src/HOL/Isar_examples/Fibonacci.thy

     = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)"
     by (simp add: add_mult_distrib2)
   finally show "?P (n + 2)" .
 qed
 
-lemma gcd_fib_Suc_eq_1: "gcd (fib n, fib (n + 1)) = 1" (is "?P n")
+lemma gcd_fib_Suc_eq_1: "gcd (fib n) (fib (n + 1)) = 1" (is "?P n")
 proof (induct n rule: fib_induct)
   show "?P 0" by simp
   show "?P 1" by simp
   fix n
   have "fib (n + 2 + 1) = fib (n + 1) + fib (n + 2)"
     by simp
-  also have "gcd (fib (n + 2), ...) = gcd (fib (n + 2), fib (n + 1))"
+  also have "gcd (fib (n + 2)) ... = gcd (fib (n + 2)) (fib (n + 1))"
     by (simp only: gcd_add2')
-  also have "... = gcd (fib (n + 1), fib (n + 1 + 1))"
+  also have "... = gcd (fib (n + 1)) (fib (n + 1 + 1))"
     by (simp add: gcd_commute)
   also assume "... = 1"
   finally show "?P (n + 2)" .
 qed
 
-lemma gcd_mult_add: "0 < n ==> gcd (n * k + m, n) = gcd (m, n)"
+lemma gcd_mult_add: "0 < n ==> gcd (n * k + m) n = gcd m n"
 proof -
   assume "0 < n"
-  then have "gcd (n * k + m, n) = gcd (n, m mod n)"
+  then have "gcd (n * k + m) n = gcd n (m mod n)"
     by (simp add: gcd_non_0 add_commute)
-  also from `0 < n` have "... = gcd (m, n)" by (simp add: gcd_non_0)
+  also from `0 < n` have "... = gcd m n" by (simp add: gcd_non_0)
   finally show ?thesis .
 qed
 
-lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)"
+lemma gcd_fib_add: "gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n)"
 proof (cases m)
   case 0
   then show ?thesis by simp
 next
   case (Suc k)
-  then have "gcd (fib m, fib (n + m)) = gcd (fib (n + k + 1), fib (k + 1))"
+  then have "gcd (fib m) (fib (n + m)) = gcd (fib (n + k + 1)) (fib (k + 1))"
     by (simp add: gcd_commute)
   also have "fib (n + k + 1)
     = fib (k + 1) * fib (n + 1) + fib k * fib n"
     by (rule fib_add)
-  also have "gcd (..., fib (k + 1)) = gcd (fib k * fib n, fib (k + 1))"
+  also have "gcd ... (fib (k + 1)) = gcd (fib k * fib n) (fib (k + 1))"
     by (simp add: gcd_mult_add)
-  also have "... = gcd (fib n, fib (k + 1))"
+  also have "... = gcd (fib n) (fib (k + 1))"
     by (simp only: gcd_fib_Suc_eq_1 gcd_mult_cancel)
-  also have "... = gcd (fib m, fib n)"
+  also have "... = gcd (fib m) (fib n)"
     using Suc by (simp add: gcd_commute)
   finally show ?thesis .
 qed
 
 lemma gcd_fib_diff:
   assumes "m <= n"
-  shows "gcd (fib m, fib (n - m)) = gcd (fib m, fib n)"
+  shows "gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
 proof -
-  have "gcd (fib m, fib (n - m)) = gcd (fib m, fib (n - m + m))"
+  have "gcd (fib m) (fib (n - m)) = gcd (fib m) (fib (n - m + m))"
     by (simp add: gcd_fib_add)
   also from `m <= n` have "n - m + m = n" by simp
   finally show ?thesis .
 qed
 
 lemma gcd_fib_mod:
   assumes "0 < m"
-  shows "gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"
+  shows "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
 proof (induct n rule: nat_less_induct)
   case (1 n) note hyp = this
   show ?case
   proof -
     have "n mod m = (if n < m then n else (n - m) mod m)"
       by (rule mod_if)
-    also have "gcd (fib m, fib ...) = gcd (fib m, fib n)"
+    also have "gcd (fib m) (fib ...) = gcd (fib m) (fib n)"
     proof (cases "n < m")
       case True then show ?thesis by simp
     next
       case False then have "m <= n" by simp
       from `0 < m` and False have "n - m < n" by simp
-      with hyp have "gcd (fib m, fib ((n - m) mod m))
-        = gcd (fib m, fib (n - m))" by simp
-      also have "... = gcd (fib m, fib n)"
+      with hyp have "gcd (fib m) (fib ((n - m) mod m))
+        = gcd (fib m) (fib (n - m))" by simp
+      also have "... = gcd (fib m) (fib n)"
         using `m <= n` by (rule gcd_fib_diff)
-      finally have "gcd (fib m, fib ((n - m) mod m)) =
-        gcd (fib m, fib n)" .
+      finally have "gcd (fib m) (fib ((n - m) mod m)) =
+        gcd (fib m) (fib n)" .
       with False show ?thesis by simp
     qed
     finally show ?thesis .
   qed
 qed
 
 
-theorem fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)" (is "?P m n")
+theorem fib_gcd: "fib (gcd m n) = gcd (fib m) (fib n)" (is "?P m n")
 proof (induct m n rule: gcd_induct)
-  fix m show "fib (gcd (m, 0)) = gcd (fib m, fib 0)" by simp
+  fix m show "fib (gcd m 0) = gcd (fib m) (fib 0)" by simp
   fix n :: nat assume n: "0 < n"
-  then have "gcd (m, n) = gcd (n, m mod n)" by (rule gcd_non_0)
-  also assume hyp: "fib ... = gcd (fib n, fib (m mod n))"
-  also from n have "... = gcd (fib n, fib m)" by (rule gcd_fib_mod)
-  also have "... = gcd (fib m, fib n)" by (rule gcd_commute)
-  finally show "fib (gcd (m, n)) = gcd (fib m, fib n)" .
+  then have "gcd m n = gcd n (m mod n)" by (rule gcd_non_0)
+  also assume hyp: "fib ... = gcd (fib n) (fib (m mod n))"
+  also from n have "... = gcd (fib n) (fib m)" by (rule gcd_fib_mod)
+  also have "... = gcd (fib m) (fib n)" by (rule gcd_commute)
+  finally show "fib (gcd m n) = gcd (fib m) (fib n)" .
 qed
 
 end