Some simple proofs
First declare the following variables : P, Q, R,
S of sort Prop.
Then prove the following theorems, using only the tactics
assumption, intros, and apply.
Lemma id_P : P -> P.
Lemma id_PP : (P->P)->(P->P).
Lemma imp_trans : (P->Q)->(Q->R)->P->R.
Lemma imp_perm : (P->Q->R)->(Q->P->R).
Lemma ignore_Q : (P->R)->P->Q->R.
Lemma delta_imp : (P->P->Q)->P->Q.
Lemma delta_impR : (P->Q)->(P->P->Q).
Lemma diamond : (P->Q)->(P->R)->(Q->R->S)->P->S.
Lemma weak_peirce : ((((P->Q)->P)->P)->Q)->Q.
Solution (without tacticals)
Solution (using some tacticals)
All these theorems can be proved with a simple call to Auto.