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{*example of arg_cong (NEW)

 @{thm[display] arg_cong[no_vars]}

 \rulename{arg_cong}

 *}

 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