Axioms are a nice way to learning logic. But it is tedious to demonstrate its theorems. They are prone to error doing in paper. That is why we need a tool to help us doing that job.
Here is a example:
Demonstrate T1 |- A -> A by
1. |- (A -> ((A -> A) -> A)) -> ((A -> (A -> A)) -> (A -> A)) by Axiom 2 with A, (A -> A), A
2. |- A -> ((A -> A) -> A) by Axiom 1 with A, (A -> A), A
3. Modus Ponens with 1, 2
4. |- A -> (A -> A) by Axiom 1 with A, A, A
5. Modus Ponens with 3, 4
We can have shorter and larger declarations For example Shorter declarations can be Modus Ponens with 1, 2 -- ID Reference, ID Reference Axiom 2 with A, (A -> A), A -- Proposition, Proposition, Proposition And Larger declarations can be A -> ((A -> A) -> A) by Axiom 1 with A, (A -> A), A (A -> ((A -> A) -> A)) -> ((A -> (A -> A)) -> (A -> A)) by Axiom 2 with A, (A -> A), A
cabal build
cabal exec ahc -- entries/test.ah