1 theory Forward = Primes:
4 Forward proof material: of, OF, THEN, simplify, rule_format.
8 SKIP most developments...
13 lemma is_gcd_commute: "is_gcd k m n = is_gcd k n m"
14 apply (auto simp add: is_gcd_def);
17 lemma gcd_commute: "gcd(m,n) = gcd(n,m)"
18 apply (rule is_gcd_unique)
20 apply (subst is_gcd_commute)
21 apply (simp add: is_gcd)
24 lemma gcd_1 [simp]: "gcd(m, Suc 0) = Suc 0"
28 lemma gcd_1_left [simp]: "gcd(Suc 0, m) = Suc 0"
29 apply (simp add: gcd_commute [of "Suc 0"])
40 lemma gcd_mult_distrib2: "k * gcd(m,n) = gcd(k*m, k*n)"
41 apply (induct_tac m n rule: gcd.induct)
42 apply (case_tac "n=0")
44 apply (case_tac "k=0")
45 apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2)
49 @{thm[display] gcd_mult_distrib2}
50 \rulename{gcd_mult_distrib2}
58 lemmas gcd_mult_0 = gcd_mult_distrib2 [of k 1];
59 lemmas gcd_mult_1 = gcd_mult_0 [simplified];
62 @{thm[display] gcd_mult_distrib2 [of _ 1]}
64 @{thm[display] gcd_mult_0}
67 @{thm[display] gcd_mult_1}
74 lemmas gcd_mult0 = gcd_mult_1 [THEN sym];
75 (*not quite right: we need ?k but this gives k*)
77 lemmas gcd_mult0' = gcd_mult_distrib2 [of k 1, simplified, THEN sym];
78 (*better in one step!*)
81 more legible, and variables properly generalized
84 lemma gcd_mult [simp]: "gcd(k, k*n) = k"
85 by (rule gcd_mult_distrib2 [of k 1, simplified, THEN sym])
88 lemmas gcd_self0 = gcd_mult [of k 1, simplified];
103 again: more legible, and variables properly generalized
106 lemma gcd_self [simp]: "gcd(k,k) = k"
107 by (rule gcd_mult [of k 1, simplified])
111 NEXT SECTION: Methods for Forward Proof
115 theorem arg_cong, useful in forward steps
116 @{thm[display] arg_cong[no_vars]}
120 lemma "2 \<le> u \<Longrightarrow> u*m \<noteq> Suc(u*n)"
123 before using arg_cong
124 @{subgoals[display,indent=0,margin=65]}
126 apply (drule_tac f="\<lambda>x. x mod u" in arg_cong)
129 @{subgoals[display,indent=0,margin=65]}
131 apply (simp add: mod_Suc)
135 have just used this rule:
136 @{thm[display] mod_Suc[no_vars]}
139 @{thm[display] mult_le_mono1[no_vars]}
140 \rulename{mult_le_mono1}
148 lemma relprime_dvd_mult:
149 "\<lbrakk> gcd(k,n)=1; k dvd m*n \<rbrakk> \<Longrightarrow> k dvd m"
150 apply (insert gcd_mult_distrib2 [of m k n])
152 apply (erule_tac t="m" in ssubst);
158 Another example of "insert"
160 @{thm[display] mod_div_equality}
161 \rulename{mod_div_equality}
164 (*MOVED to Force.thy, which now depends only on Divides.thy
165 lemma div_mult_self_is_m: "0<n \<Longrightarrow> (m*n) div n = (m::nat)"
168 lemma relprime_dvd_mult_iff: "gcd(k,n)=1 \<Longrightarrow> (k dvd m*n) = (k dvd m)";
169 by (blast intro: relprime_dvd_mult dvd_trans)
172 lemma relprime_20_81: "gcd(20,81) = 1";
173 by (simp add: gcd.simps)
178 @{thm[display] relprime_dvd_mult}
179 \rulename{relprime_dvd_mult}
181 @{thm[display] relprime_dvd_mult [OF relprime_20_81]}
183 @{thm[display] dvd_refl}
186 @{thm[display] dvd_add}
189 @{thm[display] dvd_add [OF dvd_refl dvd_refl]}
191 @{thm[display] dvd_add [OF _ dvd_refl]}
194 lemma "\<lbrakk>(z::int) < 37; 66 < 2*z; z*z \<noteq> 1225; Q(34); Q(36)\<rbrakk> \<Longrightarrow> Q(z)";
195 apply (subgoal_tac "z = 34 \<or> z = 36")
197 the tactic leaves two subgoals:
198 @{subgoals[display,indent=0,margin=65]}
201 apply (subgoal_tac "z \<noteq> 35")
203 the tactic leaves two subgoals:
204 @{subgoals[display,indent=0,margin=65]}