--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/TutorialI/Rules/Primes.thy Mon Oct 23 16:25:04 2000 +0200
@@ -0,0 +1,351 @@
+(* EXTRACT from HOL/ex/Primes.thy*)
+
+theory Primes = Main:
+consts
+ gcd :: "nat*nat=>nat" (*Euclid's algorithm *)
+
+recdef gcd "measure ((\<lambda>(m,n).n) ::nat*nat=>nat)"
+ "gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
+
+
+ML "Pretty.setmargin 64"
+ML "IsarOutput.indent := 5" (*that is, Doc/TutorialI/settings.ML*)
+
+
+text {*
+\begin{quote}
+@{thm[display]"dvd_def"}
+\rulename{dvd_def}
+\end{quote}
+*};
+
+
+(*** Euclid's Algorithm ***)
+
+lemma gcd_0 [simp]: "gcd(m,0) = m"
+ apply (simp);
+ done
+
+lemma gcd_non_0 [simp]: "0<n \<Longrightarrow> gcd(m,n) = gcd (n, m mod n)"
+ apply (simp)
+ done;
+
+declare gcd.simps [simp del];
+
+(*gcd(m,n) divides m and n. The conjunctions don't seem provable separately*)
+lemma gcd_dvd_both: "(gcd(m,n) dvd m) \<and> (gcd(m,n) dvd n)"
+ apply (induct_tac m n rule: gcd.induct)
+ apply (case_tac "n=0")
+ apply (simp_all)
+ apply (blast dest: dvd_mod_imp_dvd)
+ done
+
+
+text {*
+@{thm[display] dvd_mod_imp_dvd}
+\rulename{dvd_mod_imp_dvd}
+
+@{thm[display] dvd_trans}
+\rulename{dvd_trans}
+
+\begin{isabelle}
+proof\ (prove):\ step\ 3\isanewline
+\isanewline
+goal\ (lemma\ gcd_dvd_both):\isanewline
+gcd\ (m,\ n)\ dvd\ m\ \isasymand \ gcd\ (m,\ n)\ dvd\ n\isanewline
+\ 1.\ \isasymAnd m\ n.\ \isasymlbrakk 0\ <\ n;\ gcd\ (n,\ m\ mod\ n)\ dvd\ n\ \isasymand \ gcd\ (n,\ m\ mod\ n)\ dvd\ (m\ mod\ n)\isasymrbrakk \isanewline
+\ \ \ \ \ \ \ \ \ \ \isasymLongrightarrow \ gcd\ (n,\ m\ mod\ n)\ dvd\ m
+\end{isabelle}
+*};
+
+
+lemmas gcd_dvd1 [iff] = gcd_dvd_both [THEN conjunct1]
+lemmas gcd_dvd2 [iff] = gcd_dvd_both [THEN conjunct2];
+
+
+text {*
+\begin{quote}
+@{thm[display] gcd_dvd1}
+\rulename{gcd_dvd1}
+
+@{thm[display] gcd_dvd2}
+\rulename{gcd_dvd2}
+\end{quote}
+*};
+
+(*Maximality: for all m,n,k naturals,
+ if k divides m and k divides n then k divides gcd(m,n)*)
+lemma gcd_greatest [rule_format]:
+ "(k dvd m) \<longrightarrow> (k dvd n) \<longrightarrow> k dvd gcd(m,n)"
+ apply (induct_tac m n rule: gcd.induct)
+ apply (case_tac "n=0")
+ apply (simp_all add: dvd_mod);
+ done;
+
+theorem gcd_greatest_iff [iff]:
+ "k dvd gcd(m,n) = (k dvd m \<and> k dvd n)"
+ apply (blast intro!: gcd_greatest intro: dvd_trans);
+ done;
+
+
+constdefs
+ is_gcd :: "[nat,nat,nat]=>bool" (*gcd as a relation*)
+ "is_gcd p m n == p dvd m \<and> p dvd n \<and>
+ (ALL d. d dvd m \<and> d dvd n \<longrightarrow> d dvd p)"
+
+(*Function gcd yields the Greatest Common Divisor*)
+lemma is_gcd: "is_gcd (gcd(m,n)) m n"
+ apply (simp add: is_gcd_def gcd_greatest);
+ done
+
+(*uniqueness of GCDs*)
+lemma is_gcd_unique: "\<lbrakk> is_gcd m a b; is_gcd n a b \<rbrakk> \<Longrightarrow> m=n"
+ apply (simp add: is_gcd_def);
+ apply (blast intro: dvd_anti_sym)
+ done
+
+
+text {*
+@{thm[display] dvd_anti_sym}
+\rulename{dvd_anti_sym}
+
+\begin{isabelle}
+proof\ (prove):\ step\ 1\isanewline
+\isanewline
+goal\ (lemma\ is_gcd_unique):\isanewline
+\isasymlbrakk is_gcd\ m\ a\ b;\ is_gcd\ n\ a\ b\isasymrbrakk \ \isasymLongrightarrow \ m\ =\ n\isanewline
+\ 1.\ \isasymlbrakk m\ dvd\ a\ \isasymand \ m\ dvd\ b\ \isasymand \ (\isasymforall d.\ d\ dvd\ a\ \isasymand \ d\ dvd\ b\ \isasymlongrightarrow \ d\ dvd\ m);\isanewline
+\ \ \ \ \ \ \ n\ dvd\ a\ \isasymand \ n\ dvd\ b\ \isasymand \ (\isasymforall d.\ d\ dvd\ a\ \isasymand \ d\ dvd\ b\ \isasymlongrightarrow \ d\ dvd\ n)\isasymrbrakk \isanewline
+\ \ \ \ \isasymLongrightarrow \ m\ =\ n
+\end{isabelle}
+*};
+
+lemma gcd_assoc: "gcd(gcd(k,m),n) = gcd(k,gcd(m,n))"
+ apply (rule is_gcd_unique)
+ apply (rule is_gcd)
+ apply (simp add: is_gcd_def);
+ apply (blast intro: dvd_trans);
+ done
+
+text{*
+\begin{isabelle}
+proof\ (prove):\ step\ 3\isanewline
+\isanewline
+goal\ (lemma\ gcd_assoc):\isanewline
+gcd\ (gcd\ (k,\ m),\ n)\ =\ gcd\ (k,\ gcd\ (m,\ n))\isanewline
+\ 1.\ gcd\ (k,\ gcd\ (m,\ n))\ dvd\ k\ \isasymand \isanewline
+\ \ \ \ gcd\ (k,\ gcd\ (m,\ n))\ dvd\ m\ \isasymand \ gcd\ (k,\ gcd\ (m,\ n))\ dvd\ n
+\end{isabelle}
+*}
+
+
+lemma gcd_dvd_gcd_mult: "gcd(m,n) dvd gcd(k*m, n)"
+ apply (blast intro: dvd_trans);
+ done
+
+(*This is half of the proof (by dvd_anti_sym) of*)
+lemma gcd_mult_cancel: "gcd(k,n) = 1 \<Longrightarrow> gcd(k*m, n) = gcd(m,n)"
+ oops
+
+
+
+text{*\noindent
+Forward proof material: of, OF, THEN, simplify.
+*}
+
+text{*\noindent
+SKIP most developments...
+*}
+
+(** Commutativity **)
+
+lemma is_gcd_commute: "is_gcd k m n = is_gcd k n m"
+ apply (auto simp add: is_gcd_def);
+ done
+
+lemma gcd_commute: "gcd(m,n) = gcd(n,m)"
+ apply (rule is_gcd_unique)
+ apply (rule is_gcd)
+ apply (subst is_gcd_commute)
+ apply (simp add: is_gcd)
+ done
+
+lemma gcd_1 [simp]: "gcd(m,1) = 1"
+ apply (simp)
+ done
+
+lemma gcd_1_left [simp]: "gcd(1,m) = 1"
+ apply (simp add: gcd_commute [of 1])
+ done
+
+text{*\noindent
+as far as HERE.
+*}
+
+
+text {*
+@{thm[display] gcd_1}
+\rulename{gcd_1}
+
+@{thm[display] gcd_1_left}
+\rulename{gcd_1_left}
+*};
+
+text{*\noindent
+SKIP THIS PROOF
+*}
+
+lemma gcd_mult_distrib2: "k * gcd(m,n) = gcd(k*m, k*n)"
+ apply (induct_tac m n rule: gcd.induct)
+ apply (case_tac "n=0")
+ apply (simp)
+ apply (case_tac "k=0")
+ apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2)
+ done
+
+text {*
+@{thm[display] gcd_mult_distrib2}
+\rulename{gcd_mult_distrib2}
+*};
+
+text{*\noindent
+of, simplified
+*}
+
+
+lemmas gcd_mult_0 = gcd_mult_distrib2 [of k 1];
+lemmas gcd_mult_1 = gcd_mult_0 [simplified];
+
+text {*
+@{thm[display] gcd_mult_distrib2 [of _ 1]}
+
+@{thm[display] gcd_mult_0}
+\rulename{gcd_mult_0}
+
+@{thm[display] gcd_mult_1}
+\rulename{gcd_mult_1}
+
+@{thm[display] sym}
+\rulename{sym}
+*};
+
+lemmas gcd_mult = gcd_mult_1 [THEN sym];
+
+lemmas gcd_mult = gcd_mult_distrib2 [of k 1, simplified, THEN sym];
+ (*better in one step!*)
+
+text {*
+more legible
+*};
+
+lemma gcd_mult [simp]: "gcd(k, k*n) = k"
+ apply (rule gcd_mult_distrib2 [of k 1, simplified, THEN sym])
+ done
+
+lemmas gcd_self = gcd_mult [of k 1, simplified];
+
+
+text {*
+Rules handy with THEN
+
+@{thm[display] iffD1}
+\rulename{iffD1}
+
+@{thm[display] iffD2}
+\rulename{iffD2}
+*};
+
+
+text {*
+again: more legible
+*};
+
+lemma gcd_self [simp]: "gcd(k,k) = k"
+ apply (rule gcd_mult [of k 1, simplified])
+ done
+
+lemma relprime_dvd_mult:
+ "\<lbrakk> gcd(k,n)=1; k dvd (m*n) \<rbrakk> \<Longrightarrow> k dvd m";
+ apply (insert gcd_mult_distrib2 [of m k n])
+ apply (simp)
+ apply (erule_tac t="m" in ssubst);
+ apply (simp)
+ done
+
+
+text {*
+Another example of "insert"
+
+@{thm[display] mod_div_equality}
+\rulename{mod_div_equality}
+*};
+
+lemma div_mult_self_is_m:
+ "0<n \<Longrightarrow> (m*n) div n = (m::nat)"
+ apply (insert mod_div_equality [of "m*n" n])
+ apply (simp)
+ done
+
+lemma relprime_dvd_mult_iff: "gcd(k,n)=1 \<Longrightarrow> k dvd (m*n) = k dvd m";
+ apply (blast intro: relprime_dvd_mult dvd_trans)
+ done
+
+lemma relprime_20_81: "gcd(#20,#81) = 1";
+ apply (simp add: gcd.simps)
+ done
+
+
+text {*
+Examples of 'OF'
+
+@{thm[display] relprime_dvd_mult}
+\rulename{relprime_dvd_mult}
+
+@{thm[display] relprime_dvd_mult [OF relprime_20_81]}
+
+@{thm[display] dvd_refl}
+\rulename{dvd_refl}
+
+@{thm[display] dvd_add}
+\rulename{dvd_add}
+
+@{thm[display] dvd_add [OF dvd_refl dvd_refl]}
+
+@{thm[display] dvd_add [OF _ dvd_refl]}
+*};
+
+lemma "\<lbrakk>(z::int) < #37; #66 < #2*z; z*z \<noteq> #1225; Q(#34); Q(#36)\<rbrakk> \<Longrightarrow> Q(z)";
+apply (subgoal_tac "z = #34 \<or> z = #36")
+apply blast
+apply (subgoal_tac "z \<noteq> #35")
+apply arith
+apply force
+done
+
+text
+{*
+proof\ (prove):\ step\ 1\isanewline
+\isanewline
+goal\ (lemma):\isanewline
+\isasymlbrakk z\ <\ \#37;\ \#66\ <\ \#2\ *\ z;\ z\ *\ z\ \isasymnoteq \ \#1225;\ Q\ \#34;\ Q\ \#36\isasymrbrakk \ \isasymLongrightarrow \ Q\ z\isanewline
+\ 1.\ \isasymlbrakk z\ <\ \#37;\ \#66\ <\ \#2\ *\ z;\ z\ *\ z\ \isasymnoteq \ \#1225;\ Q\ \#34;\ Q\ \#36;\isanewline
+\ \ \ \ \ \ \ z\ =\ \#34\ \isasymor \ z\ =\ \#36\isasymrbrakk \isanewline
+\ \ \ \ \isasymLongrightarrow \ Q\ z\isanewline
+\ 2.\ \isasymlbrakk z\ <\ \#37;\ \#66\ <\ \#2\ *\ z;\ z\ *\ z\ \isasymnoteq \ \#1225;\ Q\ \#34;\ Q\ \#36\isasymrbrakk \isanewline
+\ \ \ \ \isasymLongrightarrow \ z\ =\ \#34\ \isasymor \ z\ =\ \#36
+
+
+
+proof\ (prove):\ step\ 3\isanewline
+\isanewline
+goal\ (lemma):\isanewline
+\isasymlbrakk z\ <\ \#37;\ \#66\ <\ \#2\ *\ z;\ z\ *\ z\ \isasymnoteq \ \#1225;\ Q\ \#34;\ Q\ \#36\isasymrbrakk \ \isasymLongrightarrow \ Q\ z\isanewline
+\ 1.\ \isasymlbrakk z\ <\ \#37;\ \#66\ <\ \#2\ *\ z;\ z\ *\ z\ \isasymnoteq \ \#1225;\ Q\ \#34;\ Q\ \#36;\isanewline
+\ \ \ \ \ \ \ z\ \isasymnoteq \ \#35\isasymrbrakk \isanewline
+\ \ \ \ \isasymLongrightarrow \ z\ =\ \#34\ \isasymor \ z\ =\ \#36\isanewline
+\ 2.\ \isasymlbrakk z\ <\ \#37;\ \#66\ <\ \#2\ *\ z;\ z\ *\ z\ \isasymnoteq \ \#1225;\ Q\ \#34;\ Q\ \#36\isasymrbrakk \isanewline
+\ \ \ \ \isasymLongrightarrow \ z\ \isasymnoteq \ \#35
+*}
+
+
+end