author | haftmann |
Fri, 11 Jun 2010 17:14:02 +0200 | |
changeset 37407 | 61dd8c145da7 |
parent 36268 | 65aabc2c89ae |
child 37477 | e482320bcbfe |
permissions | -rw-r--r-- |
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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] |
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subsection {* 3.1. 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 {* 3.2. Type Variables *} |
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lemma "P x \<Longrightarrow> P (THE y. P y)" |
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nitpick [verbose, expect = genuine] |
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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, expect = genuine] |
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nitpick [full_descrs, show_consts, expect = genuine] |
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nitpick [dont_specialize, full_descrs, show_consts, expect = genuine] |
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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 [expect = none] |
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nitpick [card 'a = 1\<midarrow>50, expect = none] |
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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 [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 {* 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 [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, 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 {* 3.6. 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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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, 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 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, expect = genuine] |
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oops |
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ML {* |
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(* Proof.context -> string -> (typ -> term list) -> typ -> term -> term *) |
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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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setup {* Nitpick.register_term_postprocessor @{typ my_int} my_int_postproc *} |
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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 {* 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 = 50, unary_ints, verbose, expect = potential] |
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oops |
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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, verbose, 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 = {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 = 10, show_consts, expect = genuine] |
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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 |
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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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axiomatization iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a llist" |
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204 |
|
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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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208 |
|
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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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212 |
*} |
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213 |
|
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lemma "xs \<noteq> LCons a xs" |
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215 |
nitpick [expect = genuine] |
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217 |
||
218 |
lemma "\<lbrakk>xs = LCons a xs; ys = iterates (\<lambda>b. a) b\<rbrakk> \<Longrightarrow> xs = ys" |
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219 |
nitpick [verbose, expect = genuine] |
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221 |
||
222 |
lemma "\<lbrakk>xs = LCons a xs; ys = LCons a ys\<rbrakk> \<Longrightarrow> xs = ys" |
|
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nitpick [bisim_depth = -1, show_datatypes, expect = quasi_genuine] |
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224 |
nitpick [expect = none] |
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sorry |
226 |
||
227 |
subsection {* 3.10. Boxing *} |
|
228 |
||
229 |
datatype tm = Var nat | Lam tm | App tm tm |
|
230 |
||
231 |
primrec lift where |
|
232 |
"lift (Var j) k = Var (if j < k then j else j + 1)" | |
|
233 |
"lift (Lam t) k = Lam (lift t (k + 1))" | |
|
234 |
"lift (App t u) k = App (lift t k) (lift u k)" |
|
235 |
||
236 |
primrec loose where |
|
237 |
"loose (Var j) k = (j \<ge> k)" | |
|
238 |
"loose (Lam t) k = loose t (Suc k)" | |
|
239 |
"loose (App t u) k = (loose t k \<or> loose u k)" |
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240 |
||
241 |
primrec subst\<^isub>1 where |
|
242 |
"subst\<^isub>1 \<sigma> (Var j) = \<sigma> j" | |
|
243 |
"subst\<^isub>1 \<sigma> (Lam t) = |
|
244 |
Lam (subst\<^isub>1 (\<lambda>n. case n of 0 \<Rightarrow> Var 0 | Suc m \<Rightarrow> lift (\<sigma> m) 1) t)" | |
|
245 |
"subst\<^isub>1 \<sigma> (App t u) = App (subst\<^isub>1 \<sigma> t) (subst\<^isub>1 \<sigma> u)" |
|
246 |
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247 |
lemma "\<not> loose t 0 \<Longrightarrow> subst\<^isub>1 \<sigma> t = t" |
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nitpick [verbose, expect = genuine] |
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249 |
nitpick [eval = "subst\<^isub>1 \<sigma> t", expect = genuine] |
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250 |
(* nitpick [dont_box, expect = unknown] *) |
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252 |
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253 |
primrec subst\<^isub>2 where |
|
254 |
"subst\<^isub>2 \<sigma> (Var j) = \<sigma> j" | |
|
255 |
"subst\<^isub>2 \<sigma> (Lam t) = |
|
256 |
Lam (subst\<^isub>2 (\<lambda>n. case n of 0 \<Rightarrow> Var 0 | Suc m \<Rightarrow> lift (\<sigma> m) 0) t)" | |
|
257 |
"subst\<^isub>2 \<sigma> (App t u) = App (subst\<^isub>2 \<sigma> t) (subst\<^isub>2 \<sigma> u)" |
|
258 |
||
259 |
lemma "\<not> loose t 0 \<Longrightarrow> subst\<^isub>2 \<sigma> t = t" |
|
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nitpick [card = 1\<midarrow>5, expect = none] |
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262 |
||
263 |
subsection {* 3.11. Scope Monotonicity *} |
|
264 |
||
265 |
lemma "length xs = length ys \<Longrightarrow> rev (zip xs ys) = zip xs (rev ys)" |
|
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nitpick [verbose, expect = genuine] |
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267 |
nitpick [card = 8, verbose, expect = genuine] |
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269 |
||
270 |
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, expect = none] |
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272 |
nitpick [expect = genuine] |
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274 |
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275 |
subsection {* 3.12. Inductive Properties *} |
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276 |
|
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277 |
inductive_set reach where |
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"(4\<Colon>nat) \<in> reach" | |
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279 |
"n \<in> reach \<Longrightarrow> n < 4 \<Longrightarrow> 3 * n + 1 \<in> reach" | |
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280 |
"n \<in> reach \<Longrightarrow> n + 2 \<in> reach" |
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281 |
|
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282 |
lemma "n \<in> reach \<Longrightarrow> 2 dvd n" |
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nitpick [card = 1\<midarrow>5, bits = 1\<midarrow>5, expect = none] |
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284 |
apply (induct set: reach) |
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285 |
apply auto |
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nitpick [card = 1\<midarrow>5, bits = 1\<midarrow>5, expect = none] |
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287 |
apply (thin_tac "n \<in> reach") |
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288 |
nitpick [expect = genuine] |
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|
289 |
oops |
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|
290 |
|
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291 |
lemma "n \<in> reach \<Longrightarrow> 2 dvd n \<and> n \<noteq> 0" |
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nitpick [card = 1\<midarrow>5, bits = 1\<midarrow>5, expect = none] |
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293 |
apply (induct set: reach) |
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294 |
apply auto |
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nitpick [card = 1\<midarrow>5, bits = 1\<midarrow>5, expect = none] |
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296 |
apply (thin_tac "n \<in> reach") |
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|
297 |
nitpick [expect = genuine] |
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|
298 |
oops |
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|
299 |
|
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300 |
lemma "n \<in> reach \<Longrightarrow> 2 dvd n \<and> n \<ge> 4" |
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301 |
by (induct set: reach) arith+ |
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|
302 |
|
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303 |
datatype 'a bin_tree = Leaf 'a | Branch "'a bin_tree" "'a bin_tree" |
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|
304 |
|
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|
305 |
primrec labels where |
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|
306 |
"labels (Leaf a) = {a}" | |
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307 |
"labels (Branch t u) = labels t \<union> labels u" |
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|
308 |
|
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|
309 |
primrec swap where |
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|
310 |
"swap (Leaf c) a b = |
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|
311 |
(if c = a then Leaf b else if c = b then Leaf a else Leaf c)" | |
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|
312 |
"swap (Branch t u) a b = Branch (swap t a b) (swap u a b)" |
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|
313 |
|
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|
314 |
lemma "{a, b} \<subseteq> labels t \<Longrightarrow> labels (swap t a b) = labels t" |
35671
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|
315 |
(* nitpick *) |
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|
316 |
proof (induct t) |
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|
317 |
case Leaf thus ?case by simp |
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|
318 |
next |
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319 |
case (Branch t u) thus ?case |
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320 |
(* nitpick *) |
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|
321 |
nitpick [non_std, show_all, expect = genuine] |
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|
322 |
oops |
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|
323 |
|
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324 |
lemma "labels (swap t a b) = |
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325 |
(if a \<in> labels t then |
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326 |
if b \<in> labels t then labels t else (labels t - {a}) \<union> {b} |
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|
327 |
else |
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|
328 |
if b \<in> labels t then (labels t - {b}) \<union> {a} else labels t)" |
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|
329 |
(* nitpick *) |
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330 |
proof (induct t) |
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331 |
case Leaf thus ?case by simp |
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|
332 |
next |
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|
333 |
case (Branch t u) thus ?case |
35710 | 334 |
nitpick [non_std, card = 1\<midarrow>5, expect = none] |
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|
335 |
by auto |
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|
336 |
qed |
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|
337 |
|
33197 | 338 |
section {* 4. Case Studies *} |
339 |
||
36268 | 340 |
nitpick_params [max_potential = 0] |
33197 | 341 |
|
342 |
subsection {* 4.1. A Context-Free Grammar *} |
|
343 |
||
344 |
datatype alphabet = a | b |
|
345 |
||
346 |
inductive_set S\<^isub>1 and A\<^isub>1 and B\<^isub>1 where |
|
347 |
"[] \<in> S\<^isub>1" |
|
348 |
| "w \<in> A\<^isub>1 \<Longrightarrow> b # w \<in> S\<^isub>1" |
|
349 |
| "w \<in> B\<^isub>1 \<Longrightarrow> a # w \<in> S\<^isub>1" |
|
350 |
| "w \<in> S\<^isub>1 \<Longrightarrow> a # w \<in> A\<^isub>1" |
|
351 |
| "w \<in> S\<^isub>1 \<Longrightarrow> b # w \<in> S\<^isub>1" |
|
352 |
| "\<lbrakk>v \<in> B\<^isub>1; v \<in> B\<^isub>1\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>1" |
|
353 |
||
354 |
theorem S\<^isub>1_sound: |
|
355 |
"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
|
356 |
nitpick [expect = genuine] |
33197 | 357 |
oops |
358 |
||
359 |
inductive_set S\<^isub>2 and A\<^isub>2 and B\<^isub>2 where |
|
360 |
"[] \<in> S\<^isub>2" |
|
361 |
| "w \<in> A\<^isub>2 \<Longrightarrow> b # w \<in> S\<^isub>2" |
|
362 |
| "w \<in> B\<^isub>2 \<Longrightarrow> a # w \<in> S\<^isub>2" |
|
363 |
| "w \<in> S\<^isub>2 \<Longrightarrow> a # w \<in> A\<^isub>2" |
|
364 |
| "w \<in> S\<^isub>2 \<Longrightarrow> b # w \<in> B\<^isub>2" |
|
365 |
| "\<lbrakk>v \<in> B\<^isub>2; v \<in> B\<^isub>2\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>2" |
|
366 |
||
367 |
theorem S\<^isub>2_sound: |
|
368 |
"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
|
369 |
nitpick [expect = genuine] |
33197 | 370 |
oops |
371 |
||
372 |
inductive_set S\<^isub>3 and A\<^isub>3 and B\<^isub>3 where |
|
373 |
"[] \<in> S\<^isub>3" |
|
374 |
| "w \<in> A\<^isub>3 \<Longrightarrow> b # w \<in> S\<^isub>3" |
|
375 |
| "w \<in> B\<^isub>3 \<Longrightarrow> a # w \<in> S\<^isub>3" |
|
376 |
| "w \<in> S\<^isub>3 \<Longrightarrow> a # w \<in> A\<^isub>3" |
|
377 |
| "w \<in> S\<^isub>3 \<Longrightarrow> b # w \<in> B\<^isub>3" |
|
378 |
| "\<lbrakk>v \<in> B\<^isub>3; w \<in> B\<^isub>3\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>3" |
|
379 |
||
380 |
theorem S\<^isub>3_sound: |
|
381 |
"w \<in> S\<^isub>3 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]" |
|
35710 | 382 |
nitpick [card = 1\<midarrow>6, expect = none] |
33197 | 383 |
sorry |
384 |
||
385 |
theorem S\<^isub>3_complete: |
|
386 |
"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
|
387 |
nitpick [expect = genuine] |
33197 | 388 |
oops |
389 |
||
390 |
inductive_set S\<^isub>4 and A\<^isub>4 and B\<^isub>4 where |
|
391 |
"[] \<in> S\<^isub>4" |
|
392 |
| "w \<in> A\<^isub>4 \<Longrightarrow> b # w \<in> S\<^isub>4" |
|
393 |
| "w \<in> B\<^isub>4 \<Longrightarrow> a # w \<in> S\<^isub>4" |
|
394 |
| "w \<in> S\<^isub>4 \<Longrightarrow> a # w \<in> A\<^isub>4" |
|
395 |
| "\<lbrakk>v \<in> A\<^isub>4; w \<in> A\<^isub>4\<rbrakk> \<Longrightarrow> b # v @ w \<in> A\<^isub>4" |
|
396 |
| "w \<in> S\<^isub>4 \<Longrightarrow> b # w \<in> B\<^isub>4" |
|
397 |
| "\<lbrakk>v \<in> B\<^isub>4; w \<in> B\<^isub>4\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>4" |
|
398 |
||
399 |
theorem S\<^isub>4_sound: |
|
400 |
"w \<in> S\<^isub>4 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]" |
|
35710 | 401 |
nitpick [card = 1\<midarrow>6, expect = none] |
33197 | 402 |
sorry |
403 |
||
404 |
theorem S\<^isub>4_complete: |
|
405 |
"length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] \<longrightarrow> w \<in> S\<^isub>4" |
|
35710 | 406 |
nitpick [card = 1\<midarrow>6, expect = none] |
33197 | 407 |
sorry |
408 |
||
409 |
theorem S\<^isub>4_A\<^isub>4_B\<^isub>4_sound_and_complete: |
|
410 |
"w \<in> S\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]" |
|
411 |
"w \<in> A\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] + 1" |
|
412 |
"w \<in> B\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = b] = length [x \<leftarrow> w. x = a] + 1" |
|
35710 | 413 |
nitpick [card = 1\<midarrow>6, expect = none] |
33197 | 414 |
sorry |
415 |
||
416 |
subsection {* 4.2. AA Trees *} |
|
417 |
||
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
|
418 |
datatype 'a aa_tree = \<Lambda> | N "'a\<Colon>linorder" nat "'a aa_tree" "'a aa_tree" |
33197 | 419 |
|
420 |
primrec data where |
|
421 |
"data \<Lambda> = undefined" | |
|
422 |
"data (N x _ _ _) = x" |
|
423 |
||
424 |
primrec dataset where |
|
425 |
"dataset \<Lambda> = {}" | |
|
426 |
"dataset (N x _ t u) = {x} \<union> dataset t \<union> dataset u" |
|
427 |
||
428 |
primrec level where |
|
429 |
"level \<Lambda> = 0" | |
|
430 |
"level (N _ k _ _) = k" |
|
431 |
||
432 |
primrec left where |
|
433 |
"left \<Lambda> = \<Lambda>" | |
|
434 |
"left (N _ _ t\<^isub>1 _) = t\<^isub>1" |
|
435 |
||
436 |
primrec right where |
|
437 |
"right \<Lambda> = \<Lambda>" | |
|
438 |
"right (N _ _ _ t\<^isub>2) = t\<^isub>2" |
|
439 |
||
440 |
fun wf where |
|
441 |
"wf \<Lambda> = True" | |
|
442 |
"wf (N _ k t u) = |
|
443 |
(if t = \<Lambda> then |
|
444 |
k = 1 \<and> (u = \<Lambda> \<or> (level u = 1 \<and> left u = \<Lambda> \<and> right u = \<Lambda>)) |
|
445 |
else |
|
446 |
wf t \<and> wf u \<and> u \<noteq> \<Lambda> \<and> level t < k \<and> level u \<le> k \<and> level (right u) < k)" |
|
447 |
||
448 |
fun skew where |
|
449 |
"skew \<Lambda> = \<Lambda>" | |
|
450 |
"skew (N x k t u) = |
|
451 |
(if t \<noteq> \<Lambda> \<and> k = level t then |
|
452 |
N (data t) k (left t) (N x k (right t) u) |
|
453 |
else |
|
454 |
N x k t u)" |
|
455 |
||
456 |
fun split where |
|
457 |
"split \<Lambda> = \<Lambda>" | |
|
458 |
"split (N x k t u) = |
|
459 |
(if u \<noteq> \<Lambda> \<and> k = level (right u) then |
|
460 |
N (data u) (Suc k) (N x k t (left u)) (right u) |
|
461 |
else |
|
462 |
N x k t u)" |
|
463 |
||
464 |
theorem dataset_skew_split: |
|
465 |
"dataset (skew t) = dataset t" |
|
466 |
"dataset (split t) = dataset t" |
|
35747
c910fe606829
make tests less demanding, to prevent sporadic failures
blanchet
parents:
35718
diff
changeset
|
467 |
nitpick [card = 1\<midarrow>5, expect = none] |
33197 | 468 |
sorry |
469 |
||
470 |
theorem wf_skew_split: |
|
471 |
"wf t \<Longrightarrow> skew t = t" |
|
472 |
"wf t \<Longrightarrow> split t = t" |
|
35747
c910fe606829
make tests less demanding, to prevent sporadic failures
blanchet
parents:
35718
diff
changeset
|
473 |
nitpick [card = 1\<midarrow>5, expect = none] |
33197 | 474 |
sorry |
475 |
||
476 |
primrec insort\<^isub>1 where |
|
477 |
"insort\<^isub>1 \<Lambda> x = N x 1 \<Lambda> \<Lambda>" | |
|
478 |
"insort\<^isub>1 (N y k t u) x = |
|
479 |
(* (split \<circ> skew) *) (N y k (if x < y then insort\<^isub>1 t x else t) |
|
480 |
(if x > y then insort\<^isub>1 u x else u))" |
|
481 |
||
482 |
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
|
483 |
nitpick [expect = genuine] |
33197 | 484 |
oops |
485 |
||
486 |
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
|
487 |
nitpick [eval = "insort\<^isub>1 t x", expect = genuine] |
33197 | 488 |
oops |
489 |
||
490 |
primrec insort\<^isub>2 where |
|
491 |
"insort\<^isub>2 \<Lambda> x = N x 1 \<Lambda> \<Lambda>" | |
|
492 |
"insort\<^isub>2 (N y k t u) x = |
|
493 |
(split \<circ> skew) (N y k (if x < y then insort\<^isub>2 t x else t) |
|
494 |
(if x > y then insort\<^isub>2 u x else u))" |
|
495 |
||
496 |
theorem wf_insort\<^isub>2: "wf t \<Longrightarrow> wf (insort\<^isub>2 t x)" |
|
35747
c910fe606829
make tests less demanding, to prevent sporadic failures
blanchet
parents:
35718
diff
changeset
|
497 |
nitpick [card = 1\<midarrow>5, expect = none] |
33197 | 498 |
sorry |
499 |
||
500 |
theorem dataset_insort\<^isub>2: "dataset (insort\<^isub>2 t x) = {x} \<union> dataset t" |
|
35747
c910fe606829
make tests less demanding, to prevent sporadic failures
blanchet
parents:
35718
diff
changeset
|
501 |
nitpick [card = 1\<midarrow>5, expect = none] |
33197 | 502 |
sorry |
503 |
||
504 |
end |