Small improvements for Prelude.Natural

Prelude/Natural/equal

@@ -1,15 +1,18 @@
 {-
 `equal` checks if two Naturals are equal.
 -}
+let lessThanEqual =
+        ./lessThanEqual sha256:1a5caa2b80a42b9f58fff58e47ac0d9a9946d0b2d36c54034b8ddfe3cb0f3c99
+      ? ./lessThanEqual
+
 let equal
     : Natural → Natural → Bool
-    =   λ(a : Natural)
-      → λ(b : Natural)
-      →     Natural/isZero (Natural/subtract a b)
-        &&  Natural/isZero (Natural/subtract b a)
+    = λ(a : Natural) → λ(b : Natural) → lessThanEqual a b && lessThanEqual b a
 
 let example0 = assert : equal 5 5 ≡ True
 
 let example1 = assert : equal 5 6 ≡ False
 
+let property0 = λ(n : Natural) → assert : equal n n ≡ True
+
 in  equal

Prelude/Natural/greaterThan

@@ -1,23 +1,22 @@
 {-
 `greaterThan` checks if one Natural is strictly greater than another.
 -}
-
-let Bool/not =
-        ../Bool/not sha256:723df402df24377d8a853afed08d9d69a0a6d86e2e5b2bac8960b0d4756c7dc4
-      ? ../Bool/not
-
-let lessThanEqual =
-        ./lessThanEqual sha256:1a5caa2b80a42b9f58fff58e47ac0d9a9946d0b2d36c54034b8ddfe3cb0f3c99
-      ? ./lessThanEqual
+let lessThan =
+        ./lessThan sha256:3381b66749290769badf8855d8a3f4af62e8de52d1364d838a9d1e20c94fa70c
+      ? ./lessThan
 
 let greaterThan
     : Natural → Natural → Bool
-    = λ(x : Natural) → λ(y : Natural) → Bool/not (lessThanEqual x y)
+    = λ(x : Natural) → λ(y : Natural) → lessThan y x
 
 let example0 = assert : greaterThan 5 6 ≡ False
 
 let example1 = assert : greaterThan 5 5 ≡ False
 
 let example2 = assert : greaterThan 5 4 ≡ True
 
+let property0 = λ(n : Natural) → assert : greaterThan 0 n ≡ False
+
+let property1 = λ(n : Natural) → assert : greaterThan n n ≡ False
+
 in  greaterThan

Prelude/Natural/greaterThanEqual

@@ -1,22 +1,22 @@
 {-
 `greaterThanEqual` checks if one Natural is greater than or equal to another.
 -}
-let Bool/not =
-        ../Bool/not sha256:723df402df24377d8a853afed08d9d69a0a6d86e2e5b2bac8960b0d4756c7dc4
-      ? ../Bool/not
-
-let lessThan =
-        ./lessThan sha256:d8a37951fd3f88dcc3927c20b73e6c53499f5275d949eed59bb77ea748492bf0
-      ? ./lessThan
+let lessThanEqual =
+        ./lessThanEqual sha256:1a5caa2b80a42b9f58fff58e47ac0d9a9946d0b2d36c54034b8ddfe3cb0f3c99
+      ? ./lessThanEqual
 
 let greaterThanEqual
     : Natural → Natural → Bool
-    = λ(x : Natural) → λ(y : Natural) → Bool/not (lessThan x y)
+    = λ(x : Natural) → λ(y : Natural) → lessThanEqual y x
 
 let example0 = assert : greaterThanEqual 5 6 ≡ False
 
 let example1 = assert : greaterThanEqual 5 5 ≡ True
 
-let example1 = assert : greaterThanEqual 5 4 ≡ True
+let example2 = assert : greaterThanEqual 5 4 ≡ True
+
+let property0 = λ(n : Natural) → greaterThanEqual n 0 ≡ True
+
+let property1 = λ(n : Natural) → greaterThanEqual n n ≡ True
 
 in  greaterThanEqual

Prelude/Natural/lessThan

@@ -1,23 +1,26 @@
 {-
 `lessThan` checks if one Natural is strictly less than another.
-
-Examples:
-
-```
-```
 -}
-let lessThanEqual =
-        ./lessThanEqual sha256:1a5caa2b80a42b9f58fff58e47ac0d9a9946d0b2d36c54034b8ddfe3cb0f3c99
-      ? ./lessThanEqual
+let greaterThanEqual =
+        ./greaterThanEqual sha256:30ebfab0febd7aa0ccccfdf3dc36ee6d50f0117f35dd4a9b034750b7e885a1a4
+      ? ./greaterThanEqual
+
+let Bool/not =
+        ../Bool/not sha256:723df402df24377d8a853afed08d9d69a0a6d86e2e5b2bac8960b0d4756c7dc4
+      ? ../Bool/not
 
 let lessThan
     : Natural → Natural → Bool
-    = λ(x : Natural) → λ(y : Natural) → lessThanEqual (x + 1) y
+    = λ(x : Natural) → λ(y : Natural) → Bool/not (greaterThanEqual x y)
 
 let example0 = assert : lessThan 5 6 ≡ True
 
 let example1 = assert : lessThan 5 5 ≡ False
 
 let example2 = assert : lessThan 5 4 ≡ False
 
+let property0 = λ(n : Natural) → assert : lessThan n 0 ≡ False
+
+let property1 = λ(n : Natural) → assert : lessThan n n ≡ False
+
 in  lessThan

Prelude/Natural/lessThanEqual

@@ -1,4 +1,3 @@
-
 {-
 `lessThanEqual` checks if one Natural is less than or equal to another.
 -}
@@ -12,4 +11,8 @@ let example1 = assert : lessThanEqual 5 5 ≡ True
 
 let example2 = assert : lessThanEqual 5 4 ≡ False
 
+let property0 = λ(n : Natural) → lessThanEqual 0 n ≡ True
+
+let property1 = λ(n : Natural) → lessThanEqual n n ≡ True
+
 in  lessThanEqual

Prelude/Natural/package.dhall

@@ -32,18 +32,21 @@
       ./toInteger sha256:160d2d278619f3da34a1f4f02e739a447e4f2aa5a2978c45b710515b41491e1f
     ? ./toInteger
 , lessThan =
-      ./lessThan sha256:d8a37951fd3f88dcc3927c20b73e6c53499f5275d949eed59bb77ea748492bf0
+      ./lessThan sha256:3381b66749290769badf8855d8a3f4af62e8de52d1364d838a9d1e20c94fa70c
     ? ./lessThan
 , lessThanEqual =
       ./lessThanEqual sha256:1a5caa2b80a42b9f58fff58e47ac0d9a9946d0b2d36c54034b8ddfe3cb0f3c99
     ? ./lessThanEqual
 , equal =
-      ./equal sha256:9e0e952e097a2a974ca4ed3e65bcf395cda2bb54382c8ecc3963bca50bfff124
+      ./equal sha256:7f108edfa35ddc7cebafb24dc073478e93a802e13b5bc3fd22f4768c9b066e60
     ? ./equal
 , greaterThanEqual =
-      ./greaterThanEqual sha256:8665ab478524416ecca7573488eb9b64c12a39f1ea86353b67735cf4d3a3bc4f
+      ./greaterThanEqual sha256:30ebfab0febd7aa0ccccfdf3dc36ee6d50f0117f35dd4a9b034750b7e885a1a4
     ? ./greaterThanEqual
 , greaterThan =
       ./greaterThan sha256:f702abcdfcd7ad73619b9285d7e41c3a1d017fb6b8d037cf40bd93bf30c09b2c
     ? ./greaterThan
+, subtract =
+      ./subtract sha256:b9277ac637d09142a3a3ac79137ef5955c42f8b33b6746d59db2c9d75ccdd745
+    ? ./subtract
 }

Prelude/Natural/subtract

@@ -0,0 +1,18 @@
+{-
+`subtract m n` computes `n - m`, truncating to `0` if `m > n`
+-}
+let subtract : Natural → Natural → Natural = Natural/subtract
+
+let example0 = assert : subtract 1 2 ≡ 1
+
+let example1 = assert : subtract 1 1 ≡ 0
+
+let example2 = assert : subtract 2 1 ≡ 0
+
+let property0 = λ(n : Natural) → assert : subtract 0 n ≡ n
+
+let property1 = λ(n : Natural) → assert : subtract n 0 ≡ 0
+
+let property2 = λ(n : Natural) → assert : subtract n n ≡ 0
+
+in  subtract

Prelude/package.dhall

@@ -17,7 +17,7 @@
       ./Map/package.dhall sha256:07cc274220c8bdb2c1a0c2d00d90bc1447e73e0ad2e1d72b89773e923f77e71e
     ? ./Map/package.dhall
 , Natural =
-      ./Natural/package.dhall sha256:8421079b9698b765774455b63f4561f7bfa1506ed3fa39cfb5c34a4cd1c9df4b
+      ./Natural/package.dhall sha256:4574df29e6bb82fd894c311f6c2ad9a22574756eb62f06d475aa8306d25006fc
     ? ./Natural/package.dhall
 , Optional =
       ./Optional/package.dhall sha256:36a366af67a3c26cd5d196e095d3023f18953c5b5db3a03956fa554609e5442a