diff --git a/src/format.rs b/src/format.rs
index 5364a92..67bdb29 100644
--- a/src/format.rs
+++ b/src/format.rs
@@ -5,8 +5,9 @@ impl std::fmt::Debug for crate::project::StatementKind
 		match &self
 		{
 			crate::project::StatementKind::Axiom => write!(format, "axiom"),
-			crate::project::StatementKind::Lemma => write!(format, "lemma"),
-			crate::project::StatementKind::Conjecture => write!(format, "conjecture"),
+			crate::project::StatementKind::Completion => write!(format, "completion"),
+			crate::project::StatementKind::Assumption => write!(format, "assumption"),
+			crate::project::StatementKind::Assertion => write!(format, "assertion"),
 		}
 	}
 }
diff --git a/src/format_tptp.rs b/src/format_tptp.rs
index 62dd832..aa80588 100644
--- a/src/format_tptp.rs
+++ b/src/format_tptp.rs
@@ -78,12 +78,7 @@ fn collect_predicate_declarations_in_project<'a>(project: &'a crate::Project)
 				match block
 				{
 					crate::project::Block::Whitespace(_) => None,
-					crate::project::Block::Statement(ref statement) =>
-						match statement.kind
-						{
-							crate::project::StatementKind::Axiom => Some(&statement.formula),
-							_ => None,
-						}
+					crate::project::Block::Statement(ref statement) => Some(&statement.formula),
 				});
 
 	for formula in formulas
@@ -211,12 +206,7 @@ fn collect_symbolic_constants_in_project<'a>(project: &'a crate::Project)
 				match block
 				{
 					crate::project::Block::Whitespace(_) => None,
-					crate::project::Block::Statement(ref statement) =>
-						match statement.kind
-						{
-							crate::project::StatementKind::Axiom => Some(&statement.formula),
-							_ => None,
-						}
+					crate::project::Block::Statement(ref statement) => Some(&statement.formula),
 				});
 
 	for formula in formulas
@@ -565,8 +555,9 @@ impl<'a> std::fmt::Debug for StatementKindDisplay<'a>
 		match &self.0
 		{
 			crate::project::StatementKind::Axiom => write!(format, "axiom, axiom"),
-			crate::project::StatementKind::Lemma => write!(format, "lemma, conjecture"),
-			crate::project::StatementKind::Conjecture => write!(format, "conjecture, conjecture"),
+			crate::project::StatementKind::Completion => write!(format, "completion, axiom"),
+			crate::project::StatementKind::Assumption => write!(format, "assumption, axiom"),
+			crate::project::StatementKind::Assertion => write!(format, "assertion, conjecture"),
 		}
 	}
 }
@@ -627,35 +618,31 @@ impl<'a, 'input> std::fmt::Debug for ProjectDisplay<'a, 'input>
 					crate::project::Block::Statement(ref statement) =>
 						match statement.kind
 						{
-							crate::project::StatementKind::Axiom => Some(statement),
-							_ => None,
+							crate::project::StatementKind::Axiom
+								| crate::project::StatementKind::Completion
+								| crate::project::StatementKind::Assumption => Some(statement),
+							crate::project::StatementKind::Assertion => None,
 						}
 				});
 
 		for axiom in axioms
 		{
-			write!(format, "\ntff({:?}, {:?}).", crate::project::StatementKind::Axiom.display_tptp(), axiom.formula.display_tptp())?;
+			write!(format, "\ntff({:?}, {:?}).", axiom.kind.display_tptp(), axiom.formula.display_tptp())?;
 		}
 
 		match self.conjecture.kind
 		{
-			crate::project::StatementKind::Lemma =>
+			crate::project::StatementKind::Assertion =>
 			{
-				write_title(format, "\n\n", "lemma")?;
+				write_title(format, "\n\n", "assertion")?;
 
-				write!(format, "\ntff({:?}, {:?}).", crate::project::StatementKind::Lemma.display_tptp(), self.conjecture.formula.display_tptp())?;
+				write!(format, "\ntff({:?}, {:?}).", crate::project::StatementKind::Assertion.display_tptp(), self.conjecture.formula.display_tptp())?;
 
 				Ok(())
 			},
-			crate::project::StatementKind::Conjecture =>
-			{
-				write_title(format, "\n\n", "conjecture")?;
-
-				write!(format, "\ntff({:?}, {:?}).", crate::project::StatementKind::Conjecture.display_tptp(), self.conjecture.formula.display_tptp())?;
-
-				Ok(())
-			},
-			crate::project::StatementKind::Axiom => panic!("unexpected axiom"),
+			crate::project::StatementKind::Axiom => panic!("unexpected axiom statement"),
+			crate::project::StatementKind::Completion => panic!("unexpected completion statement"),
+			crate::project::StatementKind::Assumption => panic!("unexpected assumption statement"),
 		}
 	}
 }
diff --git a/src/main.rs b/src/main.rs
index 582d7cf..ba22167 100644
--- a/src/main.rs
+++ b/src/main.rs
@@ -25,8 +25,7 @@ fn find_conjecture<'a, 'input>(project: &'a ask_dracula::Project<'input>) -> Opt
 	{
 		if let ask_dracula::project::Block::Statement(ref statement) = block
 		{
-			if statement.kind == ask_dracula::project::StatementKind::Lemma
-				|| statement.kind == ask_dracula::project::StatementKind::Conjecture
+			if statement.kind == ask_dracula::project::StatementKind::Assertion
 			{
 				return Some(statement)
 			}
@@ -42,8 +41,7 @@ fn find_conjecture_mut<'a, 'input>(project: &'a mut ask_dracula::Project<'input>
 	{
 		if let ask_dracula::project::Block::Statement(ref mut statement) = block
 		{
-			if statement.kind == ask_dracula::project::StatementKind::Lemma
-				|| statement.kind == ask_dracula::project::StatementKind::Conjecture
+			if statement.kind == ask_dracula::project::StatementKind::Assertion
 			{
 				return Some(statement)
 			}
@@ -142,7 +140,7 @@ fn main() -> Result<(), Box<dyn std::error::Error>>
 	{
 		None =>
 		{
-			eprintln!("no lemma or conjecture found, nothing to do");
+			eprintln!("nothing to prove, exiting");
 			return Ok(());
 		},
 		Some(ref conjecture) =>
@@ -166,10 +164,10 @@ fn main() -> Result<(), Box<dyn std::error::Error>>
 		},
 		ProofResult::Refutation =>
 		{
-			println!("conjecture proven");
+			println!("assertion proven");
 
 			conjecture.kind = ask_dracula::project::StatementKind::Axiom;
-			let replace_statement_kind_regex = regex::Regex::new(r"(conjecture|lemma)").unwrap();
+			let replace_statement_kind_regex = regex::Regex::new(r"(assertion)").unwrap();
 			let new_text = replace_statement_kind_regex.replace(conjecture.original_text, "axiom");
 			conjecture.original_text = &new_text;
 
diff --git a/src/parse.rs b/src/parse.rs
index 74eb48b..259ec27 100644
--- a/src/parse.rs
+++ b/src/parse.rs
@@ -42,13 +42,18 @@ fn statement_kind(i: &str) -> IResult<&str, crate::project::StatementKind>
 			),
 			map
 			(
-				tag("lemma:"),
-				|_| crate::project::StatementKind::Lemma,
+				tag("completion:"),
+				|_| crate::project::StatementKind::Completion,
 			),
 			map
 			(
-				tag("conjecture:"),
-				|_| crate::project::StatementKind::Conjecture,
+				tag("assumption:"),
+				|_| crate::project::StatementKind::Assumption,
+			),
+			map
+			(
+				tag("assertion:"),
+				|_| crate::project::StatementKind::Assertion,
 			),
 		)),
 		whitespace0,
diff --git a/src/project.rs b/src/project.rs
index a1e3c3d..20ee70b 100644
--- a/src/project.rs
+++ b/src/project.rs
@@ -2,8 +2,9 @@
 pub enum StatementKind
 {
 	Axiom,
-	Lemma,
-	Conjecture,
+	Completion,
+	Assumption,
+	Assertion,
 }
 
 pub struct Statement<'input>