doc-src/TutorialI/Rules/Forward.thy

+theory Forward = Primes:
+
+text{*\noindent
+Forward proof material: of, OF, THEN, simplify, rule_format.
+*}
+
+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"
+by (rule gcd_mult_distrib2 [of k 1, simplified, THEN sym])
+
+
+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"
+by (rule gcd_mult [of k 1, simplified])
+
+
+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)";
+by (blast intro: relprime_dvd_mult dvd_trans)
+
+
+lemma relprime_20_81: "gcd(#20,#81) = 1";
+by (simp add: gcd.simps)
+
+
+
+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