src/HOL/Nitpick_Examples/Manual_Nits.thy
author blanchet
Tue Mar 09 09:25:23 2010 +0100 (2010-03-09)
changeset 35665 ff2bf50505ab
parent 35312 99cd1f96b400
child 35671 ed2c3830d881
permissions -rw-r--r--
added "finitize" option to Nitpick + remove dependency on "Coinductive_List"
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(*  Title:      HOL/Nitpick_Examples/Manual_Nits.thy
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2009, 2010
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Examples from the Nitpick manual.
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*)
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header {* Examples from the Nitpick Manual *}
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theory Manual_Nits
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imports Main Quotient_Product RealDef
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begin
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chapter {* 3. First Steps *}
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nitpick_params [sat_solver = MiniSat_JNI, max_threads = 1, timeout = 15 s]
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subsection {* 3.1. Propositional Logic *}
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lemma "P \<longleftrightarrow> Q"
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nitpick
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apply auto
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nitpick 1
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nitpick 2
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oops
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subsection {* 3.2. Type Variables *}
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lemma "P x \<Longrightarrow> P (THE y. P y)"
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nitpick [verbose]
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oops
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subsection {* 3.3. Constants *}
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lemma "P x \<Longrightarrow> P (THE y. P y)"
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nitpick [show_consts]
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nitpick [full_descrs, show_consts]
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nitpick [dont_specialize, full_descrs, show_consts]
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oops
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lemma "\<exists>!x. P x \<Longrightarrow> P (THE y. P y)"
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nitpick
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nitpick [card 'a = 1-50]
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(* sledgehammer *)
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apply (metis the_equality)
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done
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subsection {* 3.4. Skolemization *}
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lemma "\<exists>g. \<forall>x. g (f x) = x \<Longrightarrow> \<forall>y. \<exists>x. y = f x"
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nitpick
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oops
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lemma "\<exists>x. \<forall>f. f x = x"
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nitpick
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oops
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lemma "refl r \<Longrightarrow> sym r"
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nitpick
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oops
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subsection {* 3.5. Natural Numbers and Integers *}
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lemma "\<lbrakk>i \<le> j; n \<le> (m\<Colon>int)\<rbrakk> \<Longrightarrow> i * n + j * m \<le> i * m + j * n"
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nitpick
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oops
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lemma "\<forall>n. Suc n \<noteq> n \<Longrightarrow> P"
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nitpick [card nat = 100, check_potential]
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oops
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lemma "P Suc"
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nitpick
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oops
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lemma "P (op +\<Colon>nat\<Rightarrow>nat\<Rightarrow>nat)"
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nitpick [card nat = 1]
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nitpick [card nat = 2]
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oops
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subsection {* 3.6. Inductive Datatypes *}
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lemma "hd (xs @ [y, y]) = hd xs"
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nitpick
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nitpick [show_consts, show_datatypes]
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oops
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lemma "\<lbrakk>length xs = 1; length ys = 1\<rbrakk> \<Longrightarrow> xs = ys"
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nitpick [show_datatypes]
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oops
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subsection {* 3.7. Typedefs, Records, Rationals, and Reals *}
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typedef three = "{0\<Colon>nat, 1, 2}"
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by blast
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definition A :: three where "A \<equiv> Abs_three 0"
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definition B :: three where "B \<equiv> Abs_three 1"
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definition C :: three where "C \<equiv> Abs_three 2"
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lemma "\<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P x"
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nitpick [show_datatypes]
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oops
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fun my_int_rel where
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"my_int_rel (x, y) (u, v) = (x + v = u + y)"
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quotient_type my_int = "nat \<times> nat" / my_int_rel
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by (auto simp add: equivp_def expand_fun_eq)
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definition add_raw where
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"add_raw \<equiv> \<lambda>(x, y) (u, v). (x + (u\<Colon>nat), y + (v\<Colon>nat))"
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quotient_definition "add\<Colon>my_int \<Rightarrow> my_int \<Rightarrow> my_int" is add_raw
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lemma "add x y = add x x"
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nitpick [show_datatypes]
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oops
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record point =
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  Xcoord :: int
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  Ycoord :: int
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lemma "Xcoord (p\<Colon>point) = Xcoord (q\<Colon>point)"
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nitpick [show_datatypes]
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oops
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lemma "4 * x + 3 * (y\<Colon>real) \<noteq> 1 / 2"
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nitpick [show_datatypes]
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oops
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subsection {* 3.8. Inductive and Coinductive Predicates *}
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inductive even where
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"even 0" |
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"even n \<Longrightarrow> even (Suc (Suc n))"
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lemma "\<exists>n. even n \<and> even (Suc n)"
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nitpick [card nat = 100, unary_ints, verbose]
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oops
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lemma "\<exists>n \<le> 99. even n \<and> even (Suc n)"
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nitpick [card nat = 100, unary_ints, verbose]
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oops
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inductive even' where
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"even' (0\<Colon>nat)" |
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"even' 2" |
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"\<lbrakk>even' m; even' n\<rbrakk> \<Longrightarrow> even' (m + n)"
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lemma "\<exists>n \<in> {0, 2, 4, 6, 8}. \<not> even' n"
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nitpick [card nat = 10, unary_ints, verbose, show_consts]
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oops
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lemma "even' (n - 2) \<Longrightarrow> even' n"
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nitpick [card nat = 10, show_consts]
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oops
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coinductive nats where
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"nats (x\<Colon>nat) \<Longrightarrow> nats x"
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lemma "nats = {0, 1, 2, 3, 4}"
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nitpick [card nat = 10, show_consts]
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oops
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inductive odd where
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"odd 1" |
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"\<lbrakk>odd m; even n\<rbrakk> \<Longrightarrow> odd (m + n)"
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lemma "odd n \<Longrightarrow> odd (n - 2)"
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nitpick [card nat = 10, show_consts]
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oops
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subsection {* 3.9. Coinductive Datatypes *}
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(* Lazy lists are defined in Andreas Lochbihler's "Coinductive" AFP entry. Since
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we cannot rely on its presence, we expediently provide our own axiomatization.
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The examples also work unchanged with Lochbihler's "Coinductive_List" theory. *)
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typedef 'a llist = "UNIV\<Colon>('a list + (nat \<Rightarrow> 'a)) set" by auto
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definition LNil where "LNil = Abs_llist (Inl [])"
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definition LCons where
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"LCons y ys = Abs_llist (case Rep_llist ys of
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                           Inl ys' \<Rightarrow> Inl (y # ys')
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                         | Inr f \<Rightarrow> Inr (\<lambda>n. case n of 0 \<Rightarrow> y | Suc m \<Rightarrow> f m))"
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axiomatization iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a llist"
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lemma iterates_def [nitpick_simp]:
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"iterates f a = LCons a (iterates f (f a))"
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sorry
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setup {*
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Nitpick.register_codatatype @{typ "'a llist"} ""
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    (map dest_Const [@{term LNil}, @{term LCons}])
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*}
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lemma "xs \<noteq> LCons a xs"
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nitpick
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oops
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lemma "\<lbrakk>xs = LCons a xs; ys = iterates (\<lambda>b. a) b\<rbrakk> \<Longrightarrow> xs = ys"
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nitpick [verbose]
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oops
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lemma "\<lbrakk>xs = LCons a xs; ys = LCons a ys\<rbrakk> \<Longrightarrow> xs = ys"
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nitpick [bisim_depth = -1, show_datatypes]
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nitpick
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sorry
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subsection {* 3.10. Boxing *}
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datatype tm = Var nat | Lam tm | App tm tm
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primrec lift where
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"lift (Var j) k = Var (if j < k then j else j + 1)" |
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"lift (Lam t) k = Lam (lift t (k + 1))" |
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"lift (App t u) k = App (lift t k) (lift u k)"
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primrec loose where
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"loose (Var j) k = (j \<ge> k)" |
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"loose (Lam t) k = loose t (Suc k)" |
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"loose (App t u) k = (loose t k \<or> loose u k)"
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primrec subst\<^isub>1 where
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"subst\<^isub>1 \<sigma> (Var j) = \<sigma> j" |
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"subst\<^isub>1 \<sigma> (Lam t) =
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 Lam (subst\<^isub>1 (\<lambda>n. case n of 0 \<Rightarrow> Var 0 | Suc m \<Rightarrow> lift (\<sigma> m) 1) t)" |
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"subst\<^isub>1 \<sigma> (App t u) = App (subst\<^isub>1 \<sigma> t) (subst\<^isub>1 \<sigma> u)"
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lemma "\<not> loose t 0 \<Longrightarrow> subst\<^isub>1 \<sigma> t = t"
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nitpick [verbose]
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nitpick [eval = "subst\<^isub>1 \<sigma> t"]
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(* nitpick [dont_box] *)
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oops
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primrec subst\<^isub>2 where
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"subst\<^isub>2 \<sigma> (Var j) = \<sigma> j" |
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"subst\<^isub>2 \<sigma> (Lam t) =
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 Lam (subst\<^isub>2 (\<lambda>n. case n of 0 \<Rightarrow> Var 0 | Suc m \<Rightarrow> lift (\<sigma> m) 0) t)" |
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"subst\<^isub>2 \<sigma> (App t u) = App (subst\<^isub>2 \<sigma> t) (subst\<^isub>2 \<sigma> u)"
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lemma "\<not> loose t 0 \<Longrightarrow> subst\<^isub>2 \<sigma> t = t"
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nitpick [card = 1\<midarrow>6]
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sorry
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subsection {* 3.11. Scope Monotonicity *}
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lemma "length xs = length ys \<Longrightarrow> rev (zip xs ys) = zip xs (rev ys)"
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nitpick [verbose]
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nitpick [card = 8, verbose]
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oops
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lemma "\<exists>g. \<forall>x\<Colon>'b. g (f x) = x \<Longrightarrow> \<forall>y\<Colon>'a. \<exists>x. y = f x"
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nitpick [mono]
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nitpick
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oops
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subsection {* 3.12. Inductive Properties *}
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inductive_set reach where
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"(4\<Colon>nat) \<in> reach" |
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"n \<in> reach \<Longrightarrow> n < 4 \<Longrightarrow> 3 * n + 1 \<in> reach" |
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"n \<in> reach \<Longrightarrow> n + 2 \<in> reach"
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lemma "n \<in> reach \<Longrightarrow> 2 dvd n"
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nitpick [unary_ints]
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apply (induct set: reach)
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  apply auto
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 nitpick
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 apply (thin_tac "n \<in> reach")
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 nitpick
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oops
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lemma "n \<in> reach \<Longrightarrow> 2 dvd n \<and> n \<noteq> 0"
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nitpick [unary_ints]
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apply (induct set: reach)
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  apply auto
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 nitpick
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 apply (thin_tac "n \<in> reach")
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 nitpick
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oops
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lemma "n \<in> reach \<Longrightarrow> 2 dvd n \<and> n \<ge> 4"
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by (induct set: reach) arith+
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datatype 'a bin_tree = Leaf 'a | Branch "'a bin_tree" "'a bin_tree"
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primrec labels where
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"labels (Leaf a) = {a}" |
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"labels (Branch t u) = labels t \<union> labels u"
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primrec swap where
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"swap (Leaf c) a b =
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 (if c = a then Leaf b else if c = b then Leaf a else Leaf c)" |
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"swap (Branch t u) a b = Branch (swap t a b) (swap u a b)"
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lemma "{a, b} \<subseteq> labels t \<Longrightarrow> labels (swap t a b) = labels t"
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nitpick
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proof (induct t)
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  case Leaf thus ?case by simp
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next
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  case (Branch t u) thus ?case
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  nitpick
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  nitpick [non_std, show_all]
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oops
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lemma "labels (swap t a b) =
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       (if a \<in> labels t then
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          if b \<in> labels t then labels t else (labels t - {a}) \<union> {b}
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        else
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          if b \<in> labels t then (labels t - {b}) \<union> {a} else labels t)"
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(* nitpick *)
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proof (induct t)
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  case Leaf thus ?case by simp
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next
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  case (Branch t u) thus ?case
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  nitpick [non_std, show_all]
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  by auto
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qed
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section {* 4. Case Studies *}
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nitpick_params [max_potential = 0, max_threads = 2]
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subsection {* 4.1. A Context-Free Grammar *}
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datatype alphabet = a | b
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inductive_set S\<^isub>1 and A\<^isub>1 and B\<^isub>1 where
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  "[] \<in> S\<^isub>1"
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| "w \<in> A\<^isub>1 \<Longrightarrow> b # w \<in> S\<^isub>1"
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| "w \<in> B\<^isub>1 \<Longrightarrow> a # w \<in> S\<^isub>1"
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| "w \<in> S\<^isub>1 \<Longrightarrow> a # w \<in> A\<^isub>1"
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| "w \<in> S\<^isub>1 \<Longrightarrow> b # w \<in> S\<^isub>1"
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| "\<lbrakk>v \<in> B\<^isub>1; v \<in> B\<^isub>1\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>1"
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theorem S\<^isub>1_sound:
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"w \<in> S\<^isub>1 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
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nitpick
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oops
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inductive_set S\<^isub>2 and A\<^isub>2 and B\<^isub>2 where
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  "[] \<in> S\<^isub>2"
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| "w \<in> A\<^isub>2 \<Longrightarrow> b # w \<in> S\<^isub>2"
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| "w \<in> B\<^isub>2 \<Longrightarrow> a # w \<in> S\<^isub>2"
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| "w \<in> S\<^isub>2 \<Longrightarrow> a # w \<in> A\<^isub>2"
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| "w \<in> S\<^isub>2 \<Longrightarrow> b # w \<in> B\<^isub>2"
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| "\<lbrakk>v \<in> B\<^isub>2; v \<in> B\<^isub>2\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>2"
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theorem S\<^isub>2_sound:
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"w \<in> S\<^isub>2 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
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nitpick
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oops
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inductive_set S\<^isub>3 and A\<^isub>3 and B\<^isub>3 where
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  "[] \<in> S\<^isub>3"
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| "w \<in> A\<^isub>3 \<Longrightarrow> b # w \<in> S\<^isub>3"
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| "w \<in> B\<^isub>3 \<Longrightarrow> a # w \<in> S\<^isub>3"
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| "w \<in> S\<^isub>3 \<Longrightarrow> a # w \<in> A\<^isub>3"
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| "w \<in> S\<^isub>3 \<Longrightarrow> b # w \<in> B\<^isub>3"
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| "\<lbrakk>v \<in> B\<^isub>3; w \<in> B\<^isub>3\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>3"
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theorem S\<^isub>3_sound:
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"w \<in> S\<^isub>3 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
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nitpick
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sorry
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theorem S\<^isub>3_complete:
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"length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] \<longrightarrow> w \<in> S\<^isub>3"
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nitpick
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oops
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inductive_set S\<^isub>4 and A\<^isub>4 and B\<^isub>4 where
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  "[] \<in> S\<^isub>4"
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| "w \<in> A\<^isub>4 \<Longrightarrow> b # w \<in> S\<^isub>4"
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| "w \<in> B\<^isub>4 \<Longrightarrow> a # w \<in> S\<^isub>4"
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| "w \<in> S\<^isub>4 \<Longrightarrow> a # w \<in> A\<^isub>4"
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| "\<lbrakk>v \<in> A\<^isub>4; w \<in> A\<^isub>4\<rbrakk> \<Longrightarrow> b # v @ w \<in> A\<^isub>4"
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| "w \<in> S\<^isub>4 \<Longrightarrow> b # w \<in> B\<^isub>4"
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| "\<lbrakk>v \<in> B\<^isub>4; w \<in> B\<^isub>4\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>4"
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theorem S\<^isub>4_sound:
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"w \<in> S\<^isub>4 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
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nitpick
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sorry
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theorem S\<^isub>4_complete:
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"length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] \<longrightarrow> w \<in> S\<^isub>4"
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nitpick
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sorry
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theorem S\<^isub>4_A\<^isub>4_B\<^isub>4_sound_and_complete:
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"w \<in> S\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
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"w \<in> A\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] + 1"
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"w \<in> B\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = b] = length [x \<leftarrow> w. x = a] + 1"
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nitpick
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sorry
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subsection {* 4.2. AA Trees *}
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datatype 'a aa_tree = \<Lambda> | N "'a\<Colon>linorder" nat "'a aa_tree" "'a aa_tree"
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primrec data where
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"data \<Lambda> = undefined" |
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"data (N x _ _ _) = x"
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primrec dataset where
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"dataset \<Lambda> = {}" |
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"dataset (N x _ t u) = {x} \<union> dataset t \<union> dataset u"
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primrec level where
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"level \<Lambda> = 0" |
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"level (N _ k _ _) = k"
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primrec left where
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"left \<Lambda> = \<Lambda>" |
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"left (N _ _ t\<^isub>1 _) = t\<^isub>1"
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primrec right where
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"right \<Lambda> = \<Lambda>" |
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"right (N _ _ _ t\<^isub>2) = t\<^isub>2"
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fun wf where
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"wf \<Lambda> = True" |
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"wf (N _ k t u) =
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 (if t = \<Lambda> then
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    k = 1 \<and> (u = \<Lambda> \<or> (level u = 1 \<and> left u = \<Lambda> \<and> right u = \<Lambda>))
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  else
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    wf t \<and> wf u \<and> u \<noteq> \<Lambda> \<and> level t < k \<and> level u \<le> k \<and> level (right u) < k)"
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fun skew where
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"skew \<Lambda> = \<Lambda>" |
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"skew (N x k t u) =
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 (if t \<noteq> \<Lambda> \<and> k = level t then
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    N (data t) k (left t) (N x k (right t) u)
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  else
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    N x k t u)"
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fun split where
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"split \<Lambda> = \<Lambda>" |
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"split (N x k t u) =
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 (if u \<noteq> \<Lambda> \<and> k = level (right u) then
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    N (data u) (Suc k) (N x k t (left u)) (right u)
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  else
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    N x k t u)"
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theorem dataset_skew_split:
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"dataset (skew t) = dataset t"
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"dataset (split t) = dataset t"
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   452
nitpick
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   453
sorry
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   454
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   455
theorem wf_skew_split:
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"wf t \<Longrightarrow> skew t = t"
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   457
"wf t \<Longrightarrow> split t = t"
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   458
nitpick
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   459
sorry
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   460
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   461
primrec insort\<^isub>1 where
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"insort\<^isub>1 \<Lambda> x = N x 1 \<Lambda> \<Lambda>" |
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"insort\<^isub>1 (N y k t u) x =
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 (* (split \<circ> skew) *) (N y k (if x < y then insort\<^isub>1 t x else t)
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                             (if x > y then insort\<^isub>1 u x else u))"
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   466
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   467
theorem wf_insort\<^isub>1: "wf t \<Longrightarrow> wf (insort\<^isub>1 t x)"
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   468
nitpick
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   469
oops
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   470
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   471
theorem wf_insort\<^isub>1_nat: "wf t \<Longrightarrow> wf (insort\<^isub>1 t (x\<Colon>nat))"
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   472
nitpick [eval = "insort\<^isub>1 t x"]
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   473
oops
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   474
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   475
primrec insort\<^isub>2 where
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"insort\<^isub>2 \<Lambda> x = N x 1 \<Lambda> \<Lambda>" |
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   477
"insort\<^isub>2 (N y k t u) x =
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   478
 (split \<circ> skew) (N y k (if x < y then insort\<^isub>2 t x else t)
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   479
                       (if x > y then insort\<^isub>2 u x else u))"
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   480
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   481
theorem wf_insort\<^isub>2: "wf t \<Longrightarrow> wf (insort\<^isub>2 t x)"
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   482
nitpick
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   483
sorry
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   484
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   485
theorem dataset_insort\<^isub>2: "dataset (insort\<^isub>2 t x) = {x} \<union> dataset t"
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   486
nitpick
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   487
sorry
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   488
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   489
end