author | berghofe |
Fri, 01 Jul 2005 13:54:12 +0200 | |
changeset 16633 | 208ebc9311f2 |
parent 15539 | 333a88244569 |
child 16924 | 04246269386e |
permissions | -rw-r--r-- |
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(* Title : HyperPow.thy |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 |
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*) |
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||
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header{*Exponentials on the Hyperreals*} |
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theory HyperPow |
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imports HyperArith HyperNat |
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begin |
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instance hypreal :: power .. |
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consts hpowr :: "[hypreal,nat] => hypreal" |
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primrec |
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hpowr_0: "r ^ 0 = (1::hypreal)" |
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hpowr_Suc: "r ^ (Suc n) = (r::hypreal) * (r ^ n)" |
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instance hypreal :: recpower |
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proof |
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fix z :: hypreal |
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fix n :: nat |
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show "z^0 = 1" by simp |
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show "z^(Suc n) = z * (z^n)" by simp |
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qed |
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consts |
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"pow" :: "[hypreal,hypnat] => hypreal" (infixr 80) |
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defs |
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(* hypernatural powers of hyperreals *) |
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hyperpow_def: |
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"(R::hypreal) pow (N::hypnat) == |
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Abs_hypreal(\<Union>X \<in> Rep_hypreal(R). \<Union>Y \<in> Rep_hypnat(N). |
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hyprel``{%n::nat. (X n) ^ (Y n)})" |
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lemma hrealpow_two: "(r::hypreal) ^ Suc (Suc 0) = r * r" |
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by simp |
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lemma hrealpow_two_le [simp]: "(0::hypreal) \<le> r ^ Suc (Suc 0)" |
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by (auto simp add: zero_le_mult_iff) |
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lemma hrealpow_two_le_add_order [simp]: |
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"(0::hypreal) \<le> u ^ Suc (Suc 0) + v ^ Suc (Suc 0)" |
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by (simp only: hrealpow_two_le hypreal_le_add_order) |
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lemma hrealpow_two_le_add_order2 [simp]: |
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"(0::hypreal) \<le> u ^ Suc (Suc 0) + v ^ Suc (Suc 0) + w ^ Suc (Suc 0)" |
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by (simp only: hrealpow_two_le hypreal_le_add_order) |
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lemma hypreal_add_nonneg_eq_0_iff: |
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"[| 0 \<le> x; 0 \<le> y |] ==> (x+y = 0) = (x = 0 & y = (0::hypreal))" |
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by arith |
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text{*FIXME: DELETE THESE*} |
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lemma hypreal_three_squares_add_zero_iff: |
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"(x*x + y*y + z*z = 0) = (x = 0 & y = 0 & z = (0::hypreal))" |
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apply (simp only: zero_le_square hypreal_le_add_order hypreal_add_nonneg_eq_0_iff, auto) |
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done |
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lemma hrealpow_three_squares_add_zero_iff [simp]: |
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"(x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + z ^ Suc (Suc 0) = (0::hypreal)) = |
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(x = 0 & y = 0 & z = 0)" |
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by (simp only: hypreal_three_squares_add_zero_iff hrealpow_two) |
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lemma hrabs_hrealpow_two [simp]: |
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"abs(x ^ Suc (Suc 0)) = (x::hypreal) ^ Suc (Suc 0)" |
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by (simp) |
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lemma two_hrealpow_ge_one [simp]: "(1::hypreal) \<le> 2 ^ n" |
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by (insert power_increasing [of 0 n "2::hypreal"], simp) |
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lemma two_hrealpow_gt [simp]: "hypreal_of_nat n < 2 ^ n" |
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apply (induct_tac "n") |
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apply (auto simp add: hypreal_of_nat_Suc left_distrib) |
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apply (cut_tac n = n in two_hrealpow_ge_one, arith) |
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done |
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lemma hrealpow: |
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"Abs_hypreal(hyprel``{%n. X n}) ^ m = Abs_hypreal(hyprel``{%n. (X n) ^ m})" |
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apply (induct_tac "m") |
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apply (auto simp add: hypreal_one_def hypreal_mult) |
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done |
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lemma hrealpow_sum_square_expand: |
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"(x + (y::hypreal)) ^ Suc (Suc 0) = |
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x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + (hypreal_of_nat (Suc (Suc 0)))*x*y" |
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by (simp add: right_distrib left_distrib hypreal_of_nat_Suc) |
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subsection{*Literal Arithmetic Involving Powers and Type @{typ hypreal}*} |
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15003 | 99 |
lemma hypreal_of_real_power [simp]: |
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"hypreal_of_real (x ^ n) = hypreal_of_real x ^ n" |
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by (induct_tac "n", simp_all add: nat_mult_distrib) |
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lemma power_hypreal_of_real_number_of: |
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"(number_of v :: hypreal) ^ n = hypreal_of_real ((number_of v) ^ n)" |
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by (simp only: hypreal_number_of [symmetric] hypreal_of_real_power) |
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declare power_hypreal_of_real_number_of [of _ "number_of w", standard, simp] |
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lemma hrealpow_HFinite: "x \<in> HFinite ==> x ^ n \<in> HFinite" |
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apply (induct_tac "n") |
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111 |
apply (auto intro: HFinite_mult) |
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112 |
done |
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subsection{*Powers with Hypernatural Exponents*} |
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lemma hyperpow_congruent: "(%X Y. hyprel``{%n. (X n ^ Y n)}) respects hyprel" |
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by (auto simp add: congruent_def intro!: ext, fuf+) |
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lemma hyperpow: |
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"Abs_hypreal(hyprel``{%n. X n}) pow Abs_hypnat(hypnatrel``{%n. Y n}) = |
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Abs_hypreal(hyprel``{%n. X n ^ Y n})" |
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apply (unfold hyperpow_def) |
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apply (rule_tac f = Abs_hypreal in arg_cong) |
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apply (auto intro!: lemma_hyprel_refl bexI |
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simp add: hyprel_in_hypreal [THEN Abs_hypreal_inverse] equiv_hyprel |
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hyperpow_congruent, fuf) |
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128 |
done |
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lemma hyperpow_zero: "(0::hypreal) pow (n + (1::hypnat)) = 0" |
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apply (unfold hypnat_one_def) |
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apply (simp (no_asm) add: hypreal_zero_def) |
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apply (rule_tac z = n in eq_Abs_hypnat) |
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apply (auto simp add: hyperpow hypnat_add) |
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135 |
done |
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136 |
declare hyperpow_zero [simp] |
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137 |
|
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|
138 |
lemma hyperpow_not_zero [rule_format (no_asm)]: |
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|
139 |
"r \<noteq> (0::hypreal) --> r pow n \<noteq> 0" |
14468 | 140 |
apply (simp (no_asm) add: hypreal_zero_def, cases n, cases r) |
14348
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parents:
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|
141 |
apply (auto simp add: hyperpow) |
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paulson
parents:
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|
142 |
apply (drule FreeUltrafilterNat_Compl_mem, ultra) |
14348
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|
143 |
done |
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changeset
|
144 |
|
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|
145 |
lemma hyperpow_inverse: |
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|
146 |
"r \<noteq> (0::hypreal) --> inverse(r pow n) = (inverse r) pow n" |
14468 | 147 |
apply (simp (no_asm) add: hypreal_zero_def, cases n, cases r) |
14348
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parents:
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changeset
|
148 |
apply (auto dest!: FreeUltrafilterNat_Compl_mem simp add: hypreal_inverse hyperpow) |
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|
149 |
apply (rule FreeUltrafilterNat_subset) |
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|
150 |
apply (auto dest: realpow_not_zero intro: power_inverse) |
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|
151 |
done |
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changeset
|
152 |
|
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|
153 |
lemma hyperpow_hrabs: "abs r pow n = abs (r pow n)" |
14468 | 154 |
apply (cases n, cases r) |
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|
155 |
apply (auto simp add: hypreal_hrabs hyperpow power_abs [symmetric]) |
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|
156 |
done |
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changeset
|
157 |
|
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|
158 |
lemma hyperpow_add: "r pow (n + m) = (r pow n) * (r pow m)" |
14468 | 159 |
apply (cases n, cases m, cases r) |
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Defining the type class "ringpower" and deleting superseded theorems for
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|
160 |
apply (auto simp add: hyperpow hypnat_add hypreal_mult power_add) |
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changeset
|
161 |
done |
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changeset
|
162 |
|
15003 | 163 |
lemma hyperpow_one [simp]: "r pow (1::hypnat) = r" |
14468 | 164 |
apply (unfold hypnat_one_def, cases r) |
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|
165 |
apply (auto simp add: hyperpow) |
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changeset
|
166 |
done |
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changeset
|
167 |
|
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|
168 |
lemma hyperpow_two: |
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|
169 |
"r pow ((1::hypnat) + (1::hypnat)) = r * r" |
14468 | 170 |
apply (unfold hypnat_one_def, cases r) |
14348
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|
171 |
apply (auto simp add: hyperpow hypnat_add hypreal_mult) |
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parents:
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changeset
|
172 |
done |
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changeset
|
173 |
|
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|
174 |
lemma hyperpow_gt_zero: "(0::hypreal) < r ==> 0 < r pow n" |
14468 | 175 |
apply (simp add: hypreal_zero_def, cases n, cases r) |
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changeset
|
176 |
apply (auto elim!: FreeUltrafilterNat_subset zero_less_power |
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|
177 |
simp add: hyperpow hypreal_less hypreal_le) |
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|
178 |
done |
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changeset
|
179 |
|
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|
180 |
lemma hyperpow_ge_zero: "(0::hypreal) \<le> r ==> 0 \<le> r pow n" |
14468 | 181 |
apply (simp add: hypreal_zero_def, cases n, cases r) |
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changeset
|
182 |
apply (auto elim!: FreeUltrafilterNat_subset zero_le_power |
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|
183 |
simp add: hyperpow hypreal_le) |
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changeset
|
184 |
done |
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changeset
|
185 |
|
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|
186 |
lemma hyperpow_le: "[|(0::hypreal) < x; x \<le> y|] ==> x pow n \<le> y pow n" |
14468 | 187 |
apply (simp add: hypreal_zero_def, cases n, cases x, cases y) |
14348
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paulson
parents:
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changeset
|
188 |
apply (auto simp add: hyperpow hypreal_le hypreal_less) |
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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changeset
|
189 |
apply (erule FreeUltrafilterNat_Int [THEN FreeUltrafilterNat_subset], assumption) |
14348
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parents:
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changeset
|
190 |
apply (auto intro: power_mono) |
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changeset
|
191 |
done |
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changeset
|
192 |
|
15003 | 193 |
lemma hyperpow_eq_one [simp]: "1 pow n = (1::hypreal)" |
14468 | 194 |
apply (cases n) |
14348
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|
195 |
apply (auto simp add: hypreal_one_def hyperpow) |
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parents:
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changeset
|
196 |
done |
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
197 |
|
15003 | 198 |
lemma hrabs_hyperpow_minus_one [simp]: "abs(-1 pow n) = (1::hypreal)" |
14348
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
199 |
apply (subgoal_tac "abs ((- (1::hypreal)) pow n) = (1::hypreal) ") |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
200 |
apply simp |
14468 | 201 |
apply (cases n) |
14348
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Defining the type class "ringpower" and deleting superseded theorems for
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parents:
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changeset
|
202 |
apply (auto simp add: hypreal_one_def hyperpow hypreal_minus hypreal_hrabs) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
203 |
done |
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
204 |
|
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Defining the type class "ringpower" and deleting superseded theorems for
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parents:
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changeset
|
205 |
lemma hyperpow_mult: "(r * s) pow n = (r pow n) * (s pow n)" |
14468 | 206 |
apply (cases n, cases r, cases s) |
14348
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
207 |
apply (auto simp add: hyperpow hypreal_mult power_mult_distrib) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
208 |
done |
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
209 |
|
15003 | 210 |
lemma hyperpow_two_le [simp]: "0 \<le> r pow (1 + 1)" |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
14348
diff
changeset
|
211 |
by (auto simp add: hyperpow_two zero_le_mult_iff) |
14348
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
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changeset
|
212 |
|
14371
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Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
14348
diff
changeset
|
213 |
lemma hrabs_hyperpow_two [simp]: "abs(x pow (1 + 1)) = x pow (1 + 1)" |
15003 | 214 |
by (simp add: abs_if hyperpow_two_le linorder_not_less) |
14348
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
215 |
|
15003 | 216 |
lemma hyperpow_two_hrabs [simp]: "abs(x) pow (1 + 1) = x pow (1 + 1)" |
217 |
by (simp add: hyperpow_hrabs) |
|
14348
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
218 |
|
15229 | 219 |
text{*The precondition could be weakened to @{term "0\<le>x"}*} |
220 |
lemma hypreal_mult_less_mono: |
|
221 |
"[| u<v; x<y; (0::hypreal) < v; 0 < x |] ==> u*x < v* y" |
|
222 |
by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) |
|
223 |
||
15003 | 224 |
lemma hyperpow_two_gt_one: "1 < r ==> 1 < r pow (1 + 1)" |
14348
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Defining the type class "ringpower" and deleting superseded theorems for
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changeset
|
225 |
apply (auto simp add: hyperpow_two) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
226 |
apply (rule_tac y = "1*1" in order_le_less_trans) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
14348
diff
changeset
|
227 |
apply (rule_tac [2] hypreal_mult_less_mono, auto) |
14348
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
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changeset
|
228 |
done |
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paulson
parents:
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changeset
|
229 |
|
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Defining the type class "ringpower" and deleting superseded theorems for
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|
230 |
lemma hyperpow_two_ge_one: |
15003 | 231 |
"1 \<le> r ==> 1 \<le> r pow (1 + 1)" |
232 |
by (auto dest!: order_le_imp_less_or_eq intro: hyperpow_two_gt_one order_less_imp_le) |
|
14348
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
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changeset
|
233 |
|
15003 | 234 |
lemma two_hyperpow_ge_one [simp]: "(1::hypreal) \<le> 2 pow n" |
14348
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
235 |
apply (rule_tac y = "1 pow n" in order_trans) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
14348
diff
changeset
|
236 |
apply (rule_tac [2] hyperpow_le, auto) |
14348
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
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changeset
|
237 |
done |
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
238 |
|
15003 | 239 |
lemma hyperpow_minus_one2 [simp]: |
14348
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
240 |
"-1 pow ((1 + 1)*n) = (1::hypreal)" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
241 |
apply (subgoal_tac " (- ((1::hypreal))) pow ((1 + 1)*n) = (1::hypreal) ") |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
242 |
apply simp |
14468 | 243 |
apply (simp only: hypreal_one_def, cases n) |
14435
9e22eeccf129
Conversion of Poly to Isar script, and other tidying of HOL/Hyperreal
paulson
parents:
14387
diff
changeset
|
244 |
apply (auto simp add: nat_mult_2 [symmetric] hyperpow hypnat_add hypreal_minus |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
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diff
changeset
|
245 |
left_distrib) |
14348
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Defining the type class "ringpower" and deleting superseded theorems for
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diff
changeset
|
246 |
done |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
247 |
|
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Defining the type class "ringpower" and deleting superseded theorems for
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changeset
|
248 |
lemma hyperpow_less_le: |
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paulson
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|
249 |
"[|(0::hypreal) \<le> r; r \<le> 1; n < N|] ==> r pow N \<le> r pow n" |
14468 | 250 |
apply (cases n, cases N, cases r) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
251 |
apply (auto simp add: hyperpow hypreal_le hypreal_less hypnat_less |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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|
252 |
hypreal_zero_def hypreal_one_def) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
253 |
apply (erule FreeUltrafilterNat_Int [THEN FreeUltrafilterNat_subset]) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
14348
diff
changeset
|
254 |
apply (erule FreeUltrafilterNat_Int, assumption) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
255 |
apply (auto intro: power_decreasing) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
256 |
done |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
257 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
258 |
lemma hyperpow_SHNat_le: |
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Defining the type class "ringpower" and deleting superseded theorems for
paulson
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|
259 |
"[| 0 \<le> r; r \<le> (1::hypreal); N \<in> HNatInfinite |] |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
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|
260 |
==> ALL n: Nats. r pow N \<le> r pow n" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
261 |
by (auto intro!: hyperpow_less_le simp add: HNatInfinite_iff) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
262 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
263 |
lemma hyperpow_realpow: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
264 |
"(hypreal_of_real r) pow (hypnat_of_nat n) = hypreal_of_real (r ^ n)" |
15003 | 265 |
by (simp add: hypreal_of_real_def hypnat_of_nat_eq hyperpow) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
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changeset
|
266 |
|
15003 | 267 |
lemma hyperpow_SReal [simp]: |
268 |
"(hypreal_of_real r) pow (hypnat_of_nat n) \<in> Reals" |
|
269 |
by (simp del: hypreal_of_real_power add: hyperpow_realpow SReal_def) |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
270 |
|
15003 | 271 |
|
272 |
lemma hyperpow_zero_HNatInfinite [simp]: |
|
273 |
"N \<in> HNatInfinite ==> (0::hypreal) pow N = 0" |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
14348
diff
changeset
|
274 |
by (drule HNatInfinite_is_Suc, auto) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
275 |
|
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Defining the type class "ringpower" and deleting superseded theorems for
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changeset
|
276 |
lemma hyperpow_le_le: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
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|
277 |
"[| (0::hypreal) \<le> r; r \<le> 1; n \<le> N |] ==> r pow N \<le> r pow n" |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
14348
diff
changeset
|
278 |
apply (drule order_le_less [of n, THEN iffD1]) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
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parents:
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diff
changeset
|
279 |
apply (auto intro: hyperpow_less_le) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
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changeset
|
280 |
done |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
281 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
282 |
lemma hyperpow_Suc_le_self2: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
283 |
"[| (0::hypreal) \<le> r; r < 1 |] ==> r pow (n + (1::hypnat)) \<le> r" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
284 |
apply (drule_tac n = " (1::hypnat) " in hyperpow_le_le) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
285 |
apply auto |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
286 |
done |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
287 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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changeset
|
288 |
lemma lemma_Infinitesimal_hyperpow: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
289 |
"[| x \<in> Infinitesimal; 0 < N |] ==> abs (x pow N) \<le> abs x" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
290 |
apply (unfold Infinitesimal_def) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
291 |
apply (auto intro!: hyperpow_Suc_le_self2 |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
292 |
simp add: hyperpow_hrabs [symmetric] hypnat_gt_zero_iff2 abs_ge_zero) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
293 |
done |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
294 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
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parents:
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changeset
|
295 |
lemma Infinitesimal_hyperpow: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
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parents:
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diff
changeset
|
296 |
"[| x \<in> Infinitesimal; 0 < N |] ==> x pow N \<in> Infinitesimal" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
297 |
apply (rule hrabs_le_Infinitesimal) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
14348
diff
changeset
|
298 |
apply (rule_tac [2] lemma_Infinitesimal_hyperpow, auto) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
299 |
done |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
300 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
301 |
lemma hrealpow_hyperpow_Infinitesimal_iff: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
302 |
"(x ^ n \<in> Infinitesimal) = (x pow (hypnat_of_nat n) \<in> Infinitesimal)" |
14468 | 303 |
apply (cases x) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
304 |
apply (simp add: hrealpow hyperpow hypnat_of_nat_eq) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
305 |
done |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
306 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
307 |
lemma Infinitesimal_hrealpow: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
308 |
"[| x \<in> Infinitesimal; 0 < n |] ==> x ^ n \<in> Infinitesimal" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
309 |
by (force intro!: Infinitesimal_hyperpow |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
310 |
simp add: hrealpow_hyperpow_Infinitesimal_iff |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
14348
diff
changeset
|
311 |
hypnat_of_nat_less_iff [symmetric] hypnat_of_nat_zero |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
14348
diff
changeset
|
312 |
simp del: hypnat_of_nat_less_iff) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
313 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
314 |
ML |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
315 |
{* |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
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diff
changeset
|
316 |
val hrealpow_two = thm "hrealpow_two"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
317 |
val hrealpow_two_le = thm "hrealpow_two_le"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
318 |
val hrealpow_two_le_add_order = thm "hrealpow_two_le_add_order"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
319 |
val hrealpow_two_le_add_order2 = thm "hrealpow_two_le_add_order2"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
320 |
val hypreal_add_nonneg_eq_0_iff = thm "hypreal_add_nonneg_eq_0_iff"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
321 |
val hypreal_three_squares_add_zero_iff = thm "hypreal_three_squares_add_zero_iff"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
322 |
val hrealpow_three_squares_add_zero_iff = thm "hrealpow_three_squares_add_zero_iff"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
323 |
val hrabs_hrealpow_two = thm "hrabs_hrealpow_two"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
324 |
val two_hrealpow_ge_one = thm "two_hrealpow_ge_one"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
325 |
val two_hrealpow_gt = thm "two_hrealpow_gt"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
326 |
val hrealpow = thm "hrealpow"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
327 |
val hrealpow_sum_square_expand = thm "hrealpow_sum_square_expand"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
328 |
val hypreal_of_real_power = thm "hypreal_of_real_power"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
329 |
val power_hypreal_of_real_number_of = thm "power_hypreal_of_real_number_of"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
330 |
val hrealpow_HFinite = thm "hrealpow_HFinite"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
331 |
val hyperpow_congruent = thm "hyperpow_congruent"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
332 |
val hyperpow = thm "hyperpow"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
333 |
val hyperpow_zero = thm "hyperpow_zero"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
334 |
val hyperpow_not_zero = thm "hyperpow_not_zero"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
335 |
val hyperpow_inverse = thm "hyperpow_inverse"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
336 |
val hyperpow_hrabs = thm "hyperpow_hrabs"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
337 |
val hyperpow_add = thm "hyperpow_add"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
338 |
val hyperpow_one = thm "hyperpow_one"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
339 |
val hyperpow_two = thm "hyperpow_two"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
340 |
val hyperpow_gt_zero = thm "hyperpow_gt_zero"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
341 |
val hyperpow_ge_zero = thm "hyperpow_ge_zero"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
342 |
val hyperpow_le = thm "hyperpow_le"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
343 |
val hyperpow_eq_one = thm "hyperpow_eq_one"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
344 |
val hrabs_hyperpow_minus_one = thm "hrabs_hyperpow_minus_one"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
345 |
val hyperpow_mult = thm "hyperpow_mult"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
346 |
val hyperpow_two_le = thm "hyperpow_two_le"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
347 |
val hrabs_hyperpow_two = thm "hrabs_hyperpow_two"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
348 |
val hyperpow_two_hrabs = thm "hyperpow_two_hrabs"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
349 |
val hyperpow_two_gt_one = thm "hyperpow_two_gt_one"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
350 |
val hyperpow_two_ge_one = thm "hyperpow_two_ge_one"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
351 |
val two_hyperpow_ge_one = thm "two_hyperpow_ge_one"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
352 |
val hyperpow_minus_one2 = thm "hyperpow_minus_one2"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
353 |
val hyperpow_less_le = thm "hyperpow_less_le"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
354 |
val hyperpow_SHNat_le = thm "hyperpow_SHNat_le"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
355 |
val hyperpow_realpow = thm "hyperpow_realpow"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
356 |
val hyperpow_SReal = thm "hyperpow_SReal"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
357 |
val hyperpow_zero_HNatInfinite = thm "hyperpow_zero_HNatInfinite"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
358 |
val hyperpow_le_le = thm "hyperpow_le_le"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
359 |
val hyperpow_Suc_le_self2 = thm "hyperpow_Suc_le_self2"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
360 |
val lemma_Infinitesimal_hyperpow = thm "lemma_Infinitesimal_hyperpow"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
361 |
val Infinitesimal_hyperpow = thm "Infinitesimal_hyperpow"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
362 |
val hrealpow_hyperpow_Infinitesimal_iff = thm "hrealpow_hyperpow_Infinitesimal_iff"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
363 |
val Infinitesimal_hrealpow = thm "Infinitesimal_hrealpow"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
364 |
*} |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
11713
diff
changeset
|
365 |
|
10751 | 366 |
end |