summary |
shortlog |
changelog |
graph |
tags |
branches |
files |
changeset |
file |
revisions |
annotate |
diff |
raw

src/HOL/IMP/Hoare_Examples.thy

author | huffman |

Thu Aug 11 09:11:15 2011 -0700 (2011-08-11) | |

changeset 44165 | d26a45f3c835 |

parent 43158 | 686fa0a0696e |

child 44177 | b4b5cbca2519 |

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

remove lemma stupid_ext

1 (* Author: Tobias Nipkow *)

3 theory Hoare_Examples imports Hoare Util begin

5 subsection{* Example: Sums *}

7 text{* Summing up the first @{text n} natural numbers. The sum is accumulated

8 in variable @{text 0}, the loop counter is variable @{text 1}. *}

10 abbreviation "w n ==

11 WHILE Less (V ''y'') (N n)

12 DO ( ''y'' ::= Plus (V ''y'') (N 1); ''x'' ::= Plus (V ''x'') (V ''y'') )"

14 text{* For this example we make use of some predefined functions. Function

15 @{const Setsum}, also written @{text"\<Sum>"}, sums up the elements of a set. The

16 set of numbers from @{text m} to @{text n} is written @{term "{m .. n}"}. *}

18 subsubsection{* Proof by Operational Semantics *}

20 text{* The behaviour of the loop is proved by induction: *}

22 lemma setsum_head_plus_1:

23 "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {m+1..n::int}"

24 by (subst simp_from_to) simp

26 lemma while_sum:

27 "(w n, s) \<Rightarrow> t \<Longrightarrow> t ''x'' = s ''x'' + \<Sum> {s ''y'' + 1 .. n}"

28 apply(induct "w n" s t rule: big_step_induct)

29 apply(auto simp add: setsum_head_plus_1)

30 done

32 text{* We were lucky that the proof was practically automatic, except for the

33 induction. In general, such proofs will not be so easy. The automation is

34 partly due to the right inversion rules that we set up as automatic

35 elimination rules that decompose big-step premises.

37 Now we prefix the loop with the necessary initialization: *}

39 lemma sum_via_bigstep:

40 assumes "(''x'' ::= N 0; ''y'' ::= N 0; w n, s) \<Rightarrow> t"

41 shows "t ''x'' = \<Sum> {1 .. n}"

42 proof -

43 from assms have "(w n,s(''x'':=0,''y'':=0)) \<Rightarrow> t" by auto

44 from while_sum[OF this] show ?thesis by simp

45 qed

48 subsubsection{* Proof by Hoare Logic *}

50 text{* Note that we deal with sequences of commands from right to left,

51 pulling back the postcondition towards the precondition. *}

53 lemma "\<turnstile> {\<lambda>s. 0 <= n} ''x'' ::= N 0; ''y'' ::= N 0; w n {\<lambda>s. s ''x'' = \<Sum> {1 .. n}}"

54 apply(rule hoare.Semi)

55 prefer 2

56 apply(rule While'

57 [where P = "\<lambda>s. s ''x'' = \<Sum> {1..s ''y''} \<and> 0 \<le> s ''y'' \<and> s ''y'' \<le> n"])

58 apply(rule Semi)

59 prefer 2

60 apply(rule Assign)

61 apply(rule Assign')

62 apply(fastsimp simp: atLeastAtMostPlus1_int_conv algebra_simps)

63 apply(fastsimp)

64 apply(rule Semi)

65 prefer 2

66 apply(rule Assign)

67 apply(rule Assign')

68 apply simp

69 done

71 text{* The proof is intentionally an apply skript because it merely composes

72 the rules of Hoare logic. Of course, in a few places side conditions have to

73 be proved. But since those proofs are 1-liners, a structured proof is

74 overkill. In fact, we shall learn later that the application of the Hoare

75 rules can be automated completely and all that is left for the user is to

76 provide the loop invariants and prove the side-conditions. *}

78 end