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Notes #4 - Individual steps(p3 x = 100 p4 easy{p1 read x{p2...

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Individual steps: ( p 3 x = 100) p 4 : easy. { p 1 } read x { p 2 } : easy from the read axiom. p 2 p 3 : easy from math logic. { p 3 x 6 = 100 } write x ; read x { p 3 } : { p 0 3 } write x ; read x { p 3 } where p 0 3 [ OUT ˆ x ˆ head ( IN tail ( IN ) = h 1 , 2 , . . . , 100 i ] and it is easy to see: ( p 3 x 6 = 100) p 0 3 . (Do we need x 6 = 100 on the leftside of the above implication?) 57
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To derive: { IN = h 1 , 2 , 3 , . . . , 100 i } s := 0; read x ; while x 6 = 100 do s := s + x ; read x ; end { s = Σ 99 k =0 k } Proof Outline: { IN = h 1 , 2 , 3 , . . . , 100 i } s := 0; read x ; { s = 0 x = 1 IN = h 2 , 3 , . . . , 100 i } { # IN = 100 - x ∧ ∀ k # IN.IN [ k ] = x + k 1 x 100 s = Σ x - 1 k =0 k } while x 6 = 100 do s := s + x ; read x ; end { s = Σ 99 k =0 k } A more intuitive loop invariant: IN = h x + 1 , . . . , 100 i ∧ 1 x 100 . . . We will use p 1 , p 2 , p 3 , p 4 to denote the pre-condition, the post-condition of the first read , the loop invariant, and the post-condition of the pro- gram, respectively. 58
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The individual steps: { p 1 } s := 0; read x { p 2 } : easy exercise. p 2 p 3 : easy. ( p 3 x = 100) p 4 : easy. { p 3 x 6 = 100 } s := s + x ; read x { p 3 } : From read ax., assign ax., seq. comp., { p 0 3 } s := s + x ; read x { p 3 } where p 0 3 [# tail ( IN ) = 100 - head ( IN ) ∧∀ k 100 - head ( IN ) .tail ( IN )[ k ] = head ( IN )+ k 1 head ( IN ) 100 s + x = Σ head ( IN ) - 1 k =0 k ] We can easily see [( p 3 x 6 = 100) p 0 3 ]. The clause ( head ( IN ) = x + 1) which can be derived from k 100 - x.IN [ k ] = x + 1 of p 3 , plays an important role in establishing this implication. 59
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Incompleteness : Our rule for while : p q, { q b } S { q } , ( q ∧ ¬ b ) r { p } while b do S { r } A simpler rule: p q, { q } S { q } , ( q ∧ ¬ b ) r { p } while b do S { r } We can derive: { x = 0 } while x 6 = 10 do x := x + 1 { x = 10 } Take p ( x = 0) , q true , r ( x = 10) 1. p q (easy) 2. { true } x :=
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