author | blanchet |
Tue, 17 Nov 2009 13:51:16 +0100 | |
changeset 33735 | 0c0e7b2ecf2e |
parent 33197 | de6285ebcc05 |
child 34083 | 652719832159 |
permissions | -rw-r--r-- |
33197 | 1 |
(* Title: HOL/Nitpick_Examples/Refute_Nits.thy |
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Author: Jasmin Blanchette, TU Muenchen |
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Copyright 2009 |
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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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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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33735
0c0e7b2ecf2e
use SAT solver that's available everywhere for this example
blanchet
parents:
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changeset
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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 'T' of an arbitrary relation 'P' is non-empty." *} |
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constdefs |
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"trans" :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" |
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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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"subset" :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" |
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"subset P Q \<equiv> (ALL x y. P x y \<longrightarrow> Q x y)" |
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"trans_closure" :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" |
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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\<midarrow>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 = 1-5, 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 = 1-4, 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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309 |
oops |
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310 |
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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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317 |
oops |
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318 |
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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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325 |
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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lemma "?P" |
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nitpick [expect = none] |
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apply auto |
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done |
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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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361 |
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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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366 |
oops |
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367 |
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368 |
lemma "P (A\<Colon>'a set set)" |
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nitpick [expect = genuine] |
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370 |
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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379 |
oops |
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lemma "P (op \<in>)" |
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nitpick [expect = genuine] |
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383 |
oops |
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384 |
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385 |
lemma "P (op \<in> x)" |
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nitpick [expect = genuine] |
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387 |
oops |
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388 |
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lemma "P Collect" |
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nitpick [expect = genuine] |
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391 |
oops |
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392 |
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393 |
lemma "A Un B = A Int B" |
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nitpick [expect = genuine] |
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395 |
oops |
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396 |
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397 |
lemma "(A Int B) Un C = (A Un C) Int B" |
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398 |
nitpick [expect = genuine] |
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399 |
oops |
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400 |
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401 |
lemma "Ball A P \<longrightarrow> Bex A P" |
|
402 |
nitpick [expect = genuine] |
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403 |
oops |
|
404 |
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405 |
subsubsection {* @{const undefined} *} |
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406 |
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407 |
lemma "undefined" |
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408 |
nitpick [expect = genuine] |
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409 |
oops |
|
410 |
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411 |
lemma "P undefined" |
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412 |
nitpick [expect = genuine] |
|
413 |
oops |
|
414 |
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415 |
lemma "undefined x" |
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416 |
nitpick [expect = genuine] |
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417 |
oops |
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418 |
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419 |
lemma "undefined undefined" |
|
420 |
nitpick [expect = genuine] |
|
421 |
oops |
|
422 |
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423 |
subsubsection {* @{const The} *} |
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424 |
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425 |
lemma "The P" |
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426 |
nitpick [expect = genuine] |
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427 |
oops |
|
428 |
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429 |
lemma "P The" |
|
430 |
nitpick [expect = genuine] |
|
431 |
oops |
|
432 |
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433 |
lemma "P (The P)" |
|
434 |
nitpick [expect = genuine] |
|
435 |
oops |
|
436 |
||
437 |
lemma "(THE x. x=y) = z" |
|
438 |
nitpick [expect = genuine] |
|
439 |
oops |
|
440 |
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441 |
lemma "Ex P \<longrightarrow> P (The P)" |
|
442 |
nitpick [expect = genuine] |
|
443 |
oops |
|
444 |
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445 |
subsubsection {* @{const Eps} *} |
|
446 |
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447 |
lemma "Eps P" |
|
448 |
nitpick [expect = genuine] |
|
449 |
oops |
|
450 |
||
451 |
lemma "P Eps" |
|
452 |
nitpick [expect = genuine] |
|
453 |
oops |
|
454 |
||
455 |
lemma "P (Eps P)" |
|
456 |
nitpick [expect = genuine] |
|
457 |
oops |
|
458 |
||
459 |
lemma "(SOME x. x=y) = z" |
|
460 |
nitpick [expect = genuine] |
|
461 |
oops |
|
462 |
||
463 |
lemma "Ex P \<longrightarrow> P (Eps P)" |
|
464 |
nitpick [expect = none] |
|
465 |
apply (auto simp add: someI) |
|
466 |
done |
|
467 |
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468 |
subsubsection {* Operations on Natural Numbers *} |
|
469 |
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470 |
lemma "(x\<Colon>nat) + y = 0" |
|
471 |
nitpick [expect = genuine] |
|
472 |
oops |
|
473 |
||
474 |
lemma "(x\<Colon>nat) = x + x" |
|
475 |
nitpick [expect = genuine] |
|
476 |
oops |
|
477 |
||
478 |
lemma "(x\<Colon>nat) - y + y = x" |
|
479 |
nitpick [expect = genuine] |
|
480 |
oops |
|
481 |
||
482 |
lemma "(x\<Colon>nat) = x * x" |
|
483 |
nitpick [expect = genuine] |
|
484 |
oops |
|
485 |
||
486 |
lemma "(x\<Colon>nat) < x + y" |
|
487 |
nitpick [card = 1, expect = genuine] |
|
488 |
nitpick [card = 2-5, expect = genuine] |
|
489 |
oops |
|
490 |
||
491 |
text {* \<times> *} |
|
492 |
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493 |
lemma "P (x\<Colon>'a\<times>'b)" |
|
494 |
nitpick [expect = genuine] |
|
495 |
oops |
|
496 |
||
497 |
lemma "\<forall>x\<Colon>'a\<times>'b. P x" |
|
498 |
nitpick [expect = genuine] |
|
499 |
oops |
|
500 |
||
501 |
lemma "P (x, y)" |
|
502 |
nitpick [expect = genuine] |
|
503 |
oops |
|
504 |
||
505 |
lemma "P (fst x)" |
|
506 |
nitpick [expect = genuine] |
|
507 |
oops |
|
508 |
||
509 |
lemma "P (snd x)" |
|
510 |
nitpick [expect = genuine] |
|
511 |
oops |
|
512 |
||
513 |
lemma "P Pair" |
|
514 |
nitpick [expect = genuine] |
|
515 |
oops |
|
516 |
||
517 |
lemma "prod_rec f p = f (fst p) (snd p)" |
|
518 |
nitpick [expect = none] |
|
519 |
by (case_tac p) auto |
|
520 |
||
521 |
lemma "prod_rec f (a, b) = f a b" |
|
522 |
nitpick [expect = none] |
|
523 |
by auto |
|
524 |
||
525 |
lemma "P (prod_rec f x)" |
|
526 |
nitpick [expect = genuine] |
|
527 |
oops |
|
528 |
||
529 |
lemma "P (case x of Pair a b \<Rightarrow> f a b)" |
|
530 |
nitpick [expect = genuine] |
|
531 |
oops |
|
532 |
||
533 |
subsubsection {* Subtypes (typedef), typedecl *} |
|
534 |
||
535 |
text {* A completely unspecified non-empty subset of @{typ "'a"}: *} |
|
536 |
||
537 |
typedef 'a myTdef = "insert (undefined\<Colon>'a) (undefined\<Colon>'a set)" |
|
538 |
by auto |
|
539 |
||
540 |
lemma "(x\<Colon>'a myTdef) = y" |
|
541 |
nitpick [expect = genuine] |
|
542 |
oops |
|
543 |
||
544 |
typedecl myTdecl |
|
545 |
||
546 |
typedef 'a T_bij = "{(f\<Colon>'a\<Rightarrow>'a). \<forall>y. \<exists>!x. f x = y}" |
|
547 |
by auto |
|
548 |
||
549 |
lemma "P (f\<Colon>(myTdecl myTdef) T_bij)" |
|
550 |
nitpick [expect = genuine] |
|
551 |
oops |
|
552 |
||
553 |
subsubsection {* Inductive Datatypes *} |
|
554 |
||
555 |
text {* unit *} |
|
556 |
||
557 |
lemma "P (x\<Colon>unit)" |
|
558 |
nitpick [expect = genuine] |
|
559 |
oops |
|
560 |
||
561 |
lemma "\<forall>x\<Colon>unit. P x" |
|
562 |
nitpick [expect = genuine] |
|
563 |
oops |
|
564 |
||
565 |
lemma "P ()" |
|
566 |
nitpick [expect = genuine] |
|
567 |
oops |
|
568 |
||
569 |
lemma "unit_rec u x = u" |
|
570 |
nitpick [expect = none] |
|
571 |
apply simp |
|
572 |
done |
|
573 |
||
574 |
lemma "P (unit_rec u x)" |
|
575 |
nitpick [expect = genuine] |
|
576 |
oops |
|
577 |
||
578 |
lemma "P (case x of () \<Rightarrow> u)" |
|
579 |
nitpick [expect = genuine] |
|
580 |
oops |
|
581 |
||
582 |
text {* option *} |
|
583 |
||
584 |
lemma "P (x\<Colon>'a option)" |
|
585 |
nitpick [expect = genuine] |
|
586 |
oops |
|
587 |
||
588 |
lemma "\<forall>x\<Colon>'a option. P x" |
|
589 |
nitpick [expect = genuine] |
|
590 |
oops |
|
591 |
||
592 |
lemma "P None" |
|
593 |
nitpick [expect = genuine] |
|
594 |
oops |
|
595 |
||
596 |
lemma "P (Some x)" |
|
597 |
nitpick [expect = genuine] |
|
598 |
oops |
|
599 |
||
600 |
lemma "option_rec n s None = n" |
|
601 |
nitpick [expect = none] |
|
602 |
apply simp |
|
603 |
done |
|
604 |
||
605 |
lemma "option_rec n s (Some x) = s x" |
|
606 |
nitpick [expect = none] |
|
607 |
apply simp |
|
608 |
done |
|
609 |
||
610 |
lemma "P (option_rec n s x)" |
|
611 |
nitpick [expect = genuine] |
|
612 |
oops |
|
613 |
||
614 |
lemma "P (case x of None \<Rightarrow> n | Some u \<Rightarrow> s u)" |
|
615 |
nitpick [expect = genuine] |
|
616 |
oops |
|
617 |
||
618 |
text {* + *} |
|
619 |
||
620 |
lemma "P (x\<Colon>'a+'b)" |
|
621 |
nitpick [expect = genuine] |
|
622 |
oops |
|
623 |
||
624 |
lemma "\<forall>x\<Colon>'a+'b. P x" |
|
625 |
nitpick [expect = genuine] |
|
626 |
oops |
|
627 |
||
628 |
lemma "P (Inl x)" |
|
629 |
nitpick [expect = genuine] |
|
630 |
oops |
|
631 |
||
632 |
lemma "P (Inr x)" |
|
633 |
nitpick [expect = genuine] |
|
634 |
oops |
|
635 |
||
636 |
lemma "P Inl" |
|
637 |
nitpick [expect = genuine] |
|
638 |
oops |
|
639 |
||
640 |
lemma "sum_rec l r (Inl x) = l x" |
|
641 |
nitpick [expect = none] |
|
642 |
apply simp |
|
643 |
done |
|
644 |
||
645 |
lemma "sum_rec l r (Inr x) = r x" |
|
646 |
nitpick [expect = none] |
|
647 |
apply simp |
|
648 |
done |
|
649 |
||
650 |
lemma "P (sum_rec l r x)" |
|
651 |
nitpick [expect = genuine] |
|
652 |
oops |
|
653 |
||
654 |
lemma "P (case x of Inl a \<Rightarrow> l a | Inr b \<Rightarrow> r b)" |
|
655 |
nitpick [expect = genuine] |
|
656 |
oops |
|
657 |
||
658 |
text {* Non-recursive datatypes *} |
|
659 |
||
660 |
datatype T1 = A | B |
|
661 |
||
662 |
lemma "P (x\<Colon>T1)" |
|
663 |
nitpick [expect = genuine] |
|
664 |
oops |
|
665 |
||
666 |
lemma "\<forall>x\<Colon>T1. P x" |
|
667 |
nitpick [expect = genuine] |
|
668 |
oops |
|
669 |
||
670 |
lemma "P A" |
|
671 |
nitpick [expect = genuine] |
|
672 |
oops |
|
673 |
||
674 |
lemma "P B" |
|
675 |
nitpick [expect = genuine] |
|
676 |
oops |
|
677 |
||
678 |
lemma "T1_rec a b A = a" |
|
679 |
nitpick [expect = none] |
|
680 |
apply simp |
|
681 |
done |
|
682 |
||
683 |
lemma "T1_rec a b B = b" |
|
684 |
nitpick [expect = none] |
|
685 |
apply simp |
|
686 |
done |
|
687 |
||
688 |
lemma "P (T1_rec a b x)" |
|
689 |
nitpick [expect = genuine] |
|
690 |
oops |
|
691 |
||
692 |
lemma "P (case x of A \<Rightarrow> a | B \<Rightarrow> b)" |
|
693 |
nitpick [expect = genuine] |
|
694 |
oops |
|
695 |
||
696 |
datatype 'a T2 = C T1 | D 'a |
|
697 |
||
698 |
lemma "P (x\<Colon>'a T2)" |
|
699 |
nitpick [expect = genuine] |
|
700 |
oops |
|
701 |
||
702 |
lemma "\<forall>x\<Colon>'a T2. P x" |
|
703 |
nitpick [expect = genuine] |
|
704 |
oops |
|
705 |
||
706 |
lemma "P D" |
|
707 |
nitpick [expect = genuine] |
|
708 |
oops |
|
709 |
||
710 |
lemma "T2_rec c d (C x) = c x" |
|
711 |
nitpick [expect = none] |
|
712 |
apply simp |
|
713 |
done |
|
714 |
||
715 |
lemma "T2_rec c d (D x) = d x" |
|
716 |
nitpick [expect = none] |
|
717 |
apply simp |
|
718 |
done |
|
719 |
||
720 |
lemma "P (T2_rec c d x)" |
|
721 |
nitpick [expect = genuine] |
|
722 |
oops |
|
723 |
||
724 |
lemma "P (case x of C u \<Rightarrow> c u | D v \<Rightarrow> d v)" |
|
725 |
nitpick [expect = genuine] |
|
726 |
oops |
|
727 |
||
728 |
datatype ('a, 'b) T3 = E "'a \<Rightarrow> 'b" |
|
729 |
||
730 |
lemma "P (x\<Colon>('a, 'b) T3)" |
|
731 |
nitpick [expect = genuine] |
|
732 |
oops |
|
733 |
||
734 |
lemma "\<forall>x\<Colon>('a, 'b) T3. P x" |
|
735 |
nitpick [expect = genuine] |
|
736 |
oops |
|
737 |
||
738 |
lemma "P E" |
|
739 |
nitpick [expect = genuine] |
|
740 |
oops |
|
741 |
||
742 |
lemma "T3_rec e (E x) = e x" |
|
743 |
nitpick [card = 1\<midarrow>4, expect = none] |
|
744 |
apply simp |
|
745 |
done |
|
746 |
||
747 |
lemma "P (T3_rec e x)" |
|
748 |
nitpick [expect = genuine] |
|
749 |
oops |
|
750 |
||
751 |
lemma "P (case x of E f \<Rightarrow> e f)" |
|
752 |
nitpick [expect = genuine] |
|
753 |
oops |
|
754 |
||
755 |
text {* Recursive datatypes *} |
|
756 |
||
757 |
text {* nat *} |
|
758 |
||
759 |
lemma "P (x\<Colon>nat)" |
|
760 |
nitpick [expect = genuine] |
|
761 |
oops |
|
762 |
||
763 |
lemma "\<forall>x\<Colon>nat. P x" |
|
764 |
nitpick [expect = genuine] |
|
765 |
oops |
|
766 |
||
767 |
lemma "P (Suc 0)" |
|
768 |
nitpick [expect = genuine] |
|
769 |
oops |
|
770 |
||
771 |
lemma "P Suc" |
|
772 |
nitpick [card = 1\<midarrow>7, expect = none] |
|
773 |
oops |
|
774 |
||
775 |
lemma "nat_rec zero suc 0 = zero" |
|
776 |
nitpick [expect = none] |
|
777 |
apply simp |
|
778 |
done |
|
779 |
||
780 |
lemma "nat_rec zero suc (Suc x) = suc x (nat_rec zero suc x)" |
|
781 |
nitpick [expect = none] |
|
782 |
apply simp |
|
783 |
done |
|
784 |
||
785 |
lemma "P (nat_rec zero suc x)" |
|
786 |
nitpick [expect = genuine] |
|
787 |
oops |
|
788 |
||
789 |
lemma "P (case x of 0 \<Rightarrow> zero | Suc n \<Rightarrow> suc n)" |
|
790 |
nitpick [expect = genuine] |
|
791 |
oops |
|
792 |
||
793 |
text {* 'a list *} |
|
794 |
||
795 |
lemma "P (xs\<Colon>'a list)" |
|
796 |
nitpick [expect = genuine] |
|
797 |
oops |
|
798 |
||
799 |
lemma "\<forall>xs\<Colon>'a list. P xs" |
|
800 |
nitpick [expect = genuine] |
|
801 |
oops |
|
802 |
||
803 |
lemma "P [x, y]" |
|
804 |
nitpick [expect = genuine] |
|
805 |
oops |
|
806 |
||
807 |
lemma "list_rec nil cons [] = nil" |
|
808 |
nitpick [expect = none] |
|
809 |
apply simp |
|
810 |
done |
|
811 |
||
812 |
lemma "list_rec nil cons (x#xs) = cons x xs (list_rec nil cons xs)" |
|
813 |
nitpick [expect = none] |
|
814 |
apply simp |
|
815 |
done |
|
816 |
||
817 |
lemma "P (list_rec nil cons xs)" |
|
818 |
nitpick [expect = genuine] |
|
819 |
oops |
|
820 |
||
821 |
lemma "P (case x of Nil \<Rightarrow> nil | Cons a b \<Rightarrow> cons a b)" |
|
822 |
nitpick [expect = genuine] |
|
823 |
oops |
|
824 |
||
825 |
lemma "(xs\<Colon>'a list) = ys" |
|
826 |
nitpick [expect = genuine] |
|
827 |
oops |
|
828 |
||
829 |
lemma "a # xs = b # xs" |
|
830 |
nitpick [expect = genuine] |
|
831 |
oops |
|
832 |
||
833 |
datatype BitList = BitListNil | Bit0 BitList | Bit1 BitList |
|
834 |
||
835 |
lemma "P (x\<Colon>BitList)" |
|
836 |
nitpick [expect = genuine] |
|
837 |
oops |
|
838 |
||
839 |
lemma "\<forall>x\<Colon>BitList. P x" |
|
840 |
nitpick [expect = genuine] |
|
841 |
oops |
|
842 |
||
843 |
lemma "P (Bit0 (Bit1 BitListNil))" |
|
844 |
nitpick [expect = genuine] |
|
845 |
oops |
|
846 |
||
847 |
lemma "BitList_rec nil bit0 bit1 BitListNil = nil" |
|
848 |
nitpick [expect = none] |
|
849 |
apply simp |
|
850 |
done |
|
851 |
||
852 |
lemma "BitList_rec nil bit0 bit1 (Bit0 xs) = bit0 xs (BitList_rec nil bit0 bit1 xs)" |
|
853 |
nitpick [expect = none] |
|
854 |
apply simp |
|
855 |
done |
|
856 |
||
857 |
lemma "BitList_rec nil bit0 bit1 (Bit1 xs) = bit1 xs (BitList_rec nil bit0 bit1 xs)" |
|
858 |
nitpick [expect = none] |
|
859 |
apply simp |
|
860 |
done |
|
861 |
||
862 |
lemma "P (BitList_rec nil bit0 bit1 x)" |
|
863 |
nitpick [expect = genuine] |
|
864 |
oops |
|
865 |
||
866 |
datatype 'a BinTree = Leaf 'a | Node "'a BinTree" "'a BinTree" |
|
867 |
||
868 |
lemma "P (x\<Colon>'a BinTree)" |
|
869 |
nitpick [expect = genuine] |
|
870 |
oops |
|
871 |
||
872 |
lemma "\<forall>x\<Colon>'a BinTree. P x" |
|
873 |
nitpick [expect = genuine] |
|
874 |
oops |
|
875 |
||
876 |
lemma "P (Node (Leaf x) (Leaf y))" |
|
877 |
nitpick [expect = genuine] |
|
878 |
oops |
|
879 |
||
880 |
lemma "BinTree_rec l n (Leaf x) = l x" |
|
881 |
nitpick [expect = none] |
|
882 |
apply simp |
|
883 |
done |
|
884 |
||
885 |
lemma "BinTree_rec l n (Node x y) = n x y (BinTree_rec l n x) (BinTree_rec l n y)" |
|
886 |
nitpick [card = 1\<midarrow>6, expect = none] |
|
887 |
apply simp |
|
888 |
done |
|
889 |
||
890 |
lemma "P (BinTree_rec l n x)" |
|
891 |
nitpick [expect = genuine] |
|
892 |
oops |
|
893 |
||
894 |
lemma "P (case x of Leaf a \<Rightarrow> l a | Node a b \<Rightarrow> n a b)" |
|
895 |
nitpick [expect = genuine] |
|
896 |
oops |
|
897 |
||
898 |
text {* Mutually recursive datatypes *} |
|
899 |
||
900 |
datatype 'a aexp = Number 'a | ITE "'a bexp" "'a aexp" "'a aexp" |
|
901 |
and 'a bexp = Equal "'a aexp" "'a aexp" |
|
902 |
||
903 |
lemma "P (x\<Colon>'a aexp)" |
|
904 |
nitpick [expect = genuine] |
|
905 |
oops |
|
906 |
||
907 |
lemma "\<forall>x\<Colon>'a aexp. P x" |
|
908 |
nitpick [expect = genuine] |
|
909 |
oops |
|
910 |
||
911 |
lemma "P (ITE (Equal (Number x) (Number y)) (Number x) (Number y))" |
|
912 |
nitpick [expect = genuine] |
|
913 |
oops |
|
914 |
||
915 |
lemma "P (x\<Colon>'a bexp)" |
|
916 |
nitpick [expect = genuine] |
|
917 |
oops |
|
918 |
||
919 |
lemma "\<forall>x\<Colon>'a bexp. P x" |
|
920 |
nitpick [expect = genuine] |
|
921 |
oops |
|
922 |
||
923 |
lemma "aexp_bexp_rec_1 number ite equal (Number x) = number x" |
|
924 |
nitpick [card = 1\<midarrow>4, expect = none] |
|
925 |
apply simp |
|
926 |
done |
|
927 |
||
928 |
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)" |
|
929 |
nitpick [card = 1\<midarrow>4, expect = none] |
|
930 |
apply simp |
|
931 |
done |
|
932 |
||
933 |
lemma "P (aexp_bexp_rec_1 number ite equal x)" |
|
934 |
nitpick [expect = genuine] |
|
935 |
oops |
|
936 |
||
937 |
lemma "P (case x of Number a \<Rightarrow> number a | ITE b a1 a2 \<Rightarrow> ite b a1 a2)" |
|
938 |
nitpick [expect = genuine] |
|
939 |
oops |
|
940 |
||
941 |
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)" |
|
942 |
nitpick [card = 1\<midarrow>4, expect = none] |
|
943 |
apply simp |
|
944 |
done |
|
945 |
||
946 |
lemma "P (aexp_bexp_rec_2 number ite equal x)" |
|
947 |
nitpick [expect = genuine] |
|
948 |
oops |
|
949 |
||
950 |
lemma "P (case x of Equal a1 a2 \<Rightarrow> equal a1 a2)" |
|
951 |
nitpick [expect = genuine] |
|
952 |
oops |
|
953 |
||
954 |
datatype X = A | B X | C Y |
|
955 |
and Y = D X | E Y | F |
|
956 |
||
957 |
lemma "P (x\<Colon>X)" |
|
958 |
nitpick [expect = genuine] |
|
959 |
oops |
|
960 |
||
961 |
lemma "P (y\<Colon>Y)" |
|
962 |
nitpick [expect = genuine] |
|
963 |
oops |
|
964 |
||
965 |
lemma "P (B (B A))" |
|
966 |
nitpick [expect = genuine] |
|
967 |
oops |
|
968 |
||
969 |
lemma "P (B (C F))" |
|
970 |
nitpick [expect = genuine] |
|
971 |
oops |
|
972 |
||
973 |
lemma "P (C (D A))" |
|
974 |
nitpick [expect = genuine] |
|
975 |
oops |
|
976 |
||
977 |
lemma "P (C (E F))" |
|
978 |
nitpick [expect = genuine] |
|
979 |
oops |
|
980 |
||
981 |
lemma "P (D (B A))" |
|
982 |
nitpick [expect = genuine] |
|
983 |
oops |
|
984 |
||
985 |
lemma "P (D (C F))" |
|
986 |
nitpick [expect = genuine] |
|
987 |
oops |
|
988 |
||
989 |
lemma "P (E (D A))" |
|
990 |
nitpick [expect = genuine] |
|
991 |
oops |
|
992 |
||
993 |
lemma "P (E (E F))" |
|
994 |
nitpick [expect = genuine] |
|
995 |
oops |
|
996 |
||
997 |
lemma "P (C (D (C F)))" |
|
998 |
nitpick [expect = genuine] |
|
999 |
oops |
|
1000 |
||
1001 |
lemma "X_Y_rec_1 a b c d e f A = a" |
|
1002 |
nitpick [expect = none] |
|
1003 |
apply simp |
|
1004 |
done |
|
1005 |
||
1006 |
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)" |
|
1007 |
nitpick [expect = none] |
|
1008 |
apply simp |
|
1009 |
done |
|
1010 |
||
1011 |
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)" |
|
1012 |
nitpick [expect = none] |
|
1013 |
apply simp |
|
1014 |
done |
|
1015 |
||
1016 |
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)" |
|
1017 |
nitpick [expect = none] |
|
1018 |
apply simp |
|
1019 |
done |
|
1020 |
||
1021 |
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)" |
|
1022 |
nitpick [expect = none] |
|
1023 |
apply simp |
|
1024 |
done |
|
1025 |
||
1026 |
lemma "X_Y_rec_2 a b c d e f F = f" |
|
1027 |
nitpick [expect = none] |
|
1028 |
apply simp |
|
1029 |
done |
|
1030 |
||
1031 |
lemma "P (X_Y_rec_1 a b c d e f x)" |
|
1032 |
nitpick [expect = genuine] |
|
1033 |
oops |
|
1034 |
||
1035 |
lemma "P (X_Y_rec_2 a b c d e f y)" |
|
1036 |
nitpick [expect = genuine] |
|
1037 |
oops |
|
1038 |
||
1039 |
text {* Other datatype examples *} |
|
1040 |
||
1041 |
text {* Indirect recursion is implemented via mutual recursion. *} |
|
1042 |
||
1043 |
datatype XOpt = CX "XOpt option" | DX "bool \<Rightarrow> XOpt option" |
|
1044 |
||
1045 |
lemma "P (x\<Colon>XOpt)" |
|
1046 |
nitpick [expect = genuine] |
|
1047 |
oops |
|
1048 |
||
1049 |
lemma "P (CX None)" |
|
1050 |
nitpick [expect = genuine] |
|
1051 |
oops |
|
1052 |
||
1053 |
lemma "P (CX (Some (CX None)))" |
|
1054 |
nitpick [expect = genuine] |
|
1055 |
oops |
|
1056 |
||
1057 |
lemma "XOpt_rec_1 cx dx n1 s1 n2 s2 (CX x) = cx x (XOpt_rec_2 cx dx n1 s1 n2 s2 x)" |
|
1058 |
nitpick [card = 1\<midarrow>6, expect = none] |
|
1059 |
apply simp |
|
1060 |
done |
|
1061 |
||
1062 |
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))" |
|
1063 |
nitpick [card = 1\<midarrow>4, expect = none] |
|
1064 |
apply simp |
|
1065 |
done |
|
1066 |
||
1067 |
lemma "XOpt_rec_2 cx dx n1 s1 n2 s2 None = n1" |
|
1068 |
nitpick [card = 1\<midarrow>4, expect = none] |
|
1069 |
apply simp |
|
1070 |
done |
|
1071 |
||
1072 |
lemma "XOpt_rec_2 cx dx n1 s1 n2 s2 (Some x) = s1 x (XOpt_rec_1 cx dx n1 s1 n2 s2 x)" |
|
1073 |
nitpick [card = 1\<midarrow>4, expect = none] |
|
1074 |
apply simp |
|
1075 |
done |
|
1076 |
||
1077 |
lemma "XOpt_rec_3 cx dx n1 s1 n2 s2 None = n2" |
|
1078 |
nitpick [card = 1\<midarrow>4, expect = none] |
|
1079 |
apply simp |
|
1080 |
done |
|
1081 |
||
1082 |
lemma "XOpt_rec_3 cx dx n1 s1 n2 s2 (Some x) = s2 x (XOpt_rec_1 cx dx n1 s1 n2 s2 x)" |
|
1083 |
nitpick [card = 1\<midarrow>4, expect = none] |
|
1084 |
apply simp |
|
1085 |
done |
|
1086 |
||
1087 |
lemma "P (XOpt_rec_1 cx dx n1 s1 n2 s2 x)" |
|
1088 |
nitpick [expect = genuine] |
|
1089 |
oops |
|
1090 |
||
1091 |
lemma "P (XOpt_rec_2 cx dx n1 s1 n2 s2 x)" |
|
1092 |
nitpick [expect = genuine] |
|
1093 |
oops |
|
1094 |
||
1095 |
lemma "P (XOpt_rec_3 cx dx n1 s1 n2 s2 x)" |
|
1096 |
nitpick [expect = genuine] |
|
1097 |
oops |
|
1098 |
||
1099 |
datatype 'a YOpt = CY "('a \<Rightarrow> 'a YOpt) option" |
|
1100 |
||
1101 |
lemma "P (x\<Colon>'a YOpt)" |
|
1102 |
nitpick [expect = genuine] |
|
1103 |
oops |
|
1104 |
||
1105 |
lemma "P (CY None)" |
|
1106 |
nitpick [expect = genuine] |
|
1107 |
oops |
|
1108 |
||
1109 |
lemma "P (CY (Some (\<lambda>a. CY None)))" |
|
1110 |
nitpick [expect = genuine] |
|
1111 |
oops |
|
1112 |
||
1113 |
lemma "YOpt_rec_1 cy n s (CY x) = cy x (YOpt_rec_2 cy n s x)" |
|
1114 |
nitpick [card = 1\<midarrow>2, expect = none] |
|
1115 |
apply simp |
|
1116 |
done |
|
1117 |
||
1118 |
lemma "YOpt_rec_2 cy n s None = n" |
|
1119 |
nitpick [card = 1\<midarrow>2, expect = none] |
|
1120 |
apply simp |
|
1121 |
done |
|
1122 |
||
1123 |
lemma "YOpt_rec_2 cy n s (Some x) = s x (\<lambda>a. YOpt_rec_1 cy n s (x a))" |
|
1124 |
nitpick [card = 1\<midarrow>2, expect = none] |
|
1125 |
apply simp |
|
1126 |
done |
|
1127 |
||
1128 |
lemma "P (YOpt_rec_1 cy n s x)" |
|
1129 |
nitpick [expect = genuine] |
|
1130 |
oops |
|
1131 |
||
1132 |
lemma "P (YOpt_rec_2 cy n s x)" |
|
1133 |
nitpick [expect = genuine] |
|
1134 |
oops |
|
1135 |
||
1136 |
datatype Trie = TR "Trie list" |
|
1137 |
||
1138 |
lemma "P (x\<Colon>Trie)" |
|
1139 |
nitpick [expect = genuine] |
|
1140 |
oops |
|
1141 |
||
1142 |
lemma "\<forall>x\<Colon>Trie. P x" |
|
1143 |
nitpick [expect = genuine] |
|
1144 |
oops |
|
1145 |
||
1146 |
lemma "P (TR [TR []])" |
|
1147 |
nitpick [expect = genuine] |
|
1148 |
oops |
|
1149 |
||
1150 |
lemma "Trie_rec_1 tr nil cons (TR x) = tr x (Trie_rec_2 tr nil cons x)" |
|
1151 |
nitpick [card = 1\<midarrow>6, expect = none] |
|
1152 |
apply simp |
|
1153 |
done |
|
1154 |
||
1155 |
lemma "Trie_rec_2 tr nil cons [] = nil" |
|
1156 |
nitpick [card = 1\<midarrow>6, expect = none] |
|
1157 |
apply simp |
|
1158 |
done |
|
1159 |
||
1160 |
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)" |
|
1161 |
nitpick [card = 1\<midarrow>6, expect = none] |
|
1162 |
apply simp |
|
1163 |
done |
|
1164 |
||
1165 |
lemma "P (Trie_rec_1 tr nil cons x)" |
|
1166 |
nitpick [card = 1, expect = genuine] |
|
1167 |
oops |
|
1168 |
||
1169 |
lemma "P (Trie_rec_2 tr nil cons x)" |
|
1170 |
nitpick [card = 1, expect = genuine] |
|
1171 |
oops |
|
1172 |
||
1173 |
datatype InfTree = Leaf | Node "nat \<Rightarrow> InfTree" |
|
1174 |
||
1175 |
lemma "P (x\<Colon>InfTree)" |
|
1176 |
nitpick [expect = genuine] |
|
1177 |
oops |
|
1178 |
||
1179 |
lemma "\<forall>x\<Colon>InfTree. P x" |
|
1180 |
nitpick [expect = genuine] |
|
1181 |
oops |
|
1182 |
||
1183 |
lemma "P (Node (\<lambda>n. Leaf))" |
|
1184 |
nitpick [expect = genuine] |
|
1185 |
oops |
|
1186 |
||
1187 |
lemma "InfTree_rec leaf node Leaf = leaf" |
|
1188 |
nitpick [card = 1\<midarrow>3, expect = none] |
|
1189 |
apply simp |
|
1190 |
done |
|
1191 |
||
1192 |
lemma "InfTree_rec leaf node (Node x) = node x (\<lambda>n. InfTree_rec leaf node (x n))" |
|
1193 |
nitpick [card = 1\<midarrow>3, expect = none] |
|
1194 |
apply simp |
|
1195 |
done |
|
1196 |
||
1197 |
lemma "P (InfTree_rec leaf node x)" |
|
1198 |
nitpick [expect = genuine] |
|
1199 |
oops |
|
1200 |
||
1201 |
datatype 'a lambda = Var 'a | App "'a lambda" "'a lambda" | Lam "'a \<Rightarrow> 'a lambda" |
|
1202 |
||
1203 |
lemma "P (x\<Colon>'a lambda)" |
|
1204 |
nitpick [expect = genuine] |
|
1205 |
oops |
|
1206 |
||
1207 |
lemma "\<forall>x\<Colon>'a lambda. P x" |
|
1208 |
nitpick [expect = genuine] |
|
1209 |
oops |
|
1210 |
||
1211 |
lemma "P (Lam (\<lambda>a. Var a))" |
|
1212 |
nitpick [card = 1\<midarrow>5, expect = none] |
|
1213 |
nitpick [card 'a = 4, card "'a lambda" = 5, expect = genuine] |
|
1214 |
oops |
|
1215 |
||
1216 |
lemma "lambda_rec var app lam (Var x) = var x" |
|
1217 |
nitpick [card = 1\<midarrow>3, expect = none] |
|
1218 |
apply simp |
|
1219 |
done |
|
1220 |
||
1221 |
lemma "lambda_rec var app lam (App x y) = app x y (lambda_rec var app lam x) (lambda_rec var app lam y)" |
|
1222 |
nitpick [card = 1\<midarrow>3, expect = none] |
|
1223 |
apply simp |
|
1224 |
done |
|
1225 |
||
1226 |
lemma "lambda_rec var app lam (Lam x) = lam x (\<lambda>a. lambda_rec var app lam (x a))" |
|
1227 |
nitpick [card = 1\<midarrow>3, expect = none] |
|
1228 |
apply simp |
|
1229 |
done |
|
1230 |
||
1231 |
lemma "P (lambda_rec v a l x)" |
|
1232 |
nitpick [expect = genuine] |
|
1233 |
oops |
|
1234 |
||
1235 |
text {* Taken from "Inductive datatypes in HOL", p. 8: *} |
|
1236 |
||
1237 |
datatype ('a, 'b) T = C "'a \<Rightarrow> bool" | D "'b list" |
|
1238 |
datatype 'c U = E "('c, 'c U) T" |
|
1239 |
||
1240 |
lemma "P (x\<Colon>'c U)" |
|
1241 |
nitpick [expect = genuine] |
|
1242 |
oops |
|
1243 |
||
1244 |
lemma "\<forall>x\<Colon>'c U. P x" |
|
1245 |
nitpick [expect = genuine] |
|
1246 |
oops |
|
1247 |
||
1248 |
lemma "P (E (C (\<lambda>a. True)))" |
|
1249 |
nitpick [expect = genuine] |
|
1250 |
oops |
|
1251 |
||
1252 |
lemma "U_rec_1 e c d nil cons (E x) = e x (U_rec_2 e c d nil cons x)" |
|
1253 |
nitpick [card = 1\<midarrow>3, expect = none] |
|
1254 |
apply simp |
|
1255 |
done |
|
1256 |
||
1257 |
lemma "U_rec_2 e c d nil cons (C x) = c x" |
|
1258 |
nitpick [card = 1\<midarrow>3, expect = none] |
|
1259 |
apply simp |
|
1260 |
done |
|
1261 |
||
1262 |
lemma "U_rec_2 e c d nil cons (D x) = d x (U_rec_3 e c d nil cons x)" |
|
1263 |
nitpick [card = 1\<midarrow>3, expect = none] |
|
1264 |
apply simp |
|
1265 |
done |
|
1266 |
||
1267 |
lemma "U_rec_3 e c d nil cons [] = nil" |
|
1268 |
nitpick [card = 1\<midarrow>3, expect = none] |
|
1269 |
apply simp |
|
1270 |
done |
|
1271 |
||
1272 |
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)" |
|
1273 |
nitpick [card = 1\<midarrow>3, expect = none] |
|
1274 |
apply simp |
|
1275 |
done |
|
1276 |
||
1277 |
lemma "P (U_rec_1 e c d nil cons x)" |
|
1278 |
nitpick [expect = genuine] |
|
1279 |
oops |
|
1280 |
||
1281 |
lemma "P (U_rec_2 e c d nil cons x)" |
|
1282 |
nitpick [card = 1, expect = genuine] |
|
1283 |
oops |
|
1284 |
||
1285 |
lemma "P (U_rec_3 e c d nil cons x)" |
|
1286 |
nitpick [card = 1, expect = genuine] |
|
1287 |
oops |
|
1288 |
||
1289 |
subsubsection {* Records *} |
|
1290 |
||
1291 |
record ('a, 'b) point = |
|
1292 |
xpos :: 'a |
|
1293 |
ypos :: 'b |
|
1294 |
||
1295 |
lemma "(x\<Colon>('a, 'b) point) = y" |
|
1296 |
nitpick [expect = genuine] |
|
1297 |
oops |
|
1298 |
||
1299 |
record ('a, 'b, 'c) extpoint = "('a, 'b) point" + |
|
1300 |
ext :: 'c |
|
1301 |
||
1302 |
lemma "(x\<Colon>('a, 'b, 'c) extpoint) = y" |
|
1303 |
nitpick [expect = genuine] |
|
1304 |
oops |
|
1305 |
||
1306 |
subsubsection {* Inductively Defined Sets *} |
|
1307 |
||
1308 |
inductive_set undefinedSet :: "'a set" where |
|
1309 |
"undefined \<in> undefinedSet" |
|
1310 |
||
1311 |
lemma "x \<in> undefinedSet" |
|
1312 |
nitpick [expect = genuine] |
|
1313 |
oops |
|
1314 |
||
1315 |
inductive_set evenCard :: "'a set set" |
|
1316 |
where |
|
1317 |
"{} \<in> evenCard" | |
|
1318 |
"\<lbrakk>S \<in> evenCard; x \<notin> S; y \<notin> S; x \<noteq> y\<rbrakk> \<Longrightarrow> S \<union> {x, y} \<in> evenCard" |
|
1319 |
||
1320 |
lemma "S \<in> evenCard" |
|
1321 |
nitpick [expect = genuine] |
|
1322 |
oops |
|
1323 |
||
1324 |
inductive_set |
|
1325 |
even :: "nat set" |
|
1326 |
and odd :: "nat set" |
|
1327 |
where |
|
1328 |
"0 \<in> even" | |
|
1329 |
"n \<in> even \<Longrightarrow> Suc n \<in> odd" | |
|
1330 |
"n \<in> odd \<Longrightarrow> Suc n \<in> even" |
|
1331 |
||
1332 |
lemma "n \<in> odd" |
|
1333 |
nitpick [expect = genuine] |
|
1334 |
oops |
|
1335 |
||
1336 |
consts f :: "'a \<Rightarrow> 'a" |
|
1337 |
||
1338 |
inductive_set a_even :: "'a set" and a_odd :: "'a set" where |
|
1339 |
"undefined \<in> a_even" | |
|
1340 |
"x \<in> a_even \<Longrightarrow> f x \<in> a_odd" | |
|
1341 |
"x \<in> a_odd \<Longrightarrow> f x \<in> a_even" |
|
1342 |
||
1343 |
lemma "x \<in> a_odd" |
|
1344 |
nitpick [expect = genuine] |
|
1345 |
oops |
|
1346 |
||
1347 |
subsubsection {* Examples Involving Special Functions *} |
|
1348 |
||
1349 |
lemma "card x = 0" |
|
1350 |
nitpick [expect = genuine] |
|
1351 |
oops |
|
1352 |
||
1353 |
lemma "finite x" |
|
1354 |
nitpick [expect = none] |
|
1355 |
oops |
|
1356 |
||
1357 |
lemma "xs @ [] = ys @ []" |
|
1358 |
nitpick [expect = genuine] |
|
1359 |
oops |
|
1360 |
||
1361 |
lemma "xs @ ys = ys @ xs" |
|
1362 |
nitpick [expect = genuine] |
|
1363 |
oops |
|
1364 |
||
1365 |
lemma "f (lfp f) = lfp f" |
|
1366 |
nitpick [expect = genuine] |
|
1367 |
oops |
|
1368 |
||
1369 |
lemma "f (gfp f) = gfp f" |
|
1370 |
nitpick [expect = genuine] |
|
1371 |
oops |
|
1372 |
||
1373 |
lemma "lfp f = gfp f" |
|
1374 |
nitpick [expect = genuine] |
|
1375 |
oops |
|
1376 |
||
1377 |
subsubsection {* Axiomatic Type Classes and Overloading *} |
|
1378 |
||
1379 |
text {* A type class without axioms: *} |
|
1380 |
||
1381 |
axclass classA |
|
1382 |
||
1383 |
lemma "P (x\<Colon>'a\<Colon>classA)" |
|
1384 |
nitpick [expect = genuine] |
|
1385 |
oops |
|
1386 |
||
1387 |
text {* The axiom of this type class does not contain any type variables: *} |
|
1388 |
||
1389 |
axclass classB |
|
1390 |
classB_ax: "P \<or> \<not> P" |
|
1391 |
||
1392 |
lemma "P (x\<Colon>'a\<Colon>classB)" |
|
1393 |
nitpick [expect = genuine] |
|
1394 |
oops |
|
1395 |
||
1396 |
text {* An axiom with a type variable (denoting types which have at least two elements): *} |
|
1397 |
||
1398 |
axclass classC < type |
|
1399 |
classC_ax: "\<exists>x y. x \<noteq> y" |
|
1400 |
||
1401 |
lemma "P (x\<Colon>'a\<Colon>classC)" |
|
1402 |
nitpick [expect = genuine] |
|
1403 |
oops |
|
1404 |
||
1405 |
lemma "\<exists>x y. (x\<Colon>'a\<Colon>classC) \<noteq> y" |
|
1406 |
nitpick [expect = none] |
|
1407 |
sorry |
|
1408 |
||
1409 |
text {* A type class for which a constant is defined: *} |
|
1410 |
||
1411 |
consts |
|
1412 |
classD_const :: "'a \<Rightarrow> 'a" |
|
1413 |
||
1414 |
axclass classD < type |
|
1415 |
classD_ax: "classD_const (classD_const x) = classD_const x" |
|
1416 |
||
1417 |
lemma "P (x\<Colon>'a\<Colon>classD)" |
|
1418 |
nitpick [expect = genuine] |
|
1419 |
oops |
|
1420 |
||
1421 |
text {* A type class with multiple superclasses: *} |
|
1422 |
||
1423 |
axclass classE < classC, classD |
|
1424 |
||
1425 |
lemma "P (x\<Colon>'a\<Colon>classE)" |
|
1426 |
nitpick [expect = genuine] |
|
1427 |
oops |
|
1428 |
||
1429 |
lemma "P (x\<Colon>'a\<Colon>{classB, classE})" |
|
1430 |
nitpick [expect = genuine] |
|
1431 |
oops |
|
1432 |
||
1433 |
text {* OFCLASS: *} |
|
1434 |
||
1435 |
lemma "OFCLASS('a\<Colon>type, type_class)" |
|
1436 |
nitpick [expect = none] |
|
1437 |
apply intro_classes |
|
1438 |
done |
|
1439 |
||
1440 |
lemma "OFCLASS('a\<Colon>classC, type_class)" |
|
1441 |
nitpick [expect = none] |
|
1442 |
apply intro_classes |
|
1443 |
done |
|
1444 |
||
1445 |
lemma "OFCLASS('a, classB_class)" |
|
1446 |
nitpick [expect = none] |
|
1447 |
apply intro_classes |
|
1448 |
apply simp |
|
1449 |
done |
|
1450 |
||
1451 |
lemma "OFCLASS('a\<Colon>type, classC_class)" |
|
1452 |
nitpick [expect = genuine] |
|
1453 |
oops |
|
1454 |
||
1455 |
text {* Overloading: *} |
|
1456 |
||
1457 |
consts inverse :: "'a \<Rightarrow> 'a" |
|
1458 |
||
1459 |
defs (overloaded) |
|
1460 |
inverse_bool: "inverse (b\<Colon>bool) \<equiv> \<not> b" |
|
1461 |
inverse_set: "inverse (S\<Colon>'a set) \<equiv> -S" |
|
1462 |
inverse_pair: "inverse p \<equiv> (inverse (fst p), inverse (snd p))" |
|
1463 |
||
1464 |
lemma "inverse b" |
|
1465 |
nitpick [expect = genuine] |
|
1466 |
oops |
|
1467 |
||
1468 |
lemma "P (inverse (S\<Colon>'a set))" |
|
1469 |
nitpick [expect = genuine] |
|
1470 |
oops |
|
1471 |
||
1472 |
lemma "P (inverse (p\<Colon>'a\<times>'b))" |
|
1473 |
nitpick [expect = genuine] |
|
1474 |
oops |
|
1475 |
||
1476 |
end |