From 6f4601d5bd245e99831cedb12b732d77501a16e1 Mon Sep 17 00:00:00 2001
From: Masaya Tojo <masaya@tojo.tokyo>
Date: Sun, 16 Jun 2024 18:39:20 +0900
Subject: Add usages section to README.md

---
 README.md                        | 85 ++++++++++++++++++++++++++++++++++++++++
 examples/mod7.scm                | 51 ++++++++++++++++++++++++
 examples/pythagorean-triples.scm | 11 ++++++
 3 files changed, 147 insertions(+)
 create mode 100644 examples/mod7.scm
 create mode 100644 examples/pythagorean-triples.scm

diff --git a/README.md b/README.md
index 7152615..cc112db 100644
--- a/README.md
+++ b/README.md
@@ -11,6 +11,91 @@ $ cd algebraic-structures
 $ chicken-install
 ```
 
+## Usage
+
+### Define Semigroup, Monoid and Group
+
+```scheme
+(import (algebraic-structures semigroup)
+        (algebraic-structures monoid)
+        (algebraic-structures group))
+
+(module (mod7 semigroup) = (algebraic-structures semigroup)
+  (import scheme
+          (chicken module)
+          (chicken base))
+  (export <>)
+
+  (define (<> x y)
+    (assert (integer? x))
+    (assert (integer? y))
+    (assert (not (zero? x)))
+    (assert (not (zero? y)))
+    (modulo (* x y) 7)))
+
+(module (mod7 monoid) = (algebraic-structures monoid)
+  (import scheme
+          (chicken module)
+          (chicken base))
+  (reexport (mod7 semigroup))
+  (export unit)
+
+  (define unit 1))
+
+(module (mod7 group) = (algebraic-structures group)
+  (import scheme
+          (chicken base)
+          (chicken module)
+          matchable)
+  (reexport (mod7 monoid))
+  (export inv)
+
+  (define (inv n)
+    (assert (integer? n))
+    (assert (not (zero? n)))
+    (match (modulo n 7)
+      (1 1)
+      (2 4)
+      (3 5)
+      (4 2)
+      (5 3)
+      (6 6))))
+
+(import (prefix (mod7 group) mod7:))
+```
+
+In REPL:
+
+```
+> (map (cut mod7:pow 3 <>) '(0 1 2 3 4 5 6 7 8 9 10 11))
+(1 3 2 6 4 5 1 3 2 6 4 5)
+> (mod7:fold '(1 2 3 4 5 6))
+6
+```
+
+### Monad Syntax
+
+```scheme
+(import (srfi 41)
+        (prefix (algebraic-structures stream monad) stream:)
+        (prefix (algebraic-structures stream alternative) stream:))
+
+(define (pythagorean-triples)
+  (stream:do (b <- (stream-from 1))
+             (a <- (stream-range 1 b))
+             (let c^2 = (+ (* a a) (* b b)))
+             (let-values (c r) = (exact-integer-sqrt c^2))
+             (stream:guard (zero? r))
+             (stream (list a b c))))
+```
+
+In REPL:
+
+```
+> (stream->list (stream-take 10 (pythagorean-triples)))
+((3 4 5) (6 8 10) (5 12 13) (9 12 15) (8 15 17) (12 16 20) (15 20 25) (20 21 29) (7 24 25) (10 24 26))
+```
+
 ## Supported Features
 
 - Semigroup: `(algebraic-structures semigroup)`
diff --git a/examples/mod7.scm b/examples/mod7.scm
new file mode 100644
index 0000000..cfdce61
--- /dev/null
+++ b/examples/mod7.scm
@@ -0,0 +1,51 @@
+(import (algebraic-structures semigroup)
+        (algebraic-structures monoid)
+        (algebraic-structures group)
+        (algebraic-structures monoid fold)
+        (algebraic-structures list foldable))
+
+(module (mod7 semigroup) = (algebraic-structures semigroup)
+  (import scheme
+          (chicken module)
+          (chicken base))
+  (export <>)
+
+  (define (<> x y)
+    (assert (integer? x))
+    (assert (integer? y))
+    (assert (not (zero? x)))
+    (assert (not (zero? y)))
+    (modulo (* x y) 7)))
+
+(module (mod7 monoid) = (algebraic-structures monoid)
+  (import scheme
+          (chicken module)
+          (chicken base))
+  (reexport (mod7 semigroup))
+  (export unit)
+
+  (define unit 1))
+
+(module (mod7 group) = (algebraic-structures group)
+  (import scheme
+          (chicken base)
+          (chicken module)
+          matchable)
+  (reexport (mod7 monoid))
+  (export inv)
+
+  (define (inv n)
+    (assert (integer? n))
+    (assert (not (zero? n)))
+    (match (modulo n 7)
+      (1 1)
+      (2 4)
+      (3 5)
+      (4 2)
+      (5 3)
+      (6 6))))
+
+(module (mod7 fold) = ((algebraic-structures monoid fold) (mod7 monoid) (algebraic-structures list foldable)))
+
+(import (prefix (mod7 group) mod7:)
+        (prefix (mod7 fold) mod7:))
diff --git a/examples/pythagorean-triples.scm b/examples/pythagorean-triples.scm
new file mode 100644
index 0000000..4f620af
--- /dev/null
+++ b/examples/pythagorean-triples.scm
@@ -0,0 +1,11 @@
+(import (srfi 41)
+        (prefix (algebraic-structures stream monad) stream:)
+        (prefix (algebraic-structures stream alternative) stream:))
+
+(define (pythagorean-triples)
+  (stream:do (b <- (stream-from 1))
+             (a <- (stream-range 1 b))
+             (let c^2 = (+ (* a a) (* b b)))
+             (let-values (c r) = (exact-integer-sqrt c^2))
+             (stream:guard (zero? r))
+             (stream (list a b c))))
-- 
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