author | huffman |
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permissions | -rw-r--r-- |
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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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instance star :: (zero) zero |
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star_zero_def: "0 \<equiv> star_of 0" .. |
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instance star :: (one) one |
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star_one_def: "1 \<equiv> star_of 1" .. |
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instance star :: (plus) plus |
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star_add_def: "(op +) \<equiv> *f2* (op +)" .. |
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instance star :: (times) times |
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star_mult_def: "(op *) \<equiv> *f2* (op *)" .. |
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instance star :: (minus) minus |
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star_minus_def: "uminus \<equiv> *f* uminus" |
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star_diff_def: "(op -) \<equiv> *f2* (op -)" |
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star_abs_def: "abs \<equiv> *f* abs" .. |
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instance star :: (inverse) inverse |
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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 star :: (number) number |
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star_number_def: "number_of b \<equiv> star_of (number_of b)" .. |
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instance star :: ("Divides.div") "Divides.div" |
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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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instance 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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instance star :: (ord) ord |
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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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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 |
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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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by (simp add: star_add_def) |
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lemma Standard_diff: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x - y \<in> Standard" |
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by (simp add: star_diff_def) |
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lemma Standard_minus: "x \<in> Standard \<Longrightarrow> - x \<in> Standard" |
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by (simp add: star_minus_def) |
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lemma Standard_mult: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x * y \<in> Standard" |
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by (simp add: star_mult_def) |
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lemma Standard_divide: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x / y \<in> Standard" |
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by (simp add: star_divide_def) |
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lemma Standard_inverse: "x \<in> Standard \<Longrightarrow> inverse x \<in> Standard" |
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by (simp add: star_inverse_def) |
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lemma Standard_abs: "x \<in> Standard \<Longrightarrow> abs x \<in> Standard" |
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by (simp add: star_abs_def) |
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lemma Standard_div: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x div y \<in> Standard" |
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by (simp add: star_div_def) |
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lemma Standard_mod: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x mod y \<in> Standard" |
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by (simp add: star_mod_def) |
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lemma Standard_power: "x \<in> Standard \<Longrightarrow> x ^ n \<in> Standard" |
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by (simp add: star_power_def) |
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lemmas Standard_simps [simp] = |
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Standard_zero Standard_one Standard_number_of |
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Standard_add Standard_diff Standard_minus |
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Standard_mult Standard_divide Standard_inverse |
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Standard_abs Standard_div Standard_mod |
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Standard_power |
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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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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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lemma star_of_inverse: "star_of (inverse x) = inverse (star_of x)" |
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by transfer (rule refl) |
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lemma star_of_div: "star_of (x div y) = star_of x div star_of y" |
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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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lemma star_of_power: "star_of (x ^ n) = star_of x ^ n" |
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lemma star_of_abs: "star_of (abs x) = abs (star_of x)" |
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text {* @{term star_of} preserves numerals *} |
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lemma star_of_zero: "star_of 0 = 0" |
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lemma star_of_one: "star_of 1 = 1" |
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by transfer (rule refl) |
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lemma star_of_number_of: "star_of (number_of x) = number_of x" |
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text {* @{term star_of} preserves orderings *} |
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lemma star_of_less: "(star_of x < star_of y) = (x < y)" |
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lemma star_of_le: "(star_of x \<le> star_of y) = (x \<le> y)" |
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lemma star_of_eq: "(star_of x = star_of y) = (x = y)" |
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text{*As above, for 0*} |
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lemmas star_of_0_less = star_of_less [of 0, simplified star_of_zero] |
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lemmas star_of_0_le = star_of_le [of 0, simplified star_of_zero] |
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lemmas star_of_0_eq = star_of_eq [of 0, simplified star_of_zero] |
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lemmas star_of_less_0 = star_of_less [of _ 0, simplified star_of_zero] |
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lemmas star_of_le_0 = star_of_le [of _ 0, simplified star_of_zero] |
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lemmas star_of_eq_0 = star_of_eq [of _ 0, simplified star_of_zero] |
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|
167 |
|
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168 |
text{*As above, for 1*} |
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|
169 |
|
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parents:
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|
170 |
lemmas star_of_1_less = star_of_less [of 1, simplified star_of_one] |
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parents:
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|
171 |
lemmas star_of_1_le = star_of_le [of 1, simplified star_of_one] |
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huffman
parents:
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changeset
|
172 |
lemmas star_of_1_eq = star_of_eq [of 1, simplified star_of_one] |
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huffman
parents:
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changeset
|
173 |
|
4910cf8c0cd2
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huffman
parents:
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changeset
|
174 |
lemmas star_of_less_1 = star_of_less [of _ 1, simplified star_of_one] |
4910cf8c0cd2
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huffman
parents:
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changeset
|
175 |
lemmas star_of_le_1 = star_of_le [of _ 1, simplified star_of_one] |
4910cf8c0cd2
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huffman
parents:
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changeset
|
176 |
lemmas star_of_eq_1 = star_of_eq [of _ 1, simplified star_of_one] |
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huffman
parents:
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changeset
|
177 |
|
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parents:
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changeset
|
178 |
text{*As above, for numerals*} |
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parents:
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changeset
|
179 |
|
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parents:
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changeset
|
180 |
lemmas star_of_number_less = |
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huffman
parents:
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changeset
|
181 |
star_of_less [of "number_of w", standard, simplified star_of_number_of] |
4910cf8c0cd2
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huffman
parents:
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changeset
|
182 |
lemmas star_of_number_le = |
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huffman
parents:
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changeset
|
183 |
star_of_le [of "number_of w", standard, simplified star_of_number_of] |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
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changeset
|
184 |
lemmas star_of_number_eq = |
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huffman
parents:
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changeset
|
185 |
star_of_eq [of "number_of w", standard, simplified star_of_number_of] |
4910cf8c0cd2
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huffman
parents:
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changeset
|
186 |
|
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
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changeset
|
187 |
lemmas star_of_less_number = |
4910cf8c0cd2
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huffman
parents:
17296
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changeset
|
188 |
star_of_less [of _ "number_of w", standard, simplified star_of_number_of] |
4910cf8c0cd2
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huffman
parents:
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changeset
|
189 |
lemmas star_of_le_number = |
4910cf8c0cd2
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huffman
parents:
17296
diff
changeset
|
190 |
star_of_le [of _ "number_of w", standard, simplified star_of_number_of] |
4910cf8c0cd2
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huffman
parents:
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changeset
|
191 |
lemmas star_of_eq_number = |
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huffman
parents:
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changeset
|
192 |
star_of_eq [of _ "number_of w", standard, simplified star_of_number_of] |
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huffman
parents:
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changeset
|
193 |
|
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parents:
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changeset
|
194 |
lemmas star_of_simps [simp] = |
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huffman
parents:
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changeset
|
195 |
star_of_add star_of_diff star_of_minus |
4910cf8c0cd2
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huffman
parents:
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changeset
|
196 |
star_of_mult star_of_divide star_of_inverse |
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added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
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changeset
|
197 |
star_of_div star_of_mod |
4910cf8c0cd2
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huffman
parents:
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changeset
|
198 |
star_of_power star_of_abs |
4910cf8c0cd2
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parents:
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changeset
|
199 |
star_of_zero star_of_one star_of_number_of |
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huffman
parents:
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changeset
|
200 |
star_of_less star_of_le star_of_eq |
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parents:
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changeset
|
201 |
star_of_0_less star_of_0_le star_of_0_eq |
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parents:
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changeset
|
202 |
star_of_less_0 star_of_le_0 star_of_eq_0 |
4910cf8c0cd2
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huffman
parents:
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changeset
|
203 |
star_of_1_less star_of_1_le star_of_1_eq |
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huffman
parents:
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changeset
|
204 |
star_of_less_1 star_of_le_1 star_of_eq_1 |
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huffman
parents:
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changeset
|
205 |
star_of_number_less star_of_number_le star_of_number_eq |
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huffman
parents:
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changeset
|
206 |
star_of_less_number star_of_le_number star_of_eq_number |
4910cf8c0cd2
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huffman
parents:
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changeset
|
207 |
|
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
208 |
subsection {* Ordering and lattice classes *} |
17296 | 209 |
|
210 |
instance star :: (order) order |
|
211 |
apply (intro_classes) |
|
22316 | 212 |
apply (transfer, rule order_less_le) |
17296 | 213 |
apply (transfer, rule order_refl) |
214 |
apply (transfer, erule (1) order_trans) |
|
215 |
apply (transfer, erule (1) order_antisym) |
|
216 |
done |
|
217 |
||
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
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changeset
|
218 |
instance star :: (lower_semilattice) lower_semilattice |
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|
219 |
star_inf_def [transfer_unfold]: "inf \<equiv> *f2* inf" |
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changeset
|
220 |
by default (transfer star_inf_def, auto)+ |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
221 |
|
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
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changeset
|
222 |
instance star :: (upper_semilattice) upper_semilattice |
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changeset
|
223 |
star_sup_def [transfer_unfold]: "sup \<equiv> *f2* sup" |
8a86fd2a1bf0
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haftmann
parents:
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diff
changeset
|
224 |
by default (transfer star_sup_def, auto)+ |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
225 |
|
8a86fd2a1bf0
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haftmann
parents:
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diff
changeset
|
226 |
instance star :: (lattice) lattice .. |
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changeset
|
227 |
|
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changeset
|
228 |
instance star :: (distrib_lattice) distrib_lattice |
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changeset
|
229 |
by default (transfer, auto simp add: sup_inf_distrib1) |
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haftmann
parents:
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diff
changeset
|
230 |
|
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haftmann
parents:
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changeset
|
231 |
lemma Standard_inf [simp]: |
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parents:
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changeset
|
232 |
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> inf x y \<in> Standard" |
8a86fd2a1bf0
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parents:
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changeset
|
233 |
by (simp add: star_inf_def) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
234 |
|
8a86fd2a1bf0
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haftmann
parents:
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changeset
|
235 |
lemma Standard_sup [simp]: |
8a86fd2a1bf0
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haftmann
parents:
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diff
changeset
|
236 |
"\<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
|
237 |
by (simp add: star_sup_def) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
238 |
|
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
239 |
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
|
240 |
by transfer (rule refl) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
241 |
|
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
242 |
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
|
243 |
by transfer (rule refl) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
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diff
changeset
|
244 |
|
17296 | 245 |
instance star :: (linorder) linorder |
246 |
by (intro_classes, transfer, rule linorder_linear) |
|
247 |
||
20720 | 248 |
lemma star_max_def [transfer_unfold]: "max = *f2* max" |
249 |
apply (rule ext, rule ext) |
|
250 |
apply (unfold max_def, transfer, fold max_def) |
|
251 |
apply (rule refl) |
|
252 |
done |
|
253 |
||
254 |
lemma star_min_def [transfer_unfold]: "min = *f2* min" |
|
255 |
apply (rule ext, rule ext) |
|
256 |
apply (unfold min_def, transfer, fold min_def) |
|
257 |
apply (rule refl) |
|
258 |
done |
|
259 |
||
260 |
lemma Standard_max [simp]: |
|
261 |
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> max x y \<in> Standard" |
|
262 |
by (simp add: star_max_def) |
|
263 |
||
264 |
lemma Standard_min [simp]: |
|
265 |
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> min x y \<in> Standard" |
|
266 |
by (simp add: star_min_def) |
|
267 |
||
268 |
lemma star_of_max [simp]: "star_of (max x y) = max (star_of x) (star_of y)" |
|
269 |
by transfer (rule refl) |
|
270 |
||
271 |
lemma star_of_min [simp]: "star_of (min x y) = min (star_of x) (star_of y)" |
|
272 |
by transfer (rule refl) |
|
273 |
||
17296 | 274 |
|
17332
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
275 |
subsection {* Ordered group classes *} |
17296 | 276 |
|
277 |
instance star :: (semigroup_add) semigroup_add |
|
278 |
by (intro_classes, transfer, rule add_assoc) |
|
279 |
||
280 |
instance star :: (ab_semigroup_add) ab_semigroup_add |
|
281 |
by (intro_classes, transfer, rule add_commute) |
|
282 |
||
283 |
instance star :: (semigroup_mult) semigroup_mult |
|
284 |
by (intro_classes, transfer, rule mult_assoc) |
|
285 |
||
286 |
instance star :: (ab_semigroup_mult) ab_semigroup_mult |
|
287 |
by (intro_classes, transfer, rule mult_commute) |
|
288 |
||
289 |
instance star :: (comm_monoid_add) comm_monoid_add |
|
22384
33a46e6c7f04
prefix of class interpretation not mandatory any longer
haftmann
parents:
22316
diff
changeset
|
290 |
by (intro_classes, transfer, rule comm_monoid_add_class.zero_plus.add_0) |
17296 | 291 |
|
292 |
instance star :: (monoid_mult) monoid_mult |
|
293 |
apply (intro_classes) |
|
294 |
apply (transfer, rule mult_1_left) |
|
295 |
apply (transfer, rule mult_1_right) |
|
296 |
done |
|
297 |
||
298 |
instance star :: (comm_monoid_mult) comm_monoid_mult |
|
299 |
by (intro_classes, transfer, rule mult_1) |
|
300 |
||
301 |
instance star :: (cancel_semigroup_add) cancel_semigroup_add |
|
302 |
apply (intro_classes) |
|
303 |
apply (transfer, erule add_left_imp_eq) |
|
304 |
apply (transfer, erule add_right_imp_eq) |
|
305 |
done |
|
306 |
||
307 |
instance star :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add |
|
308 |
by (intro_classes, transfer, rule add_imp_eq) |
|
309 |
||
310 |
instance star :: (ab_group_add) ab_group_add |
|
311 |
apply (intro_classes) |
|
312 |
apply (transfer, rule left_minus) |
|
313 |
apply (transfer, rule diff_minus) |
|
314 |
done |
|
315 |
||
316 |
instance star :: (pordered_ab_semigroup_add) pordered_ab_semigroup_add |
|
317 |
by (intro_classes, transfer, rule add_left_mono) |
|
318 |
||
319 |
instance star :: (pordered_cancel_ab_semigroup_add) pordered_cancel_ab_semigroup_add .. |
|
320 |
||
321 |
instance star :: (pordered_ab_semigroup_add_imp_le) pordered_ab_semigroup_add_imp_le |
|
322 |
by (intro_classes, transfer, rule add_le_imp_le_left) |
|
323 |
||
324 |
instance star :: (pordered_ab_group_add) pordered_ab_group_add .. |
|
325 |
instance star :: (ordered_cancel_ab_semigroup_add) ordered_cancel_ab_semigroup_add .. |
|
326 |
instance star :: (lordered_ab_group_meet) lordered_ab_group_meet .. |
|
327 |
instance star :: (lordered_ab_group_meet) lordered_ab_group_meet .. |
|
328 |
instance star :: (lordered_ab_group) lordered_ab_group .. |
|
329 |
||
330 |
instance star :: (lordered_ab_group_abs) lordered_ab_group_abs |
|
17332
4910cf8c0cd2
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huffman
parents:
17296
diff
changeset
|
331 |
by (intro_classes, transfer, rule abs_lattice) |
17296 | 332 |
|
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
333 |
subsection {* Ring and field classes *} |
17296 | 334 |
|
335 |
instance star :: (semiring) semiring |
|
336 |
apply (intro_classes) |
|
337 |
apply (transfer, rule left_distrib) |
|
338 |
apply (transfer, rule right_distrib) |
|
339 |
done |
|
340 |
||
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20720
diff
changeset
|
341 |
instance star :: (semiring_0) semiring_0 |
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20720
diff
changeset
|
342 |
by intro_classes (transfer, simp)+ |
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20720
diff
changeset
|
343 |
|
17296 | 344 |
instance star :: (semiring_0_cancel) semiring_0_cancel .. |
345 |
||
346 |
instance star :: (comm_semiring) comm_semiring |
|
347 |
by (intro_classes, transfer, rule distrib) |
|
348 |
||
349 |
instance star :: (comm_semiring_0) comm_semiring_0 .. |
|
350 |
instance star :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. |
|
351 |
||
20633 | 352 |
instance star :: (zero_neq_one) zero_neq_one |
17296 | 353 |
by (intro_classes, transfer, rule zero_neq_one) |
354 |
||
355 |
instance star :: (semiring_1) semiring_1 .. |
|
356 |
instance star :: (comm_semiring_1) comm_semiring_1 .. |
|
357 |
||
20633 | 358 |
instance star :: (no_zero_divisors) no_zero_divisors |
17296 | 359 |
by (intro_classes, transfer, rule no_zero_divisors) |
360 |
||
361 |
instance star :: (semiring_1_cancel) semiring_1_cancel .. |
|
362 |
instance star :: (comm_semiring_1_cancel) comm_semiring_1_cancel .. |
|
363 |
instance star :: (ring) ring .. |
|
364 |
instance star :: (comm_ring) comm_ring .. |
|
365 |
instance star :: (ring_1) ring_1 .. |
|
366 |
instance star :: (comm_ring_1) comm_ring_1 .. |
|
22992 | 367 |
instance star :: (ring_no_zero_divisors) ring_no_zero_divisors .. |
368 |
instance star :: (dom) dom .. |
|
17296 | 369 |
instance star :: (idom) idom .. |
370 |
||
20540 | 371 |
instance star :: (division_ring) division_ring |
372 |
apply (intro_classes) |
|
373 |
apply (transfer, erule left_inverse) |
|
374 |
apply (transfer, erule right_inverse) |
|
375 |
done |
|
376 |
||
17296 | 377 |
instance star :: (field) field |
378 |
apply (intro_classes) |
|
379 |
apply (transfer, erule left_inverse) |
|
380 |
apply (transfer, rule divide_inverse) |
|
381 |
done |
|
382 |
||
383 |
instance star :: (division_by_zero) division_by_zero |
|
384 |
by (intro_classes, transfer, rule inverse_zero) |
|
385 |
||
386 |
instance star :: (pordered_semiring) pordered_semiring |
|
387 |
apply (intro_classes) |
|
388 |
apply (transfer, erule (1) mult_left_mono) |
|
389 |
apply (transfer, erule (1) mult_right_mono) |
|
390 |
done |
|
391 |
||
392 |
instance star :: (pordered_cancel_semiring) pordered_cancel_semiring .. |
|
393 |
||
394 |
instance star :: (ordered_semiring_strict) ordered_semiring_strict |
|
395 |
apply (intro_classes) |
|
396 |
apply (transfer, erule (1) mult_strict_left_mono) |
|
397 |
apply (transfer, erule (1) mult_strict_right_mono) |
|
398 |
done |
|
399 |
||
400 |
instance star :: (pordered_comm_semiring) pordered_comm_semiring |
|
22384
33a46e6c7f04
prefix of class interpretation not mandatory any longer
haftmann
parents:
22316
diff
changeset
|
401 |
by (intro_classes, transfer, rule mult_mono1_class.times_zero_less_eq_less.mult_mono) |
17296 | 402 |
|
403 |
instance star :: (pordered_cancel_comm_semiring) pordered_cancel_comm_semiring .. |
|
404 |
||
405 |
instance star :: (ordered_comm_semiring_strict) ordered_comm_semiring_strict |
|
22518 | 406 |
by (intro_classes, transfer, rule ordered_comm_semiring_strict_class.plus_less_eq_less_zero_times.mult_strict_mono) |
17296 | 407 |
|
408 |
instance star :: (pordered_ring) pordered_ring .. |
|
409 |
instance star :: (lordered_ring) lordered_ring .. |
|
410 |
||
20633 | 411 |
instance star :: (abs_if) abs_if |
17296 | 412 |
by (intro_classes, transfer, rule abs_if) |
413 |
||
414 |
instance star :: (ordered_ring_strict) ordered_ring_strict .. |
|
415 |
instance star :: (pordered_comm_ring) pordered_comm_ring .. |
|
416 |
||
417 |
instance star :: (ordered_semidom) ordered_semidom |
|
418 |
by (intro_classes, transfer, rule zero_less_one) |
|
419 |
||
420 |
instance star :: (ordered_idom) ordered_idom .. |
|
421 |
instance star :: (ordered_field) ordered_field .. |
|
422 |
||
17332
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
423 |
subsection {* Power classes *} |
17296 | 424 |
|
425 |
text {* |
|
426 |
Proving the class axiom @{thm [source] power_Suc} for type |
|
427 |
@{typ "'a star"} is a little tricky, because it quantifies |
|
428 |
over values of type @{typ nat}. The transfer principle does |
|
429 |
not handle quantification over non-star types in general, |
|
430 |
but we can work around this by fixing an arbitrary @{typ nat} |
|
431 |
value, and then applying the transfer principle. |
|
432 |
*} |
|
433 |
||
434 |
instance star :: (recpower) recpower |
|
435 |
proof |
|
436 |
show "\<And>a::'a star. a ^ 0 = 1" |
|
437 |
by transfer (rule power_0) |
|
438 |
next |
|
439 |
fix n show "\<And>a::'a star. a ^ Suc n = a * a ^ n" |
|
440 |
by transfer (rule power_Suc) |
|
441 |
qed |
|
442 |
||
17332
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
443 |
subsection {* Number classes *} |
17296 | 444 |
|
20720 | 445 |
lemma star_of_nat_def [transfer_unfold]: "of_nat n = star_of (of_nat n)" |
446 |
by (induct_tac n, simp_all) |
|
447 |
||
448 |
lemma Standard_of_nat [simp]: "of_nat n \<in> Standard" |
|
449 |
by (simp add: star_of_nat_def) |
|
17296 | 450 |
|
17332
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
451 |
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
|
452 |
by transfer (rule refl) |
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
453 |
|
20720 | 454 |
lemma star_of_int_def [transfer_unfold]: "of_int z = star_of (of_int z)" |
455 |
by (rule_tac z=z in int_diff_cases, simp) |
|
456 |
||
457 |
lemma Standard_of_int [simp]: "of_int z \<in> Standard" |
|
458 |
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
|
459 |
|
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17296
diff
changeset
|
460 |
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
|
461 |
by transfer (rule refl) |
17296 | 462 |
|
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
|
463 |
instance star :: (ring_char_0) ring_char_0 |
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
|
464 |
by (intro_classes, simp only: star_of_int_def star_of_eq of_int_eq_iff) |
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
|
465 |
|
17296 | 466 |
instance star :: (number_ring) number_ring |
467 |
by (intro_classes, simp only: star_number_def star_of_int_def number_of_eq) |
|
468 |
||
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
469 |
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
|
470 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
471 |
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
|
472 |
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
|
473 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
474 |
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
|
475 |
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
|
476 |
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
|
477 |
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
|
478 |
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
|
479 |
done |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
480 |
|
17296 | 481 |
end |