SET_SPEC_CONV

pred_setLib.SET_SPEC_CONV : conv

Axiom-scheme of specification for set abstractions.

The conversion SET_SPEC_CONV expects its term argument to be an assertion of the form t IN {E | P}. Given such a term, the conversion returns a theorem that defines the condition under which this membership assertion holds. When E is just a variable v, the conversion returns:

   |- t IN {v | P} = P[t/v]

and when E is not a variable but some other expression, the theorem returned is:

   |- t IN {E | P} = ?x1...xn. (t = E) /\ P

where x1, …, xn are the variables that occur free both in the expression E and in the proposition P.

Example

> pred_setLib.SET_SPEC_CONV ``12 IN {n | n > N}``;
val it = ⊢ 12 ∈ {n | n > N} ⇔ 12 > N: thm

> pred_setLib.SET_SPEC_CONV ``p IN {(n,m) | n < m}``;
val it = ⊢ p ∈ {(n,m) | n < m} ⇔ ∃n m. p = (n,m) ∧ n < m: thm

Failure

Fails if applied to a term that is not of the form t IN {E | P}.