From 35c4714a8cc38b2a8106b54b87684483b5a94f62 Mon Sep 17 00:00:00 2001
From: Masaya Tojo <masaya@tojo.tokyo>
Date: Fri, 18 Dec 2020 09:41:23 +0900
Subject: wip58

---
 vikalpa/the-little-prover.scm | 104 ++++++++++++++++++++++++++----------------
 1 file changed, 64 insertions(+), 40 deletions(-)

(limited to 'vikalpa/the-little-prover.scm')

diff --git a/vikalpa/the-little-prover.scm b/vikalpa/the-little-prover.scm
index 338843b..fe7735e 100644
--- a/vikalpa/the-little-prover.scm
+++ b/vikalpa/the-little-prover.scm
@@ -17,24 +17,53 @@
 ;;; along with Vikalpa.  If not, see <http://www.gnu.org/licenses/>.
 
 (define-module (vikalpa the-little-prover)
-  #:export (the-little-prover)
+  #:export (atom? nat? the-little-prover)
   #:use-module (vikalpa))
 
-(define (size x)
-  (if (pair? x)
-      (+ (size (car x))
-         (size (cdr x)))
-      0))
+(define (atom? x)
+  (not (pair? x)))
+
+(define (nat? x)
+  (and (integer? x)
+       (<= 0 x)))
 
 (define-system the-little-prover ()
-  (define-function atom? (x)
-    (not (pair? x)))
-  (define-builtin-function size (x))
-  (define-builtin-function + (x y))
-  (define-builtin-function natural? (x))
-  (define-builtin-function < (x y))
-  (define-totality-claim natural? natural? <)
-  
+  (define-core-function atom? (x) atom?)
+  (define-core-function nat? (x) nat?)
+  (define-core-function < (x y)
+    (lambda (x y)
+      (if (number? x)
+          (if (number? y)
+              (< x y)
+              (< x 0))
+          (if (number? y)
+              (< 0 y)
+              #f))))
+  (set-measure-predicate nat?)
+  (set-measure-less-than <)
+  (define-core-function + (x y)
+    (lambda (x y)
+      (if (number? x)
+          (if (number? y)
+              (+ x y)
+              x)
+          (if (number? y)
+              y
+              0))))
+  (define-core-function cons (x y) cons)
+  (define-core-function car (x)
+    (lambda (x)
+      (if (atom? x) '() (car x))))
+  (define-core-function cdr (x)
+    (lambda (x)
+      (if (atom? x) '() (cdr x))))
+  (define-trivial-total-function size (x)
+    (if (atom? x)
+        0
+        (+ 1
+           (+ (size (car x))
+              (size (cdr x))))))
+
   ;; Axioms of Equal
   (define-axiom equal-same (x)
     (equal? (equal? x x) #t))
@@ -68,8 +97,8 @@
     (if x #t (equal? (if x y z) z)))
 
   ;; Axioms of Size
-  (define-axiom natural?/size (x)
-    (equal? (natural? (size x)) #t))
+  (define-axiom nat/size (x)
+    (equal? (nat? (size x)) #t))
   (define-axiom size/car (x)
     (if (atom? x)
         #t
@@ -81,8 +110,8 @@
 
   ;; Axioms of `+` and `<`
   (define-axiom identity-+ (x)
-    (if (natural? x)
-        (equal? (+ (+ x y) z))
+    (if (nat? x)
+        (equal? (+ 0 x) x)
         #t))
   (define-axiom commute-+ (x y)
     (equal? (+ x y) (+ y x)))
@@ -94,23 +123,16 @@
             (equal? (< '0 (+ x y)) #t)
             #t)
         #t))
-  (define-axiom natural?/+ (x y)
-    (if (natural? x)
-        (if (natural? y)
-            (equal? (natural? (+ x y)) #t)
+  (define-axiom nat/+ (x y)
+    (if (nat? x)
+        (if (nat? y)
+            (equal? (nat? (+ x y)) #t)
             #t)
         #t))
   (define-axiom common-addends-< (x y z)
-    (equal? (< (+ x z) (+ y z) (< x y))))
-
-  ;; Axioms for Vikalpa
-  (define-axiom natural?/0 ()
-    (equal? (natural? '0) #t))
-  (define-axiom natural?/succ (x)
-    (if (natural? x)
-        (equal? (natural? (succ x)) #t)
-        #t))
-
+    (equal? (< (+ x z) (+ y z))
+            (< x y)))
+  
   ;; Prelude
   (define-function list-induction (x)
     (if (atom? x)
@@ -118,22 +140,24 @@
         (cons (car x)
               (list-induction (cdr x)))))
 
-  (define-proof list-induction
-    (natural? (size x))
-    (((2 3) size/cdr)
-     ((2) if-same)
-     (() if-same)))
-  
   (define-function star-induction (x)
     (if (atom? x)
         x
         (cons (star-induction (car x))
               (star-induction (cdr x)))))
 
+  (define-proof list-induction
+    (size x)
+    (((2 3) size/cdr)
+     ((2) if-same)
+     ((1) nat/size)
+     (() if-true)))
+
   (define-proof star-induction
-    (natural? (size x))
+    (size x)
     (((2 3 1) size/car)
      ((2 3 2) size/cdr)
      ((2 3) if-true)
      ((2) if-same)
-     (() if-same))))
+     ((1) nat/size)
+     (() if-true))))
-- 
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