summary |
shortlog |
changelog |
graph |
tags |
bookmarks |
branches |
files |
changeset |
file |
latest |
revisions |
annotate |
diff |
comparison |
raw |
help

src/HOL/IMP/Hoare_Examples.thy

author | wenzelm |

Sat, 07 Apr 2012 16:41:59 +0200 | |

changeset 47389 | e8552cba702d |

parent 45015 | fdac1e9880eb |

child 47818 | 151d137f1095 |

permissions | -rw-r--r-- |

explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions;
tuned;

(* Author: Tobias Nipkow *) theory Hoare_Examples imports Hoare begin subsection{* Example: Sums *} text{* Summing up the first @{text n} natural numbers. The sum is accumulated in variable @{text 0}, the loop counter is variable @{text 1}. *} abbreviation "w n == WHILE Less (V ''y'') (N n) DO ( ''y'' ::= Plus (V ''y'') (N 1); ''x'' ::= Plus (V ''x'') (V ''y'') )" text{* For this example we make use of some predefined functions. Function @{const Setsum}, also written @{text"\<Sum>"}, sums up the elements of a set. The set of numbers from @{text m} to @{text n} is written @{term "{m .. n}"}. *} subsubsection{* Proof by Operational Semantics *} text{* The behaviour of the loop is proved by induction: *} lemma setsum_head_plus_1: "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {m+1..n::int}" by (subst simp_from_to) simp lemma while_sum: "(w n, s) \<Rightarrow> t \<Longrightarrow> t ''x'' = s ''x'' + \<Sum> {s ''y'' + 1 .. n}" apply(induction "w n" s t rule: big_step_induct) apply(auto simp add: setsum_head_plus_1) done text{* We were lucky that the proof was practically automatic, except for the induction. In general, such proofs will not be so easy. The automation is partly due to the right inversion rules that we set up as automatic elimination rules that decompose big-step premises. Now we prefix the loop with the necessary initialization: *} lemma sum_via_bigstep: assumes "(''x'' ::= N 0; ''y'' ::= N 0; w n, s) \<Rightarrow> t" shows "t ''x'' = \<Sum> {1 .. n}" proof - from assms have "(w n,s(''x'':=0,''y'':=0)) \<Rightarrow> t" by auto from while_sum[OF this] show ?thesis by simp qed subsubsection{* Proof by Hoare Logic *} text{* Note that we deal with sequences of commands from right to left, pulling back the postcondition towards the precondition. *} lemma "\<turnstile> {\<lambda>s. 0 <= n} ''x'' ::= N 0; ''y'' ::= N 0; w n {\<lambda>s. s ''x'' = \<Sum> {1 .. n}}" apply(rule hoare.Semi) prefer 2 apply(rule While' [where P = "\<lambda>s. s ''x'' = \<Sum> {1..s ''y''} \<and> 0 \<le> s ''y'' \<and> s ''y'' \<le> n"]) apply(rule Semi) prefer 2 apply(rule Assign) apply(rule Assign') apply(fastforce simp: atLeastAtMostPlus1_int_conv algebra_simps) apply(fastforce) apply(rule Semi) prefer 2 apply(rule Assign) apply(rule Assign') apply simp done text{* The proof is intentionally an apply skript because it merely composes the rules of Hoare logic. Of course, in a few places side conditions have to be proved. But since those proofs are 1-liners, a structured proof is overkill. In fact, we shall learn later that the application of the Hoare rules can be automated completely and all that is left for the user is to provide the loop invariants and prove the side-conditions. *} end