Implementing Sets with Lists
Associativity of union

We want to prove

S1,S2,S3:Set (S1 U (S2 U S3) = (S1 U S2) U S3) .
We use attribute coinduction to prove this is a behavorial property of the input specification.
• First, we prove the attribute coherence of the input specification.
• Introduce the lemma coherence(SET) .
• This goal is true because coherence(SET) is in the hypothesis.
• Then we prove S1,S2,S3:Set N:Nat N in S1 U (S2 U S3) = N in (S1 U S2) U S3
Implementing Sets with Lists

The empty set is the unit of union.

We want to prove

S1:Set (empty U S1 = S1) .
We use attribute coinduction to prove this is a behavorial property of the input specification.
Implementing Sets with Lists

Idemopotency of union

We want to prove

S1:Set (S1 U S1 = S1) .
We use attribute coinduction to prove this is a behavorial property of the input specification.
Implementing Sets with Lists

Commutativity of union

We want to prove

S1,S2:Set (S1 U S2 = S2 U S1) .
We use attribute coinduction to prove this is a behavorial property of the input specification.
• First, we prove the attribute coherence of the input specification.
• Introduce the lemma coherence(SET) .
• This goal is true because coherence(SET) is in the hypothesis.
• Then we prove S1,S2:Set (N:Nat ( N in S1 U S2 = N in S2 U S1 ))