theory Hoare_Examples imports Hoare begin

text{* Improves proof automation for negative numerals: *}

lemma add_neg1R[simp]:

"x + -1 = x - (1 :: int)"

by arith

lemma add_neg_numeralR[simp]:

"x + neg_numeral n = (x::'a::neg_numeral) - numeral(n)"

by (simp only: diff_minus_eq_add[symmetric] minus_neg_numeral)

text{* Summing up the first @{text x} natural numbers in variable @{text y}. *}

fun sum :: "int => int" where

"sum i = (if i ≤ 0 then 0 else sum (i - 1) + i)"

lemma sum_simps[simp]:

"0 < i ==> sum i = sum (i - 1) + i"

"i ≤ 0 ==> sum i = 0"

by(simp_all)

declare sum.simps[simp del]

abbreviation "wsum ==

WHILE Less (N 0) (V ''x'')

DO (''y'' ::= Plus (V ''y'') (V ''x'');;

''x'' ::= Plus (V ''x'') (N -1))"

subsubsection{* Proof by Operational Semantics *}

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

lemma while_sum:

"(wsum, s) => t ==> t ''y'' = s ''y'' + sum(s ''x'')"

apply(induction wsum s t rule: big_step_induct)

apply(auto)

done

text{* We were lucky that the proof was 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 "(''y'' ::= N 0;; wsum, s) => t"

shows "t ''y'' = sum (s ''x'')"

proof -

from assms have "(wsum,s(''y'':=0)) => 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> {λs. s ''x'' = n} ''y'' ::= N 0;; wsum {λs. s ''y'' = sum n}"

apply(rule Seq)

prefer 2

apply(rule While' [where P = "λs. (s ''y'' = sum n - sum(s ''x''))"])

apply(rule Seq)

prefer 2

apply(rule Assign)

apply(rule Assign')

apply simp

apply(simp)

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