| We_proceed_by_induction_on (_, term, term1) -> "we proceed by induction on" ^ term_pp term ^ "to prove" ^ term_pp term1
| Byinduction (_, term, ident) -> "by induction hypothesis we know" ^ term_pp term ^ "(" ^ ident ^ ")"
| Thesisbecomes (_, term) -> "the thesis becomes " ^ term_pp term
| ExistsElim (_, term0, ident, term, ident1, term1) -> "by " ^ term_pp term0 ^ "let " ^ ident ^ ":" ^ term_pp term ^ "such that " ^ term_pp term1 ^ "(" ^ ident1 ^ ")"
| AndElim (_, term, ident1, term1, ident2, term2) -> "by " ^ term_pp term ^ "we have " ^ term_pp term1 ^ " (" ^ ident1 ^ ") " ^ "and " ^ term_pp term2 ^ " (" ^ ident2 ^ ")"
| We_proceed_by_induction_on (_, term, term1) -> "we proceed by induction on" ^ term_pp term ^ "to prove" ^ term_pp term1
| Byinduction (_, term, ident) -> "by induction hypothesis we know" ^ term_pp term ^ "(" ^ ident ^ ")"
| Thesisbecomes (_, term) -> "the thesis becomes " ^ term_pp term
| ExistsElim (_, term0, ident, term, ident1, term1) -> "by " ^ term_pp term0 ^ "let " ^ ident ^ ":" ^ term_pp term ^ "such that " ^ term_pp term1 ^ "(" ^ ident1 ^ ")"
| AndElim (_, term, ident1, term1, ident2, term2) -> "by " ^ term_pp term ^ "we have " ^ term_pp term1 ^ " (" ^ ident1 ^ ") " ^ "and " ^ term_pp term2 ^ " (" ^ ident2 ^ ")"