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 {*

@{thm[display] gcd_mult}

\rulename{gcd_mult}

@{thm[display] gcd_self0}

\rulename{gcd_self0}

*};

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])

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

apply simp

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

apply (erule_tac t="m" in ssubst);

apply simp

done

text {*

@{thm[display] relprime_dvd_mult}

\rulename{relprime_dvd_mult}

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 (auto intro: relprime_dvd_mult elim: dvdE)

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