1. Relating Static and Dynamic Semantics
COS 441
Princeton University
Fall 2004

2. Motivations
We want to know that when evaluating certain well-formed programs certain errors never occur
Example
Transition semantics for -calculus is “stuck” when applied to expressions with free variables in it
So if {} ` E ok then E should never be “stuck”

3. Formal Statement isFinal(e) = e 2 F steps(e) = 9 e’. e  e’
stuck(e) = :(steps(e) or isFinal(e))
Soundness Theorem:
If {} ` E ok and E * E’ then :stuck(E’)

4. Formal Statement isFinal(e) = e 2 F steps(e) = 9 e’. e  e’
stuck(e) = :(steps(e) or isFinal(e))
Soundness Theorem:
If {} ` E ok and E * E’ then
(steps(E’) or isFinal(E’))

5. Proof: Soundness Theorem
By induction on derivations of * with Preservation and Progress Lemmas
Preservation Lemma:
If {} ` E ok and E  E’ then {} ` E’ ok
Progress Lemma:
If {} ` E ok then (steps(E) or isFinal(E))

6. Warning!!
The remainder of the lecture consists of a series of tedious proofs
Take that swig of coffee now
Slides will be on web-site
Last set of tedious proofs in lecture
I’ll assign them as homework from now on! ;)
What we discuss today is a template for Assignment 3

7. Proof by Induction over *
S * S Z*
S * S’’
S  S’ S’ * S’’ S*
To show 8 e,e’ P(e,e’) we must show
case Z*: IH(E,E)
case S*: If E  E’ and IH(E’,E’’) then IH(E,E’’)
IH(e,e’) = If {} ` e ok and e * e’ then
(steps(e’) or isFinal(e’))

8. Proof: Soundness Theorem
case Z*: IH(E,E)

9. Proof: Soundness Theorem
case Z*: If {} ` E ok and E * E then (steps(E) or isFinal(E))

10. Proof: Soundness Theorem
case Z*: (steps(E) or isFinal(E))
{} ` E ok and E * E by assumption

11. Proof: Soundness Theorem
case Z*:
{} ` E ok and E * E by assumption
2. (steps(E) or isFinal(E))
by ??

12. Proof: Soundness Theorem
case Z*:
{} ` E ok and E * E by assumption
2. (steps(E) or isFinal(E))
by Progress Lemma with (1)

13. Proof: Soundness Theorem
case S*: If E  E’ and IH(E’,E’’) then IH(E,E’’)

14. Proof: Soundness Theorem
case S*: IH(E,E’’)
1. E  E’ and IH(E’,E’’) by assumption

15. Proof: Soundness Theorem
case S*: If {} ` E ok and E * E’’ then
(steps(E’’) or isFinal(E’’))
E  E’ and IH(E’,E’’) by assumption

16. Proof: Soundness Theorem
case S*: (steps(E’’) or isFinal(E’’))
E  E’ and IH(E’,E’’) by assumption
{} ` E ok and E * E’’ by assumption
` E’ ok by Preservation with (2,1)
E’ * E’’ by inversion of S* and (2)
(steps(E) or isFinal(E’’))
by IH with (3, 4)

17. Proof: Soundness Theorem
case S*: (steps(E’’) or isFinal(E’’))
E  E’ and IH(E’,E’’) by assumption
{} ` E ok and E * E’’ by assumption
{} ` E’ ok by ?? E’ * E’’ by inversion of S* and (2)
(steps(E) or isFinal(E’’))
by IH with (3, 4)

18. Proof: Soundness Theorem
case S*: (steps(E’’) or isFinal(E’’))
E  E’ and IH(E’,E’’) by assumption
{} ` E ok and E * E’’ by assumption
{} ` E’ ok by Preservation with (2,1)
E’ * E’’ by inversion of S* and (2)
(steps(E) or isFinal(E’’))
by IH with (3, 4)

19. Proof: Soundness Theorem
case S*: (steps(E’’) or isFinal(E’’))
E  E’ and IH(E’,E’’) by assumption
{} ` E ok and E * E’’ by assumption
{} ` E’ ok by Preservation with (2,1)
E’ * E’’ by ??
(steps(E’’) or isFinal(E’’))
by IH with (3, 4)

20. Proof: Soundness Theorem
case S*: (steps(E’’) or isFinal(E’’))
E  E’ and IH(E’,E’’) by assumption
{} ` E ok and E * E’’ by assumption
{} ` E’ ok by Preservation with (2,1)
E’ * E’’ by inversion of S* and (2)
(steps(E’’) or isFinal(E’’))
by IH with (3, 4)

21. Proof: Soundness Theorem
case S*:
E  E’ and IH(E’,E’’) by assumption
{} ` E ok and E * E’’ by assumption
{} ` E’ ok by Preservation with (2,1)
E’ * E’’ by inversion of S* and (2)
(steps(E’’) or isFinal(E’’))
by ??

22. Proof: Soundness Theorem
case S*:
E  E’ and IH(E’,E’’) by assumption
{} ` E ok and E * E’’ by assumption
{} ` E’ ok by Preservation with (2,1)
E’ * E’’ by inversion of S* and (2)
(steps(E’’) or isFinal(E’’))
by IH(E’,E’’) with (3, 4)

23. Notes About our Proof
Note our Proof works for any single step relation ()
Specific details of step function factored into Progress and Preservation lemmas
Need to refer to the static and dynamic semantics of the step relation to prove Progress and Preservation Lemmas

24. Static Semantics for -calculus
Names x 2 …
Expressions e
::=
lam(x.e) | apply(e1,e2)| x
 ` apply(E1,E2)ok
 ` E1 ok  ` E2 ok
ok-A
 ` lam(X.E)ok
 [ {X} ` E ok X  
ok-L
 ` X ok
X 2 
ok-V

25. Dynamic Semantics for -calculus=
{ E | 9.  ` E ok } I =
{ E | {} ` E ok } F =
{ x.e | {} ` x.e ok }
((x.e1) (y.e2))  [xÃ(y.e2)] e1 A1
((x.e1) e2)  ((x.e1) e’2)
e2  e’2 A2
(e1 e2)  (e’1 e2)
e1  e’1 A3

26. Proof: Preservation Lemma
Proof by induction on the derivations of E  E’
case A1: IH(((X.E1) (Y.E2)),[X Ã (Y.E2)] E1)
case A2: If IH(E2,E’2) then
IH(((X.E1) E2)),((X.E1) E’2))
case A3: If IH(E1,E’1) then
IH((E1 E2)),(E’1 E2))
IH(e,e’) = If {} ` e ok and e  e’ then {} ` e’ ok

27. Proof: Preservation Lemma
case A1: If {} ` ((X.E1) (Y.E1)) ok and ((X.E1) (Y.E1))  [X Ã (Y.E2)] E1 then {} ` [X Ã (Y.E2)] E1 ok

28. Proof: Preservation Lemma
case A1: {} ` [X Ã (Y.E2)] E1 ok
{} ` ((X.E1) (Y.E2)) ok and ((X.E1) (Y.E2))  [X Ã (Y.E2)] E1 by assumption
{} ` (X.E1) ok and {} ` (Y.E2) ok by inversion of ok-A and (1)
{} [ {X} ` E1 ok by inversion of ok-L and (2)
{} ` [X Ã (Y.E2)] E1 ok by Substitution Lemma with (3) and (2)

29. Proof: Preservation Lemma
case A1: {} ` [X Ã (Y.E2)] E1 ok
{} ` ((X.E1) (Y.E2)) ok and ((X.E1) (Y.E2))  [X Ã (Y.E2)] E1 by assumption
{} ` (X.E1) ok and {} ` (Y.E2) ok by ??
{} [ {X} ` E1 ok by inversion of ok-L and (2)
{} ` [X Ã (Y.E2)] E1 ok by Substitution Lemma with (3) and (2)

30. Proof: Preservation Lemma
case A1: {} ` [X Ã (Y.E2)] E1 ok
{} ` ((X.E1) (Y.E2)) ok and ((X.E1) (Y.E2))  [X Ã (Y.E2)] E1 by assumption
{} ` (X.E1) ok and {} ` (Y.E2) ok by inversion of ok-A and (1)
{} [ {X} ` E1 ok by inversion of ok-L and (2)
{} ` [X Ã (Y.E2)] E1 ok by Substitution Lemma with (3) and (2)

31. Proof: Preservation Lemma
case A1: {} ` [X Ã (Y.E2)] E1 ok
{} ` ((X.E1) (Y.E2)) ok and ((X.E1) (Y.E2))  [X Ã (Y.E2)] E1 by assumption
{} ` (X.E1) ok and {} ` (Y.E2) ok by inversion of ok-A and (1)
{} [ {X} ` E1 ok by ??
{} ` [X Ã (Y.E2)] E1 ok by Substitution Lemma with (3) and (2)

32. Proof: Preservation Lemma
case A1: {} ` [X Ã (Y.E2)] E1 ok
{} ` ((X.E1) (Y.E2)) ok and ((X.E1) (Y.E2))  [X Ã (Y.E2)] E1 by assumption
{} ` (X.E1) ok and {} ` (Y.E2) ok by inversion of ok-A and (1)
{} [ {X} ` E1 ok by inversion of ok-L and (2)
{} ` [X Ã (Y.E2)] E1 ok by Substitution Lemma with (3) and (2)

33. Proof: Preservation Lemma
case A1: {} ` [X Ã (Y.E2)] E1 ok
{} ` ((X.E1) (Y.E2)) ok and ((X.E1) (Y.E2))  [X Ã (Y.E2)] E1 by assumption
{} ` (X.E1) ok and {} ` (Y.E2) ok by inversion of ok-A and (1)
{} [ {X} ` E1 ok by inversion of ok-L and (2)
{} ` [X Ã (Y.E2)] E1 ok by ??

34. Proof: Preservation Lemma
case A1: {} ` [X Ã (Y.E2)] E1 ok
{} ` ((X.E1) (Y.E2)) ok and ((X.E1) (Y.E2))  [X Ã (Y.E2)] E1 by assumption
{} ` (X.E1) ok and {} ` (Y.E2) ok by inversion of ok-A and (1)
{} [ {X} ` E1 ok by inversion of ok-L and (2)
{} ` [X Ã (Y.E2)] E1 ok by Substitution Lemma with (3) and (2)

35. Substitution Lemma Proof by induction on the derivations of  ` E ok
If  [ {X} ` E ok and {} ` E’ ok then
 ` [XÃE’]E ok
case ok-V: …
case ok-L: …
case ok-A: …
IH(env,e) = If env [ {X} ` e ok and {} ` E’ ok then env ` [XÃE’]e ok

36. Substitution Proof by induction on the derivations of  ` E ok
If  [ {X} ` E ok and {} ` E’ ok then
 ` [XÃE’]E ok
case ok-V: If X 2  then IH(,X)
case ok-L: If IH( [ {X}, E) and X   then IH(,(X.E))
case ok-A: If IH(,E1) and IH(,E2) then IH(,(E1 E2))
IH(env,e) = If env [ {X} ` e ok and {} ` E’ ok then env ` [XÃE’]e ok

37. Proof: Substitution case ok-V: 1. X 2  by assumption
2.  [ {Y} ` X ok and {} ` E’ ok by assumption
3.  ` [YÃE’]X ok by cases
case X = Y:
3.1. [YÃE’]X = E’ by def of subst.
3.2.  ` E’ ok by (2)
3.3.  ` [YÃE’]X ok by (3.1) and (3.2)
case X  Y:
3.1. [YÃE’]X = X by def of subst.
3.2.  ` X ok by ok-V and (1)
3.3.  ` [YÃE’]X ok by (3.1) and (3.2)

38. Proof: Substitution
case ok-L: If IH( [ {X}, E) and X   then IH(,(X.E)) …

39. Proof: Substitution
case ok-A: If IH(,E1) and IH(,E2) then IH(,(E1 E2)) …

40. Proof: Preservation Lemma
case A2: If IH(E2,E’2) then
IH(((X.E1) E2)),((X.E1) E’2))

41. Proof: Preservation Lemma
case A2: IH(((X.E1) E2)),((X.E1) E’2))
IH(E2,E’2) by assumption

42. Proof: Preservation Lemma
case A2: If {} ` ((X.E1) E2)) ok and
((X.E1) E2))  ((X.E1) E’2) then
{} ` ((X.E1) E’2) ok
IH(E2,E’2) by assumption

43. Proof: Preservation Lemma
case A2: {} ` ((X.E1) E’2) ok
IH(E2,E’2) by assumption
{} ` ((X.E1) E2)) ok and ((X.E1) E2))  ((X.E1) E’2) by assumption
{} ` (X.E1) ok and {} ` E2 ok by inversion of ok-A and (2)
E2  E’2 by inversion of A2
{} ` E’2 ok by IH(E2,E’2) with (3) and (4)
{} ` ((X.E1) E’2) ok by ok-A with (3) and (5)

44. Proof: Preservation Lemma
case A2: {} ` ((X.E1) E’2) ok
IH(E2,E’2) by assumption
{} ` ((X.E1) E2)) ok and ((X.E1) E2))  ((X.E1) E’2) by assumption
{} ` (X.E1) ok and {} ` E2 ok by ??
E2  E’2 by inversion of A2
{} ` E’2 ok by IH(E2,E’2) with (3) and (4)
{} ` ((X.E1) E’2) ok by ok-A with (3) and (5)

45. Proof: Preservation Lemma
case A2: {} ` ((X.E1) E’2) ok
IH(E2,E’2) by assumption
{} ` ((X.E1) E2)) ok and ((X.E1) E2))  ((X.E1) E’2) by assumption
{} ` (X.E1) ok and {} ` E2 ok by inversion of ok-A and (2)
E2  E’2 by inversion of A2
{} ` E’2 ok by IH(E2,E’2) with (3) and (4)
{} ` ((X.E1) E’2) ok by ok-A with (3) and (5)

46. Proof: Preservation Lemma
case A2: {} ` ((X.E1) E’2) ok
IH(E2,E’2) by assumption
{} ` ((X.E1) E2)) ok and ((X.E1) E2))  ((X.E1) E’2) by assumption
{} ` (X.E1) ok and {} ` E2 ok by inversion of ok-A and (2)
E2  E’2 by ??
{} ` E’2 ok by IH(E2,E’2) with (3) and (4)
{} ` ((X.E1) E’2) ok by ok-A with (3) and (5)

47. Proof: Preservation Lemma
case A2: {} ` ((X.E1) E’2) ok
IH(E2,E’2) by assumption
{} ` ((X.E1) E2)) ok and ((X.E1) E2))  ((X.E1) E’2) by assumption
{} ` (X.E1) ok and {} ` E2 ok by inversion of ok-A and (2)
E2  E’2 by inversion of A2 and (2)
{} ` E’2 ok by IH(E2,E’2) with (3) and (4)
{} ` ((X.E1) E’2) ok by ok-A with (3) and (5)

48. Proof: Preservation Lemma
case A2: {} ` ((X.E1) E’2) ok
IH(E2,E’2) by assumption
{} ` ((X.E1) E2)) ok and ((X.E1) E2))  ((X.E1) E’2) by assumption
{} ` (X.E1) ok and {} ` E2 ok by inversion of ok-A and (2)
E2  E’2 by inversion of A2 and (2)
{} ` E’2 ok by ??
{} ` ((X.E1) E’2) ok by ok-A with (3) and (5)

49. Proof: Preservation Lemma
case A2: {} ` ((X.E1) E’2) ok
IH(E2,E’2) by assumption
{} ` ((X.E1) E2)) ok and ((X.E1) E2))  ((X.E1) E’2) by assumption
{} ` (X.E1) ok and {} ` E2 ok by inversion of ok-A and (2)
E2  E’2 by inversion of A2 and (2)
{} ` E’2 ok by IH(E2,E’2) with (3) and (4)
{} ` ((X.E1) E’2) ok by ok-A with (3) and (5)

50. Proof: Preservation Lemma
case A2: {} ` ((X.E1) E’2) ok
IH(E2,E’2) by assumption
{} ` ((X.E1) E2)) ok and ((X.E1) E2))  ((X.E1) E’2) by assumption
{} ` (X.E1) ok and {} ` E2 ok by inversion of ok-A and (2)
E2  E’2 by inversion of A2 and (2)
{} ` E’2 ok by IH(E2,E’2) with (3) and (4)
{} ` ((X.E1) E’2) ok by ??

51. Proof: Preservation Lemma
case A2: {} ` ((X.E1) E’2) ok
IH(E2,E’2) by assumption
{} ` ((X.E1) E2)) ok and ((X.E1) E2))  ((X.E1) E’2) by assumption
{} ` (X.E1) ok and {} ` E2 ok by inversion of ok-A and (2)
E2  E’2 by inversion of A2 and (2)
{} ` E’2 ok by IH(E2,E’2) with (3) and (4)
{} ` ((X.E1) E’2) ok by ok-A with (3) and (5)

52. Proof: Preservation Lemma
case A3: If IH(E1,E’1) then
IH(E1 E2)),(E’1 E2))

53. Proof: Preservation Lemma
case A3: IH((E1 E2)),((E’1 E2))
IH(E1,E’1) by assumption

54. Proof: Preservation Lemma
case A3: If {} ` (E1 E2) ok and
(E1 E2)  (E’1 E2) then {} ` (E’1 E2) ok
IH(E1,E’1) by assumption

55. Proof: Preservation Lemma
case A3: {} ` (E’1 E2) ok
IH(E1,E’1) by assumption
{} ` (E1 E2) ok and (E1 E2)  (E’1 E’2) by assumption
{} ` E1 ok and {} ` E2 ok by inversion of ok-A and (2)
E1  E’1 by inversion of A3 and (2)
{} ` E’1 ok by IH(E1,E’1) with (3) and (4)
{} ` (E’1 E2) ok by ok-A with (5) and (3)

56. Proof: Preservation Lemma
case A3: {} ` (E’1 E2) ok
IH(E1,E’1) by assumption
{} ` (E1 E2) ok and (E1 E2)  (E’1 E’2) by assumption
{} ` E1 ok and {} ` E2 ok by inversion of ok-A and (2)
E1  E’1 by inversion of A3 and (2)
{} ` E’1 ok by IH(E1,E’1) with (3) and (4)
{} ` (E’1 E2) ok by ok-A with (5) and (3)

57. Proof: Preservation Lemma
case A3: {} ` (E’1 E2) ok
IH(E1,E’1) by assumption
{} ` (E1 E2) ok and (E1 E2)  (E’1 E’2) by assumption
{} ` E1 ok and {} ` E2 ok by inversion of ok-A and (2)
E1  E’1 by inversion of A3 and (2)
{} ` E’1 ok by IH(E1,E’1) with (3) and (4)
{} ` (E’1 E2) ok by ok-A with (5) and (3)

58. Proof: Preservation Lemma
case A3: {} ` (E’1 E2) ok
IH(E1,E’1) by assumption
{} ` (E1 E2) ok and (E1 E2)  (E’1 E’2) by assumption
{} ` E1 ok and {} ` E2 ok by inversion of ok-A and (2)
E1  E’1 by inversion of A3 and (2)
{} ` E’1 ok by IH(E1,E’1) with (3) and (4)
{} ` (E’1 E2) ok by ok-A with (5) and (3)

59. Proof: Preservation Lemma
case A3: {} ` (E’1 E2) ok
IH(E1,E’1) by assumption
{} ` (E1 E2) ok and (E1 E2)  (E’1 E’2) by assumption
{} ` E1 ok and {} ` E2 ok by inversion of ok-A and (2)
E1  E’1 by inversion of A3 and (2)
{} ` E’1 ok by IH(E1,E’1) with (3) and (4)
{} ` (E’1 E2) ok by ok-A with (5) and (3)

60. Progress Lemma Proof by induction on the derivations of  ` E ok
case ok-V: If X 2  then IH(,X)
case ok-L: If IH( [ {X}, E) and X   then IH(,(X.E))
case ok-A: If IH(,E1) and IH(,E2) then IH(,(E1 E2))
IH(env,e) = If env = {} and env ` e ok then (steps(e) or isFinal(e))

61. Proof: Progress Lemma
case ok-V: If X 2  then IH(,X)

62. Proof: Progress Lemma
case ok-V: IH(,X)
X 2  by assumption

63. Proof: Progress Lemma case ok-V: If = {} and ` X ok then
(steps(X) or isFinal(X))
X 2  by assumption

64. Proof: Progress Lemma case ok-V: If = {} and {} ` X ok then
(steps(X) or isFinal(X))
X 2  by assumption

65. Proof: Progress Lemma case ok-V: steps(X) or isFinal(X)
X 2  by assumption
= {} and {} ` X ok by assumption
X 2 {} by (1) and (2)
(steps(X) or isFinal(X)) by contradiction implied by (3)

66. Proof: Progress Lemma case ok-V: steps(X) or isFinal(X)
X 2  by assumption
= {} and {} ` X ok by assumption
X 2 {} by ??
(steps(X) or isFinal(X)) by contradiction implied by (3)

67. Proof: Progress Lemma case ok-V: steps(X) or isFinal(X)
X 2  by assumption
= {} and {} ` X ok by assumption
X 2 {} by (1) and (2)
(steps(X) or isFinal(X)) by contradiction implied by (3)

68. Proof: Progress Lemma case ok-V: (steps(X) or isFinal(X))
X 2  by assumption
= {} and {} ` X ok by assumption
X 2 {} by (2) and invert-ok-V
steps(X) or isFinal(X) by ??

69. Proof: Progress Lemma case ok-V: (steps(X) or isFinal(X))
X 2  by assumption
= {} and {} ` X ok by assumption
X 2 {} by (2) and invert-ok-V
steps(X) or isFinal(X) by contradiction implied by (3)

70. Proof: Progress Lemma
case ok-L: If IH( [ {X}, E) and X   then IH(,(X.E))

71. Proof: Progress Lemma case ok-L: IH(,(X.E))
IH( [ {X}, E) and X   by assumption

72. Proof: Progress Lemma
case ok-L: If  = {} and  ` (X.E) ok then (steps((X.E)) or isFinal((X.E)))
IH( [ {X}, E) and X   by assumption

73. Proof: Progress Lemma case ok-L: steps((X.E)) or isFinal((X.E))
IH( [ {X}, E) and X   by assumption
 = {} and  ` (X.E) ok by assumption
{} ` (X.E) ok by (2)
(X.E) 2 F by definition of F and (3)
isFinal((X.E)) by definition of isFinal and (4)
steps((X.E)) or isFinal((X.E)) by (5)

74. Proof: Progress Lemma case ok-L: steps((X.E)) or isFinal((X.E))
IH( [ {X}, E) and X   by assumption
 = {} and  ` (X.E) ok by assumption
{} ` (X.E) ok by ??
(X.E) 2 F by definition of F and (3)
isFinal((X.E)) by definition of isFinal and (4)
steps((X.E)) or isFinal((X.E)) by (5)

75. Proof: Progress Lemma case ok-L: steps((X.E)) or isFinal((X.E))
IH( [ {X}, E) and X   by assumption
 = {} and  ` (X.E) ok by assumption
{} ` (X.E) ok by (2)
(X.E) 2 F by definition of F and (3)
isFinal((X.E)) by definition of isFinal and (4)
steps((X.E)) or isFinal((X.E)) by (5)

76. Proof: Progress Lemma case ok-L: steps((X.E)) or isFinal((X.E))
IH( [ {X}, E) and X   by assumption
 = {} and  ` (X.E) ok by assumption
{} ` (X.E) ok by (2)
(X.E) 2 F by ??
isFinal((X.E)) by definition of isFinal and (4)
steps((X.E)) or isFinal((X.E)) by (5)

77. Proof: Progress Lemma case ok-L: steps((X.E)) or isFinal((X.E))
IH( [ {X}, E) and X   by assumption
 = {} and  ` (X.E) ok by assumption
{} ` (X.E) ok by (2)
(X.E) 2 F by definition of F and (3)
isFinal((X.E)) by definition of isFinal and (4)
steps((X.E)) or isFinal((X.E)) by (5)

78. Proof: Progress Lemma case ok-L: steps((X.E)) or isFinal((X.E))
IH( [ {X}, E) and X   by assumption
 = {} and  ` (X.E) ok by assumption
{} ` (X.E) ok by (2)
(X.E) 2 F by definition of F and (3)
isFinal((X.E)) by ??
steps((X.E)) or isFinal((X.E)) by (5)

79. Proof: Progress Lemma case ok-L: steps((X.E)) or isFinal((X.E))
IH( [ {X}, E) and X   by assumption
 = {} and  ` (X.E) ok by assumption
{} ` (X.E) ok by (2)
(X.E) 2 F by definition of F and (3)
isFinal((X.E)) by definition of isFinal and (4)
steps((X.E)) or isFinal((X.E)) by (5)

80. Proof: Progress Lemma case ok-L: steps((X.E)) or isFinal((X.E))
IH( [ {X}, E) and X   by assumption
 = {} and  ` (X.E) ok by assumption
{} ` (X.E) ok by (2)
(X.E) 2 F by definition of F and (3)
isFinal((X.E)) by definition of isFinal and (4)
steps((X.E)) or isFinal((X.E)) by ??

81. Proof: Progress Lemma case ok-L: steps((X.E)) or isFinal((X.E))
IH( [ {X}, E) and X   by assumption
 = {} and  ` (X.E) ok by assumption
{} ` (X.E) ok by (2)
(X.E) 2 F by definition of F and (3)
isFinal((X.E)) by definition of isFinal and (4)
steps((X.E)) or isFinal((X.E)) by (5)

82. Proof: Progress Lemma
case ok-A: If IH(,E1) and IH(,E2) then IH(,(E1 E2))

83. Proof: Progress Lemma
case ok-A: IH(,(E1 E2))
IH(,E1) and IH(,E2)

84. Proof: Progress Lemma
case ok-A: If  = {} and ` (E1 E2) ok then (steps((E1 E2)) or isFinal((E1 E2)))
IH(,E1) and IH(,E2) by assumption

85. Proof: Progress Lemma case ok-A: steps((E1 E2)) or isFinal((E1 E2))
IH(,E1) and IH(,E2) by assumption
 = {} and ` (E1 E2) ok by assumption
{} ` (E1 E2) ok by (2)
{} ` E1 ok and {} ` E2 ok by inversion of ok-A
9 e. (E1 E2)  e by induction on (E1 E2)  e …
steps((E1 E2)) by definition of steps and (5)
7. steps((E1 E2)) or isFinal((E1 E2)) by (6)

86. Proof: Progress Lemma case ok-A: steps((E1 E2)) or isFinal((E1 E2))
IH(,E1) and IH(,E2) by assumption
 = {} and ` (E1 E2) ok by assumption
{} ` (E1 E2) ok by (2)
{} ` E1 ok and {} ` E2 ok by inversion of ok-A
9 e. (E1 E2)  e by induction on (E1 E2)  e …
steps((E1 E2)) by definition of steps and (5)
7. steps((E1 E2)) or isFinal((E1 E2)) by (6)

87. Proof: Progress Lemma case ok-A: steps((E1 E2)) or isFinal((E1 E2))
IH(,E1) and IH(,E2) by assumption
 = {} and ` (E1 E2) ok by assumption
{} ` (E1 E2) ok by (2)
{} ` E1 ok and {} ` E2 ok by inversion of ok-A
9 e. (E1 E2)  e by induction on (E1 E2)  e …
steps((E1 E2)) by definition of steps and (5)
7. steps((E1 E2)) or isFinal((E1 E2)) by (6)

88. Proof: Progress Lemma case ok-A: steps((E1 E2)) or isFinal((E1 E2))
IH(,E1) and IH(,E2) by assumption
 = {} and ` (E1 E2) ok by assumption
{} ` (E1 E2) ok by (2)
{} ` E1 ok and {} ` E2 ok by inversion of ok-A
9 e. (E1 E2)  e by cases …
steps((E1 E2)) by definition of steps and (5)
7. steps((E1 E2)) or isFinal((E1 E2)) by (6)

89. Proof: Progress Lemma case ok-A: steps((E1 E2)) or isFinal((E1 E2))
IH(,E1) and IH(,E2) by assumption
 = {} and ` (E1 E2) ok by assumption
{} ` (E1 E2) ok by (2)
{} ` E1 ok and {} ` E2 ok by inversion of ok-A
9 e. (E1 E2)  e by cases …
steps((E1 E2)) by definition of steps and (5)
7. steps((E1 E2)) or isFinal((E1 E2)) by (6)

90. Proof: Progress Lemma case ok-A: steps((E1 E2)) or isFinal((E1 E2))
IH(,E1) and IH(,E2) by assumption
 = {} and ` (E1 E2) ok by assumption
{} ` (E1 E2) ok by (2)
{} ` E1 ok and {} ` E2 ok by inversion of ok-A
9 e. (E1 E2)  e by cases …
steps((E1 E2)) by definition of steps and (5)
7. steps((E1 E2)) or isFinal((E1 E2)) by (6)

91. Proof: Progress Lemma 5. 9 e. (E1 E2)  e by cases (E1 E2)
case E1 = (X’.E’) and E2 = (X’’.E’’):
5.1. (E1 E2)  [X’ Ã(X’’.E’’) ] E’ by A1
case E1 = (X’.E’) and E2  F:
5.1. E2  E’2 by IH({},E2) with (4) and E2  F
5.2. (E1 E2)  (E1 E’2) by A2 with (5.1)
case E1  F :
5.1. E1  E’1 by IH({},E1) with (4) and E1  F
5.2. (E’1 E2)  (E’1 E2) by A3 with (5.1)

92. Proof: Progress Lemma 5. 9 e. (E1 E2)  e by cases (E1 E2)
case E1 = (X’.E’) and E2 = (X’’.E’’):
5.1. (E1 E2)  [X’ Ã(X’’.E’’) ] E’ by ??
case E1 = (X’.E’) and E2  F:
5.1. E2  E’2 by IH({},E2) with (4) and E2  F
5.2. (E1 E2)  (E1 E’2) by A2 with (5.1)
case E1  F :
5.1. E1  E’1 by IH({},E1) with (4) and E1  F
5.2. (E’1 E2)  (E’1 E2) by A3 with (5.1)

93. Proof: Progress Lemma 5. 9 e. (E1 E2)  e by cases (E1 E2)
case E1 = (X’.E’) and E2 = (X’’.E’’):
5.1. (E1 E2)  [X’ Ã(X’’.E’’) ] E’ by A1
case E1 = (X’.E’) and E2  F:
5.1. E2  E’2 by IH({},E2) with (4) and E2  F
5.2. (E1 E2)  (E1 E’2) by A2 with (5.1)
case E1  F :
5.1. E1  E’1 by IH({},E1) with (4) and E1  F
5.2. (E’1 E2)  (E’1 E2) by A3 with (5.1)

94. Proof: Progress Lemma 5. 9 e. (E1 E2)  e by cases (E1 E2)
case E1 = (X’.E’) and E2 = (X’’.E’’):
5.1. (E1 E2)  [X’ Ã(X’’.E’’) ] E’ by A1
case E1 = (X’.E’) and E2  F:
5.1. E2  E’2 by ??
5.2. (E1 E2)  (E1 E’2)
case E1  F :
5.1. E1  E’1 by IH({},E1) with (4) and E1  F
5.2. (E’1 E2)  (E’1 E2) by A3 with (5.1)

95. Proof: Progress Lemma 5. 9 e. (E1 E2)  e by cases (E1 E2)
case E1 = (X’.E’) and E2 = (X’’.E’’):
5.1. (E1 E2)  [X’ Ã(X’’.E’’) ] E’ by A1
case E1 = (X’.E’) and E2  F:
5.1. E2  E’2 by IH({},E2) with (4) and E2  F
5.2. (E1 E2)  (E1 E’2) by ??
case E1  F :
5.1. E1  E’1 by IH({},E1) with (4) and E1  F
5.2. (E’1 E2)  (E’1 E2) by A3 with (5.1)

96. Proof: Progress Lemma 5. 9 e. (E1 E2)  e by cases (E1 E2)
case E1 = (X’.E’) and E2 = (X’’.E’’):
5.1. (E1 E2)  [X’ Ã(X’’.E’’) ] E’ by A1
case E1 = (X’.E’) and E2  F:
5.1. E2  E’2 by IH({},E2) with (4) and E2  F
5.2. (E1 E2)  (E1 E’2) by A2 with (5.1)
case E1  F :
5.1. E1  E’1 by IH({},E1) with (4) and E1  F
5.2. (E’1 E2)  (E’1 E2) by A3 with (5.1)

97. Proof: Progress Lemma 5. 9 e. (E1 E2)  e by cases (E1 E2)
case E1 = (X’.E’) and E2 = (X’’.E’’):
5.1. (E1 E2)  [X’ Ã(X’’.E’’) ] E’ by A1
case E1 = (X’.E’) and E2  F:
5.1. E2  E’2 by IH({},E2) with (4) and E2  F
5.2. (E1 E2)  (E1 E’2) by A2 with (5.1)
case E1  F :
5.1. E1  E’1 by ??
5.2. (E’1 E2)  (E’1 E2)

98. Proof: Progress Lemma 5. 9 e. (E1 E2)  e by cases (E1 E2)
case E1 = (X’.E’) and E2 = (X’’.E’’):
5.1. (E1 E2)  [X’ Ã(X’’.E’’) ] E’ by A1
case E1 = (X’.E’) and E2  F:
5.1. E2  E’2 by IH({},E2) with (4) and E2  F
5.2. (E1 E2)  (E1 E’2) by A2 with (5.1)
case E1  F :
5.1. E1  E’1 by IH({},E1) with (4) and E1  F
5.2. (E’1 E2)  (E’1 E2) by ??

99. Proof: Progress Lemma 5. 9 e. (E1 E2)  e by cases (E1 E2)
case E1 = (X’.E’) and E2 = (X’’.E’’):
5.1. (E1 E2)  [X’ Ã(X’’.E’’) ] E’ by A1
case E1 = (X’.E’) and E2  F:
5.1. E2  E’2 by IH({},E2) with (4) and E2  F
5.2. (E1 E2)  (E1 E’2) by A2 with (5.1)
case E1  F :
5.1. E1  E’1 by IH({},E1) with (4) and E1  F
5.2. (E’1 E2)  (E’1 E2) by A3 with (5.1)

100. Summary Soundness Theorem: If {} ` E ok and E * E’ then :stuck(E’)
Preservation Lemma:
If {} ` E ok and E  E’ then {} ` E’ ok
Progress Lemma:
If {} ` E ok then (steps(E) or isFinal(E))
Substitution Lemma:
If  [ {X} ` E ok and {} ` E’ ok then
 ` [XÃE’]E ok

101. Summary
Soundness follows from Preservation and Progress by induction on the ?? relation
Soundness means well formed programs don’t get “stuck”

102. Summary
Soundness follows from Preservation and Progress by induction on the * relation
Soundness means well formed programs don’t get “stuck”

103. Summary
Soundness follows from Preservation and Progress by induction on the * relation
Soundness means well formed programs don’t get “stuck”
Preservation follows by induction on the ?? relation

104. Summary
Soundness follows from Preservation and Progress by induction on the * relation
Soundness means well formed programs don’t get “stuck”
Preservation follows by induction on the  relation

105. Summary
Soundness follows from Preservation and Progress by induction on the * relation
Soundness means well formed programs don’t get “stuck”
Preservation follows by induction on the  relation
Progress follows by induction on the wellformedness relation ??

106. Summary
Soundness follows from Preservation and Progress by induction on the * relation
Soundness means well formed programs don’t get “stuck”
Preservation follows by induction on the  relation
Progress follows by induction on the wellformedness relation ( ` E ok)

107. Lesson Learned High-level structure of soundness proof
All soundness for SOS semantics proofs are basically the same
The details vary in small but important ways
Proofs are straightforward but tedious
Details easy to get confused if not organized
Someone ought to automate these proofs or at least their checking
See Twelf, Coq, Isabella/HOL … etc.