author | wenzelm |
Mon, 09 Mar 2009 11:57:48 +0100 | |
changeset 30382 | 910290f04692 |
parent 27658 | 674496eb5965 |
child 45617 | cc0800432333 |
permissions | -rw-r--r-- |
16417 | 1 |
theory Forward imports Primes begin |
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text{*\noindent |
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Forward proof material: of, OF, THEN, simplify, rule_format. |
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*} |
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text{*\noindent |
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SKIP most developments... |
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*} |
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(** Commutativity **) |
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lemma is_gcd_commute: "is_gcd k m n = is_gcd k n m" |
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apply (auto simp add: is_gcd_def); |
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done |
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lemma gcd_commute: "gcd m n = gcd n m" |
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apply (rule is_gcd_unique) |
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apply (rule is_gcd) |
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apply (subst is_gcd_commute) |
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apply (simp add: is_gcd) |
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done |
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lemma gcd_1 [simp]: "gcd m (Suc 0) = Suc 0" |
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apply simp |
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done |
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lemma gcd_1_left [simp]: "gcd (Suc 0) m = Suc 0" |
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apply (simp add: gcd_commute [of "Suc 0"]) |
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done |
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text{*\noindent |
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as far as HERE. |
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*} |
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text{*\noindent |
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SKIP THIS PROOF |
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*} |
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lemma gcd_mult_distrib2: "k * gcd m n = gcd (k*m) (k*n)" |
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apply (induct_tac m n rule: gcd.induct) |
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apply (case_tac "n=0") |
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apply simp |
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apply (case_tac "k=0") |
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apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2) |
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done |
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text {* |
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@{thm[display] gcd_mult_distrib2} |
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\rulename{gcd_mult_distrib2} |
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*}; |
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text{*\noindent |
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of, simplified |
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*} |
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lemmas gcd_mult_0 = gcd_mult_distrib2 [of k 1]; |
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lemmas gcd_mult_1 = gcd_mult_0 [simplified]; |
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lemmas where1 = gcd_mult_distrib2 [where m=1] |
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lemmas where2 = gcd_mult_distrib2 [where m=1 and k=1] |
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lemmas where3 = gcd_mult_distrib2 [where m=1 and k="j+k"] |
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text {* |
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example using ``of'': |
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@{thm[display] gcd_mult_distrib2 [of _ 1]} |
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example using ``where'': |
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@{thm[display] gcd_mult_distrib2 [where m=1]} |
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example using ``where'', ``and'': |
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@{thm[display] gcd_mult_distrib2 [where m=1 and k="j+k"]} |
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@{thm[display] gcd_mult_0} |
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\rulename{gcd_mult_0} |
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@{thm[display] gcd_mult_1} |
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\rulename{gcd_mult_1} |
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@{thm[display] sym} |
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\rulename{sym} |
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*}; |
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lemmas gcd_mult0 = gcd_mult_1 [THEN sym]; |
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(*not quite right: we need ?k but this gives k*) |
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lemmas gcd_mult0' = gcd_mult_distrib2 [of k 1, simplified, THEN sym]; |
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(*better in one step!*) |
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text {* |
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more legible, and variables properly generalized |
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*}; |
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lemma gcd_mult [simp]: "gcd k (k*n) = k" |
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by (rule gcd_mult_distrib2 [of k 1, simplified, THEN sym]) |
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lemmas gcd_self0 = gcd_mult [of k 1, simplified]; |
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text {* |
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@{thm[display] gcd_mult} |
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\rulename{gcd_mult} |
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@{thm[display] gcd_self0} |
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\rulename{gcd_self0} |
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*}; |
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text {* |
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Rules handy with THEN |
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@{thm[display] iffD1} |
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\rulename{iffD1} |
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@{thm[display] iffD2} |
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\rulename{iffD2} |
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*}; |
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text {* |
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again: more legible, and variables properly generalized |
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*}; |
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lemma gcd_self [simp]: "gcd k k = k" |
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by (rule gcd_mult [of k 1, simplified]) |
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text{* |
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NEXT SECTION: Methods for Forward Proof |
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NEW |
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theorem arg_cong, useful in forward steps |
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@{thm[display] arg_cong[no_vars]} |
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\rulename{arg_cong} |
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*} |
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lemma "2 \<le> u \<Longrightarrow> u*m \<noteq> Suc(u*n)" |
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apply (intro notI) |
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txt{* |
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before using arg_cong |
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@{subgoals[display,indent=0,margin=65]} |
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*}; |
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apply (drule_tac f="\<lambda>x. x mod u" in arg_cong) |
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txt{* |
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after using arg_cong |
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@{subgoals[display,indent=0,margin=65]} |
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*}; |
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apply (simp add: mod_Suc) |
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done |
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text{* |
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have just used this rule: |
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@{thm[display] mod_Suc[no_vars]} |
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\rulename{mod_Suc} |
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@{thm[display] mult_le_mono1[no_vars]} |
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\rulename{mult_le_mono1} |
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*} |
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text{* |
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example of "insert" |
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*} |
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lemma relprime_dvd_mult: |
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"\<lbrakk> gcd k n = 1; k dvd m*n \<rbrakk> \<Longrightarrow> k dvd m" |
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apply (insert gcd_mult_distrib2 [of m k n]) |
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txt{*@{subgoals[display,indent=0,margin=65]}*} |
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apply simp |
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txt{*@{subgoals[display,indent=0,margin=65]}*} |
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apply (erule_tac t="m" in ssubst); |
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apply simp |
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done |
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text {* |
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@{thm[display] relprime_dvd_mult} |
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\rulename{relprime_dvd_mult} |
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Another example of "insert" |
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@{thm[display] mod_div_equality} |
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\rulename{mod_div_equality} |
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*}; |
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(*MOVED to Force.thy, which now depends only on Divides.thy |
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lemma div_mult_self_is_m: "0<n \<Longrightarrow> (m*n) div n = (m::nat)" |
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*) |
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lemma relprime_dvd_mult_iff: "gcd k n = 1 \<Longrightarrow> (k dvd m*n) = (k dvd m)"; |
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by (auto intro: relprime_dvd_mult elim: dvdE) |
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lemma relprime_20_81: "gcd 20 81 = 1"; |
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by (simp add: gcd.simps) |
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text {* |
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Examples of 'OF' |
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|
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@{thm[display] relprime_dvd_mult} |
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\rulename{relprime_dvd_mult} |
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@{thm[display] relprime_dvd_mult [OF relprime_20_81]} |
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@{thm[display] dvd_refl} |
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\rulename{dvd_refl} |
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@{thm[display] dvd_add} |
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\rulename{dvd_add} |
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@{thm[display] dvd_add [OF dvd_refl dvd_refl]} |
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|
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@{thm[display] dvd_add [OF _ dvd_refl]} |
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*}; |
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|
11711 | 219 |
lemma "\<lbrakk>(z::int) < 37; 66 < 2*z; z*z \<noteq> 1225; Q(34); Q(36)\<rbrakk> \<Longrightarrow> Q(z)"; |
220 |
apply (subgoal_tac "z = 34 \<or> z = 36") |
|
10958 | 221 |
txt{* |
222 |
the tactic leaves two subgoals: |
|
223 |
@{subgoals[display,indent=0,margin=65]} |
|
224 |
*}; |
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10846
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apply blast |
11711 | 226 |
apply (subgoal_tac "z \<noteq> 35") |
10958 | 227 |
txt{* |
228 |
the tactic leaves two subgoals: |
|
229 |
@{subgoals[display,indent=0,margin=65]} |
|
230 |
*}; |
|
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231 |
apply arith |
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apply force |
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done |
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|
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235 |
|
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236 |
end |