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(* Title : HOL/Hyperreal/StarClasses.thy |
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ID : $Id$ |
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Author : Brian Huffman |
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*) |
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header {* Class Instances *} |
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theory StarClasses |
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imports StarDef |
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begin |
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subsection {* Syntactic classes *} |
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instantiation star :: (zero) zero |
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definition |
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star_zero_def: "0 \<equiv> star_of 0" |
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instance .. |
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end |
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instantiation star :: (one) one |
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definition |
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star_one_def: "1 \<equiv> star_of 1" |
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instance .. |
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end |
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instantiation star :: (plus) plus |
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definition |
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star_add_def: "(op +) \<equiv> *f2* (op +)" |
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instance .. |
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end |
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instantiation star :: (times) times |
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definition |
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star_mult_def: "(op *) \<equiv> *f2* (op *)" |
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instance .. |
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end |
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instantiation star :: (minus) minus |
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definition |
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star_minus_def: "uminus \<equiv> *f* uminus" |
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definition |
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star_diff_def: "(op -) \<equiv> *f2* (op -)" |
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instance .. |
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end |
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instantiation star :: (abs) abs |
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definition |
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star_abs_def: "abs \<equiv> *f* abs" |
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instance .. |
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end |
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instantiation star :: (sgn) sgn |
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definition |
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star_sgn_def: "sgn \<equiv> *f* sgn" |
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instance .. |
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end |
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instantiation star :: (inverse) inverse |
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definition |
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star_divide_def: "(op /) \<equiv> *f2* (op /)" |
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star_inverse_def: "inverse \<equiv> *f* inverse" |
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instance .. |
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end |
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instantiation star :: (number) number |
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star_number_def: "number_of b \<equiv> star_of (number_of b)" |
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instance .. |
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end |
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instantiation star :: (Divides.div) Divides.div |
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definition |
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star_div_def: "(op div) \<equiv> *f2* (op div)" |
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star_mod_def: "(op mod) \<equiv> *f2* (op mod)" |
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end |
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instantiation star :: (power) power |
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star_power_def: "(op ^) \<equiv> \<lambda>x n. ( *f* (\<lambda>x. x ^ n)) x" |
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instantiation star :: (ord) ord |
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definition |
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star_le_def: "(op \<le>) \<equiv> *p2* (op \<le>)" |
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star_less_def: "(op <) \<equiv> *p2* (op <)" |
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end |
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lemmas star_class_defs [transfer_unfold] = |
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star_zero_def star_one_def star_number_def |
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star_add_def star_diff_def star_minus_def |
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star_mult_def star_divide_def star_inverse_def |
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star_le_def star_less_def star_abs_def star_sgn_def |
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star_div_def star_mod_def star_power_def |
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text {* Class operations preserve standard elements *} |
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lemma Standard_zero: "0 \<in> Standard" |
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by (simp add: star_zero_def) |
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lemma Standard_one: "1 \<in> Standard" |
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by (simp add: star_one_def) |
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lemma Standard_number_of: "number_of b \<in> Standard" |
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by (simp add: star_number_def) |
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lemma Standard_add: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x + y \<in> Standard" |
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165 |
by (simp add: star_add_def) |
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166 |
||
167 |
lemma Standard_diff: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x - y \<in> Standard" |
|
168 |
by (simp add: star_diff_def) |
|
169 |
||
170 |
lemma Standard_minus: "x \<in> Standard \<Longrightarrow> - x \<in> Standard" |
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171 |
by (simp add: star_minus_def) |
|
172 |
||
173 |
lemma Standard_mult: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x * y \<in> Standard" |
|
174 |
by (simp add: star_mult_def) |
|
175 |
||
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lemma Standard_divide: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x / y \<in> Standard" |
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177 |
by (simp add: star_divide_def) |
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178 |
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179 |
lemma Standard_inverse: "x \<in> Standard \<Longrightarrow> inverse x \<in> Standard" |
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180 |
by (simp add: star_inverse_def) |
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181 |
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182 |
lemma Standard_abs: "x \<in> Standard \<Longrightarrow> abs x \<in> Standard" |
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183 |
by (simp add: star_abs_def) |
|
184 |
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185 |
lemma Standard_div: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x div y \<in> Standard" |
|
186 |
by (simp add: star_div_def) |
|
187 |
||
188 |
lemma Standard_mod: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x mod y \<in> Standard" |
|
189 |
by (simp add: star_mod_def) |
|
190 |
||
191 |
lemma Standard_power: "x \<in> Standard \<Longrightarrow> x ^ n \<in> Standard" |
|
192 |
by (simp add: star_power_def) |
|
193 |
||
194 |
lemmas Standard_simps [simp] = |
|
195 |
Standard_zero Standard_one Standard_number_of |
|
196 |
Standard_add Standard_diff Standard_minus |
|
197 |
Standard_mult Standard_divide Standard_inverse |
|
198 |
Standard_abs Standard_div Standard_mod |
|
199 |
Standard_power |
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200 |
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text {* @{term star_of} preserves class operations *} |
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lemma star_of_add: "star_of (x + y) = star_of x + star_of y" |
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by transfer (rule refl) |
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lemma star_of_diff: "star_of (x - y) = star_of x - star_of y" |
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by transfer (rule refl) |
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lemma star_of_minus: "star_of (-x) = - star_of x" |
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by transfer (rule refl) |
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|
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lemma star_of_mult: "star_of (x * y) = star_of x * star_of y" |
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by transfer (rule refl) |
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lemma star_of_divide: "star_of (x / y) = star_of x / star_of y" |
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by transfer (rule refl) |
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217 |
|
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lemma star_of_inverse: "star_of (inverse x) = inverse (star_of x)" |
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by transfer (rule refl) |
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|
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lemma star_of_div: "star_of (x div y) = star_of x div star_of y" |
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by transfer (rule refl) |
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223 |
|
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lemma star_of_mod: "star_of (x mod y) = star_of x mod star_of y" |
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by transfer (rule refl) |
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226 |
|
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lemma star_of_power: "star_of (x ^ n) = star_of x ^ n" |
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by transfer (rule refl) |
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229 |
|
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lemma star_of_abs: "star_of (abs x) = abs (star_of x)" |
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231 |
by transfer (rule refl) |
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232 |
|
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233 |
text {* @{term star_of} preserves numerals *} |
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|
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lemma star_of_zero: "star_of 0 = 0" |
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236 |
by transfer (rule refl) |
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237 |
|
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238 |
lemma star_of_one: "star_of 1 = 1" |
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239 |
by transfer (rule refl) |
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240 |
|
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241 |
lemma star_of_number_of: "star_of (number_of x) = number_of x" |
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242 |
by transfer (rule refl) |
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243 |
|
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244 |
text {* @{term star_of} preserves orderings *} |
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245 |
|
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246 |
lemma star_of_less: "(star_of x < star_of y) = (x < y)" |
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247 |
by transfer (rule refl) |
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248 |
|
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249 |
lemma star_of_le: "(star_of x \<le> star_of y) = (x \<le> y)" |
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250 |
by transfer (rule refl) |
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251 |
|
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252 |
lemma star_of_eq: "(star_of x = star_of y) = (x = y)" |
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253 |
by transfer (rule refl) |
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|
254 |
|
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255 |
text{*As above, for 0*} |
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|
256 |
|
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257 |
lemmas star_of_0_less = star_of_less [of 0, simplified star_of_zero] |
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258 |
lemmas star_of_0_le = star_of_le [of 0, simplified star_of_zero] |
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259 |
lemmas star_of_0_eq = star_of_eq [of 0, simplified star_of_zero] |
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260 |
|
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261 |
lemmas star_of_less_0 = star_of_less [of _ 0, simplified star_of_zero] |
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262 |
lemmas star_of_le_0 = star_of_le [of _ 0, simplified star_of_zero] |
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263 |
lemmas star_of_eq_0 = star_of_eq [of _ 0, simplified star_of_zero] |
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264 |
|
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265 |
text{*As above, for 1*} |
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266 |
|
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267 |
lemmas star_of_1_less = star_of_less [of 1, simplified star_of_one] |
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268 |
lemmas star_of_1_le = star_of_le [of 1, simplified star_of_one] |
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269 |
lemmas star_of_1_eq = star_of_eq [of 1, simplified star_of_one] |
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270 |
|
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271 |
lemmas star_of_less_1 = star_of_less [of _ 1, simplified star_of_one] |
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272 |
lemmas star_of_le_1 = star_of_le [of _ 1, simplified star_of_one] |
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273 |
lemmas star_of_eq_1 = star_of_eq [of _ 1, simplified star_of_one] |
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274 |
|
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275 |
text{*As above, for numerals*} |
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276 |
|
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277 |
lemmas star_of_number_less = |
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278 |
star_of_less [of "number_of w", standard, simplified star_of_number_of] |
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279 |
lemmas star_of_number_le = |
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280 |
star_of_le [of "number_of w", standard, simplified star_of_number_of] |
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281 |
lemmas star_of_number_eq = |
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282 |
star_of_eq [of "number_of w", standard, simplified star_of_number_of] |
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283 |
|
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284 |
lemmas star_of_less_number = |
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285 |
star_of_less [of _ "number_of w", standard, simplified star_of_number_of] |
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286 |
lemmas star_of_le_number = |
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287 |
star_of_le [of _ "number_of w", standard, simplified star_of_number_of] |
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288 |
lemmas star_of_eq_number = |
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289 |
star_of_eq [of _ "number_of w", standard, simplified star_of_number_of] |
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290 |
|
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291 |
lemmas star_of_simps [simp] = |
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292 |
star_of_add star_of_diff star_of_minus |
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293 |
star_of_mult star_of_divide star_of_inverse |
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|
294 |
star_of_div star_of_mod |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
295 |
star_of_power star_of_abs |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
296 |
star_of_zero star_of_one star_of_number_of |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
297 |
star_of_less star_of_le star_of_eq |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
298 |
star_of_0_less star_of_0_le star_of_0_eq |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
299 |
star_of_less_0 star_of_le_0 star_of_eq_0 |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
300 |
star_of_1_less star_of_1_le star_of_1_eq |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
301 |
star_of_less_1 star_of_le_1 star_of_eq_1 |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
302 |
star_of_number_less star_of_number_le star_of_number_eq |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
303 |
star_of_less_number star_of_le_number star_of_eq_number |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
304 |
|
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
305 |
subsection {* Ordering and lattice classes *} |
17296 | 306 |
|
307 |
instance star :: (order) order |
|
308 |
apply (intro_classes) |
|
22316 | 309 |
apply (transfer, rule order_less_le) |
17296 | 310 |
apply (transfer, rule order_refl) |
311 |
apply (transfer, erule (1) order_trans) |
|
312 |
apply (transfer, erule (1) order_antisym) |
|
313 |
done |
|
314 |
||
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
315 |
instantiation star :: (lower_semilattice) lower_semilattice |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
316 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
317 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
318 |
definition |
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
319 |
star_inf_def [transfer_unfold]: "inf \<equiv> *f2* inf" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
320 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
321 |
instance |
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
322 |
by default (transfer star_inf_def, auto)+ |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
323 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
324 |
end |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
325 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
326 |
instantiation star :: (upper_semilattice) upper_semilattice |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
327 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
328 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
329 |
definition |
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
330 |
star_sup_def [transfer_unfold]: "sup \<equiv> *f2* sup" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
331 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
332 |
instance |
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
333 |
by default (transfer star_sup_def, auto)+ |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
334 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
335 |
end |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25304
diff
changeset
|
336 |
|
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
337 |
instance star :: (lattice) lattice .. |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
338 |
|
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
339 |
instance star :: (distrib_lattice) distrib_lattice |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
340 |
by default (transfer, auto simp add: sup_inf_distrib1) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
341 |
|
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
342 |
lemma Standard_inf [simp]: |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
343 |
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> inf x y \<in> Standard" |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
344 |
by (simp add: star_inf_def) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
345 |
|
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
346 |
lemma Standard_sup [simp]: |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
347 |
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> sup x y \<in> Standard" |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
348 |
by (simp add: star_sup_def) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
349 |
|
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
350 |
lemma star_of_inf [simp]: "star_of (inf x y) = inf (star_of x) (star_of y)" |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
351 |
by transfer (rule refl) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
352 |
|
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
353 |
lemma star_of_sup [simp]: "star_of (sup x y) = sup (star_of x) (star_of y)" |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
354 |
by transfer (rule refl) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
355 |
|
17296 | 356 |
instance star :: (linorder) linorder |
357 |
by (intro_classes, transfer, rule linorder_linear) |
|
358 |
||
20720 | 359 |
lemma star_max_def [transfer_unfold]: "max = *f2* max" |
360 |
apply (rule ext, rule ext) |
|
361 |
apply (unfold max_def, transfer, fold max_def) |
|
362 |
apply (rule refl) |
|
363 |
done |
|
364 |
||
365 |
lemma star_min_def [transfer_unfold]: "min = *f2* min" |
|
366 |
apply (rule ext, rule ext) |
|
367 |
apply (unfold min_def, transfer, fold min_def) |
|
368 |
apply (rule refl) |
|
369 |
done |
|
370 |
||
371 |
lemma Standard_max [simp]: |
|
372 |
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> max x y \<in> Standard" |
|
373 |
by (simp add: star_max_def) |
|
374 |
||
375 |
lemma Standard_min [simp]: |
|
376 |
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> min x y \<in> Standard" |
|
377 |
by (simp add: star_min_def) |
|
378 |
||
379 |
lemma star_of_max [simp]: "star_of (max x y) = max (star_of x) (star_of y)" |
|
380 |
by transfer (rule refl) |
|
381 |
||
382 |
lemma star_of_min [simp]: "star_of (min x y) = min (star_of x) (star_of y)" |
|
383 |
by transfer (rule refl) |
|
384 |
||
17296 | 385 |
|
17332
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
386 |
subsection {* Ordered group classes *} |
17296 | 387 |
|
388 |
instance star :: (semigroup_add) semigroup_add |
|
389 |
by (intro_classes, transfer, rule add_assoc) |
|
390 |
||
391 |
instance star :: (ab_semigroup_add) ab_semigroup_add |
|
392 |
by (intro_classes, transfer, rule add_commute) |
|
393 |
||
394 |
instance star :: (semigroup_mult) semigroup_mult |
|
395 |
by (intro_classes, transfer, rule mult_assoc) |
|
396 |
||
397 |
instance star :: (ab_semigroup_mult) ab_semigroup_mult |
|
398 |
by (intro_classes, transfer, rule mult_commute) |
|
399 |
||
400 |
instance star :: (comm_monoid_add) comm_monoid_add |
|
22384
33a46e6c7f04
prefix of class interpretation not mandatory any longer
haftmann
parents:
22316
diff
changeset
|
401 |
by (intro_classes, transfer, rule comm_monoid_add_class.zero_plus.add_0) |
17296 | 402 |
|
403 |
instance star :: (monoid_mult) monoid_mult |
|
404 |
apply (intro_classes) |
|
405 |
apply (transfer, rule mult_1_left) |
|
406 |
apply (transfer, rule mult_1_right) |
|
407 |
done |
|
408 |
||
409 |
instance star :: (comm_monoid_mult) comm_monoid_mult |
|
410 |
by (intro_classes, transfer, rule mult_1) |
|
411 |
||
412 |
instance star :: (cancel_semigroup_add) cancel_semigroup_add |
|
413 |
apply (intro_classes) |
|
414 |
apply (transfer, erule add_left_imp_eq) |
|
415 |
apply (transfer, erule add_right_imp_eq) |
|
416 |
done |
|
417 |
||
418 |
instance star :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add |
|
419 |
by (intro_classes, transfer, rule add_imp_eq) |
|
420 |
||
421 |
instance star :: (ab_group_add) ab_group_add |
|
422 |
apply (intro_classes) |
|
423 |
apply (transfer, rule left_minus) |
|
424 |
apply (transfer, rule diff_minus) |
|
425 |
done |
|
426 |
||
427 |
instance star :: (pordered_ab_semigroup_add) pordered_ab_semigroup_add |
|
428 |
by (intro_classes, transfer, rule add_left_mono) |
|
429 |
||
430 |
instance star :: (pordered_cancel_ab_semigroup_add) pordered_cancel_ab_semigroup_add .. |
|
431 |
||
432 |
instance star :: (pordered_ab_semigroup_add_imp_le) pordered_ab_semigroup_add_imp_le |
|
433 |
by (intro_classes, transfer, rule add_le_imp_le_left) |
|
434 |
||
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25230
diff
changeset
|
435 |
instance star :: (pordered_comm_monoid_add) pordered_comm_monoid_add .. |
17296 | 436 |
instance star :: (pordered_ab_group_add) pordered_ab_group_add .. |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25230
diff
changeset
|
437 |
|
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25230
diff
changeset
|
438 |
instance star :: (pordered_ab_group_add_abs) pordered_ab_group_add_abs |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25230
diff
changeset
|
439 |
by intro_classes (transfer, |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25230
diff
changeset
|
440 |
simp add: abs_ge_self abs_leI abs_triangle_ineq)+ |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25230
diff
changeset
|
441 |
|
17296 | 442 |
instance star :: (ordered_cancel_ab_semigroup_add) ordered_cancel_ab_semigroup_add .. |
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25230
diff
changeset
|
443 |
instance star :: (lordered_ab_group_add_meet) lordered_ab_group_add_meet .. |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25230
diff
changeset
|
444 |
instance star :: (lordered_ab_group_add_meet) lordered_ab_group_add_meet .. |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25230
diff
changeset
|
445 |
instance star :: (lordered_ab_group_add) lordered_ab_group_add .. |
17296 | 446 |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25230
diff
changeset
|
447 |
instance star :: (lordered_ab_group_add_abs) lordered_ab_group_add_abs |
17332
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
448 |
by (intro_classes, transfer, rule abs_lattice) |
17296 | 449 |
|
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
450 |
subsection {* Ring and field classes *} |
17296 | 451 |
|
452 |
instance star :: (semiring) semiring |
|
453 |
apply (intro_classes) |
|
454 |
apply (transfer, rule left_distrib) |
|
455 |
apply (transfer, rule right_distrib) |
|
456 |
done |
|
457 |
||
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20720
diff
changeset
|
458 |
instance star :: (semiring_0) semiring_0 |
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20720
diff
changeset
|
459 |
by intro_classes (transfer, simp)+ |
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20720
diff
changeset
|
460 |
|
17296 | 461 |
instance star :: (semiring_0_cancel) semiring_0_cancel .. |
462 |
||
24742
73b8b42a36b6
removal of some "ref"s from res_axioms.ML; a side-effect is that the ordering
paulson
parents:
24506
diff
changeset
|
463 |
instance star :: (comm_semiring) comm_semiring |
73b8b42a36b6
removal of some "ref"s from res_axioms.ML; a side-effect is that the ordering
paulson
parents:
24506
diff
changeset
|
464 |
by (intro_classes, transfer, rule left_distrib) |
17296 | 465 |
|
466 |
instance star :: (comm_semiring_0) comm_semiring_0 .. |
|
467 |
instance star :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. |
|
468 |
||
20633 | 469 |
instance star :: (zero_neq_one) zero_neq_one |
17296 | 470 |
by (intro_classes, transfer, rule zero_neq_one) |
471 |
||
472 |
instance star :: (semiring_1) semiring_1 .. |
|
473 |
instance star :: (comm_semiring_1) comm_semiring_1 .. |
|
474 |
||
20633 | 475 |
instance star :: (no_zero_divisors) no_zero_divisors |
17296 | 476 |
by (intro_classes, transfer, rule no_zero_divisors) |
477 |
||
478 |
instance star :: (semiring_1_cancel) semiring_1_cancel .. |
|
479 |
instance star :: (comm_semiring_1_cancel) comm_semiring_1_cancel .. |
|
480 |
instance star :: (ring) ring .. |
|
481 |
instance star :: (comm_ring) comm_ring .. |
|
482 |
instance star :: (ring_1) ring_1 .. |
|
483 |
instance star :: (comm_ring_1) comm_ring_1 .. |
|
22992 | 484 |
instance star :: (ring_no_zero_divisors) ring_no_zero_divisors .. |
23551
84f0996a530b
rename class dom to ring_1_no_zero_divisors (cf. HOL/Ring_and_Field.thy 1.84 by huffman);
wenzelm
parents:
23282
diff
changeset
|
485 |
instance star :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors .. |
17296 | 486 |
instance star :: (idom) idom .. |
487 |
||
20540 | 488 |
instance star :: (division_ring) division_ring |
489 |
apply (intro_classes) |
|
490 |
apply (transfer, erule left_inverse) |
|
491 |
apply (transfer, erule right_inverse) |
|
492 |
done |
|
493 |
||
17296 | 494 |
instance star :: (field) field |
495 |
apply (intro_classes) |
|
496 |
apply (transfer, erule left_inverse) |
|
497 |
apply (transfer, rule divide_inverse) |
|
498 |
done |
|
499 |
||
500 |
instance star :: (division_by_zero) division_by_zero |
|
501 |
by (intro_classes, transfer, rule inverse_zero) |
|
502 |
||
503 |
instance star :: (pordered_semiring) pordered_semiring |
|
504 |
apply (intro_classes) |
|
505 |
apply (transfer, erule (1) mult_left_mono) |
|
506 |
apply (transfer, erule (1) mult_right_mono) |
|
507 |
done |
|
508 |
||
509 |
instance star :: (pordered_cancel_semiring) pordered_cancel_semiring .. |
|
510 |
||
511 |
instance star :: (ordered_semiring_strict) ordered_semiring_strict |
|
512 |
apply (intro_classes) |
|
513 |
apply (transfer, erule (1) mult_strict_left_mono) |
|
514 |
apply (transfer, erule (1) mult_strict_right_mono) |
|
515 |
done |
|
516 |
||
517 |
instance star :: (pordered_comm_semiring) pordered_comm_semiring |
|
25230 | 518 |
by (intro_classes, transfer, rule mult_mono1_class.times_zero_less_eq_less.mult_mono1) |
17296 | 519 |
|
520 |
instance star :: (pordered_cancel_comm_semiring) pordered_cancel_comm_semiring .. |
|
521 |
||
522 |
instance star :: (ordered_comm_semiring_strict) ordered_comm_semiring_strict |
|
25208 | 523 |
by (intro_classes, transfer, rule ordered_comm_semiring_strict_class.plus_times_zero_less_eq_less.mult_strict_mono) |
17296 | 524 |
|
525 |
instance star :: (pordered_ring) pordered_ring .. |
|
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25230
diff
changeset
|
526 |
instance star :: (pordered_ring_abs) pordered_ring_abs |
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25230
diff
changeset
|
527 |
by intro_classes (transfer, rule abs_eq_mult) |
17296 | 528 |
instance star :: (lordered_ring) lordered_ring .. |
529 |
||
20633 | 530 |
instance star :: (abs_if) abs_if |
17296 | 531 |
by (intro_classes, transfer, rule abs_if) |
532 |
||
24506 | 533 |
instance star :: (sgn_if) sgn_if |
534 |
by (intro_classes, transfer, rule sgn_if) |
|
535 |
||
17296 | 536 |
instance star :: (ordered_ring_strict) ordered_ring_strict .. |
537 |
instance star :: (pordered_comm_ring) pordered_comm_ring .. |
|
538 |
||
539 |
instance star :: (ordered_semidom) ordered_semidom |
|
540 |
by (intro_classes, transfer, rule zero_less_one) |
|
541 |
||
542 |
instance star :: (ordered_idom) ordered_idom .. |
|
543 |
instance star :: (ordered_field) ordered_field .. |
|
544 |
||
17332
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
545 |
subsection {* Power classes *} |
17296 | 546 |
|
547 |
text {* |
|
548 |
Proving the class axiom @{thm [source] power_Suc} for type |
|
549 |
@{typ "'a star"} is a little tricky, because it quantifies |
|
550 |
over values of type @{typ nat}. The transfer principle does |
|
551 |
not handle quantification over non-star types in general, |
|
552 |
but we can work around this by fixing an arbitrary @{typ nat} |
|
553 |
value, and then applying the transfer principle. |
|
554 |
*} |
|
555 |
||
556 |
instance star :: (recpower) recpower |
|
557 |
proof |
|
558 |
show "\<And>a::'a star. a ^ 0 = 1" |
|
559 |
by transfer (rule power_0) |
|
560 |
next |
|
561 |
fix n show "\<And>a::'a star. a ^ Suc n = a * a ^ n" |
|
562 |
by transfer (rule power_Suc) |
|
563 |
qed |
|
564 |
||
17332
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
565 |
subsection {* Number classes *} |
17296 | 566 |
|
20720 | 567 |
lemma star_of_nat_def [transfer_unfold]: "of_nat n = star_of (of_nat n)" |
25230 | 568 |
by (induct n, simp_all) |
20720 | 569 |
|
570 |
lemma Standard_of_nat [simp]: "of_nat n \<in> Standard" |
|
571 |
by (simp add: star_of_nat_def) |
|
17296 | 572 |
|
17332
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
573 |
lemma star_of_of_nat [simp]: "star_of (of_nat n) = of_nat n" |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
574 |
by transfer (rule refl) |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
575 |
|
20720 | 576 |
lemma star_of_int_def [transfer_unfold]: "of_int z = star_of (of_int z)" |
577 |
by (rule_tac z=z in int_diff_cases, simp) |
|
578 |
||
579 |
lemma Standard_of_int [simp]: "of_int z \<in> Standard" |
|
580 |
by (simp add: star_of_int_def) |
|
17332
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
581 |
|
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
582 |
lemma star_of_of_int [simp]: "star_of (of_int z) = of_int z" |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
583 |
by transfer (rule refl) |
17296 | 584 |
|
23282
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
22993
diff
changeset
|
585 |
instance star :: (semiring_char_0) semiring_char_0 |
24195 | 586 |
by intro_classes (simp only: star_of_nat_def star_of_eq of_nat_eq_iff) |
23282
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
22993
diff
changeset
|
587 |
|
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
22993
diff
changeset
|
588 |
instance star :: (ring_char_0) ring_char_0 .. |
22911
2f5e8d70a179
new axclass ring_char_0 for rings with characteristic 0, used for of_int_eq_iff and related lemmas
huffman
parents:
22518
diff
changeset
|
589 |
|
17296 | 590 |
instance star :: (number_ring) number_ring |
591 |
by (intro_classes, simp only: star_number_def star_of_int_def number_of_eq) |
|
592 |
||
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
593 |
subsection {* Finite class *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
594 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
595 |
lemma starset_finite: "finite A \<Longrightarrow> *s* A = star_of ` A" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
596 |
by (erule finite_induct, simp_all) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
597 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
598 |
instance star :: (finite) finite |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
599 |
apply (intro_classes) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
600 |
apply (subst starset_UNIV [symmetric]) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
601 |
apply (subst starset_finite [OF finite]) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
602 |
apply (rule finite_imageI [OF finite]) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
603 |
done |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
604 |
|
17296 | 605 |
end |