more lemmas on absolute value and subtraction

theories/algebra.why

@@ -32,6 +32,9 @@ theory Group
   clone Assoc with type t = t, function op = op
 
   axiom Inv_def : forall x:t. op x (inv x) = unit
+
+  lemma Inv_unit : forall x y:t. op x (inv y) = unit -> x = y
+
 end
 
 theory CommutativeGroup

theories/int.why

@@ -19,8 +19,12 @@ theory Abs
 
   function abs (x:int) : int = if x >= 0 then x else -x
 
+  lemma Abs_le: forall x y:int. abs x <= y <-> -y <= x <= y
+
   lemma Abs_pos: forall x:int. abs x >= 0
 
+  lemma Abs_zero: forall x:int. abs x = 0 -> x = 0
+
 end
 
 theory MinMax

theories/real.why

@@ -11,6 +11,8 @@ theory Real
   clone export algebra.OrderedField with type t = real,
     function zero = zero, function one = one, predicate (<=) = (<=)
 
+  lemma sub_zero: forall x y:real. x - y = 0.0 -> x = y
+
 end
 
 theory RealInfix
@@ -40,6 +42,8 @@ theory Abs
 
   lemma Abs_pos: forall x:real. abs x >= 0.0
 
+  lemma Abs_zero: forall x:real. abs x = 0.0 -> x = 0.0
+
 end
 
 theory MinMax