author | nipkow |
Thu, 15 Mar 2001 15:05:51 +0100 | |
changeset 11210 | 33300d16a63a |
parent 10958 | fd582f0d649b |
child 11407 | 138919f1a135 |
permissions | -rw-r--r-- |
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1 |
theory Forward = Primes: |
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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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8 |
SKIP most developments... |
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9 |
*} |
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(** Commutativity **) |
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|
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13 |
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); |
15 |
done |
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|
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lemma gcd_commute: "gcd(m,n) = gcd(n,m)" |
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apply (rule is_gcd_unique) |
19 |
apply (rule is_gcd) |
|
20 |
apply (subst is_gcd_commute) |
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apply (simp add: is_gcd) |
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22 |
done |
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|
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lemma gcd_1 [simp]: "gcd(m,1) = 1" |
10958 | 25 |
apply simp |
26 |
done |
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lemma gcd_1_left [simp]: "gcd(1,m) = 1" |
10958 | 29 |
apply (simp add: gcd_commute [of 1]) |
30 |
done |
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31 |
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text{*\noindent |
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33 |
as far as HERE. |
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*} |
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text {* |
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38 |
@{thm[display] gcd_1} |
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39 |
\rulename{gcd_1} |
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@{thm[display] gcd_1_left} |
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\rulename{gcd_1_left} |
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*}; |
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|
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text{*\noindent |
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46 |
SKIP THIS PROOF |
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47 |
*} |
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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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|
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text {* |
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58 |
@{thm[display] gcd_mult_distrib2} |
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\rulename{gcd_mult_distrib2} |
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*}; |
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61 |
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text{*\noindent |
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63 |
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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text {* |
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71 |
@{thm[display] gcd_mult_distrib2 [of _ 1]} |
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@{thm[display] gcd_mult_0} |
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74 |
\rulename{gcd_mult_0} |
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76 |
@{thm[display] gcd_mult_1} |
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\rulename{gcd_mult_1} |
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78 |
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@{thm[display] sym} |
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\rulename{sym} |
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81 |
*}; |
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82 |
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lemmas gcd_mult = gcd_mult_1 [THEN sym]; |
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84 |
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lemmas gcd_mult = 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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89 |
more legible |
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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_self = gcd_mult [of k 1, simplified]; |
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text {* |
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100 |
Rules handy with THEN |
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101 |
|
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102 |
@{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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107 |
*}; |
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108 |
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109 |
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text {* |
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111 |
again: more legible |
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112 |
*}; |
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114 |
lemma gcd_self [simp]: "gcd(k,k) = k" |
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by (rule gcd_mult [of k 1, simplified]) |
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117 |
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10958 | 118 |
text{* |
119 |
NEXT SECTION: Methods for Forward Proof |
|
120 |
||
121 |
NEW |
|
122 |
||
123 |
theorem arg_cong, useful in forward steps |
|
124 |
@{thm[display] arg_cong[no_vars]} |
|
125 |
\rulename{arg_cong} |
|
126 |
*} |
|
127 |
||
128 |
lemma "#2 \<le> u \<Longrightarrow> u*m \<noteq> Suc(u*n)" |
|
129 |
apply intro |
|
130 |
txt{* |
|
131 |
before using arg_cong |
|
132 |
@{subgoals[display,indent=0,margin=65]} |
|
133 |
*}; |
|
134 |
apply (drule_tac f="\<lambda>x. x mod u" in arg_cong) |
|
135 |
txt{* |
|
136 |
after using arg_cong |
|
137 |
@{subgoals[display,indent=0,margin=65]} |
|
138 |
*}; |
|
139 |
apply (simp add: mod_Suc) |
|
140 |
done |
|
141 |
||
142 |
text{* |
|
143 |
have just used this rule: |
|
144 |
@{thm[display] mod_Suc[no_vars]} |
|
145 |
\rulename{mod_Suc} |
|
146 |
||
147 |
@{thm[display] mult_le_mono1[no_vars]} |
|
148 |
\rulename{mult_le_mono1} |
|
149 |
*} |
|
150 |
||
151 |
||
152 |
text{* |
|
153 |
example of "insert" |
|
154 |
*} |
|
155 |
||
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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]) |
10958 | 159 |
apply simp |
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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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164 |
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text {* |
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166 |
Another example of "insert" |
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167 |
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168 |
@{thm[display] mod_div_equality} |
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\rulename{mod_div_equality} |
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170 |
*}; |
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171 |
|
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lemma div_mult_self_is_m: |
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"0<n \<Longrightarrow> (m*n) div n = (m::nat)" |
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apply (insert mod_div_equality [of "m*n" n]) |
10958 | 175 |
apply simp |
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done |
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177 |
|
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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 (blast intro: relprime_dvd_mult dvd_trans) |
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180 |
|
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181 |
|
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lemma relprime_20_81: "gcd(#20,#81) = 1"; |
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183 |
by (simp add: gcd.simps) |
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184 |
|
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185 |
text {* |
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186 |
Examples of 'OF' |
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|
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188 |
@{thm[display] relprime_dvd_mult} |
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\rulename{relprime_dvd_mult} |
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190 |
|
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191 |
@{thm[display] relprime_dvd_mult [OF relprime_20_81]} |
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192 |
|
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193 |
@{thm[display] dvd_refl} |
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194 |
\rulename{dvd_refl} |
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195 |
|
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@{thm[display] dvd_add} |
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\rulename{dvd_add} |
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198 |
|
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199 |
@{thm[display] dvd_add [OF dvd_refl dvd_refl]} |
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200 |
|
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201 |
@{thm[display] dvd_add [OF _ dvd_refl]} |
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202 |
*}; |
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203 |
|
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lemma "\<lbrakk>(z::int) < #37; #66 < #2*z; z*z \<noteq> #1225; Q(#34); Q(#36)\<rbrakk> \<Longrightarrow> Q(z)"; |
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apply (subgoal_tac "z = #34 \<or> z = #36") |
10958 | 206 |
txt{* |
207 |
the tactic leaves two subgoals: |
|
208 |
@{subgoals[display,indent=0,margin=65]} |
|
209 |
*}; |
|
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210 |
apply blast |
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apply (subgoal_tac "z \<noteq> #35") |
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txt{* |
213 |
the tactic leaves two subgoals: |
|
214 |
@{subgoals[display,indent=0,margin=65]} |
|
215 |
*}; |
|
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216 |
apply arith |
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217 |
apply force |
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218 |
done |
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219 |
|
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220 |
|
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221 |
end |