3.12. Inductive Properties *}

inductive_set reach where

"(4\<Colon>nat) \<in> reach" |

"n \<in> reach \<Longrightarrow> n < 4 \<Longrightarrow> 3 * n + 1 \<in> reach" |

"n \<in> reach \<Longrightarrow> n + 2 \<in> reach"

lemma "n \<in> reach \<Longrightarrow> 2 dvd n"

nitpick

apply (induct set: reach)

  apply auto

 nitpick

 apply (thin_tac "n \<in> reach")

 nitpick

oops

lemma "n \<in> reach \<Longrightarrow> 2 dvd n \<and> n \<noteq> 0"

nitpick

apply (induct set: reach)

  apply auto

 nitpick

 apply (thin_tac "n \<in> reach")

 nitpick

oops

lemma "n \<in> reach \<Longrightarrow> 2 dvd n \<and> n \<ge> 4"

by (induct set: reach) arith+

datatype 'a bin_tree = Leaf 'a | Branch "'a bin_tree" "'a bin_tree"

primrec labels where

"labels (Leaf a) = {a}" |

"labels (Branch t u) = labels t \<union> labels u"

primrec swap where

"swap (Leaf c) a b =

 (if c = a then Leaf b else if c = b then Leaf a else Leaf c)" |

"swap (Branch t u) a b = Branch (swap t a b) (swap u a b)"

lemma "\<lbrakk>a \<in> labels t; b \<in> labels t; a \<noteq> b\<rbrakk> \<Longrightarrow> labels (swap t a b) = labels t"

nitpick

proof (induct t)

  case Leaf thus ?case by simp

next

  case (Branch t u) thus ?case

  nitpick

  nitpick [non_std "'a bin_tree", show_consts]

oops

lemma "labels (swap t a b) =

       (if a \<in> labels t then

          if b \<in> labels t then labels t else (labels t - {a}) \<union> {b}

        else

          if b \<in> labels t then (labels t - {b}) \<union> {a} else labels t)"

nitpick

proof (induct t)

  case Leaf thus ?case by simp

next

  case (Branch t u) thus ?case

  nitpick [non_std "'a bin_tree", show_consts]

  by auto

qed

section {* 4. Case Studies *}

nitpick_params [max_potential = 0, max_threads = 2]

subsection {* 4.1. A 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"

theorem S\<^isub>1_sound:

"w \<in> S\<^isub>1 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"

nitpick

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"

theorem S\<^isub>2_sound:

"w \<in> S\<^isub>2 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"

nitpick

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"

theorem S\<^isub>3_sound:

"w \<in> S\<^isub>3 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"

nitpick

sorry

theorem S\<^isub>3_complete:

"length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] \<longrightarrow> w \<in> S\<^isub>3"

nitpick

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]"

nitpick

sorry

theorem S\<^isub>4_complete:

"length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] \<longrightarrow> w \<in> S\<^isub>4"

nitpick

sorry

theorem S\<^isub>4_A\<^isub>4_B\<^isub>4_sound_and_complete:

"w \<in> S\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"

"w \<in> A\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] + 1"

"w \<in> B\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = b] = length [x \<leftarrow> w. x = a] + 1"

nitpick

sorry

subsection {* 4.2. AA Trees *}

datatype 'a aa_tree = \<Lambda> | N "'a\<Colon>linorder" nat "'a aa_tree" "'a aa_tree"

primrec data where

"data \<Lambda> = undefined" |

"data (N x _ _ _) = x"

primrec dataset where

"dataset \<Lambda> = {}" |

"dataset (N x _ t u) = {x} \<union> dataset t \<union> dataset u"

primrec level where

"level \<Lambda> = 0" |

"level (N _ k _ _) = k"

primrec left where

"left \<Lambda> = \<Lambda>" |

"left (N _ _ t\<^isub>1 _) = t\<^isub>1"

primrec right where

"right \<Lambda> = \<Lambda>" |

"right (N _ _ _ t\<^isub>2) = t\<^isub>2"

fun wf where

"wf \<Lambda> = True" |

"wf (N _ k t u) =

 (if t = \<Lambda> then

    k = 1 \<and> (u = \<Lambda> \<or> (level u = 1 \<and> left u = \<Lambda> \<and> right u = \<Lambda>))

  else

    wf t \<and> wf u \<and> u \<noteq> \<Lambda> \<and> level t < k \<and> level u \<le> k \<and> level (right u) < k)"

fun skew where

"skew \<Lambda> = \<Lambda>" |

"skew (N x k t u) =

 (if t \<noteq> \<Lambda> \<and> k = level t then

    N (data t) k (left t) (N x k (right t) u)

  else

    N x k t u)"

fun split where

"split \<Lambda> = \<Lambda>" |

"split (N x k t u) =

 (if u \<noteq> \<Lambda> \<and> k = level (right u) then

    N (data u) (Suc k) (N x k t (left u)) (right u)

  else

    N x k t u)"

theorem dataset_skew_split:

"dataset (skew t) = dataset t"

"dataset (split t) = dataset t"

nitpick

sorry

theorem wf_skew_split:

"wf t \<Longrightarrow> skew t = t"

"wf t \<Longrightarrow> split t = t"

nitpick

sorry

primrec insort\<^isub>1 where

"insort\<^isub>1 \<Lambda> x = N x 1 \<Lambda> \<Lambda>" |

"insort\<^isub>1 (N y k t u) x =

 (* (split \<circ> skew) *) (N y k (if x < y then insort\<^isub>1 t x else t)

                             (if x > y then insort\<^isub>1 u x else u))"

theorem wf_insort\<^isub>1: "wf t \<Longrightarrow> wf (insort\<^isub>1 t x)"

nitpick

oops

theorem wf_insort\<^isub>1_nat: "wf t \<Longrightarrow> wf (insort\<^isub>1 t (x\<Colon>nat))"

nitpick [eval = "insort\<^isub>1 t x"]

oops

primrec insort\<^isub>2 where

"insort\<^isub>2 \<Lambda> x = N x 1 \<Lambda> \<Lambda>" |

"insort\<^isub>2 (N y k t u) x =

 (split \<circ> skew) (N y k (if x < y then insort\<^isub>2 t x else t)

                       (if x > y then insort\<^isub>2 u x else u))"

theorem wf_insort\<^isub>2: "wf t \<Longrightarrow> wf (insort\<^isub>2 t x)"

nitpick

sorry

theorem dataset_insort\<^isub>2: "dataset (insort\<^isub>2 t x) = {x} \<union> dataset t"

nitpick

sorry

end