src/HOL/ex/InductiveInvariant_examples.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/InductiveInvariant_examples.thy	Wed Oct 22 10:52:36 2003 +0200
@@ -0,0 +1,127 @@
+theory InductiveInvariant_examples = InductiveInvariant :
+
+(** Authors: Sava Krsti\'{c} and John Matthews **)
+(**    Date: Oct 17, 2003                      **)
+
+text "A simple example showing how to use an inductive invariant
+      to solve termination conditions generated by recdef on
+      nested recursive function definitions."
+
+consts g :: "nat => nat"
+
+recdef (permissive) g "less_than"
+  "g 0 = 0"
+  "g (Suc n) = g (g n)"
+
+text "We can prove the unsolved termination condition for
+      g by showing it is an inductive invariant."
+
+recdef_tc g_tc[simp]: g
+apply (rule allI)
+apply (rule_tac x=n in tfl_indinv_wfrec [OF g_def])
+apply (auto simp add: indinv_def split: nat.split)
+apply (frule_tac x=nat in spec)
+apply (drule_tac x="f nat" in spec)
+by auto
+
+
+text "This declaration invokes Isabelle's simplifier to
+      remove any termination conditions before adding
+      g's rules to the simpset."
+declare g.simps [simplified, simp]
+
+
+text "This is an example where the termination condition generated
+      by recdef is not itself an inductive invariant."
+
+consts g' :: "nat => nat"
+recdef (permissive) g' "less_than"
+  "g' 0 = 0"
+  "g' (Suc n) = g' n + g' (g' n)"
+
+thm g'.simps
+
+
+text "The strengthened inductive invariant is as follows
+      (this invariant also works for the first example above):"
+
+lemma g'_inv: "g' n = 0"
+thm tfl_indinv_wfrec [OF g'_def]
+apply (rule_tac x=n in tfl_indinv_wfrec [OF g'_def])
+by (auto simp add: indinv_def split: nat.split)
+
+recdef_tc g'_tc[simp]: g'
+by (simp add: g'_inv)
+
+text "Now we can remove the termination condition from
+      the rules for g' ."
+thm g'.simps [simplified]
+
+
+text {* Sometimes a recursive definition is partial, that is, it
+        is only meant to be invoked on "good" inputs. As a contrived
+        example, we will define a new version of g that is only
+        well defined for even inputs greater than zero. *}
+
+consts g_even :: "nat => nat"
+recdef (permissive) g_even "less_than"
+  "g_even (Suc (Suc 0)) = 3"
+  "g_even n = g_even (g_even (n - 2) - 1)"
+
+
+text "We can prove a conditional version of the unsolved termination
+      condition for @{term g_even} by proving a stronger inductive invariant."
+
+lemma g_even_indinv: "\<exists>k. n = Suc (Suc (2*k)) ==> g_even n = 3"
+apply (rule_tac D="{n. \<exists>k. n = Suc (Suc (2*k))}" and x=n in tfl_indinv_on_wfrec [OF g_even_def])
+apply (auto simp add: indinv_on_def split: nat.split)
+by (case_tac ka, auto)
+
+
+text "Now we can prove that the second recursion equation for @{term g_even}
+      holds, provided that n is an even number greater than two."
+
+theorem g_even_n: "\<exists>k. n = 2*k + 4 ==> g_even n = g_even (g_even (n - 2) - 1)"
+apply (subgoal_tac "(\<exists>k. n - 2 = 2*k + 2) & (\<exists>k. n = 2*k + 2)")
+by (auto simp add: g_even_indinv, arith)
+
+
+text "McCarthy's ninety-one function. This function requires a
+      non-standard measure to prove termination."
+
+consts ninety_one :: "nat => nat"
+recdef (permissive) ninety_one "measure (%n. 101 - n)"
+  "ninety_one x = (if 100 < x
+                     then x - 10
+                     else (ninety_one (ninety_one (x+11))))"
+
+text "To discharge the termination condition, we will prove
+      a strengthened inductive invariant:
+         S x y == x < y + 11"
+
+lemma ninety_one_inv: "n < ninety_one n + 11"
+apply (rule_tac x=n in tfl_indinv_wfrec [OF ninety_one_def])
+apply force
+apply (auto simp add: indinv_def measure_def inv_image_def)
+apply (frule_tac x="x+11" in spec)
+apply (frule_tac x="f (x + 11)" in spec)
+by arith
+
+text "Proving the termination condition using the
+      strengthened inductive invariant."
+
+recdef_tc ninety_one_tc[rule_format]: ninety_one
+apply clarify
+by (cut_tac n="x+11" in ninety_one_inv, arith)
+
+text "Now we can remove the termination condition from
+      the simplification rule for @{term ninety_one}."
+
+theorem def_ninety_one:
+"ninety_one x = (if 100 < x
+                   then x - 10
+                   else ninety_one (ninety_one (x+11)))"
+by (subst ninety_one.simps,
+    simp add: ninety_one_tc measure_def inv_image_def)
+
+end
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