src/Doc/Tutorial/Rules/Forward.thy

+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
+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