# HG changeset patch
# User paulson
# Date 1111764057 -3600
# Node ID 9f912f8fd2df1986cfea383949add2e3b6316008
# Parent  7a108ae4c7981aac227567006171f9b4b4e9155a
tidied up

diff -r 7a108ae4c798 -r 9f912f8fd2df src/HOL/Library/Primes.thy
--- a/src/HOL/Library/Primes.thy	Fri Mar 25 14:14:01 2005 +0100
+++ b/src/HOL/Library/Primes.thy	Fri Mar 25 16:20:57 2005 +0100
@@ -210,11 +210,13 @@
   done
 
 lemma gcd_add2 [simp]: "gcd (m, m + n) = gcd (m, n)"
-  apply (rule gcd_commute [THEN trans])
-  apply (subst add_commute)
-  apply (simp add: gcd_add1)
-  apply (rule gcd_commute)
-  done
+proof -
+  have "gcd (m, m + n) = gcd (m + n, m)" by (rule gcd_commute) 
+  also have "... = gcd (n + m, m)" by (simp add: add_commute)
+  also have "... = gcd (n, m)" by simp
+  also have  "... = gcd (m, n)" by (rule gcd_commute) 
+  finally show ?thesis .
+qed
 
 lemma gcd_add2' [simp]: "gcd (m, n + m) = gcd (m, n)"
   apply (subst add_commute)
@@ -223,7 +225,7 @@
 
 lemma gcd_add_mult: "gcd (m, k * m + n) = gcd (m, n)"
   apply (induct k)
-   apply (simp_all add: gcd_add2 add_assoc)
+   apply (simp_all add: add_assoc)
   done
 
 
@@ -235,8 +237,8 @@
     apply (rule_tac n = k in relprime_dvd_mult)
      apply (simp add: gcd_assoc)
      apply (simp add: gcd_commute)
-    apply (simp_all add: mult_commute gcd_dvd1 gcd_dvd2)
-  apply (blast intro: gcd_dvd1 dvd_trans)
+    apply (simp_all add: mult_commute)
+  apply (blast intro: dvd_trans)
   done
 
 end
diff -r 7a108ae4c798 -r 9f912f8fd2df src/HOL/NumberTheory/Fib.thy
--- a/src/HOL/NumberTheory/Fib.thy	Fri Mar 25 14:14:01 2005 +0100
+++ b/src/HOL/NumberTheory/Fib.thy	Fri Mar 25 16:20:57 2005 +0100
@@ -1,5 +1,4 @@
-(*  Title:      HOL/NumberTheory/Fib.thy
-    ID:         $Id$
+(*  ID:         $Id$
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1997  University of Cambridge
 *)
@@ -17,10 +16,10 @@
 *}
 
 consts fib :: "nat => nat"
-recdef fib  less_than
-  zero: "fib 0  = 0"
-  one:  "fib (Suc 0) = Suc 0"
-  Suc_Suc: "fib (Suc (Suc x)) = fib x + fib (Suc x)"
+recdef fib  "measure (\<lambda>x. x)"
+   zero:    "fib 0  = 0"
+   one:     "fib (Suc 0) = Suc 0"
+   Suc_Suc: "fib (Suc (Suc x)) = fib x + fib (Suc x)"
 
 text {*
   \medskip The difficulty in these proofs is to ensure that the
@@ -29,11 +28,12 @@
   to the Simplifier and are applied very selectively at first.
 *}
 
+text{*We disable @{text fib.Suc_Suc} for simplification ...*}
 declare fib.Suc_Suc [simp del]
 
+text{*...then prove a version that has a more restrictive pattern.*}
 lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))"
-  apply (rule fib.Suc_Suc)
-  done
+  by (rule fib.Suc_Suc)
 
 
 text {* \medskip Concrete Mathematics, page 280 *}
@@ -42,45 +42,53 @@
   apply (induct n rule: fib.induct)
     prefer 3
     txt {* simplify the LHS just enough to apply the induction hypotheses *}
-    apply (simp add: fib.Suc_Suc [of "Suc (m + n)", standard])
-    apply (simp_all (no_asm_simp) add: fib.Suc_Suc add_mult_distrib add_mult_distrib2)
+    apply (simp add: fib_Suc3)
+    apply (simp_all add: fib.Suc_Suc add_mult_distrib2)
     done
 
-lemma fib_Suc_neq_0 [simp]: "fib (Suc n) \<noteq> 0"
+lemma fib_Suc_neq_0: "fib (Suc n) \<noteq> 0"
   apply (induct n rule: fib.induct)
     apply (simp_all add: fib.Suc_Suc)
   done
 
-lemma [simp]: "0 < fib (Suc n)"
-  apply (simp add: neq0_conv [symmetric])
-  done
+lemma fib_Suc_gr_0: "0 < fib (Suc n)"
+  by (insert fib_Suc_neq_0 [of n], simp)  
 
 lemma fib_gr_0: "0 < n ==> 0 < fib n"
-  apply (rule not0_implies_Suc [THEN exE])
-   apply auto
-  done
+  by (case_tac n, auto simp add: fib_Suc_gr_0) 
 
 
 text {*
-  \medskip Concrete Mathematics, page 278: Cassini's identity.  It is
-  much easier to prove using integers!
+  \medskip Concrete Mathematics, page 278: Cassini's identity.  The proof is
+  much easier using integers, not natural numbers!
 *}
 
-lemma fib_Cassini:
+lemma fib_Cassini_int:
  "int (fib (Suc (Suc n)) * fib n) =
   (if n mod 2 = 0 then int (fib (Suc n) * fib (Suc n)) - 1
    else int (fib (Suc n) * fib (Suc n)) + 1)"
   apply (induct n rule: fib.induct)
     apply (simp add: fib.Suc_Suc)
    apply (simp add: fib.Suc_Suc mod_Suc)
-  apply (simp add: fib.Suc_Suc
-    add_mult_distrib add_mult_distrib2 mod_Suc zmult_int [symmetric] zmult_ac)
-  apply (subgoal_tac "x mod 2 < 2", arith)
-  apply simp
+  apply (simp add: fib.Suc_Suc add_mult_distrib add_mult_distrib2
+                   mod_Suc zmult_int [symmetric])
+  apply presburger
+  done
+
+text{*We now obtain a version for the natural numbers via the coercion 
+   function @{term int}.*}
+theorem fib_Cassini:
+ "fib (Suc (Suc n)) * fib n =
+  (if n mod 2 = 0 then fib (Suc n) * fib (Suc n) - 1
+   else fib (Suc n) * fib (Suc n) + 1)"
+  apply (rule int_int_eq [THEN iffD1]) 
+  apply (simp add: fib_Cassini_int) 
+  apply (subst zdiff_int [symmetric]) 
+   apply (insert fib_Suc_gr_0 [of n], simp_all)
   done
 
 
-text {* \medskip Towards Law 6.111 of Concrete Mathematics *}
+text {* \medskip Toward Law 6.111 of Concrete Mathematics *}
 
 lemma gcd_fib_Suc_eq_1: "gcd (fib n, fib (Suc n)) = Suc 0"
   apply (induct n rule: fib.induct)
@@ -90,35 +98,29 @@
   done
 
 lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)"
-  apply (simp (no_asm) add: gcd_commute [of "fib m"])
-  apply (case_tac "m = 0")
-   apply simp
-  apply (clarify dest!: not0_implies_Suc)
+  apply (simp add: gcd_commute [of "fib m"])
+  apply (case_tac m)
+   apply simp 
   apply (simp add: fib_add)
-  apply (simp add: add_commute gcd_non_0)
-  apply (simp add: gcd_non_0 [symmetric])
+  apply (simp add: add_commute gcd_non_0 [OF fib_Suc_gr_0])
+  apply (simp add: gcd_non_0 [OF fib_Suc_gr_0, symmetric])
   apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)
   done
 
 lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)"
-  apply (rule gcd_fib_add [symmetric, THEN trans])
-  apply simp
-  done
+  by (simp add: gcd_fib_add [symmetric, of _ "n-m"]) 
 
 lemma gcd_fib_mod: "0 < m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"
   apply (induct n rule: nat_less_induct)
-  apply (subst mod_if)
-  apply (simp add: gcd_fib_diff mod_geq not_less_iff_le)
+  apply (simp add: mod_if gcd_fib_diff mod_geq)
   done
 
 lemma fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)"  -- {* Law 6.111 *}
   apply (induct m n rule: gcd_induct)
-   apply simp
-  apply (simp add: gcd_non_0)
-  apply (simp add: gcd_commute gcd_fib_mod)
+  apply (simp_all add: gcd_non_0 gcd_commute gcd_fib_mod)
   done
 
-lemma fib_mult_eq_setsum:
+theorem fib_mult_eq_setsum:
     "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
   apply (induct n rule: fib.induct)
     apply (auto simp add: atMost_Suc fib.Suc_Suc)