author | wenzelm |
Wed, 10 Oct 2012 15:21:26 +0200 | |
changeset 49754 | acafcac41690 |
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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 2009-2011 |
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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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(* The "expect" arguments to Nitpick in this theory and the other example |
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theories are there so that the example can also serve as a regression test |
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suite. *) |
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theory Manual_Nits |
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imports Main "~~/src/HOL/Library/Quotient_Product" RealDef |
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begin |
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chapter {* 2. First Steps *} |
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nitpick_params [sat_solver = MiniSat_JNI, max_threads = 1, timeout = 240] |
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subsection {* 2.1. Propositional Logic *} |
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lemma "P \<longleftrightarrow> Q" |
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nitpick [expect = genuine] |
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apply auto |
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nitpick [expect = genuine] 1 |
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nitpick [expect = genuine] 2 |
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oops |
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subsection {* 2.2. Type Variables *} |
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lemma "x \<in> A \<Longrightarrow> (THE y. y \<in> A) \<in> A" |
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nitpick [verbose, expect = genuine] |
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oops |
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subsection {* 2.3. Constants *} |
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lemma "x \<in> A \<Longrightarrow> (THE y. y \<in> A) \<in> A" |
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nitpick [show_consts, expect = genuine] |
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nitpick [dont_specialize, show_consts, expect = genuine] |
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oops |
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lemma "\<exists>!x. x \<in> A \<Longrightarrow> (THE y. y \<in> A) \<in> A" |
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nitpick [expect = none] |
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nitpick [card 'a = 1\<emdash>50, expect = none] |
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(* sledgehammer *) |
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by (metis the_equality) |
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subsection {* 2.4. 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 [expect = genuine] |
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oops |
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lemma "\<exists>x. \<forall>f. f x = x" |
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nitpick [expect = genuine] |
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oops |
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lemma "refl r \<Longrightarrow> sym r" |
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nitpick [expect = genuine] |
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oops |
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subsection {* 2.5. 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 [expect = genuine] |
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nitpick [binary_ints, bits = 16, expect = genuine] |
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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, tac_timeout = 5, expect = genuine] |
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oops |
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lemma "P Suc" |
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nitpick [expect = none] |
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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, expect = genuine] |
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nitpick [card nat = 2, expect = none] |
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oops |
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subsection {* 2.6. Inductive Datatypes *} |
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lemma "hd (xs @ [y, y]) = hd xs" |
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nitpick [expect = genuine] |
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nitpick [show_consts, show_datatypes, expect = genuine] |
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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, expect = genuine] |
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subsection {* 2.7. 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>A \<in> X; B \<in> X\<rbrakk> \<Longrightarrow> c \<in> X" |
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nitpick [show_datatypes, expect = genuine] |
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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 fun_eq_iff) |
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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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unfolding add_raw_def by auto |
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lemma "add x y = add x x" |
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nitpick [show_datatypes, expect = genuine] |
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oops |
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ML {* |
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fun my_int_postproc _ _ _ T (Const _ $ (Const _ $ t1 $ t2)) = |
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HOLogic.mk_number T (snd (HOLogic.dest_number t1) |
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- snd (HOLogic.dest_number t2)) |
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| my_int_postproc _ _ _ _ t = t |
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*} |
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declaration {* |
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Nitpick_Model.register_term_postprocessor @{typ my_int} my_int_postproc |
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*} |
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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, expect = genuine] |
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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, expect = genuine] |
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oops |
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subsection {* 2.8. 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 = 50, unary_ints, verbose, expect = potential] |
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lemma "\<exists>n \<le> 49. even n \<and> even (Suc n)" |
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nitpick [card nat = 50, unary_ints, expect = genuine] |
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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, expect = genuine] |
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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, expect = genuine] |
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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 = (\<lambda>n. n \<in> {0, 1, 2, 3, 4})" |
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nitpick [card nat = 10, show_consts, expect = genuine] |
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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 = 4, show_consts, expect = genuine] |
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oops |
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subsection {* 2.9. 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 |
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axiomatization. The examples also work unchanged with Lochbihler's |
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"Coinductive_List" theory. *) |
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(* BEGIN LAZY LIST SETUP *) |
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definition "llist = (UNIV\<Colon>('a list + (nat \<Rightarrow> 'a)) set)" |
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typedef (open) 'a llist = "llist\<Colon>('a list + (nat \<Rightarrow> 'a)) set" |
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unfolding llist_def by auto |
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definition LNil where |
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"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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212 |
|
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213 |
axiomatization iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a llist" |
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214 |
|
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215 |
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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218 |
|
38288 | 219 |
declaration {* |
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Nitpick_HOL.register_codatatype @{typ "'a llist"} "" |
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(map dest_Const [@{term LNil}, @{term LCons}]) |
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*} |
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(* END LAZY LIST SETUP *) |
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224 |
|
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lemma "xs \<noteq> LCons a xs" |
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226 |
nitpick [expect = genuine] |
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228 |
||
229 |
lemma "\<lbrakk>xs = LCons a xs; ys = iterates (\<lambda>b. a) b\<rbrakk> \<Longrightarrow> xs = ys" |
|
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230 |
nitpick [verbose, expect = genuine] |
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232 |
||
233 |
lemma "\<lbrakk>xs = LCons a xs; ys = LCons a ys\<rbrakk> \<Longrightarrow> xs = ys" |
|
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234 |
nitpick [bisim_depth = -1, show_datatypes, expect = quasi_genuine] |
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nitpick [card = 1\<emdash>5, expect = none] |
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sorry |
237 |
||
45053 | 238 |
subsection {* 2.10. Boxing *} |
33197 | 239 |
|
240 |
datatype tm = Var nat | Lam tm | App tm tm |
|
241 |
||
242 |
primrec lift where |
|
243 |
"lift (Var j) k = Var (if j < k then j else j + 1)" | |
|
244 |
"lift (Lam t) k = Lam (lift t (k + 1))" | |
|
245 |
"lift (App t u) k = App (lift t k) (lift u k)" |
|
246 |
||
247 |
primrec loose where |
|
248 |
"loose (Var j) k = (j \<ge> k)" | |
|
249 |
"loose (Lam t) k = loose t (Suc k)" | |
|
250 |
"loose (App t u) k = (loose t k \<or> loose u k)" |
|
251 |
||
252 |
primrec subst\<^isub>1 where |
|
253 |
"subst\<^isub>1 \<sigma> (Var j) = \<sigma> j" | |
|
254 |
"subst\<^isub>1 \<sigma> (Lam t) = |
|
255 |
Lam (subst\<^isub>1 (\<lambda>n. case n of 0 \<Rightarrow> Var 0 | Suc m \<Rightarrow> lift (\<sigma> m) 1) t)" | |
|
256 |
"subst\<^isub>1 \<sigma> (App t u) = App (subst\<^isub>1 \<sigma> t) (subst\<^isub>1 \<sigma> u)" |
|
257 |
||
258 |
lemma "\<not> loose t 0 \<Longrightarrow> subst\<^isub>1 \<sigma> t = t" |
|
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nitpick [verbose, expect = genuine] |
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260 |
nitpick [eval = "subst\<^isub>1 \<sigma> t", expect = genuine] |
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261 |
(* nitpick [dont_box, expect = unknown] *) |
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oops |
263 |
||
264 |
primrec subst\<^isub>2 where |
|
265 |
"subst\<^isub>2 \<sigma> (Var j) = \<sigma> j" | |
|
266 |
"subst\<^isub>2 \<sigma> (Lam t) = |
|
267 |
Lam (subst\<^isub>2 (\<lambda>n. case n of 0 \<Rightarrow> Var 0 | Suc m \<Rightarrow> lift (\<sigma> m) 0) t)" | |
|
268 |
"subst\<^isub>2 \<sigma> (App t u) = App (subst\<^isub>2 \<sigma> t) (subst\<^isub>2 \<sigma> u)" |
|
269 |
||
270 |
lemma "\<not> loose t 0 \<Longrightarrow> subst\<^isub>2 \<sigma> t = t" |
|
42959 | 271 |
nitpick [card = 1\<emdash>5, expect = none] |
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sorry |
273 |
||
45053 | 274 |
subsection {* 2.11. Scope Monotonicity *} |
33197 | 275 |
|
276 |
lemma "length xs = length ys \<Longrightarrow> rev (zip xs ys) = zip xs (rev ys)" |
|
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277 |
nitpick [verbose, expect = genuine] |
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279 |
||
280 |
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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281 |
nitpick [mono, expect = none] |
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282 |
nitpick [expect = genuine] |
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284 |
||
45053 | 285 |
subsection {* 2.12. Inductive Properties *} |
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286 |
|
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287 |
inductive_set reach where |
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"(4\<Colon>nat) \<in> reach" | |
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289 |
"n \<in> reach \<Longrightarrow> n < 4 \<Longrightarrow> 3 * n + 1 \<in> reach" | |
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290 |
"n \<in> reach \<Longrightarrow> n + 2 \<in> reach" |
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|
291 |
|
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292 |
lemma "n \<in> reach \<Longrightarrow> 2 dvd n" |
38184 | 293 |
(* nitpick *) |
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294 |
apply (induct set: reach) |
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295 |
apply auto |
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nitpick [card = 1\<emdash>4, bits = 1\<emdash>4, expect = none] |
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|
297 |
apply (thin_tac "n \<in> reach") |
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|
298 |
nitpick [expect = genuine] |
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|
299 |
oops |
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|
300 |
|
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|
301 |
lemma "n \<in> reach \<Longrightarrow> 2 dvd n \<and> n \<noteq> 0" |
38184 | 302 |
(* nitpick *) |
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|
303 |
apply (induct set: reach) |
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|
304 |
apply auto |
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nitpick [card = 1\<emdash>4, bits = 1\<emdash>4, expect = none] |
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|
306 |
apply (thin_tac "n \<in> reach") |
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|
307 |
nitpick [expect = genuine] |
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|
308 |
oops |
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|
309 |
|
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|
310 |
lemma "n \<in> reach \<Longrightarrow> 2 dvd n \<and> n \<ge> 4" |
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|
311 |
by (induct set: reach) arith+ |
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|
312 |
|
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|
313 |
datatype 'a bin_tree = Leaf 'a | Branch "'a bin_tree" "'a bin_tree" |
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|
314 |
|
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|
315 |
primrec labels where |
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|
316 |
"labels (Leaf a) = {a}" | |
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|
317 |
"labels (Branch t u) = labels t \<union> labels u" |
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|
318 |
|
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|
319 |
primrec swap where |
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|
320 |
"swap (Leaf c) a b = |
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|
321 |
(if c = a then Leaf b else if c = b then Leaf a else Leaf c)" | |
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|
322 |
"swap (Branch t u) a b = Branch (swap t a b) (swap u a b)" |
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|
323 |
|
35180
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|
324 |
lemma "{a, b} \<subseteq> labels t \<Longrightarrow> labels (swap t a b) = labels t" |
35671
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|
325 |
(* nitpick *) |
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|
326 |
proof (induct t) |
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|
327 |
case Leaf thus ?case by simp |
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|
328 |
next |
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|
329 |
case (Branch t u) thus ?case |
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|
330 |
(* nitpick *) |
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|
331 |
nitpick [non_std, show_all, expect = genuine] |
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|
332 |
oops |
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|
333 |
|
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|
334 |
lemma "labels (swap t a b) = |
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|
335 |
(if a \<in> labels t then |
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|
336 |
if b \<in> labels t then labels t else (labels t - {a}) \<union> {b} |
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|
337 |
else |
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|
338 |
if b \<in> labels t then (labels t - {b}) \<union> {a} else labels t)" |
35309
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|
339 |
(* nitpick *) |
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|
340 |
proof (induct t) |
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|
341 |
case Leaf thus ?case by simp |
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|
342 |
next |
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|
343 |
case (Branch t u) thus ?case |
42959 | 344 |
nitpick [non_std, card = 1\<emdash>4, expect = none] |
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|
345 |
by auto |
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|
346 |
qed |
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|
347 |
|
45053 | 348 |
section {* 3. Case Studies *} |
33197 | 349 |
|
36268 | 350 |
nitpick_params [max_potential = 0] |
33197 | 351 |
|
45053 | 352 |
subsection {* 3.1. A Context-Free Grammar *} |
33197 | 353 |
|
354 |
datatype alphabet = a | b |
|
355 |
||
356 |
inductive_set S\<^isub>1 and A\<^isub>1 and B\<^isub>1 where |
|
357 |
"[] \<in> S\<^isub>1" |
|
358 |
| "w \<in> A\<^isub>1 \<Longrightarrow> b # w \<in> S\<^isub>1" |
|
359 |
| "w \<in> B\<^isub>1 \<Longrightarrow> a # w \<in> S\<^isub>1" |
|
360 |
| "w \<in> S\<^isub>1 \<Longrightarrow> a # w \<in> A\<^isub>1" |
|
361 |
| "w \<in> S\<^isub>1 \<Longrightarrow> b # w \<in> S\<^isub>1" |
|
362 |
| "\<lbrakk>v \<in> B\<^isub>1; v \<in> B\<^isub>1\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>1" |
|
363 |
||
364 |
theorem S\<^isub>1_sound: |
|
365 |
"w \<in> S\<^isub>1 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]" |
|
35671
ed2c3830d881
improved Nitpick's precision for "card" and "setsum" + fix incorrect outcome code w.r.t. "bisim_depth = -1"
blanchet
parents:
35665
diff
changeset
|
366 |
nitpick [expect = genuine] |
33197 | 367 |
oops |
368 |
||
369 |
inductive_set S\<^isub>2 and A\<^isub>2 and B\<^isub>2 where |
|
370 |
"[] \<in> S\<^isub>2" |
|
371 |
| "w \<in> A\<^isub>2 \<Longrightarrow> b # w \<in> S\<^isub>2" |
|
372 |
| "w \<in> B\<^isub>2 \<Longrightarrow> a # w \<in> S\<^isub>2" |
|
373 |
| "w \<in> S\<^isub>2 \<Longrightarrow> a # w \<in> A\<^isub>2" |
|
374 |
| "w \<in> S\<^isub>2 \<Longrightarrow> b # w \<in> B\<^isub>2" |
|
375 |
| "\<lbrakk>v \<in> B\<^isub>2; v \<in> B\<^isub>2\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>2" |
|
376 |
||
377 |
theorem S\<^isub>2_sound: |
|
378 |
"w \<in> S\<^isub>2 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]" |
|
35671
ed2c3830d881
improved Nitpick's precision for "card" and "setsum" + fix incorrect outcome code w.r.t. "bisim_depth = -1"
blanchet
parents:
35665
diff
changeset
|
379 |
nitpick [expect = genuine] |
33197 | 380 |
oops |
381 |
||
382 |
inductive_set S\<^isub>3 and A\<^isub>3 and B\<^isub>3 where |
|
383 |
"[] \<in> S\<^isub>3" |
|
384 |
| "w \<in> A\<^isub>3 \<Longrightarrow> b # w \<in> S\<^isub>3" |
|
385 |
| "w \<in> B\<^isub>3 \<Longrightarrow> a # w \<in> S\<^isub>3" |
|
386 |
| "w \<in> S\<^isub>3 \<Longrightarrow> a # w \<in> A\<^isub>3" |
|
387 |
| "w \<in> S\<^isub>3 \<Longrightarrow> b # w \<in> B\<^isub>3" |
|
388 |
| "\<lbrakk>v \<in> B\<^isub>3; w \<in> B\<^isub>3\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>3" |
|
389 |
||
390 |
theorem S\<^isub>3_sound: |
|
391 |
"w \<in> S\<^isub>3 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]" |
|
42959 | 392 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 393 |
sorry |
394 |
||
395 |
theorem S\<^isub>3_complete: |
|
396 |
"length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] \<longrightarrow> w \<in> S\<^isub>3" |
|
35671
ed2c3830d881
improved Nitpick's precision for "card" and "setsum" + fix incorrect outcome code w.r.t. "bisim_depth = -1"
blanchet
parents:
35665
diff
changeset
|
397 |
nitpick [expect = genuine] |
33197 | 398 |
oops |
399 |
||
400 |
inductive_set S\<^isub>4 and A\<^isub>4 and B\<^isub>4 where |
|
401 |
"[] \<in> S\<^isub>4" |
|
402 |
| "w \<in> A\<^isub>4 \<Longrightarrow> b # w \<in> S\<^isub>4" |
|
403 |
| "w \<in> B\<^isub>4 \<Longrightarrow> a # w \<in> S\<^isub>4" |
|
404 |
| "w \<in> S\<^isub>4 \<Longrightarrow> a # w \<in> A\<^isub>4" |
|
405 |
| "\<lbrakk>v \<in> A\<^isub>4; w \<in> A\<^isub>4\<rbrakk> \<Longrightarrow> b # v @ w \<in> A\<^isub>4" |
|
406 |
| "w \<in> S\<^isub>4 \<Longrightarrow> b # w \<in> B\<^isub>4" |
|
407 |
| "\<lbrakk>v \<in> B\<^isub>4; w \<in> B\<^isub>4\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>4" |
|
408 |
||
409 |
theorem S\<^isub>4_sound: |
|
410 |
"w \<in> S\<^isub>4 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]" |
|
42959 | 411 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 412 |
sorry |
413 |
||
414 |
theorem S\<^isub>4_complete: |
|
415 |
"length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] \<longrightarrow> w \<in> S\<^isub>4" |
|
42959 | 416 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 417 |
sorry |
418 |
||
419 |
theorem S\<^isub>4_A\<^isub>4_B\<^isub>4_sound_and_complete: |
|
420 |
"w \<in> S\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]" |
|
421 |
"w \<in> A\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] + 1" |
|
422 |
"w \<in> B\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = b] = length [x \<leftarrow> w. x = a] + 1" |
|
42959 | 423 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 424 |
sorry |
425 |
||
45053 | 426 |
subsection {* 3.2. AA Trees *} |
33197 | 427 |
|
34982
7b8c366e34a2
added support for nonstandard models to Nitpick (based on an idea by Koen Claessen) and did other fixes to Nitpick
blanchet
parents:
34126
diff
changeset
|
428 |
datatype 'a aa_tree = \<Lambda> | N "'a\<Colon>linorder" nat "'a aa_tree" "'a aa_tree" |
33197 | 429 |
|
430 |
primrec data where |
|
431 |
"data \<Lambda> = undefined" | |
|
432 |
"data (N x _ _ _) = x" |
|
433 |
||
434 |
primrec dataset where |
|
435 |
"dataset \<Lambda> = {}" | |
|
436 |
"dataset (N x _ t u) = {x} \<union> dataset t \<union> dataset u" |
|
437 |
||
438 |
primrec level where |
|
439 |
"level \<Lambda> = 0" | |
|
440 |
"level (N _ k _ _) = k" |
|
441 |
||
442 |
primrec left where |
|
443 |
"left \<Lambda> = \<Lambda>" | |
|
444 |
"left (N _ _ t\<^isub>1 _) = t\<^isub>1" |
|
445 |
||
446 |
primrec right where |
|
447 |
"right \<Lambda> = \<Lambda>" | |
|
448 |
"right (N _ _ _ t\<^isub>2) = t\<^isub>2" |
|
449 |
||
450 |
fun wf where |
|
451 |
"wf \<Lambda> = True" | |
|
452 |
"wf (N _ k t u) = |
|
453 |
(if t = \<Lambda> then |
|
454 |
k = 1 \<and> (u = \<Lambda> \<or> (level u = 1 \<and> left u = \<Lambda> \<and> right u = \<Lambda>)) |
|
455 |
else |
|
456 |
wf t \<and> wf u \<and> u \<noteq> \<Lambda> \<and> level t < k \<and> level u \<le> k \<and> level (right u) < k)" |
|
457 |
||
458 |
fun skew where |
|
459 |
"skew \<Lambda> = \<Lambda>" | |
|
460 |
"skew (N x k t u) = |
|
461 |
(if t \<noteq> \<Lambda> \<and> k = level t then |
|
462 |
N (data t) k (left t) (N x k (right t) u) |
|
463 |
else |
|
464 |
N x k t u)" |
|
465 |
||
466 |
fun split where |
|
467 |
"split \<Lambda> = \<Lambda>" | |
|
468 |
"split (N x k t u) = |
|
469 |
(if u \<noteq> \<Lambda> \<and> k = level (right u) then |
|
470 |
N (data u) (Suc k) (N x k t (left u)) (right u) |
|
471 |
else |
|
472 |
N x k t u)" |
|
473 |
||
474 |
theorem dataset_skew_split: |
|
475 |
"dataset (skew t) = dataset t" |
|
476 |
"dataset (split t) = dataset t" |
|
42959 | 477 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 478 |
sorry |
479 |
||
480 |
theorem wf_skew_split: |
|
481 |
"wf t \<Longrightarrow> skew t = t" |
|
482 |
"wf t \<Longrightarrow> split t = t" |
|
42959 | 483 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 484 |
sorry |
485 |
||
486 |
primrec insort\<^isub>1 where |
|
487 |
"insort\<^isub>1 \<Lambda> x = N x 1 \<Lambda> \<Lambda>" | |
|
488 |
"insort\<^isub>1 (N y k t u) x = |
|
489 |
(* (split \<circ> skew) *) (N y k (if x < y then insort\<^isub>1 t x else t) |
|
490 |
(if x > y then insort\<^isub>1 u x else u))" |
|
491 |
||
492 |
theorem wf_insort\<^isub>1: "wf t \<Longrightarrow> wf (insort\<^isub>1 t x)" |
|
35671
ed2c3830d881
improved Nitpick's precision for "card" and "setsum" + fix incorrect outcome code w.r.t. "bisim_depth = -1"
blanchet
parents:
35665
diff
changeset
|
493 |
nitpick [expect = genuine] |
33197 | 494 |
oops |
495 |
||
496 |
theorem wf_insort\<^isub>1_nat: "wf t \<Longrightarrow> wf (insort\<^isub>1 t (x\<Colon>nat))" |
|
35671
ed2c3830d881
improved Nitpick's precision for "card" and "setsum" + fix incorrect outcome code w.r.t. "bisim_depth = -1"
blanchet
parents:
35665
diff
changeset
|
497 |
nitpick [eval = "insort\<^isub>1 t x", expect = genuine] |
33197 | 498 |
oops |
499 |
||
500 |
primrec insort\<^isub>2 where |
|
501 |
"insort\<^isub>2 \<Lambda> x = N x 1 \<Lambda> \<Lambda>" | |
|
502 |
"insort\<^isub>2 (N y k t u) x = |
|
503 |
(split \<circ> skew) (N y k (if x < y then insort\<^isub>2 t x else t) |
|
504 |
(if x > y then insort\<^isub>2 u x else u))" |
|
505 |
||
506 |
theorem wf_insort\<^isub>2: "wf t \<Longrightarrow> wf (insort\<^isub>2 t x)" |
|
42959 | 507 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 508 |
sorry |
509 |
||
510 |
theorem dataset_insort\<^isub>2: "dataset (insort\<^isub>2 t x) = {x} \<union> dataset t" |
|
42959 | 511 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 512 |
sorry |
513 |
||
514 |
end |