Dan Grossman; Graduate Programming Languages; Lecture 3 Proofs

Note: It is instructive to write out proofs "live" in class.

Theorem: For all H and e there is exactly one c such that
  H;e V c.

Proof: By induction on the structure of e.

  Case: e is a constant: Then the only rule that applies is Const.
  And exactly one constant, namely e, can be produced.

  Case: e is a variable: Then the only rule that applies is Var.
  And exactly one constant, namely H(e), can be produced.

  Case: e is e1+e2 for some e1 and e2: Then the only rule that 
  applies is Add.  And e1 and e2 are smaller expressions.  So
  by induction there is exactly one c1 and exactly one c2 such that
  H;e1 V c1 and H;e2 V c2.  Blue-plus produces exactly one constant
  for any c1 and c2.

  Case: e is e1*e2 for some e1 and e2: Then the only rule that 
  applies is Mult.  And e1 and e2 are smaller expressions.  So
  by induction there is exactly one c1 and exactly one c2 such that
  H;e1 V c1 and H;e2 V c2.  Blue-times produces exactly one constant
  for any c1 and c2.