theory Forward imports Primes begin

 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 (Suc 0) = Suc 0"

 apply simp

 done

 lemma gcd_1_left [simp]: "gcd (Suc 0) m = Suc 0"

 apply (simp add: gcd_commute [of "Suc 0"])

 done

 text{*\noindent

 as far as HERE.

 *}

 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];

 lemmas where1 = gcd_mult_distrib2 [where m=1]

 lemmas where2 = gcd_mult_distrib2 [where m=1 and k=1]

 lemmas where3 = gcd_mult_distrib2 [where m=1 and k="j+k"]

 text {*

 example using ``of'':

 @{thm[display] gcd_mult_distrib2 [of _ 1]}

 example using ``where'':

 @{thm[display] gcd_mult_distrib2 [where m=1]}

 example using ``where'', ``and'':

 @{thm[display] gcd_mult_distrib2 [where m=1 and k="j+k"]}

 @{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_mult0 = gcd_mult_1 [THEN sym];

       (*not quite right: we need ?k but this gives k*)

 lemmas gcd_mult0' = gcd_mult_distrib2 [of k 1, simplified, THEN sym];

       (*better in one step!*)

 text {*

 more legible, and variables properly generalized

 *};

 lemma gcd_mult [simp]: "gcd k (k*n) = k"

 by (rule gcd_mult_distrib2 [of k 1, simplified, THEN sym])

 lemmas gcd_self0 = 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, and variables properly generalized

 *};

 lemma gcd_self [simp]: "gcd k k = k"

 by (rule gcd_mult [of k 1, simplified])

 text{*

 NEXT SECTION: Methods for Forward Proof

 NEW

 theorem arg_cong, useful in forward steps

 @{thm[display] arg_cong[no_vars]}

 \rulename{arg_cong}

 *}

 lemma "2 \<le> u \<Longrightarrow> u*m \<noteq> Suc(u*n)"

 apply (intro notI)

 txt{*

 before using arg_cong

 @{subgoals[display,indent=0,margin=65]}

 *};

 apply (drule_tac f="\<lambda>x. x mod u" in arg_cong)

 txt{*

 after using arg_cong

 @{subgoals[display,indent=0,margin=65]}

 *};

 apply (simp add: mod_Suc)

 done

 text{*

 have just used this rule:

 @{thm[display] mod_Suc[no_vars]}

 \rulename{mod_Suc}

 @{thm[display] mult_le_mono1[no_vars]}

 \rulename{mult_le_mono1}

 *}

 text{*

 example of "insert"

 *}

 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}

 *};

 (*MOVED to Force.thy, which now depends only on Divides.thy

 lemma div_mult_self_is_m: "0<n \<Longrightarrow> (m*n) div n = (m::nat)"

 *)

 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")

 txt{*

 the tactic leaves two subgoals:

 @{subgoals[display,indent=0,margin=65]}

 *};

 apply blast

 apply (subgoal_tac "z \<noteq> 35")

 txt{*

 the tactic leaves two subgoals:

 @{subgoals[display,indent=0,margin=65]}

 *};

 apply arith

 apply force

 done

 end