Founding the newly structured GF2.0 cvs archive.

grammars/logic/ArithmEng.gf

@@ -0,0 +1,40 @@
+concrete ArithmEng of Arithm = LogicEng ** open LogicResEng in {
+
+lin
+  Nat = {s = nomReg "number"} ;
+  zero = ss "zero" ;
+  succ = fun1 "successor" ;
+
+  EqNat = adj2 ["equal to"] ;
+  LtNat = adj2 ["smaller than"] ;
+  Div   = adj2 ["divisible by"] ;
+  Even  = adj1 "even" ;
+  Odd   = adj1 "odd" ;
+  Prime = adj1 "prime" ;
+
+  one = ss "one" ;
+  two = ss "two" ;
+  sum = fun2 "sum" ;
+  prod = fun2 "product" ;
+
+  evax1 = ss ["by the first axiom of evenness , zero is even"] ;
+  evax2 n c = {s = 
+    c.s ++ [". By the second axiom of evenness , the successor of"] ++ 
+    n.s ++ ["is odd"]} ;
+  evax3 n c = {s = 
+    c.s ++ [". By the third axiom of evenness , the successor of"] ++ 
+    n.s ++ ["is even"]} ;
+  eqax1 = ss ["by the first axiom of equality , zero is equal to zero"] ;
+  eqax2 m n c = {s = 
+    c.s ++ [". By the second axiom of equality , the successor of"] ++ m.s ++ 
+    ["is equal to the successor of"] ++ n.s} ;
+  IndNat C d e = {s = 
+    ["we proceed by induction . For the basis ,"] ++ d.s ++ 
+    [". For the induction step, consider a number"] ++ C.$0 ++ 
+    ["and assume"] ++ C.s ++ "(" ++ e.$1 ++ ")" ++ "." ++ e.s ++ 
+    ["Hence, for all numbers"] ++ C.$0 ++ "," ++ C.s} ;
+
+  ex1 = ss ["The first theorem and its proof ."] ;
+
+} ;
+