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Hoare Logic 2

Hoare Logic 2. slides by Chris Osborn. Hoare Triple. P { …code…} Q. P[e/x] { x := e } P. P { C 1 } R. R { C 2 } Q. P { C 1 ; C 2 } Q. P ∧ ¬ b { C 2 } Q. P ∧ b { C 1 } Q. P { if b then C 1 else C 2 } Q. While Rule. P ∧ b { C } P. P {While b C} P ∧ ¬ b.

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Hoare Logic 2

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  1. Hoare Logic 2 slides by Chris Osborn

  2. Hoare Triple P { …code…} Q

  3. P[e/x] { x := e } P P { C1 } R R { C2 } Q P { C1; C2 } Q P ∧¬ b { C2 } Q P ∧ b { C1 } Q P { if b then C1 else C2 } Q

  4. While Rule P ∧ b { C } P P {While b C} P ∧¬ b (P is a loop invariant)

  5. Rule of Consequence P  Pʹ Pʹ { C } Qʹ Qʹ Q P { C } Q

  6. Sample Proofs • sum of n • fibonacci • list append • list reverse • termination

  7. Sum of n x = 0 & y = 0 { While y < n y := y + 1; x := x + y } x = 1 + ... + n P ≡ x = 1 + ... + y ∧ y ≤ n

  8. x = 0 ∧ y = 0  x = 1 + ... + y ∧ y ≤ n ✔ ✔ x = 1 + ... + y ∧ y ≤ n ∧¬(y < n)  x = 1 + ... + n x = 1 + ... + y ∧ y ≤ n ∧ y < n  ? ✔ x + y + 1 = (1 + ... + (y + 1))∧ y + 1 ≤ n {y := y + 1} x + y = 1 + ... + y ∧ y ≤ n {x := x + y} x = 1 + ... + y ∧ y ≤ n ? {y := y + 1; x := x + y} x = 1 + ... + y ∧ y ≤ n x = 1 + ... + y ∧ y ≤ n ∧ y < n {y := y + 1; x := x + y} x = 1 + ... + y ∧ y ≤ n x = 1 + ... + y ∧ y ≤ n {While y < n ...} x = 1 + ... + y ∧ y ≤ n ∧¬(y < n) x = 0 ∧ y = 0 {While ...} x = 1 + ... + n

  9. Fibonacci x = 0 & y = 1 & z = 1 & 1 ≤ n { While z < n y := x + y; x := y – x; z := z + 1 } y = fib n P ≡ y = fib z ∧ x = fib (z-1) ∧ z ≤ n

  10. ✔ x = 0 ∧ y = 1 ∧ z = 0 ∧ 1 ≤ n y = fib z ∧ x = fib (z-1) ∧ z ≤ n ✔ y = fib z ∧ x = fib (z-1) ∧ z ≤ n ∧¬(z < n)  y = fib n y = fib z ∧ x = fib (z-1) ∧ z ≤ n ∧ z < n ? ✔ x+y = fib (z+1) ∧ x+y-x = fib (z+1-1) ∧ z + 1 ≤ n {y := x + y} y = fib (z+1) ∧ y-x = fib (z+1-1) ∧ z + 1 ≤ n {x := y - x} y = fib (z+1) ∧ x = fib (z+1-1) ∧ z + 1 ≤ n {z := z + 1} y = fib z ∧ x = fib (z-1) ∧ z ≤ n ? {y := x + y; x := y – x; z := z + 1} y = fib z ∧ x = fib (z-1) ∧ z ≤ n y = fib z ∧ x = fib (z-1) ∧ z ≤ n ∧ z < n{y := x + y; x := y – x; z := z + 1} y = fib z ∧ x = fib (z-1) ∧ z ≤ n y = fib z ∧ x = fib (z-1) ∧ z ≤ n{While z < n ...} y = fib z ∧ x = fib (z-1) ∧ z ≤ n ∧¬(z < n) x = 0 ∧ y = 1 ∧ z = 0 ∧ 1 ≤ n {While ...} y = fib n

  11. List length x = lst & y = 0 { While x ≠ [] x := tl x; y := y + 1 } y = len lst P ≡ len lst = y + len x

  12. ✔ x = lst ∧ y = 0  len lst = y + len x ✔ len lst = y + len x ∧¬(x ≠ [])  y = len lst len lst = y + len x ∧ x ≠ []  ? ✔ len lst = y + 1 + len(tl x) {x := tl x} len lst = y + 1 + len x {y := y + 1} len lst = y + len x ? {x := tl x; y := y + 1} len lst = y + len x len lst = y + len x ∧ x ≠ [] {x := tl x; y := y + 1} len lst = y + len x len lst = y + len x {While x ≠ [] ...} len lst = y + len x ∧¬(x ≠ []) x = lst ∧ y = 0 {While ...} y = len lst

  13. List reverse x = lst & y = [] { While x ≠ [] y := hd x :: y; x := tl x } y = rev lst P ≡ lst = rev y @ x

  14. ✔ x = lst ∧ y = [] lst = rev y @ x lst = rev y @ x ∧ ¬(x ≠ [])  y = rev lst ✔ lst = rev y @ x ∧ x ≠ []  ? ✔ lst = rev (hd x @ y) @ (tl x) {y := hd x @ y} lst = rev y @ (tl x) {x := tl x} lst = rev y @ x ? {y := hd x @ y; x := tl x} lst = rev y @ x lst = rev y @ x ∧ x ≠ [] {y := hd x @ y; x := tl x} lst = rev y @ x lst = rev y @ x {While x ≠ [] ...} lst = rev y @ x ∧¬(x ≠ []) x = lst ∧ y = [] {While ...} y = rev lst

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