author | blanchet |
Wed, 12 Feb 2014 08:35:56 +0100 | |
changeset 55413 | a8e96847523c |
parent 55395 | 4e79187f847e |
child 55415 | 05f5fdb8d093 |
permissions | -rw-r--r-- |
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(* Title: HOL/Nitpick_Examples/Refute_Nits.thy |
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Author: Jasmin Blanchette, TU Muenchen |
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Copyright 2009-2011 |
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Refute examples adapted to Nitpick. |
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*) |
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header {* Refute Examples Adapted to Nitpick *} |
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theory Refute_Nits |
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imports Main |
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begin |
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nitpick_params [verbose, card = 1\<emdash>6, max_potential = 0, |
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sat_solver = MiniSat_JNI, max_threads = 1, timeout = 240] |
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lemma "P \<and> Q" |
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apply (rule conjI) |
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nitpick [expect = genuine] 1 |
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nitpick [expect = genuine] 2 |
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nitpick [expect = genuine] |
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nitpick [card = 5, expect = genuine] |
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nitpick [sat_solver = SAT4J, expect = genuine] 2 |
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oops |
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subsection {* Examples and Test Cases *} |
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subsubsection {* Propositional logic *} |
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lemma "True" |
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nitpick [expect = none] |
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apply auto |
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done |
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lemma "False" |
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nitpick [expect = genuine] |
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oops |
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lemma "P" |
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nitpick [expect = genuine] |
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oops |
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lemma "\<not> P" |
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nitpick [expect = genuine] |
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oops |
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lemma "P \<and> Q" |
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nitpick [expect = genuine] |
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oops |
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lemma "P \<or> Q" |
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nitpick [expect = genuine] |
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oops |
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lemma "P \<longrightarrow> Q" |
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nitpick [expect = genuine] |
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oops |
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lemma "(P\<Colon>bool) = Q" |
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nitpick [expect = genuine] |
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oops |
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lemma "(P \<or> Q) \<longrightarrow> (P \<and> Q)" |
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nitpick [expect = genuine] |
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oops |
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subsubsection {* Predicate logic *} |
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lemma "P x y z" |
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nitpick [expect = genuine] |
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oops |
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lemma "P x y \<longrightarrow> P y x" |
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nitpick [expect = genuine] |
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oops |
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lemma "P (f (f x)) \<longrightarrow> P x \<longrightarrow> P (f x)" |
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nitpick [expect = genuine] |
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oops |
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subsubsection {* Equality *} |
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lemma "P = True" |
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nitpick [expect = genuine] |
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oops |
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lemma "P = False" |
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nitpick [expect = genuine] |
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oops |
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lemma "x = y" |
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nitpick [expect = genuine] |
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oops |
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lemma "f x = g x" |
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nitpick [expect = genuine] |
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oops |
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lemma "(f\<Colon>'a\<Rightarrow>'b) = g" |
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nitpick [expect = genuine] |
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oops |
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lemma "(f\<Colon>('d\<Rightarrow>'d)\<Rightarrow>('c\<Rightarrow>'d)) = g" |
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nitpick [expect = genuine] |
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oops |
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lemma "distinct [a, b]" |
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nitpick [expect = genuine] |
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apply simp |
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nitpick [expect = genuine] |
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oops |
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subsubsection {* First-Order Logic *} |
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lemma "\<exists>x. P x" |
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nitpick [expect = genuine] |
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oops |
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lemma "\<forall>x. P x" |
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nitpick [expect = genuine] |
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oops |
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lemma "\<exists>!x. P x" |
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nitpick [expect = genuine] |
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oops |
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lemma "Ex P" |
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nitpick [expect = genuine] |
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oops |
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lemma "All P" |
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nitpick [expect = genuine] |
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oops |
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lemma "Ex1 P" |
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nitpick [expect = genuine] |
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oops |
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lemma "(\<exists>x. P x) \<longrightarrow> (\<forall>x. P x)" |
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nitpick [expect = genuine] |
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oops |
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lemma "(\<forall>x. \<exists>y. P x y) \<longrightarrow> (\<exists>y. \<forall>x. P x y)" |
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nitpick [expect = genuine] |
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oops |
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lemma "(\<exists>x. P x) \<longrightarrow> (\<exists>!x. P x)" |
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nitpick [expect = genuine] |
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oops |
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text {* A true statement (also testing names of free and bound variables being identical) *} |
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lemma "(\<forall>x y. P x y \<longrightarrow> P y x) \<longrightarrow> (\<forall>x. P x y) \<longrightarrow> P y x" |
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nitpick [expect = none] |
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apply fast |
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done |
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text {* "A type has at most 4 elements." *} |
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lemma "\<not> distinct [a, b, c, d, e]" |
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nitpick [expect = genuine] |
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apply simp |
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nitpick [expect = genuine] |
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oops |
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lemma "distinct [a, b, c, d]" |
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nitpick [expect = genuine] |
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apply simp |
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nitpick [expect = genuine] |
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oops |
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text {* "Every reflexive and symmetric relation is transitive." *} |
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lemma "\<lbrakk>\<forall>x. P x x; \<forall>x y. P x y \<longrightarrow> P y x\<rbrakk> \<Longrightarrow> P x y \<longrightarrow> P y z \<longrightarrow> P x z" |
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nitpick [expect = genuine] |
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oops |
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text {* The ``Drinker's theorem'' *} |
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lemma "\<exists>x. f x = g x \<longrightarrow> f = g" |
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nitpick [expect = none] |
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apply (auto simp add: ext) |
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done |
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text {* And an incorrect version of it *} |
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lemma "(\<exists>x. f x = g x) \<longrightarrow> f = g" |
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nitpick [expect = genuine] |
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oops |
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text {* "Every function has a fixed point." *} |
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lemma "\<exists>x. f x = x" |
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nitpick [expect = genuine] |
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oops |
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text {* "Function composition is commutative." *} |
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lemma "f (g x) = g (f x)" |
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nitpick [expect = genuine] |
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oops |
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text {* "Two functions that are equivalent wrt.\ the same predicate 'P' are equal." *} |
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lemma "((P\<Colon>('a\<Rightarrow>'b)\<Rightarrow>bool) f = P g) \<longrightarrow> (f x = g x)" |
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nitpick [expect = genuine] |
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oops |
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subsubsection {* Higher-Order Logic *} |
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lemma "\<exists>P. P" |
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nitpick [expect = none] |
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apply auto |
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done |
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lemma "\<forall>P. P" |
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nitpick [expect = genuine] |
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oops |
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lemma "\<exists>!P. P" |
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nitpick [expect = none] |
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apply auto |
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done |
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lemma "\<exists>!P. P x" |
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nitpick [expect = genuine] |
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oops |
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lemma "P Q \<or> Q x" |
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nitpick [expect = genuine] |
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oops |
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lemma "x \<noteq> All" |
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nitpick [expect = genuine] |
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oops |
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lemma "x \<noteq> Ex" |
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nitpick [expect = genuine] |
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oops |
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lemma "x \<noteq> Ex1" |
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nitpick [expect = genuine] |
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oops |
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text {* ``The transitive closure of an arbitrary relation is non-empty.'' *} |
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definition "trans" :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where |
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"trans P \<equiv> (ALL x y z. P x y \<longrightarrow> P y z \<longrightarrow> P x z)" |
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definition |
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"subset" :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where |
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"subset P Q \<equiv> (ALL x y. P x y \<longrightarrow> Q x y)" |
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definition |
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"trans_closure" :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where |
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"trans_closure P Q \<equiv> (subset Q P) \<and> (trans P) \<and> (ALL R. subset Q R \<longrightarrow> trans R \<longrightarrow> subset P R)" |
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lemma "trans_closure T P \<longrightarrow> (\<exists>x y. T x y)" |
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nitpick [expect = genuine] |
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oops |
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text {* ``The union of transitive closures is equal to the transitive closure of unions.'' *} |
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lemma "(\<forall>x y. (P x y \<or> R x y) \<longrightarrow> T x y) \<longrightarrow> trans T \<longrightarrow> (\<forall>Q. (\<forall>x y. (P x y \<or> R x y) \<longrightarrow> Q x y) \<longrightarrow> trans Q \<longrightarrow> subset T Q) |
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\<longrightarrow> trans_closure TP P |
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\<longrightarrow> trans_closure TR R |
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\<longrightarrow> (T x y = (TP x y \<or> TR x y))" |
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nitpick [expect = genuine] |
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oops |
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text {* ``Every surjective function is invertible.'' *} |
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lemma "(\<forall>y. \<exists>x. y = f x) \<longrightarrow> (\<exists>g. \<forall>x. g (f x) = x)" |
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nitpick [expect = genuine] |
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oops |
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text {* ``Every invertible function is surjective.'' *} |
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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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text {* ``Every point is a fixed point of some function.'' *} |
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lemma "\<exists>f. f x = x" |
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nitpick [card = 1\<emdash>7, expect = none] |
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apply (rule_tac x = "\<lambda>x. x" in exI) |
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apply simp |
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done |
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text {* Axiom of Choice: first an incorrect version *} |
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lemma "(\<forall>x. \<exists>y. P x y) \<longrightarrow> (\<exists>!f. \<forall>x. P x (f x))" |
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nitpick [expect = genuine] |
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oops |
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text {* And now two correct ones *} |
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lemma "(\<forall>x. \<exists>y. P x y) \<longrightarrow> (\<exists>f. \<forall>x. P x (f x))" |
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nitpick [card = 1-4, expect = none] |
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apply (simp add: choice) |
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done |
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lemma "(\<forall>x. \<exists>!y. P x y) \<longrightarrow> (\<exists>!f. \<forall>x. P x (f x))" |
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nitpick [card = 1-3, expect = none] |
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apply auto |
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apply (simp add: ex1_implies_ex choice) |
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apply (fast intro: ext) |
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done |
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subsubsection {* Metalogic *} |
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lemma "\<And>x. P x" |
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nitpick [expect = genuine] |
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oops |
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lemma "f x \<equiv> g x" |
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nitpick [expect = genuine] |
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oops |
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lemma "P \<Longrightarrow> Q" |
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nitpick [expect = genuine] |
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oops |
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lemma "\<lbrakk>P; Q; R\<rbrakk> \<Longrightarrow> S" |
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nitpick [expect = genuine] |
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oops |
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lemma "(x \<equiv> all) \<Longrightarrow> False" |
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nitpick [expect = genuine] |
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oops |
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lemma "(x \<equiv> (op \<equiv>)) \<Longrightarrow> False" |
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nitpick [expect = genuine] |
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oops |
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lemma "(x \<equiv> (op \<Longrightarrow>)) \<Longrightarrow> False" |
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nitpick [expect = genuine] |
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oops |
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subsubsection {* Schematic Variables *} |
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schematic_lemma "?P" |
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nitpick [expect = none] |
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apply auto |
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done |
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schematic_lemma "x = ?y" |
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nitpick [expect = none] |
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apply auto |
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done |
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subsubsection {* Abstractions *} |
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lemma "(\<lambda>x. x) = (\<lambda>x. y)" |
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nitpick [expect = genuine] |
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oops |
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lemma "(\<lambda>f. f x) = (\<lambda>f. True)" |
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nitpick [expect = genuine] |
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oops |
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lemma "(\<lambda>x. x) = (\<lambda>y. y)" |
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nitpick [expect = none] |
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apply simp |
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done |
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subsubsection {* Sets *} |
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lemma "P (A\<Colon>'a set)" |
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nitpick [expect = genuine] |
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oops |
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lemma "P (A\<Colon>'a set set)" |
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nitpick [expect = genuine] |
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oops |
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lemma "{x. P x} = {y. P y}" |
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nitpick [expect = none] |
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apply simp |
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done |
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lemma "x \<in> {x. P x}" |
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nitpick [expect = genuine] |
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385 |
oops |
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387 |
lemma "P (op \<in>)" |
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nitpick [expect = genuine] |
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389 |
oops |
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lemma "P (op \<in> x)" |
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nitpick [expect = genuine] |
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393 |
oops |
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395 |
lemma "P Collect" |
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396 |
nitpick [expect = genuine] |
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397 |
oops |
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398 |
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399 |
lemma "A Un B = A Int B" |
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nitpick [expect = genuine] |
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401 |
oops |
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403 |
lemma "(A Int B) Un C = (A Un C) Int B" |
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nitpick [expect = genuine] |
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oops |
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407 |
lemma "Ball A P \<longrightarrow> Bex A P" |
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408 |
nitpick [expect = genuine] |
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409 |
oops |
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411 |
subsubsection {* @{const undefined} *} |
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lemma "undefined" |
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nitpick [expect = genuine] |
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415 |
oops |
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417 |
lemma "P undefined" |
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418 |
nitpick [expect = genuine] |
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oops |
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lemma "undefined x" |
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422 |
nitpick [expect = genuine] |
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423 |
oops |
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425 |
lemma "undefined undefined" |
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426 |
nitpick [expect = genuine] |
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427 |
oops |
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subsubsection {* @{const The} *} |
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lemma "The P" |
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nitpick [expect = genuine] |
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oops |
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lemma "P The" |
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nitpick [expect = genuine] |
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oops |
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lemma "P (The P)" |
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440 |
nitpick [expect = genuine] |
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441 |
oops |
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443 |
lemma "(THE x. x=y) = z" |
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nitpick [expect = genuine] |
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445 |
oops |
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lemma "Ex P \<longrightarrow> P (The P)" |
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448 |
nitpick [expect = genuine] |
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449 |
oops |
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subsubsection {* @{const Eps} *} |
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lemma "Eps P" |
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nitpick [expect = genuine] |
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oops |
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lemma "P Eps" |
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458 |
nitpick [expect = genuine] |
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oops |
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lemma "P (Eps P)" |
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nitpick [expect = genuine] |
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oops |
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464 |
||
465 |
lemma "(SOME x. x=y) = z" |
|
466 |
nitpick [expect = genuine] |
|
467 |
oops |
|
468 |
||
469 |
lemma "Ex P \<longrightarrow> P (Eps P)" |
|
470 |
nitpick [expect = none] |
|
471 |
apply (auto simp add: someI) |
|
472 |
done |
|
473 |
||
474 |
subsubsection {* Operations on Natural Numbers *} |
|
475 |
||
476 |
lemma "(x\<Colon>nat) + y = 0" |
|
477 |
nitpick [expect = genuine] |
|
478 |
oops |
|
479 |
||
480 |
lemma "(x\<Colon>nat) = x + x" |
|
481 |
nitpick [expect = genuine] |
|
482 |
oops |
|
483 |
||
484 |
lemma "(x\<Colon>nat) - y + y = x" |
|
485 |
nitpick [expect = genuine] |
|
486 |
oops |
|
487 |
||
488 |
lemma "(x\<Colon>nat) = x * x" |
|
489 |
nitpick [expect = genuine] |
|
490 |
oops |
|
491 |
||
492 |
lemma "(x\<Colon>nat) < x + y" |
|
493 |
nitpick [card = 1, expect = genuine] |
|
494 |
oops |
|
495 |
||
496 |
text {* \<times> *} |
|
497 |
||
498 |
lemma "P (x\<Colon>'a\<times>'b)" |
|
499 |
nitpick [expect = genuine] |
|
500 |
oops |
|
501 |
||
502 |
lemma "\<forall>x\<Colon>'a\<times>'b. P x" |
|
503 |
nitpick [expect = genuine] |
|
504 |
oops |
|
505 |
||
506 |
lemma "P (x, y)" |
|
507 |
nitpick [expect = genuine] |
|
508 |
oops |
|
509 |
||
510 |
lemma "P (fst x)" |
|
511 |
nitpick [expect = genuine] |
|
512 |
oops |
|
513 |
||
514 |
lemma "P (snd x)" |
|
515 |
nitpick [expect = genuine] |
|
516 |
oops |
|
517 |
||
518 |
lemma "P Pair" |
|
519 |
nitpick [expect = genuine] |
|
520 |
oops |
|
521 |
||
522 |
lemma "P (case x of Pair a b \<Rightarrow> f a b)" |
|
523 |
nitpick [expect = genuine] |
|
524 |
oops |
|
525 |
||
526 |
subsubsection {* Subtypes (typedef), typedecl *} |
|
527 |
||
528 |
text {* A completely unspecified non-empty subset of @{typ "'a"}: *} |
|
529 |
||
45694
4a8743618257
prefer typedef without extra definition and alternative name;
wenzelm
parents:
45035
diff
changeset
|
530 |
definition "myTdef = insert (undefined::'a) (undefined::'a set)" |
4a8743618257
prefer typedef without extra definition and alternative name;
wenzelm
parents:
45035
diff
changeset
|
531 |
|
49834 | 532 |
typedef 'a myTdef = "myTdef :: 'a set" |
45694
4a8743618257
prefer typedef without extra definition and alternative name;
wenzelm
parents:
45035
diff
changeset
|
533 |
unfolding myTdef_def by auto |
33197 | 534 |
|
535 |
lemma "(x\<Colon>'a myTdef) = y" |
|
536 |
nitpick [expect = genuine] |
|
537 |
oops |
|
538 |
||
539 |
typedecl myTdecl |
|
540 |
||
45694
4a8743618257
prefer typedef without extra definition and alternative name;
wenzelm
parents:
45035
diff
changeset
|
541 |
definition "T_bij = {(f::'a\<Rightarrow>'a). \<forall>y. \<exists>!x. f x = y}" |
4a8743618257
prefer typedef without extra definition and alternative name;
wenzelm
parents:
45035
diff
changeset
|
542 |
|
49834 | 543 |
typedef 'a T_bij = "T_bij :: ('a \<Rightarrow> 'a) set" |
45694
4a8743618257
prefer typedef without extra definition and alternative name;
wenzelm
parents:
45035
diff
changeset
|
544 |
unfolding T_bij_def by auto |
33197 | 545 |
|
546 |
lemma "P (f\<Colon>(myTdecl myTdef) T_bij)" |
|
547 |
nitpick [expect = genuine] |
|
548 |
oops |
|
549 |
||
550 |
subsubsection {* Inductive Datatypes *} |
|
551 |
||
552 |
text {* unit *} |
|
553 |
||
554 |
lemma "P (x\<Colon>unit)" |
|
555 |
nitpick [expect = genuine] |
|
556 |
oops |
|
557 |
||
558 |
lemma "\<forall>x\<Colon>unit. P x" |
|
559 |
nitpick [expect = genuine] |
|
560 |
oops |
|
561 |
||
562 |
lemma "P ()" |
|
563 |
nitpick [expect = genuine] |
|
564 |
oops |
|
565 |
||
566 |
lemma "P (case x of () \<Rightarrow> u)" |
|
567 |
nitpick [expect = genuine] |
|
568 |
oops |
|
569 |
||
570 |
text {* option *} |
|
571 |
||
572 |
lemma "P (x\<Colon>'a option)" |
|
573 |
nitpick [expect = genuine] |
|
574 |
oops |
|
575 |
||
576 |
lemma "\<forall>x\<Colon>'a option. P x" |
|
577 |
nitpick [expect = genuine] |
|
578 |
oops |
|
579 |
||
580 |
lemma "P None" |
|
581 |
nitpick [expect = genuine] |
|
582 |
oops |
|
583 |
||
584 |
lemma "P (Some x)" |
|
585 |
nitpick [expect = genuine] |
|
586 |
oops |
|
587 |
||
55413
a8e96847523c
adapted theories to '{case,rec}_{list,option}' names
blanchet
parents:
55395
diff
changeset
|
588 |
lemma "rec_option n s None = n" |
33197 | 589 |
nitpick [expect = none] |
590 |
apply simp |
|
591 |
done |
|
592 |
||
55413
a8e96847523c
adapted theories to '{case,rec}_{list,option}' names
blanchet
parents:
55395
diff
changeset
|
593 |
lemma "rec_option n s (Some x) = s x" |
33197 | 594 |
nitpick [expect = none] |
595 |
apply simp |
|
596 |
done |
|
597 |
||
55413
a8e96847523c
adapted theories to '{case,rec}_{list,option}' names
blanchet
parents:
55395
diff
changeset
|
598 |
lemma "P (rec_option n s x)" |
33197 | 599 |
nitpick [expect = genuine] |
600 |
oops |
|
601 |
||
602 |
lemma "P (case x of None \<Rightarrow> n | Some u \<Rightarrow> s u)" |
|
603 |
nitpick [expect = genuine] |
|
604 |
oops |
|
605 |
||
606 |
text {* + *} |
|
607 |
||
608 |
lemma "P (x\<Colon>'a+'b)" |
|
609 |
nitpick [expect = genuine] |
|
610 |
oops |
|
611 |
||
612 |
lemma "\<forall>x\<Colon>'a+'b. P x" |
|
613 |
nitpick [expect = genuine] |
|
614 |
oops |
|
615 |
||
616 |
lemma "P (Inl x)" |
|
617 |
nitpick [expect = genuine] |
|
618 |
oops |
|
619 |
||
620 |
lemma "P (Inr x)" |
|
621 |
nitpick [expect = genuine] |
|
622 |
oops |
|
623 |
||
624 |
lemma "P Inl" |
|
625 |
nitpick [expect = genuine] |
|
626 |
oops |
|
627 |
||
628 |
lemma "P (case x of Inl a \<Rightarrow> l a | Inr b \<Rightarrow> r b)" |
|
629 |
nitpick [expect = genuine] |
|
630 |
oops |
|
631 |
||
632 |
text {* Non-recursive datatypes *} |
|
633 |
||
634 |
datatype T1 = A | B |
|
635 |
||
636 |
lemma "P (x\<Colon>T1)" |
|
637 |
nitpick [expect = genuine] |
|
638 |
oops |
|
639 |
||
640 |
lemma "\<forall>x\<Colon>T1. P x" |
|
641 |
nitpick [expect = genuine] |
|
642 |
oops |
|
643 |
||
644 |
lemma "P A" |
|
645 |
nitpick [expect = genuine] |
|
646 |
oops |
|
647 |
||
648 |
lemma "P B" |
|
649 |
nitpick [expect = genuine] |
|
650 |
oops |
|
651 |
||
652 |
lemma "T1_rec a b A = a" |
|
653 |
nitpick [expect = none] |
|
654 |
apply simp |
|
655 |
done |
|
656 |
||
657 |
lemma "T1_rec a b B = b" |
|
658 |
nitpick [expect = none] |
|
659 |
apply simp |
|
660 |
done |
|
661 |
||
662 |
lemma "P (T1_rec a b x)" |
|
663 |
nitpick [expect = genuine] |
|
664 |
oops |
|
665 |
||
666 |
lemma "P (case x of A \<Rightarrow> a | B \<Rightarrow> b)" |
|
667 |
nitpick [expect = genuine] |
|
668 |
oops |
|
669 |
||
670 |
datatype 'a T2 = C T1 | D 'a |
|
671 |
||
672 |
lemma "P (x\<Colon>'a T2)" |
|
673 |
nitpick [expect = genuine] |
|
674 |
oops |
|
675 |
||
676 |
lemma "\<forall>x\<Colon>'a T2. P x" |
|
677 |
nitpick [expect = genuine] |
|
678 |
oops |
|
679 |
||
680 |
lemma "P D" |
|
681 |
nitpick [expect = genuine] |
|
682 |
oops |
|
683 |
||
684 |
lemma "T2_rec c d (C x) = c x" |
|
685 |
nitpick [expect = none] |
|
686 |
apply simp |
|
687 |
done |
|
688 |
||
689 |
lemma "T2_rec c d (D x) = d x" |
|
690 |
nitpick [expect = none] |
|
691 |
apply simp |
|
692 |
done |
|
693 |
||
694 |
lemma "P (T2_rec c d x)" |
|
695 |
nitpick [expect = genuine] |
|
696 |
oops |
|
697 |
||
698 |
lemma "P (case x of C u \<Rightarrow> c u | D v \<Rightarrow> d v)" |
|
699 |
nitpick [expect = genuine] |
|
700 |
oops |
|
701 |
||
702 |
datatype ('a, 'b) T3 = E "'a \<Rightarrow> 'b" |
|
703 |
||
704 |
lemma "P (x\<Colon>('a, 'b) T3)" |
|
705 |
nitpick [expect = genuine] |
|
706 |
oops |
|
707 |
||
708 |
lemma "\<forall>x\<Colon>('a, 'b) T3. P x" |
|
709 |
nitpick [expect = genuine] |
|
710 |
oops |
|
711 |
||
712 |
lemma "P E" |
|
713 |
nitpick [expect = genuine] |
|
714 |
oops |
|
715 |
||
716 |
lemma "T3_rec e (E x) = e x" |
|
42959 | 717 |
nitpick [card = 1\<emdash>4, expect = none] |
33197 | 718 |
apply simp |
719 |
done |
|
720 |
||
721 |
lemma "P (T3_rec e x)" |
|
722 |
nitpick [expect = genuine] |
|
723 |
oops |
|
724 |
||
725 |
lemma "P (case x of E f \<Rightarrow> e f)" |
|
726 |
nitpick [expect = genuine] |
|
727 |
oops |
|
728 |
||
729 |
text {* Recursive datatypes *} |
|
730 |
||
731 |
text {* nat *} |
|
732 |
||
733 |
lemma "P (x\<Colon>nat)" |
|
734 |
nitpick [expect = genuine] |
|
735 |
oops |
|
736 |
||
737 |
lemma "\<forall>x\<Colon>nat. P x" |
|
738 |
nitpick [expect = genuine] |
|
739 |
oops |
|
740 |
||
741 |
lemma "P (Suc 0)" |
|
742 |
nitpick [expect = genuine] |
|
743 |
oops |
|
744 |
||
745 |
lemma "P Suc" |
|
42959 | 746 |
nitpick [card = 1\<emdash>7, expect = none] |
33197 | 747 |
oops |
748 |
||
749 |
lemma "nat_rec zero suc 0 = zero" |
|
750 |
nitpick [expect = none] |
|
751 |
apply simp |
|
752 |
done |
|
753 |
||
754 |
lemma "nat_rec zero suc (Suc x) = suc x (nat_rec zero suc x)" |
|
755 |
nitpick [expect = none] |
|
756 |
apply simp |
|
757 |
done |
|
758 |
||
759 |
lemma "P (nat_rec zero suc x)" |
|
760 |
nitpick [expect = genuine] |
|
761 |
oops |
|
762 |
||
763 |
lemma "P (case x of 0 \<Rightarrow> zero | Suc n \<Rightarrow> suc n)" |
|
764 |
nitpick [expect = genuine] |
|
765 |
oops |
|
766 |
||
767 |
text {* 'a list *} |
|
768 |
||
769 |
lemma "P (xs\<Colon>'a list)" |
|
770 |
nitpick [expect = genuine] |
|
771 |
oops |
|
772 |
||
773 |
lemma "\<forall>xs\<Colon>'a list. P xs" |
|
774 |
nitpick [expect = genuine] |
|
775 |
oops |
|
776 |
||
777 |
lemma "P [x, y]" |
|
778 |
nitpick [expect = genuine] |
|
779 |
oops |
|
780 |
||
55413
a8e96847523c
adapted theories to '{case,rec}_{list,option}' names
blanchet
parents:
55395
diff
changeset
|
781 |
lemma "rec_list nil cons [] = nil" |
42959 | 782 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 783 |
apply simp |
784 |
done |
|
785 |
||
55413
a8e96847523c
adapted theories to '{case,rec}_{list,option}' names
blanchet
parents:
55395
diff
changeset
|
786 |
lemma "rec_list nil cons (x#xs) = cons x xs (rec_list nil cons xs)" |
42959 | 787 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 788 |
apply simp |
789 |
done |
|
790 |
||
55413
a8e96847523c
adapted theories to '{case,rec}_{list,option}' names
blanchet
parents:
55395
diff
changeset
|
791 |
lemma "P (rec_list nil cons xs)" |
33197 | 792 |
nitpick [expect = genuine] |
793 |
oops |
|
794 |
||
795 |
lemma "P (case x of Nil \<Rightarrow> nil | Cons a b \<Rightarrow> cons a b)" |
|
796 |
nitpick [expect = genuine] |
|
797 |
oops |
|
798 |
||
799 |
lemma "(xs\<Colon>'a list) = ys" |
|
800 |
nitpick [expect = genuine] |
|
801 |
oops |
|
802 |
||
803 |
lemma "a # xs = b # xs" |
|
804 |
nitpick [expect = genuine] |
|
805 |
oops |
|
806 |
||
807 |
datatype BitList = BitListNil | Bit0 BitList | Bit1 BitList |
|
808 |
||
809 |
lemma "P (x\<Colon>BitList)" |
|
810 |
nitpick [expect = genuine] |
|
811 |
oops |
|
812 |
||
813 |
lemma "\<forall>x\<Colon>BitList. P x" |
|
814 |
nitpick [expect = genuine] |
|
815 |
oops |
|
816 |
||
817 |
lemma "P (Bit0 (Bit1 BitListNil))" |
|
818 |
nitpick [expect = genuine] |
|
819 |
oops |
|
820 |
||
821 |
lemma "BitList_rec nil bit0 bit1 BitListNil = nil" |
|
822 |
nitpick [expect = none] |
|
823 |
apply simp |
|
824 |
done |
|
825 |
||
826 |
lemma "BitList_rec nil bit0 bit1 (Bit0 xs) = bit0 xs (BitList_rec nil bit0 bit1 xs)" |
|
827 |
nitpick [expect = none] |
|
828 |
apply simp |
|
829 |
done |
|
830 |
||
831 |
lemma "BitList_rec nil bit0 bit1 (Bit1 xs) = bit1 xs (BitList_rec nil bit0 bit1 xs)" |
|
832 |
nitpick [expect = none] |
|
833 |
apply simp |
|
834 |
done |
|
835 |
||
836 |
lemma "P (BitList_rec nil bit0 bit1 x)" |
|
837 |
nitpick [expect = genuine] |
|
838 |
oops |
|
839 |
||
840 |
datatype 'a BinTree = Leaf 'a | Node "'a BinTree" "'a BinTree" |
|
841 |
||
842 |
lemma "P (x\<Colon>'a BinTree)" |
|
843 |
nitpick [expect = genuine] |
|
844 |
oops |
|
845 |
||
846 |
lemma "\<forall>x\<Colon>'a BinTree. P x" |
|
847 |
nitpick [expect = genuine] |
|
848 |
oops |
|
849 |
||
850 |
lemma "P (Node (Leaf x) (Leaf y))" |
|
851 |
nitpick [expect = genuine] |
|
852 |
oops |
|
853 |
||
854 |
lemma "BinTree_rec l n (Leaf x) = l x" |
|
855 |
nitpick [expect = none] |
|
856 |
apply simp |
|
857 |
done |
|
858 |
||
859 |
lemma "BinTree_rec l n (Node x y) = n x y (BinTree_rec l n x) (BinTree_rec l n y)" |
|
42959 | 860 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 861 |
apply simp |
862 |
done |
|
863 |
||
864 |
lemma "P (BinTree_rec l n x)" |
|
865 |
nitpick [expect = genuine] |
|
866 |
oops |
|
867 |
||
868 |
lemma "P (case x of Leaf a \<Rightarrow> l a | Node a b \<Rightarrow> n a b)" |
|
869 |
nitpick [expect = genuine] |
|
870 |
oops |
|
871 |
||
872 |
text {* Mutually recursive datatypes *} |
|
873 |
||
874 |
datatype 'a aexp = Number 'a | ITE "'a bexp" "'a aexp" "'a aexp" |
|
875 |
and 'a bexp = Equal "'a aexp" "'a aexp" |
|
876 |
||
877 |
lemma "P (x\<Colon>'a aexp)" |
|
878 |
nitpick [expect = genuine] |
|
879 |
oops |
|
880 |
||
881 |
lemma "\<forall>x\<Colon>'a aexp. P x" |
|
882 |
nitpick [expect = genuine] |
|
883 |
oops |
|
884 |
||
885 |
lemma "P (ITE (Equal (Number x) (Number y)) (Number x) (Number y))" |
|
886 |
nitpick [expect = genuine] |
|
887 |
oops |
|
888 |
||
889 |
lemma "P (x\<Colon>'a bexp)" |
|
890 |
nitpick [expect = genuine] |
|
891 |
oops |
|
892 |
||
893 |
lemma "\<forall>x\<Colon>'a bexp. P x" |
|
894 |
nitpick [expect = genuine] |
|
895 |
oops |
|
896 |
||
897 |
lemma "aexp_bexp_rec_1 number ite equal (Number x) = number x" |
|
42959 | 898 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 899 |
apply simp |
900 |
done |
|
901 |
||
902 |
lemma "aexp_bexp_rec_1 number ite equal (ITE x y z) = ite x y z (aexp_bexp_rec_2 number ite equal x) (aexp_bexp_rec_1 number ite equal y) (aexp_bexp_rec_1 number ite equal z)" |
|
42959 | 903 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 904 |
apply simp |
905 |
done |
|
906 |
||
907 |
lemma "P (aexp_bexp_rec_1 number ite equal x)" |
|
908 |
nitpick [expect = genuine] |
|
909 |
oops |
|
910 |
||
911 |
lemma "P (case x of Number a \<Rightarrow> number a | ITE b a1 a2 \<Rightarrow> ite b a1 a2)" |
|
912 |
nitpick [expect = genuine] |
|
913 |
oops |
|
914 |
||
915 |
lemma "aexp_bexp_rec_2 number ite equal (Equal x y) = equal x y (aexp_bexp_rec_1 number ite equal x) (aexp_bexp_rec_1 number ite equal y)" |
|
42959 | 916 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 917 |
apply simp |
918 |
done |
|
919 |
||
920 |
lemma "P (aexp_bexp_rec_2 number ite equal x)" |
|
921 |
nitpick [expect = genuine] |
|
922 |
oops |
|
923 |
||
924 |
lemma "P (case x of Equal a1 a2 \<Rightarrow> equal a1 a2)" |
|
925 |
nitpick [expect = genuine] |
|
926 |
oops |
|
927 |
||
928 |
datatype X = A | B X | C Y |
|
929 |
and Y = D X | E Y | F |
|
930 |
||
931 |
lemma "P (x\<Colon>X)" |
|
932 |
nitpick [expect = genuine] |
|
933 |
oops |
|
934 |
||
935 |
lemma "P (y\<Colon>Y)" |
|
936 |
nitpick [expect = genuine] |
|
937 |
oops |
|
938 |
||
939 |
lemma "P (B (B A))" |
|
940 |
nitpick [expect = genuine] |
|
941 |
oops |
|
942 |
||
943 |
lemma "P (B (C F))" |
|
944 |
nitpick [expect = genuine] |
|
945 |
oops |
|
946 |
||
947 |
lemma "P (C (D A))" |
|
948 |
nitpick [expect = genuine] |
|
949 |
oops |
|
950 |
||
951 |
lemma "P (C (E F))" |
|
952 |
nitpick [expect = genuine] |
|
953 |
oops |
|
954 |
||
955 |
lemma "P (D (B A))" |
|
956 |
nitpick [expect = genuine] |
|
957 |
oops |
|
958 |
||
959 |
lemma "P (D (C F))" |
|
960 |
nitpick [expect = genuine] |
|
961 |
oops |
|
962 |
||
963 |
lemma "P (E (D A))" |
|
964 |
nitpick [expect = genuine] |
|
965 |
oops |
|
966 |
||
967 |
lemma "P (E (E F))" |
|
968 |
nitpick [expect = genuine] |
|
969 |
oops |
|
970 |
||
971 |
lemma "P (C (D (C F)))" |
|
972 |
nitpick [expect = genuine] |
|
973 |
oops |
|
974 |
||
975 |
lemma "X_Y_rec_1 a b c d e f A = a" |
|
42959 | 976 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 977 |
apply simp |
978 |
done |
|
979 |
||
980 |
lemma "X_Y_rec_1 a b c d e f (B x) = b x (X_Y_rec_1 a b c d e f x)" |
|
42959 | 981 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 982 |
apply simp |
983 |
done |
|
984 |
||
985 |
lemma "X_Y_rec_1 a b c d e f (C y) = c y (X_Y_rec_2 a b c d e f y)" |
|
42959 | 986 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 987 |
apply simp |
988 |
done |
|
989 |
||
990 |
lemma "X_Y_rec_2 a b c d e f (D x) = d x (X_Y_rec_1 a b c d e f x)" |
|
42959 | 991 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 992 |
apply simp |
993 |
done |
|
994 |
||
995 |
lemma "X_Y_rec_2 a b c d e f (E y) = e y (X_Y_rec_2 a b c d e f y)" |
|
42959 | 996 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 997 |
apply simp |
998 |
done |
|
999 |
||
1000 |
lemma "X_Y_rec_2 a b c d e f F = f" |
|
42959 | 1001 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 1002 |
apply simp |
1003 |
done |
|
1004 |
||
1005 |
lemma "P (X_Y_rec_1 a b c d e f x)" |
|
1006 |
nitpick [expect = genuine] |
|
1007 |
oops |
|
1008 |
||
1009 |
lemma "P (X_Y_rec_2 a b c d e f y)" |
|
1010 |
nitpick [expect = genuine] |
|
1011 |
oops |
|
1012 |
||
1013 |
text {* Other datatype examples *} |
|
1014 |
||
1015 |
text {* Indirect recursion is implemented via mutual recursion. *} |
|
1016 |
||
1017 |
datatype XOpt = CX "XOpt option" | DX "bool \<Rightarrow> XOpt option" |
|
1018 |
||
1019 |
lemma "P (x\<Colon>XOpt)" |
|
1020 |
nitpick [expect = genuine] |
|
1021 |
oops |
|
1022 |
||
1023 |
lemma "P (CX None)" |
|
1024 |
nitpick [expect = genuine] |
|
1025 |
oops |
|
1026 |
||
1027 |
lemma "P (CX (Some (CX None)))" |
|
1028 |
nitpick [expect = genuine] |
|
1029 |
oops |
|
1030 |
||
1031 |
lemma "XOpt_rec_1 cx dx n1 s1 n2 s2 (CX x) = cx x (XOpt_rec_2 cx dx n1 s1 n2 s2 x)" |
|
42959 | 1032 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 1033 |
apply simp |
1034 |
done |
|
1035 |
||
1036 |
lemma "XOpt_rec_1 cx dx n1 s1 n2 s2 (DX x) = dx x (\<lambda>b. XOpt_rec_3 cx dx n1 s1 n2 s2 (x b))" |
|
42959 | 1037 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 1038 |
apply simp |
1039 |
done |
|
1040 |
||
1041 |
lemma "XOpt_rec_2 cx dx n1 s1 n2 s2 None = n1" |
|
42959 | 1042 |
nitpick [card = 1\<emdash>4, expect = none] |
33197 | 1043 |
apply simp |
1044 |
done |
|
1045 |
||
1046 |
lemma "XOpt_rec_2 cx dx n1 s1 n2 s2 (Some x) = s1 x (XOpt_rec_1 cx dx n1 s1 n2 s2 x)" |
|
42959 | 1047 |
nitpick [card = 1\<emdash>4, expect = none] |
33197 | 1048 |
apply simp |
1049 |
done |
|
1050 |
||
1051 |
lemma "XOpt_rec_3 cx dx n1 s1 n2 s2 None = n2" |
|
42959 | 1052 |
nitpick [card = 1\<emdash>4, expect = none] |
33197 | 1053 |
apply simp |
1054 |
done |
|
1055 |
||
1056 |
lemma "XOpt_rec_3 cx dx n1 s1 n2 s2 (Some x) = s2 x (XOpt_rec_1 cx dx n1 s1 n2 s2 x)" |
|
42959 | 1057 |
nitpick [card = 1\<emdash>4, expect = none] |
33197 | 1058 |
apply simp |
1059 |
done |
|
1060 |
||
1061 |
lemma "P (XOpt_rec_1 cx dx n1 s1 n2 s2 x)" |
|
1062 |
nitpick [expect = genuine] |
|
1063 |
oops |
|
1064 |
||
1065 |
lemma "P (XOpt_rec_2 cx dx n1 s1 n2 s2 x)" |
|
1066 |
nitpick [expect = genuine] |
|
1067 |
oops |
|
1068 |
||
1069 |
lemma "P (XOpt_rec_3 cx dx n1 s1 n2 s2 x)" |
|
1070 |
nitpick [expect = genuine] |
|
1071 |
oops |
|
1072 |
||
1073 |
datatype 'a YOpt = CY "('a \<Rightarrow> 'a YOpt) option" |
|
1074 |
||
1075 |
lemma "P (x\<Colon>'a YOpt)" |
|
1076 |
nitpick [expect = genuine] |
|
1077 |
oops |
|
1078 |
||
1079 |
lemma "P (CY None)" |
|
1080 |
nitpick [expect = genuine] |
|
1081 |
oops |
|
1082 |
||
1083 |
lemma "P (CY (Some (\<lambda>a. CY None)))" |
|
1084 |
nitpick [expect = genuine] |
|
1085 |
oops |
|
1086 |
||
1087 |
lemma "YOpt_rec_1 cy n s (CY x) = cy x (YOpt_rec_2 cy n s x)" |
|
42959 | 1088 |
nitpick [card = 1\<emdash>2, expect = none] |
33197 | 1089 |
apply simp |
1090 |
done |
|
1091 |
||
1092 |
lemma "YOpt_rec_2 cy n s None = n" |
|
42959 | 1093 |
nitpick [card = 1\<emdash>2, expect = none] |
33197 | 1094 |
apply simp |
1095 |
done |
|
1096 |
||
1097 |
lemma "YOpt_rec_2 cy n s (Some x) = s x (\<lambda>a. YOpt_rec_1 cy n s (x a))" |
|
42959 | 1098 |
nitpick [card = 1\<emdash>2, expect = none] |
33197 | 1099 |
apply simp |
1100 |
done |
|
1101 |
||
1102 |
lemma "P (YOpt_rec_1 cy n s x)" |
|
1103 |
nitpick [expect = genuine] |
|
1104 |
oops |
|
1105 |
||
1106 |
lemma "P (YOpt_rec_2 cy n s x)" |
|
1107 |
nitpick [expect = genuine] |
|
1108 |
oops |
|
1109 |
||
1110 |
datatype Trie = TR "Trie list" |
|
1111 |
||
1112 |
lemma "P (x\<Colon>Trie)" |
|
1113 |
nitpick [expect = genuine] |
|
1114 |
oops |
|
1115 |
||
1116 |
lemma "\<forall>x\<Colon>Trie. P x" |
|
1117 |
nitpick [expect = genuine] |
|
1118 |
oops |
|
1119 |
||
1120 |
lemma "P (TR [TR []])" |
|
1121 |
nitpick [expect = genuine] |
|
1122 |
oops |
|
1123 |
||
1124 |
lemma "Trie_rec_1 tr nil cons (TR x) = tr x (Trie_rec_2 tr nil cons x)" |
|
42959 | 1125 |
nitpick [card = 1\<emdash>4, expect = none] |
33197 | 1126 |
apply simp |
1127 |
done |
|
1128 |
||
1129 |
lemma "Trie_rec_2 tr nil cons [] = nil" |
|
42959 | 1130 |
nitpick [card = 1\<emdash>4, expect = none] |
33197 | 1131 |
apply simp |
1132 |
done |
|
1133 |
||
1134 |
lemma "Trie_rec_2 tr nil cons (x#xs) = cons x xs (Trie_rec_1 tr nil cons x) (Trie_rec_2 tr nil cons xs)" |
|
42959 | 1135 |
nitpick [card = 1\<emdash>4, expect = none] |
33197 | 1136 |
apply simp |
1137 |
done |
|
1138 |
||
1139 |
lemma "P (Trie_rec_1 tr nil cons x)" |
|
1140 |
nitpick [card = 1, expect = genuine] |
|
1141 |
oops |
|
1142 |
||
1143 |
lemma "P (Trie_rec_2 tr nil cons x)" |
|
1144 |
nitpick [card = 1, expect = genuine] |
|
1145 |
oops |
|
1146 |
||
1147 |
datatype InfTree = Leaf | Node "nat \<Rightarrow> InfTree" |
|
1148 |
||
1149 |
lemma "P (x\<Colon>InfTree)" |
|
1150 |
nitpick [expect = genuine] |
|
1151 |
oops |
|
1152 |
||
1153 |
lemma "\<forall>x\<Colon>InfTree. P x" |
|
1154 |
nitpick [expect = genuine] |
|
1155 |
oops |
|
1156 |
||
1157 |
lemma "P (Node (\<lambda>n. Leaf))" |
|
1158 |
nitpick [expect = genuine] |
|
1159 |
oops |
|
1160 |
||
1161 |
lemma "InfTree_rec leaf node Leaf = leaf" |
|
42959 | 1162 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 1163 |
apply simp |
1164 |
done |
|
1165 |
||
1166 |
lemma "InfTree_rec leaf node (Node x) = node x (\<lambda>n. InfTree_rec leaf node (x n))" |
|
42959 | 1167 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 1168 |
apply simp |
1169 |
done |
|
1170 |
||
1171 |
lemma "P (InfTree_rec leaf node x)" |
|
1172 |
nitpick [expect = genuine] |
|
1173 |
oops |
|
1174 |
||
1175 |
datatype 'a lambda = Var 'a | App "'a lambda" "'a lambda" | Lam "'a \<Rightarrow> 'a lambda" |
|
1176 |
||
1177 |
lemma "P (x\<Colon>'a lambda)" |
|
1178 |
nitpick [expect = genuine] |
|
1179 |
oops |
|
1180 |
||
1181 |
lemma "\<forall>x\<Colon>'a lambda. P x" |
|
1182 |
nitpick [expect = genuine] |
|
1183 |
oops |
|
1184 |
||
1185 |
lemma "P (Lam (\<lambda>a. Var a))" |
|
42959 | 1186 |
nitpick [card = 1\<emdash>5, expect = none] |
33197 | 1187 |
nitpick [card 'a = 4, card "'a lambda" = 5, expect = genuine] |
1188 |
oops |
|
1189 |
||
1190 |
lemma "lambda_rec var app lam (Var x) = var x" |
|
42959 | 1191 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 1192 |
apply simp |
1193 |
done |
|
1194 |
||
1195 |
lemma "lambda_rec var app lam (App x y) = app x y (lambda_rec var app lam x) (lambda_rec var app lam y)" |
|
42959 | 1196 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 1197 |
apply simp |
1198 |
done |
|
1199 |
||
1200 |
lemma "lambda_rec var app lam (Lam x) = lam x (\<lambda>a. lambda_rec var app lam (x a))" |
|
42959 | 1201 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 1202 |
apply simp |
1203 |
done |
|
1204 |
||
1205 |
lemma "P (lambda_rec v a l x)" |
|
1206 |
nitpick [expect = genuine] |
|
1207 |
oops |
|
1208 |
||
1209 |
text {* Taken from "Inductive datatypes in HOL", p. 8: *} |
|
1210 |
||
1211 |
datatype ('a, 'b) T = C "'a \<Rightarrow> bool" | D "'b list" |
|
1212 |
datatype 'c U = E "('c, 'c U) T" |
|
1213 |
||
1214 |
lemma "P (x\<Colon>'c U)" |
|
1215 |
nitpick [expect = genuine] |
|
1216 |
oops |
|
1217 |
||
1218 |
lemma "\<forall>x\<Colon>'c U. P x" |
|
1219 |
nitpick [expect = genuine] |
|
1220 |
oops |
|
1221 |
||
1222 |
lemma "P (E (C (\<lambda>a. True)))" |
|
1223 |
nitpick [expect = genuine] |
|
1224 |
oops |
|
1225 |
||
1226 |
lemma "U_rec_1 e c d nil cons (E x) = e x (U_rec_2 e c d nil cons x)" |
|
42959 | 1227 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 1228 |
apply simp |
1229 |
done |
|
1230 |
||
1231 |
lemma "U_rec_2 e c d nil cons (C x) = c x" |
|
42959 | 1232 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 1233 |
apply simp |
1234 |
done |
|
1235 |
||
1236 |
lemma "U_rec_2 e c d nil cons (D x) = d x (U_rec_3 e c d nil cons x)" |
|
42959 | 1237 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 1238 |
apply simp |
1239 |
done |
|
1240 |
||
1241 |
lemma "U_rec_3 e c d nil cons [] = nil" |
|
42959 | 1242 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 1243 |
apply simp |
1244 |
done |
|
1245 |
||
1246 |
lemma "U_rec_3 e c d nil cons (x#xs) = cons x xs (U_rec_1 e c d nil cons x) (U_rec_3 e c d nil cons xs)" |
|
42959 | 1247 |
nitpick [card = 1\<emdash>3, expect = none] |
33197 | 1248 |
apply simp |
1249 |
done |
|
1250 |
||
1251 |
lemma "P (U_rec_1 e c d nil cons x)" |
|
1252 |
nitpick [expect = genuine] |
|
1253 |
oops |
|
1254 |
||
1255 |
lemma "P (U_rec_2 e c d nil cons x)" |
|
1256 |
nitpick [card = 1, expect = genuine] |
|
1257 |
oops |
|
1258 |
||
1259 |
lemma "P (U_rec_3 e c d nil cons x)" |
|
1260 |
nitpick [card = 1, expect = genuine] |
|
1261 |
oops |
|
1262 |
||
1263 |
subsubsection {* Records *} |
|
1264 |
||
1265 |
record ('a, 'b) point = |
|
1266 |
xpos :: 'a |
|
1267 |
ypos :: 'b |
|
1268 |
||
1269 |
lemma "(x\<Colon>('a, 'b) point) = y" |
|
1270 |
nitpick [expect = genuine] |
|
1271 |
oops |
|
1272 |
||
1273 |
record ('a, 'b, 'c) extpoint = "('a, 'b) point" + |
|
1274 |
ext :: 'c |
|
1275 |
||
1276 |
lemma "(x\<Colon>('a, 'b, 'c) extpoint) = y" |
|
1277 |
nitpick [expect = genuine] |
|
1278 |
oops |
|
1279 |
||
1280 |
subsubsection {* Inductively Defined Sets *} |
|
1281 |
||
1282 |
inductive_set undefinedSet :: "'a set" where |
|
1283 |
"undefined \<in> undefinedSet" |
|
1284 |
||
1285 |
lemma "x \<in> undefinedSet" |
|
1286 |
nitpick [expect = genuine] |
|
1287 |
oops |
|
1288 |
||
1289 |
inductive_set evenCard :: "'a set set" |
|
1290 |
where |
|
1291 |
"{} \<in> evenCard" | |
|
1292 |
"\<lbrakk>S \<in> evenCard; x \<notin> S; y \<notin> S; x \<noteq> y\<rbrakk> \<Longrightarrow> S \<union> {x, y} \<in> evenCard" |
|
1293 |
||
1294 |
lemma "S \<in> evenCard" |
|
1295 |
nitpick [expect = genuine] |
|
1296 |
oops |
|
1297 |
||
1298 |
inductive_set |
|
1299 |
even :: "nat set" |
|
1300 |
and odd :: "nat set" |
|
1301 |
where |
|
1302 |
"0 \<in> even" | |
|
1303 |
"n \<in> even \<Longrightarrow> Suc n \<in> odd" | |
|
1304 |
"n \<in> odd \<Longrightarrow> Suc n \<in> even" |
|
1305 |
||
1306 |
lemma "n \<in> odd" |
|
1307 |
nitpick [expect = genuine] |
|
1308 |
oops |
|
1309 |
||
1310 |
consts f :: "'a \<Rightarrow> 'a" |
|
1311 |
||
1312 |
inductive_set a_even :: "'a set" and a_odd :: "'a set" where |
|
1313 |
"undefined \<in> a_even" | |
|
1314 |
"x \<in> a_even \<Longrightarrow> f x \<in> a_odd" | |
|
1315 |
"x \<in> a_odd \<Longrightarrow> f x \<in> a_even" |
|
1316 |
||
1317 |
lemma "x \<in> a_odd" |
|
1318 |
nitpick [expect = genuine] |
|
1319 |
oops |
|
1320 |
||
1321 |
subsubsection {* Examples Involving Special Functions *} |
|
1322 |
||
1323 |
lemma "card x = 0" |
|
1324 |
nitpick [expect = genuine] |
|
1325 |
oops |
|
1326 |
||
1327 |
lemma "finite x" |
|
1328 |
nitpick [expect = none] |
|
1329 |
oops |
|
1330 |
||
1331 |
lemma "xs @ [] = ys @ []" |
|
1332 |
nitpick [expect = genuine] |
|
1333 |
oops |
|
1334 |
||
1335 |
lemma "xs @ ys = ys @ xs" |
|
1336 |
nitpick [expect = genuine] |
|
1337 |
oops |
|
1338 |
||
1339 |
lemma "f (lfp f) = lfp f" |
|
35284
9edc2bd6d2bd
enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents:
35087
diff
changeset
|
1340 |
nitpick [card = 2, expect = genuine] |
33197 | 1341 |
oops |
1342 |
||
1343 |
lemma "f (gfp f) = gfp f" |
|
35284
9edc2bd6d2bd
enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents:
35087
diff
changeset
|
1344 |
nitpick [card = 2, expect = genuine] |
33197 | 1345 |
oops |
1346 |
||
1347 |
lemma "lfp f = gfp f" |
|
35284
9edc2bd6d2bd
enabled Nitpick's support for quotient types + shortened the Nitpick tests a bit
blanchet
parents:
35087
diff
changeset
|
1348 |
nitpick [card = 2, expect = genuine] |
33197 | 1349 |
oops |
1350 |
||
1351 |
subsubsection {* Axiomatic Type Classes and Overloading *} |
|
1352 |
||
1353 |
text {* A type class without axioms: *} |
|
1354 |
||
35338 | 1355 |
class classA |
33197 | 1356 |
|
1357 |
lemma "P (x\<Colon>'a\<Colon>classA)" |
|
1358 |
nitpick [expect = genuine] |
|
1359 |
oops |
|
1360 |
||
1361 |
text {* An axiom with a type variable (denoting types which have at least two elements): *} |
|
1362 |
||
35338 | 1363 |
class classC = |
1364 |
assumes classC_ax: "\<exists>x y. x \<noteq> y" |
|
33197 | 1365 |
|
1366 |
lemma "P (x\<Colon>'a\<Colon>classC)" |
|
1367 |
nitpick [expect = genuine] |
|
1368 |
oops |
|
1369 |
||
1370 |
lemma "\<exists>x y. (x\<Colon>'a\<Colon>classC) \<noteq> y" |
|
1371 |
nitpick [expect = none] |
|
1372 |
sorry |
|
1373 |
||
1374 |
text {* A type class for which a constant is defined: *} |
|
1375 |
||
35338 | 1376 |
class classD = |
1377 |
fixes classD_const :: "'a \<Rightarrow> 'a" |
|
1378 |
assumes classD_ax: "classD_const (classD_const x) = classD_const x" |
|
33197 | 1379 |
|
1380 |
lemma "P (x\<Colon>'a\<Colon>classD)" |
|
1381 |
nitpick [expect = genuine] |
|
1382 |
oops |
|
1383 |
||
1384 |
text {* A type class with multiple superclasses: *} |
|
1385 |
||
35338 | 1386 |
class classE = classC + classD |
33197 | 1387 |
|
1388 |
lemma "P (x\<Colon>'a\<Colon>classE)" |
|
1389 |
nitpick [expect = genuine] |
|
1390 |
oops |
|
1391 |
||
1392 |
text {* OFCLASS: *} |
|
1393 |
||
1394 |
lemma "OFCLASS('a\<Colon>type, type_class)" |
|
1395 |
nitpick [expect = none] |
|
1396 |
apply intro_classes |
|
1397 |
done |
|
1398 |
||
1399 |
lemma "OFCLASS('a\<Colon>classC, type_class)" |
|
1400 |
nitpick [expect = none] |
|
1401 |
apply intro_classes |
|
1402 |
done |
|
1403 |
||
1404 |
lemma "OFCLASS('a\<Colon>type, classC_class)" |
|
1405 |
nitpick [expect = genuine] |
|
1406 |
oops |
|
1407 |
||
1408 |
text {* Overloading: *} |
|
1409 |
||
1410 |
consts inverse :: "'a \<Rightarrow> 'a" |
|
1411 |
||
1412 |
defs (overloaded) |
|
1413 |
inverse_bool: "inverse (b\<Colon>bool) \<equiv> \<not> b" |
|
1414 |
inverse_set: "inverse (S\<Colon>'a set) \<equiv> -S" |
|
1415 |
inverse_pair: "inverse p \<equiv> (inverse (fst p), inverse (snd p))" |
|
1416 |
||
1417 |
lemma "inverse b" |
|
1418 |
nitpick [expect = genuine] |
|
1419 |
oops |
|
1420 |
||
1421 |
lemma "P (inverse (S\<Colon>'a set))" |
|
1422 |
nitpick [expect = genuine] |
|
1423 |
oops |
|
1424 |
||
1425 |
lemma "P (inverse (p\<Colon>'a\<times>'b))" |
|
1426 |
nitpick [expect = genuine] |
|
1427 |
oops |
|
1428 |
||
1429 |
end |