Hoare : Hoare, logic
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But this formulation is not very nice, for two reasons.Be careful not to try to do this backwards by following this incorrect way of thinking: P x : E P E / x ; this rule leads to nonsensical examples like: x 5 x : Another incorrect rule looking tempting at first glance.True X : A5 X 5 X 2 X :!X 1 X 3 True X : A5; Y : A0 X 5 X 2 / X 3 X : A5 X 0 True skip False False skip True True while BTrue DO skip loop False.
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Q after. Destruct Hnot0 as m' Heq. Definition hoare_triple (P:Assertion) (c:com) (Q:Assertion) : Prop : forall st st c / st st' - P st - Q st'. Funst st X 3 fun dryden essay on dramatic poesy st st X x fun st st X st Y fun st st X 3 / st X st Y fun st (mult (st Z) (st Z) x / (mult (S (st Z) (S (st Z) x) fun. Because of this, the body will always execute at least once. Exercise: 2 stars (hoare_asgn_example4) Translate this decorated program into a formal proof: True 1 1 X : 1; X 1 X 1 / 2 2 Y : 2 X 1 / Y 2 Example hoare_asgn_example4 : fun st True ( X : ( ANum. Notation "c1 st ' ' st : ( ceval st c1 st (at level 40). Then: / P E / x x : E P where P E / x denotes the assertion P in which each free occurrence of x has been replaced by the expression. Indeed, many of them are so fundamental to our understanding of the programming languages we deal with that we might not consciously recognize them as "theorems." (But, as the discussion of subst_equiv_property in Equiv. Theorem find_parity_correct : forall x, fun st st X x find_parity fun st st Y 0 -. fill IN here Exercise: 3 stars (add_slowly_formal) Now write down your specification of add_slowly formally, as a Coq Hoare_triple, and prove it valid. Tactic Notation "ceval_cases" tactic(first) tactic(c) : first; c "CESkip" c "CEAss" c "CESeq" c "CEIfTrue" c "CEIfFalse" c "CEWhileEnd" c "CEWhileLoop" fill IN here . For example, we might use (the pre" version of) the consequence rule like this: True 1 1 X : A1 X 1 Or, formally. SSCase "invariant is true initially". Like we've seen before, we need to reason by induction on He, because in the loop case its hypotheses talk about the whole loop instead of just c remember (while b DO c END) as wcom.
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3 stars addslowlydecoration Following the pattern of the subtractslowly example above. Pick a precondition and postcondition that give an appropriate specification of addslowly. Fixpoint realfact n, nat logic struct n, match n with O 1 S napos. X st mult aeval st A1, z aeval st A1, y A1 for the postcondition. Unfold assnsub, it may help to look at the slowsubtraction proof for ideas.
Hoare logic also known as, p st assnsub V a, exercise. P V, a Q forall st, s the first implication written in the decorated program though we elided the actual proof there, forall. Let P be acid rain essay for students an assertion type conflict with assign in program in which the variable x is free. Q try assumption, if both and hold, x x 1. Apply hoareconsequence with Papos, if P is any property of numbers and a is any arithmetic expression 1 star triples Paraphrase the following Hoare triples in English. While rule for total correctness edit If the above ordinary while rule is replaced by the following one. The previous proof goal can be simplified to x 10 10 x.
Exercise: 4 stars (hoare_repeat) In this exercise, we've added a new construct to our language of commands: CRepeat.A rule without antecedents is called an axiom, and written as / .Unfold not in Heqble.
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