2 theory termination = examples:
6 When a function~$f$ is defined via \isacommand{recdef}, Isabelle tries to prove
7 its termination with the help of the user-supplied measure. Each of the examples
8 above is simple enough that Isabelle can automatically prove that the
9 argument's measure decreases in each recursive call. As a result,
10 $f$@{text".simps"} will contain the defining equations (or variants derived
11 from them) as theorems. For example, look (via \isacommand{thm}) at
12 @{thm[source]sep.simps} and @{thm[source]sep1.simps} to see that they define
13 the same function. What is more, those equations are automatically declared as
16 Isabelle may fail to prove the termination condition for some
17 recursive call. Let us try the following artificial function:*}
19 consts f :: "nat\<times>nat \<Rightarrow> nat"
20 recdef (*<*)(permissive)(*>*)f "measure(\<lambda>(x,y). x-y)"
21 "f(x,y) = (if x \<le> y then x else f(x,y+1))"
25 \REMARK{error or warning? change this part? rename g to f?}
26 message showing you what it was unable to prove. You will then
27 have to prove it as a separate lemma before you attempt the definition
28 of your function once more. In our case the required lemma is the obvious one:
31 lemma termi_lem: "\<not> x \<le> y \<Longrightarrow> x - Suc y < x - y"
34 It was not proved automatically because of the awkward behaviour of subtraction
35 on type @{typ"nat"}. This requires more arithmetic than is tried by default:
42 Because \isacommand{recdef}'s termination prover involves simplification,
43 we include in our second attempt a hint: the \attrdx{recdef_simp} attribute
44 says to use @{thm[source]termi_lem} as
45 a simplification rule.
48 consts g :: "nat\<times>nat \<Rightarrow> nat"
49 recdef g "measure(\<lambda>(x,y). x-y)"
50 "g(x,y) = (if x \<le> y then x else g(x,y+1))"
51 (hints recdef_simp: termi_lem)
54 This time everything works fine. Now @{thm[source]g.simps} contains precisely
55 the stated recursion equation for @{term g}, which has been stored as a
56 simplification rule. Thus we can automatically prove results such as this one:
59 theorem "g(1,0) = g(1,1)"
64 More exciting theorems require induction, which is discussed below.
66 If the termination proof requires a new lemma that is of general use, you can
67 turn it permanently into a simplification rule, in which case the above
68 \isacommand{hint} is not necessary. But our @{thm[source]termi_lem} is not
69 sufficiently general to warrant this distinction.
71 The attentive reader may wonder why we chose to call our function @{term g}
72 rather than @{term f} the second time around. The reason is that, despite
73 the failed termination proof, the definition of @{term f} did not
74 fail, and thus we could not define it a second time. However, all theorems
75 about @{term f}, for example @{thm[source]f.simps}, carry as a precondition
76 the unproved termination condition. Moreover, the theorems
77 @{thm[source]f.simps} are not stored as simplification rules.
78 However, this mechanism
79 allows a delayed proof of termination: instead of proving
80 @{thm[source]termi_lem} up front, we could prove
81 it later on and then use it to remove the preconditions from the theorems
82 about @{term f}. In most cases this is more cumbersome than proving things
84 \REMARK{FIXME, with one exception: nested recursion.}