author | haftmann |
Tue, 16 Oct 2007 23:12:45 +0200 | |
changeset 25062 | af5ef0d4d655 |
parent 24748 | ee0a0eb6b738 |
child 25078 | a1ddc5206cb1 |
permissions | -rw-r--r-- |
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(* Title: HOL/Ring_and_Field.thy |
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ID: $Id$ |
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Author: Gertrud Bauer, Steven Obua, Tobias Nipkow, Lawrence C Paulson, and Markus Wenzel, |
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with contributions by Jeremy Avigad |
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*) |
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header {* (Ordered) Rings and Fields *} |
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theory Ring_and_Field |
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imports OrderedGroup |
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begin |
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text {* |
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The theory of partially ordered rings is taken from the books: |
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\begin{itemize} |
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\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 |
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\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 |
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\end{itemize} |
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Most of the used notions can also be looked up in |
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\begin{itemize} |
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\item \url{http://www.mathworld.com} by Eric Weisstein et. al. |
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\item \emph{Algebra I} by van der Waerden, Springer. |
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\end{itemize} |
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*} |
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class semiring = ab_semigroup_add + semigroup_mult + |
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assumes left_distrib: "(a + b) * c = a * c + b * c" |
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assumes right_distrib: "a * (b + c) = a * b + a * c" |
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class mult_zero = times + zero + |
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assumes mult_zero_left [simp]: "0 * a = 0" |
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assumes mult_zero_right [simp]: "a * 0 = 0" |
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class semiring_0 = semiring + comm_monoid_add + mult_zero |
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class semiring_0_cancel = semiring + comm_monoid_add + cancel_ab_semigroup_add |
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instance semiring_0_cancel \<subseteq> semiring_0 |
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proof |
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fix a :: 'a |
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have "0 * a + 0 * a = 0 * a + 0" |
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by (simp add: left_distrib [symmetric]) |
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thus "0 * a = 0" |
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by (simp only: add_left_cancel) |
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|
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have "a * 0 + a * 0 = a * 0 + 0" |
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by (simp add: right_distrib [symmetric]) |
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thus "a * 0 = 0" |
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by (simp only: add_left_cancel) |
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qed |
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult + |
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assumes distrib: "(a + b) * c = a * c + b * c" |
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instance comm_semiring \<subseteq> semiring |
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proof |
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fix a b c :: 'a |
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show "(a + b) * c = a * c + b * c" by (simp add: distrib) |
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have "a * (b + c) = (b + c) * a" by (simp add: mult_ac) |
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also have "... = b * a + c * a" by (simp only: distrib) |
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also have "... = a * b + a * c" by (simp add: mult_ac) |
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finally show "a * (b + c) = a * b + a * c" by blast |
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qed |
64 |
||
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero |
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instance comm_semiring_0 \<subseteq> semiring_0 .. |
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class comm_semiring_0_cancel = comm_semiring + comm_monoid_add + cancel_ab_semigroup_add |
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instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel .. |
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||
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instance comm_semiring_0_cancel \<subseteq> comm_semiring_0 .. |
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class zero_neq_one = zero + one + |
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assumes zero_neq_one [simp]: "0 \<noteq> 1" |
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult |
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class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult |
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(*previously almost_semiring*) |
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instance comm_semiring_1 \<subseteq> semiring_1 .. |
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class no_zero_divisors = zero + times + |
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assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0" |
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class semiring_1_cancel = semiring + comm_monoid_add + zero_neq_one |
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+ cancel_ab_semigroup_add + monoid_mult |
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instance semiring_1_cancel \<subseteq> semiring_0_cancel .. |
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||
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instance semiring_1_cancel \<subseteq> semiring_1 .. |
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class comm_semiring_1_cancel = comm_semiring + comm_monoid_add + comm_monoid_mult |
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+ zero_neq_one + cancel_ab_semigroup_add |
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instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel .. |
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instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel .. |
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||
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instance comm_semiring_1_cancel \<subseteq> comm_semiring_1 .. |
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class ring = semiring + ab_group_add |
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instance ring \<subseteq> semiring_0_cancel .. |
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class comm_ring = comm_semiring + ab_group_add |
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instance comm_ring \<subseteq> ring .. |
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instance comm_ring \<subseteq> comm_semiring_0_cancel .. |
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class ring_1 = ring + zero_neq_one + monoid_mult |
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instance ring_1 \<subseteq> semiring_1_cancel .. |
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class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult |
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(*previously ring*) |
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instance comm_ring_1 \<subseteq> ring_1 .. |
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instance comm_ring_1 \<subseteq> comm_semiring_1_cancel .. |
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class ring_no_zero_divisors = ring + no_zero_divisors |
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class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors |
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class idom = comm_ring_1 + no_zero_divisors |
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instance idom \<subseteq> ring_1_no_zero_divisors .. |
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class division_ring = ring_1 + inverse + |
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assumes left_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1" |
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assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1" |
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instance division_ring \<subseteq> ring_1_no_zero_divisors |
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proof |
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fix a b :: 'a |
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assume a: "a \<noteq> 0" and b: "b \<noteq> 0" |
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show "a * b \<noteq> 0" |
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proof |
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assume ab: "a * b = 0" |
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hence "0 = inverse a * (a * b) * inverse b" |
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by simp |
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also have "\<dots> = (inverse a * a) * (b * inverse b)" |
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by (simp only: mult_assoc) |
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also have "\<dots> = 1" |
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using a b by simp |
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finally show False |
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by simp |
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qed |
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qed |
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class field = comm_ring_1 + inverse + |
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assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1" |
157 |
assumes divide_inverse: "a / b = a * inverse b" |
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instance field \<subseteq> division_ring |
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proof |
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fix a :: 'a |
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assume "a \<noteq> 0" |
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thus "inverse a * a = 1" by (rule field_inverse) |
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thus "a * inverse a = 1" by (simp only: mult_commute) |
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qed |
166 |
||
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instance field \<subseteq> idom .. |
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class division_by_zero = zero + inverse + |
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assumes inverse_zero [simp]: "inverse 0 = 0" |
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23389 | 172 |
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subsection {* Distribution rules *} |
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text{*For the @{text combine_numerals} simproc*} |
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lemma combine_common_factor: |
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"a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)" |
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by (simp add: left_distrib add_ac) |
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lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)" |
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apply (rule equals_zero_I) |
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apply (simp add: left_distrib [symmetric]) |
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done |
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|
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lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)" |
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apply (rule equals_zero_I) |
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apply (simp add: right_distrib [symmetric]) |
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done |
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lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)" |
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191 |
by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) |
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|
192 |
|
14365
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|
193 |
lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)" |
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|
194 |
by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) |
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paulson
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|
195 |
|
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|
196 |
lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)" |
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|
197 |
by (simp add: right_distrib diff_minus |
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|
198 |
minus_mult_left [symmetric] minus_mult_right [symmetric]) |
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|
199 |
|
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|
200 |
lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)" |
14738 | 201 |
by (simp add: left_distrib diff_minus |
202 |
minus_mult_left [symmetric] minus_mult_right [symmetric]) |
|
14265
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|
203 |
|
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|
204 |
lemmas ring_distribs = |
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|
205 |
right_distrib left_distrib left_diff_distrib right_diff_distrib |
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nipkow
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|
206 |
|
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|
207 |
text{*This list of rewrites simplifies ring terms by multiplying |
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|
208 |
everything out and bringing sums and products into a canonical form |
f4b83f03cac9
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nipkow
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|
209 |
(by ordered rewriting). As a result it decides ring equalities but |
f4b83f03cac9
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|
210 |
also helps with inequalities. *} |
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|
211 |
lemmas ring_simps = group_simps ring_distribs |
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|
212 |
|
22390 | 213 |
class mult_mono = times + zero + ord + |
25062 | 214 |
assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b" |
215 |
assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c" |
|
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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|
216 |
|
22390 | 217 |
class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add |
21199
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changeset
|
218 |
|
22390 | 219 |
class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add |
22987
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instance division_ring < no_zero_divisors; clean up field instance proofs
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parents:
22842
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|
220 |
+ semiring + comm_monoid_add + cancel_ab_semigroup_add |
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221 |
|
14940 | 222 |
instance pordered_cancel_semiring \<subseteq> semiring_0_cancel .. |
223 |
||
21199
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diff
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|
224 |
instance pordered_cancel_semiring \<subseteq> pordered_semiring .. |
2d83f93c3580
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|
225 |
|
23521 | 226 |
class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono |
227 |
||
228 |
instance ordered_semiring \<subseteq> pordered_cancel_semiring .. |
|
229 |
||
22390 | 230 |
class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + |
25062 | 231 |
assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
232 |
assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c" |
|
14341
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|
233 |
|
14940 | 234 |
instance ordered_semiring_strict \<subseteq> semiring_0_cancel .. |
235 |
||
23521 | 236 |
instance ordered_semiring_strict \<subseteq> ordered_semiring |
23550 | 237 |
proof |
238 |
fix a b c :: 'a |
|
239 |
assume A: "a \<le> b" "0 \<le> c" |
|
240 |
from A show "c * a \<le> c * b" |
|
241 |
unfolding order_le_less |
|
242 |
using mult_strict_left_mono by auto |
|
243 |
from A show "a * c \<le> b * c" |
|
244 |
unfolding order_le_less |
|
245 |
using mult_strict_right_mono by auto |
|
246 |
qed |
|
14270 | 247 |
|
22390 | 248 |
class mult_mono1 = times + zero + ord + |
25062 | 249 |
assumes mult_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b" |
14270 | 250 |
|
22390 | 251 |
class pordered_comm_semiring = comm_semiring_0 |
252 |
+ pordered_ab_semigroup_add + mult_mono1 |
|
14270 | 253 |
|
22390 | 254 |
class pordered_cancel_comm_semiring = comm_semiring_0_cancel |
255 |
+ pordered_ab_semigroup_add + mult_mono1 |
|
21199
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* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
256 |
|
14738 | 257 |
instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring .. |
14270 | 258 |
|
22390 | 259 |
class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add + |
25062 | 260 |
assumes mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
14265
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
261 |
|
14738 | 262 |
instance pordered_comm_semiring \<subseteq> pordered_semiring |
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
263 |
proof |
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20633
diff
changeset
|
264 |
fix a b c :: 'a |
23550 | 265 |
assume "a \<le> b" "0 \<le> c" |
266 |
thus "c * a \<le> c * b" by (rule mult_mono) |
|
267 |
thus "a * c \<le> b * c" by (simp only: mult_commute) |
|
21199
2d83f93c3580
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krauss
parents:
20633
diff
changeset
|
268 |
qed |
14265
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
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diff
changeset
|
269 |
|
14738 | 270 |
instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring .. |
14265
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paulson
parents:
diff
changeset
|
271 |
|
14738 | 272 |
instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict |
23550 | 273 |
proof |
274 |
fix a b c :: 'a |
|
275 |
assume "a < b" "0 < c" |
|
276 |
thus "c * a < c * b" by (rule mult_strict_mono) |
|
277 |
thus "a * c < b * c" by (simp only: mult_commute) |
|
278 |
qed |
|
14272
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changeset
|
279 |
|
14738 | 280 |
instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring |
23550 | 281 |
proof |
282 |
fix a b c :: 'a |
|
283 |
assume "a \<le> b" "0 \<le> c" |
|
284 |
thus "c * a \<le> c * b" |
|
285 |
unfolding order_le_less |
|
286 |
using mult_strict_mono by auto |
|
287 |
qed |
|
14272
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Tidying of the integer development; towards removing the
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diff
changeset
|
288 |
|
22390 | 289 |
class pordered_ring = ring + pordered_cancel_semiring |
14270 | 290 |
|
14738 | 291 |
instance pordered_ring \<subseteq> pordered_ab_group_add .. |
14270 | 292 |
|
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
293 |
class lordered_ring = pordered_ring + lordered_ab_group_abs |
14270 | 294 |
|
14940 | 295 |
instance lordered_ring \<subseteq> lordered_ab_group_meet .. |
296 |
||
297 |
instance lordered_ring \<subseteq> lordered_ab_group_join .. |
|
298 |
||
23879 | 299 |
class abs_if = minus + ord + zero + abs + |
25062 | 300 |
assumes abs_if: "abs a = (if a < 0 then (uminus a) else a)" |
14270 | 301 |
|
24506 | 302 |
class sgn_if = sgn + zero + one + minus + ord + |
25062 | 303 |
assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else uminus 1)" |
24506 | 304 |
|
23521 | 305 |
(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors. |
306 |
Basically, ordered_ring + no_zero_divisors = ordered_ring_strict. |
|
307 |
*) |
|
308 |
class ordered_ring = ring + ordered_semiring + lordered_ab_group + abs_if |
|
14270 | 309 |
|
23550 | 310 |
instance ordered_ring \<subseteq> lordered_ring |
311 |
proof |
|
312 |
fix x :: 'a |
|
313 |
show "\<bar>x\<bar> = sup x (- x)" |
|
314 |
by (simp only: abs_if sup_eq_if) |
|
315 |
qed |
|
23521 | 316 |
|
24506 | 317 |
class ordered_ring_strict = |
318 |
ring + ordered_semiring_strict + lordered_ab_group + abs_if |
|
23521 | 319 |
|
320 |
instance ordered_ring_strict \<subseteq> ordered_ring .. |
|
14270 | 321 |
|
22390 | 322 |
class pordered_comm_ring = comm_ring + pordered_comm_semiring |
14270 | 323 |
|
23527 | 324 |
instance pordered_comm_ring \<subseteq> pordered_ring .. |
325 |
||
23073 | 326 |
instance pordered_comm_ring \<subseteq> pordered_cancel_comm_semiring .. |
327 |
||
22390 | 328 |
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict + |
329 |
(*previously ordered_semiring*) |
|
25062 | 330 |
assumes zero_less_one [simp]: "0 < 1" |
14270 | 331 |
|
24422 | 332 |
lemma pos_add_strict: |
333 |
fixes a b c :: "'a\<Colon>ordered_semidom" |
|
334 |
shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c" |
|
335 |
using add_strict_mono [of 0 a b c] by simp |
|
336 |
||
24506 | 337 |
class ordered_idom = |
338 |
comm_ring_1 + |
|
339 |
ordered_comm_semiring_strict + |
|
340 |
lordered_ab_group + |
|
341 |
abs_if + sgn_if |
|
22390 | 342 |
(*previously ordered_ring*) |
14270 | 343 |
|
14738 | 344 |
instance ordered_idom \<subseteq> ordered_ring_strict .. |
14272
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Tidying of the integer development; towards removing the
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parents:
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diff
changeset
|
345 |
|
23073 | 346 |
instance ordered_idom \<subseteq> pordered_comm_ring .. |
347 |
||
22390 | 348 |
class ordered_field = field + ordered_idom |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
349 |
|
24515
d4dc5dc2db98
linorder_neqE_ordered_idom: proper proof, avoid illegal schematic type vars;
wenzelm
parents:
24506
diff
changeset
|
350 |
lemma linorder_neqE_ordered_idom: |
d4dc5dc2db98
linorder_neqE_ordered_idom: proper proof, avoid illegal schematic type vars;
wenzelm
parents:
24506
diff
changeset
|
351 |
fixes x y :: "'a :: ordered_idom" |
d4dc5dc2db98
linorder_neqE_ordered_idom: proper proof, avoid illegal schematic type vars;
wenzelm
parents:
24506
diff
changeset
|
352 |
assumes "x \<noteq> y" obtains "x < y" | "y < x" |
d4dc5dc2db98
linorder_neqE_ordered_idom: proper proof, avoid illegal schematic type vars;
wenzelm
parents:
24506
diff
changeset
|
353 |
using assms by (rule linorder_neqE) |
15923 | 354 |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
355 |
lemma eq_add_iff1: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
356 |
"(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
357 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
358 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
359 |
lemma eq_add_iff2: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
360 |
"(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
361 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
362 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
363 |
lemma less_add_iff1: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
364 |
"(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
365 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
366 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
367 |
lemma less_add_iff2: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
368 |
"(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
369 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
370 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
371 |
lemma le_add_iff1: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
372 |
"(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
373 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
374 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
375 |
lemma le_add_iff2: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
376 |
"(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
377 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
378 |
|
23389 | 379 |
|
14270 | 380 |
subsection {* Ordering Rules for Multiplication *} |
381 |
||
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
382 |
lemma mult_left_le_imp_le: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
383 |
"[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
384 |
by (force simp add: mult_strict_left_mono linorder_not_less [symmetric]) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
385 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
386 |
lemma mult_right_le_imp_le: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
387 |
"[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
388 |
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric]) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
389 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
390 |
lemma mult_left_less_imp_less: |
23521 | 391 |
"[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
392 |
by (force simp add: mult_left_mono linorder_not_le [symmetric]) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
393 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
394 |
lemma mult_right_less_imp_less: |
23521 | 395 |
"[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
396 |
by (force simp add: mult_right_mono linorder_not_le [symmetric]) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
397 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
398 |
lemma mult_strict_left_mono_neg: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
399 |
"[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
400 |
apply (drule mult_strict_left_mono [of _ _ "-c"]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
401 |
apply (simp_all add: minus_mult_left [symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
402 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
403 |
|
14738 | 404 |
lemma mult_left_mono_neg: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
405 |
"[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::pordered_ring)" |
14738 | 406 |
apply (drule mult_left_mono [of _ _ "-c"]) |
407 |
apply (simp_all add: minus_mult_left [symmetric]) |
|
408 |
done |
|
409 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
410 |
lemma mult_strict_right_mono_neg: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
411 |
"[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
412 |
apply (drule mult_strict_right_mono [of _ _ "-c"]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
413 |
apply (simp_all add: minus_mult_right [symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
414 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
415 |
|
14738 | 416 |
lemma mult_right_mono_neg: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
417 |
"[|b \<le> a; c \<le> 0|] ==> a * c \<le> (b::'a::pordered_ring) * c" |
14738 | 418 |
apply (drule mult_right_mono [of _ _ "-c"]) |
419 |
apply (simp) |
|
420 |
apply (simp_all add: minus_mult_right [symmetric]) |
|
421 |
done |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
422 |
|
23389 | 423 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
424 |
subsection{* Products of Signs *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
425 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
426 |
lemma mult_pos_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
427 |
by (drule mult_strict_left_mono [of 0 b], auto) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
428 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
429 |
lemma mult_nonneg_nonneg: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b" |
14738 | 430 |
by (drule mult_left_mono [of 0 b], auto) |
431 |
||
432 |
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0" |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
433 |
by (drule mult_strict_left_mono [of b 0], auto) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
434 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
435 |
lemma mult_nonneg_nonpos: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0" |
14738 | 436 |
by (drule mult_left_mono [of b 0], auto) |
437 |
||
438 |
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" |
|
439 |
by (drule mult_strict_right_mono[of b 0], auto) |
|
440 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
441 |
lemma mult_nonneg_nonpos2: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" |
14738 | 442 |
by (drule mult_right_mono[of b 0], auto) |
443 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
444 |
lemma mult_neg_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
445 |
by (drule mult_strict_right_mono_neg, auto) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
446 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
447 |
lemma mult_nonpos_nonpos: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b" |
14738 | 448 |
by (drule mult_right_mono_neg[of a 0 b ], auto) |
449 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
450 |
lemma zero_less_mult_pos: |
14738 | 451 |
"[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)" |
21328 | 452 |
apply (cases "b\<le>0") |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
453 |
apply (auto simp add: order_le_less linorder_not_less) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
454 |
apply (drule_tac mult_pos_neg [of a b]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
455 |
apply (auto dest: order_less_not_sym) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
456 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
457 |
|
14738 | 458 |
lemma zero_less_mult_pos2: |
459 |
"[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)" |
|
21328 | 460 |
apply (cases "b\<le>0") |
14738 | 461 |
apply (auto simp add: order_le_less linorder_not_less) |
462 |
apply (drule_tac mult_pos_neg2 [of a b]) |
|
463 |
apply (auto dest: order_less_not_sym) |
|
464 |
done |
|
465 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
466 |
lemma zero_less_mult_iff: |
14738 | 467 |
"((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
468 |
apply (auto simp add: order_le_less linorder_not_less mult_pos_pos |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
469 |
mult_neg_neg) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
470 |
apply (blast dest: zero_less_mult_pos) |
14738 | 471 |
apply (blast dest: zero_less_mult_pos2) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
472 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
473 |
|
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
474 |
lemma mult_eq_0_iff [simp]: |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
475 |
fixes a b :: "'a::ring_no_zero_divisors" |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
476 |
shows "(a * b = 0) = (a = 0 \<or> b = 0)" |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
477 |
by (cases "a = 0 \<or> b = 0", auto dest: no_zero_divisors) |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
478 |
|
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
479 |
instance ordered_ring_strict \<subseteq> ring_no_zero_divisors |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
480 |
apply intro_classes |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
481 |
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
482 |
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+ |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
483 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
484 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
485 |
lemma zero_le_mult_iff: |
14738 | 486 |
"((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
487 |
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
488 |
zero_less_mult_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
489 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
490 |
lemma mult_less_0_iff: |
14738 | 491 |
"(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
492 |
apply (insert zero_less_mult_iff [of "-a" b]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
493 |
apply (force simp add: minus_mult_left[symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
494 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
495 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
496 |
lemma mult_le_0_iff: |
14738 | 497 |
"(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
498 |
apply (insert zero_le_mult_iff [of "-a" b]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
499 |
apply (force simp add: minus_mult_left[symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
500 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
501 |
|
14738 | 502 |
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
503 |
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos) |
14738 | 504 |
|
505 |
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> (0::_::pordered_cancel_semiring)" |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
506 |
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2) |
14738 | 507 |
|
23095 | 508 |
lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a" |
509 |
by (simp add: zero_le_mult_iff linorder_linear) |
|
510 |
||
511 |
lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))" |
|
512 |
by (simp add: not_less) |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
513 |
|
14738 | 514 |
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom} |
515 |
theorems available to members of @{term ordered_idom} *} |
|
516 |
||
517 |
instance ordered_idom \<subseteq> ordered_semidom |
|
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
518 |
proof |
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
519 |
have "(0::'a) \<le> 1*1" by (rule zero_le_square) |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
520 |
thus "(0::'a) < 1" by (simp add: order_le_less) |
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
521 |
qed |
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
522 |
|
14738 | 523 |
instance ordered_idom \<subseteq> idom .. |
524 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
525 |
text{*All three types of comparision involving 0 and 1 are covered.*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
526 |
|
17085 | 527 |
lemmas one_neq_zero = zero_neq_one [THEN not_sym] |
528 |
declare one_neq_zero [simp] |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
529 |
|
14738 | 530 |
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
531 |
by (rule zero_less_one [THEN order_less_imp_le]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
532 |
|
14738 | 533 |
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0" |
534 |
by (simp add: linorder_not_le) |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
535 |
|
14738 | 536 |
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0" |
537 |
by (simp add: linorder_not_less) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
538 |
|
23389 | 539 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
540 |
subsection{*More Monotonicity*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
541 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
542 |
text{*Strict monotonicity in both arguments*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
543 |
lemma mult_strict_mono: |
14738 | 544 |
"[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)" |
21328 | 545 |
apply (cases "c=0") |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
546 |
apply (simp add: mult_pos_pos) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
547 |
apply (erule mult_strict_right_mono [THEN order_less_trans]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
548 |
apply (force simp add: order_le_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
549 |
apply (erule mult_strict_left_mono, assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
550 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
551 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
552 |
text{*This weaker variant has more natural premises*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
553 |
lemma mult_strict_mono': |
14738 | 554 |
"[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
555 |
apply (rule mult_strict_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
556 |
apply (blast intro: order_le_less_trans)+ |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
557 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
558 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
559 |
lemma mult_mono: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
560 |
"[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] |
14738 | 561 |
==> a * c \<le> b * (d::'a::pordered_semiring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
562 |
apply (erule mult_right_mono [THEN order_trans], assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
563 |
apply (erule mult_left_mono, assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
564 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
565 |
|
21258 | 566 |
lemma mult_mono': |
567 |
"[|a \<le> b; c \<le> d; 0 \<le> a; 0 \<le> c|] |
|
568 |
==> a * c \<le> b * (d::'a::pordered_semiring)" |
|
569 |
apply (rule mult_mono) |
|
570 |
apply (fast intro: order_trans)+ |
|
571 |
done |
|
572 |
||
14738 | 573 |
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
574 |
apply (insert mult_strict_mono [of 1 m 1 n]) |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
575 |
apply (simp add: order_less_trans [OF zero_less_one]) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
576 |
done |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
577 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
578 |
lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
579 |
c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
580 |
apply (subgoal_tac "a * c < b * c") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
581 |
apply (erule order_less_le_trans) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
582 |
apply (erule mult_left_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
583 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
584 |
apply (erule mult_strict_right_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
585 |
apply assumption |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
586 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
587 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
588 |
lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
589 |
c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
590 |
apply (subgoal_tac "a * c <= b * c") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
591 |
apply (erule order_le_less_trans) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
592 |
apply (erule mult_strict_left_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
593 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
594 |
apply (erule mult_right_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
595 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
596 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
597 |
|
23389 | 598 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
599 |
subsection{*Cancellation Laws for Relationships With a Common Factor*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
600 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
601 |
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"}, |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
602 |
also with the relations @{text "\<le>"} and equality.*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
603 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
604 |
text{*These ``disjunction'' versions produce two cases when the comparison is |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
605 |
an assumption, but effectively four when the comparison is a goal.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
606 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
607 |
lemma mult_less_cancel_right_disj: |
14738 | 608 |
"(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))" |
21328 | 609 |
apply (cases "c = 0") |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
610 |
apply (auto simp add: linorder_neq_iff mult_strict_right_mono |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
611 |
mult_strict_right_mono_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
612 |
apply (auto simp add: linorder_not_less |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
613 |
linorder_not_le [symmetric, of "a*c"] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
614 |
linorder_not_le [symmetric, of a]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
615 |
apply (erule_tac [!] notE) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
616 |
apply (auto simp add: order_less_imp_le mult_right_mono |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
617 |
mult_right_mono_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
618 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
619 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
620 |
lemma mult_less_cancel_left_disj: |
14738 | 621 |
"(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))" |
21328 | 622 |
apply (cases "c = 0") |
14738 | 623 |
apply (auto simp add: linorder_neq_iff mult_strict_left_mono |
624 |
mult_strict_left_mono_neg) |
|
625 |
apply (auto simp add: linorder_not_less |
|
626 |
linorder_not_le [symmetric, of "c*a"] |
|
627 |
linorder_not_le [symmetric, of a]) |
|
628 |
apply (erule_tac [!] notE) |
|
629 |
apply (auto simp add: order_less_imp_le mult_left_mono |
|
630 |
mult_left_mono_neg) |
|
631 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
632 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
633 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
634 |
text{*The ``conjunction of implication'' lemmas produce two cases when the |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
635 |
comparison is a goal, but give four when the comparison is an assumption.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
636 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
637 |
lemma mult_less_cancel_right: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
638 |
fixes c :: "'a :: ordered_ring_strict" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
639 |
shows "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
640 |
by (insert mult_less_cancel_right_disj [of a c b], auto) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
641 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
642 |
lemma mult_less_cancel_left: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
643 |
fixes c :: "'a :: ordered_ring_strict" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
644 |
shows "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
645 |
by (insert mult_less_cancel_left_disj [of c a b], auto) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
646 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
647 |
lemma mult_le_cancel_right: |
14738 | 648 |
"(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
649 |
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
650 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
651 |
lemma mult_le_cancel_left: |
14738 | 652 |
"(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
653 |
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
654 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
655 |
lemma mult_less_imp_less_left: |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
656 |
assumes less: "c*a < c*b" and nonneg: "0 \<le> c" |
14738 | 657 |
shows "a < (b::'a::ordered_semiring_strict)" |
14377 | 658 |
proof (rule ccontr) |
659 |
assume "~ a < b" |
|
660 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
23389 | 661 |
hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono) |
14377 | 662 |
with this and less show False |
663 |
by (simp add: linorder_not_less [symmetric]) |
|
664 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
665 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
666 |
lemma mult_less_imp_less_right: |
14738 | 667 |
assumes less: "a*c < b*c" and nonneg: "0 <= c" |
668 |
shows "a < (b::'a::ordered_semiring_strict)" |
|
669 |
proof (rule ccontr) |
|
670 |
assume "~ a < b" |
|
671 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
23389 | 672 |
hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono) |
14738 | 673 |
with this and less show False |
674 |
by (simp add: linorder_not_less [symmetric]) |
|
675 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
676 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
677 |
text{*Cancellation of equalities with a common factor*} |
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
678 |
lemma mult_cancel_right [simp,noatp]: |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
679 |
fixes a b c :: "'a::ring_no_zero_divisors" |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
680 |
shows "(a * c = b * c) = (c = 0 \<or> a = b)" |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
681 |
proof - |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
682 |
have "(a * c = b * c) = ((a - b) * c = 0)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
683 |
by (simp add: ring_distribs) |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
684 |
thus ?thesis |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
685 |
by (simp add: disj_commute) |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
686 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
687 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
688 |
lemma mult_cancel_left [simp,noatp]: |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
689 |
fixes a b c :: "'a::ring_no_zero_divisors" |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
690 |
shows "(c * a = c * b) = (c = 0 \<or> a = b)" |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
691 |
proof - |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
692 |
have "(c * a = c * b) = (c * (a - b) = 0)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
693 |
by (simp add: ring_distribs) |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
694 |
thus ?thesis |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
695 |
by simp |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
696 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
697 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
698 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
699 |
subsubsection{*Special Cancellation Simprules for Multiplication*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
700 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
701 |
text{*These also produce two cases when the comparison is a goal.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
702 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
703 |
lemma mult_le_cancel_right1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
704 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
705 |
shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
706 |
by (insert mult_le_cancel_right [of 1 c b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
707 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
708 |
lemma mult_le_cancel_right2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
709 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
710 |
shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
711 |
by (insert mult_le_cancel_right [of a c 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
712 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
713 |
lemma mult_le_cancel_left1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
714 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
715 |
shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
716 |
by (insert mult_le_cancel_left [of c 1 b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
717 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
718 |
lemma mult_le_cancel_left2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
719 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
720 |
shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
721 |
by (insert mult_le_cancel_left [of c a 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
722 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
723 |
lemma mult_less_cancel_right1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
724 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
725 |
shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
726 |
by (insert mult_less_cancel_right [of 1 c b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
727 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
728 |
lemma mult_less_cancel_right2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
729 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
730 |
shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
731 |
by (insert mult_less_cancel_right [of a c 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
732 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
733 |
lemma mult_less_cancel_left1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
734 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
735 |
shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
736 |
by (insert mult_less_cancel_left [of c 1 b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
737 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
738 |
lemma mult_less_cancel_left2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
739 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
740 |
shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
741 |
by (insert mult_less_cancel_left [of c a 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
742 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
743 |
lemma mult_cancel_right1 [simp]: |
23544 | 744 |
fixes c :: "'a :: ring_1_no_zero_divisors" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
745 |
shows "(c = b*c) = (c = 0 | b=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
746 |
by (insert mult_cancel_right [of 1 c b], force) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
747 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
748 |
lemma mult_cancel_right2 [simp]: |
23544 | 749 |
fixes c :: "'a :: ring_1_no_zero_divisors" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
750 |
shows "(a*c = c) = (c = 0 | a=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
751 |
by (insert mult_cancel_right [of a c 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
752 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
753 |
lemma mult_cancel_left1 [simp]: |
23544 | 754 |
fixes c :: "'a :: ring_1_no_zero_divisors" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
755 |
shows "(c = c*b) = (c = 0 | b=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
756 |
by (insert mult_cancel_left [of c 1 b], force) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
757 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
758 |
lemma mult_cancel_left2 [simp]: |
23544 | 759 |
fixes c :: "'a :: ring_1_no_zero_divisors" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
760 |
shows "(c*a = c) = (c = 0 | a=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
761 |
by (insert mult_cancel_left [of c a 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
762 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
763 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
764 |
text{*Simprules for comparisons where common factors can be cancelled.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
765 |
lemmas mult_compare_simps = |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
766 |
mult_le_cancel_right mult_le_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
767 |
mult_le_cancel_right1 mult_le_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
768 |
mult_le_cancel_left1 mult_le_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
769 |
mult_less_cancel_right mult_less_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
770 |
mult_less_cancel_right1 mult_less_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
771 |
mult_less_cancel_left1 mult_less_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
772 |
mult_cancel_right mult_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
773 |
mult_cancel_right1 mult_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
774 |
mult_cancel_left1 mult_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
775 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
776 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
777 |
subsection {* Fields *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
778 |
|
14288 | 779 |
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" |
780 |
proof |
|
781 |
assume neq: "b \<noteq> 0" |
|
782 |
{ |
|
783 |
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) |
|
784 |
also assume "a / b = 1" |
|
785 |
finally show "a = b" by simp |
|
786 |
next |
|
787 |
assume "a = b" |
|
788 |
with neq show "a / b = 1" by (simp add: divide_inverse) |
|
789 |
} |
|
790 |
qed |
|
791 |
||
792 |
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" |
|
793 |
by (simp add: divide_inverse) |
|
794 |
||
23398 | 795 |
lemma divide_self[simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1" |
14288 | 796 |
by (simp add: divide_inverse) |
797 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
798 |
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
799 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
800 |
|
15228 | 801 |
lemma divide_self_if [simp]: |
802 |
"a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)" |
|
803 |
by (simp add: divide_self) |
|
804 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
805 |
lemma divide_zero_left [simp]: "0/a = (0::'a::field)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
806 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
807 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
808 |
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
809 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
810 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
811 |
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
812 |
by (simp add: divide_inverse ring_distribs) |
14293 | 813 |
|
23482 | 814 |
(* what ordering?? this is a straight instance of mult_eq_0_iff |
14270 | 815 |
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement |
816 |
of an ordering.*} |
|
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
817 |
lemma field_mult_eq_0_iff [simp]: |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
818 |
"(a*b = (0::'a::division_ring)) = (a = 0 | b = 0)" |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
819 |
by simp |
23482 | 820 |
*) |
23496 | 821 |
(* subsumed by mult_cancel lemmas on ring_no_zero_divisors |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
822 |
text{*Cancellation of equalities with a common factor*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
823 |
lemma field_mult_cancel_right_lemma: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
824 |
assumes cnz: "c \<noteq> (0::'a::division_ring)" |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
825 |
and eq: "a*c = b*c" |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
826 |
shows "a=b" |
14377 | 827 |
proof - |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
828 |
have "(a * c) * inverse c = (b * c) * inverse c" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
829 |
by (simp add: eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
830 |
thus "a=b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
831 |
by (simp add: mult_assoc cnz) |
14377 | 832 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
833 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
834 |
lemma field_mult_cancel_right [simp]: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
835 |
"(a*c = b*c) = (c = (0::'a::division_ring) | a=b)" |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
836 |
by simp |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
837 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
838 |
lemma field_mult_cancel_left [simp]: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
839 |
"(c*a = c*b) = (c = (0::'a::division_ring) | a=b)" |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
840 |
by simp |
23496 | 841 |
*) |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
842 |
lemma nonzero_imp_inverse_nonzero: |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
843 |
"a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)" |
14377 | 844 |
proof |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
845 |
assume ianz: "inverse a = 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
846 |
assume "a \<noteq> 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
847 |
hence "1 = a * inverse a" by simp |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
848 |
also have "... = 0" by (simp add: ianz) |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
849 |
finally have "1 = (0::'a::division_ring)" . |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
850 |
thus False by (simp add: eq_commute) |
14377 | 851 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
852 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
853 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
854 |
subsection{*Basic Properties of @{term inverse}*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
855 |
|
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
856 |
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::division_ring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
857 |
apply (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
858 |
apply (blast dest: nonzero_imp_inverse_nonzero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
859 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
860 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
861 |
lemma inverse_nonzero_imp_nonzero: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
862 |
"inverse a = 0 ==> a = (0::'a::division_ring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
863 |
apply (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
864 |
apply (blast dest: nonzero_imp_inverse_nonzero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
865 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
866 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
867 |
lemma inverse_nonzero_iff_nonzero [simp]: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
868 |
"(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
869 |
by (force dest: inverse_nonzero_imp_nonzero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
870 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
871 |
lemma nonzero_inverse_minus_eq: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
872 |
assumes [simp]: "a\<noteq>0" |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
873 |
shows "inverse(-a) = -inverse(a::'a::division_ring)" |
14377 | 874 |
proof - |
875 |
have "-a * inverse (- a) = -a * - inverse a" |
|
876 |
by simp |
|
877 |
thus ?thesis |
|
23496 | 878 |
by (simp only: mult_cancel_left, simp) |
14377 | 879 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
880 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
881 |
lemma inverse_minus_eq [simp]: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
882 |
"inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})" |
14377 | 883 |
proof cases |
884 |
assume "a=0" thus ?thesis by (simp add: inverse_zero) |
|
885 |
next |
|
886 |
assume "a\<noteq>0" |
|
887 |
thus ?thesis by (simp add: nonzero_inverse_minus_eq) |
|
888 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
889 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
890 |
lemma nonzero_inverse_eq_imp_eq: |
14269 | 891 |
assumes inveq: "inverse a = inverse b" |
892 |
and anz: "a \<noteq> 0" |
|
893 |
and bnz: "b \<noteq> 0" |
|
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
894 |
shows "a = (b::'a::division_ring)" |
14377 | 895 |
proof - |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
896 |
have "a * inverse b = a * inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
897 |
by (simp add: inveq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
898 |
hence "(a * inverse b) * b = (a * inverse a) * b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
899 |
by simp |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
900 |
thus "a = b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
901 |
by (simp add: mult_assoc anz bnz) |
14377 | 902 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
903 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
904 |
lemma inverse_eq_imp_eq: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
905 |
"inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})" |
21328 | 906 |
apply (cases "a=0 | b=0") |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
907 |
apply (force dest!: inverse_zero_imp_zero |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
908 |
simp add: eq_commute [of "0::'a"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
909 |
apply (force dest!: nonzero_inverse_eq_imp_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
910 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
911 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
912 |
lemma inverse_eq_iff_eq [simp]: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
913 |
"(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))" |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
914 |
by (force dest!: inverse_eq_imp_eq) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
915 |
|
14270 | 916 |
lemma nonzero_inverse_inverse_eq: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
917 |
assumes [simp]: "a \<noteq> 0" |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
918 |
shows "inverse(inverse (a::'a::division_ring)) = a" |
14270 | 919 |
proof - |
920 |
have "(inverse (inverse a) * inverse a) * a = a" |
|
921 |
by (simp add: nonzero_imp_inverse_nonzero) |
|
922 |
thus ?thesis |
|
923 |
by (simp add: mult_assoc) |
|
924 |
qed |
|
925 |
||
926 |
lemma inverse_inverse_eq [simp]: |
|
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
927 |
"inverse(inverse (a::'a::{division_ring,division_by_zero})) = a" |
14270 | 928 |
proof cases |
929 |
assume "a=0" thus ?thesis by simp |
|
930 |
next |
|
931 |
assume "a\<noteq>0" |
|
932 |
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) |
|
933 |
qed |
|
934 |
||
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
935 |
lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)" |
14270 | 936 |
proof - |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
937 |
have "inverse 1 * 1 = (1::'a::division_ring)" |
14270 | 938 |
by (rule left_inverse [OF zero_neq_one [symmetric]]) |
939 |
thus ?thesis by simp |
|
940 |
qed |
|
941 |
||
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
942 |
lemma inverse_unique: |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
943 |
assumes ab: "a*b = 1" |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
944 |
shows "inverse a = (b::'a::division_ring)" |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
945 |
proof - |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
946 |
have "a \<noteq> 0" using ab by auto |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
947 |
moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
948 |
ultimately show ?thesis by (simp add: mult_assoc [symmetric]) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
949 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
950 |
|
14270 | 951 |
lemma nonzero_inverse_mult_distrib: |
952 |
assumes anz: "a \<noteq> 0" |
|
953 |
and bnz: "b \<noteq> 0" |
|
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
954 |
shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)" |
14270 | 955 |
proof - |
956 |
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" |
|
23482 | 957 |
by (simp add: anz bnz) |
14270 | 958 |
hence "inverse(a*b) * a = inverse(b)" |
959 |
by (simp add: mult_assoc bnz) |
|
960 |
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" |
|
961 |
by simp |
|
962 |
thus ?thesis |
|
963 |
by (simp add: mult_assoc anz) |
|
964 |
qed |
|
965 |
||
966 |
text{*This version builds in division by zero while also re-orienting |
|
967 |
the right-hand side.*} |
|
968 |
lemma inverse_mult_distrib [simp]: |
|
969 |
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" |
|
970 |
proof cases |
|
971 |
assume "a \<noteq> 0 & b \<noteq> 0" |
|
22993 | 972 |
thus ?thesis |
973 |
by (simp add: nonzero_inverse_mult_distrib mult_commute) |
|
14270 | 974 |
next |
975 |
assume "~ (a \<noteq> 0 & b \<noteq> 0)" |
|
22993 | 976 |
thus ?thesis |
977 |
by force |
|
14270 | 978 |
qed |
979 |
||
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
980 |
lemma division_ring_inverse_add: |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
981 |
"[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|] |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
982 |
==> inverse a + inverse b = inverse a * (a+b) * inverse b" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
983 |
by (simp add: ring_simps) |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
984 |
|
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
985 |
lemma division_ring_inverse_diff: |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
986 |
"[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|] |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
987 |
==> inverse a - inverse b = inverse a * (b-a) * inverse b" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
988 |
by (simp add: ring_simps) |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
989 |
|
14270 | 990 |
text{*There is no slick version using division by zero.*} |
991 |
lemma inverse_add: |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
992 |
"[|a \<noteq> 0; b \<noteq> 0|] |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
993 |
==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)" |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
994 |
by (simp add: division_ring_inverse_add mult_ac) |
14270 | 995 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
996 |
lemma inverse_divide [simp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
997 |
"inverse (a/b) = b / (a::'a::{field,division_by_zero})" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
998 |
by (simp add: divide_inverse mult_commute) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
999 |
|
23389 | 1000 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1001 |
subsection {* Calculations with fractions *} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1002 |
|
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1003 |
text{* There is a whole bunch of simp-rules just for class @{text |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1004 |
field} but none for class @{text field} and @{text nonzero_divides} |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1005 |
because the latter are covered by a simproc. *} |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1006 |
|
24427 | 1007 |
lemma nonzero_mult_divide_mult_cancel_left[simp,noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1008 |
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)" |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1009 |
proof - |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1010 |
have "(c*a)/(c*b) = c * a * (inverse b * inverse c)" |
23482 | 1011 |
by (simp add: divide_inverse nonzero_inverse_mult_distrib) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1012 |
also have "... = a * inverse b * (inverse c * c)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1013 |
by (simp only: mult_ac) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1014 |
also have "... = a * inverse b" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1015 |
by simp |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1016 |
finally show ?thesis |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1017 |
by (simp add: divide_inverse) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1018 |
qed |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1019 |
|
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1020 |
lemma mult_divide_mult_cancel_left: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1021 |
"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})" |
21328 | 1022 |
apply (cases "b = 0") |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1023 |
apply (simp_all add: nonzero_mult_divide_mult_cancel_left) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1024 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1025 |
|
24427 | 1026 |
lemma nonzero_mult_divide_mult_cancel_right [noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1027 |
"[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)" |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1028 |
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) |
14321 | 1029 |
|
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1030 |
lemma mult_divide_mult_cancel_right: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1031 |
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" |
21328 | 1032 |
apply (cases "b = 0") |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1033 |
apply (simp_all add: nonzero_mult_divide_mult_cancel_right) |
14321 | 1034 |
done |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1035 |
|
14284
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
1036 |
lemma divide_1 [simp]: "a/1 = (a::'a::field)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1037 |
by (simp add: divide_inverse) |
14284
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
1038 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1039 |
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1040 |
by (simp add: divide_inverse mult_assoc) |
14288 | 1041 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1042 |
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1043 |
by (simp add: divide_inverse mult_ac) |
14288 | 1044 |
|
23482 | 1045 |
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left |
1046 |
||
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1047 |
lemma divide_divide_eq_right [simp,noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1048 |
"a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1049 |
by (simp add: divide_inverse mult_ac) |
14288 | 1050 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1051 |
lemma divide_divide_eq_left [simp,noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1052 |
"(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1053 |
by (simp add: divide_inverse mult_assoc) |
14288 | 1054 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1055 |
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1056 |
x / y + w / z = (x * z + w * y) / (y * z)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1057 |
apply (subgoal_tac "x / y = (x * z) / (y * z)") |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1058 |
apply (erule ssubst) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1059 |
apply (subgoal_tac "w / z = (w * y) / (y * z)") |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1060 |
apply (erule ssubst) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1061 |
apply (rule add_divide_distrib [THEN sym]) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1062 |
apply (subst mult_commute) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1063 |
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym]) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1064 |
apply assumption |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1065 |
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym]) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1066 |
apply assumption |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1067 |
done |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1068 |
|
23389 | 1069 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1070 |
subsubsection{*Special Cancellation Simprules for Division*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1071 |
|
24427 | 1072 |
lemma mult_divide_mult_cancel_left_if[simp,noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1073 |
fixes c :: "'a :: {field,division_by_zero}" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1074 |
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)" |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1075 |
by (simp add: mult_divide_mult_cancel_left) |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1076 |
|
24427 | 1077 |
lemma nonzero_mult_divide_cancel_right[simp,noatp]: |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1078 |
"b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)" |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1079 |
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1080 |
|
24427 | 1081 |
lemma nonzero_mult_divide_cancel_left[simp,noatp]: |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1082 |
"a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)" |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1083 |
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1084 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1085 |
|
24427 | 1086 |
lemma nonzero_divide_mult_cancel_right[simp,noatp]: |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1087 |
"\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)" |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1088 |
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1089 |
|
24427 | 1090 |
lemma nonzero_divide_mult_cancel_left[simp,noatp]: |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1091 |
"\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)" |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1092 |
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1093 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1094 |
|
24427 | 1095 |
lemma nonzero_mult_divide_mult_cancel_left2[simp,noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1096 |
"[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)" |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1097 |
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac) |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1098 |
|
24427 | 1099 |
lemma nonzero_mult_divide_mult_cancel_right2[simp,noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1100 |
"[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)" |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1101 |
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac) |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1102 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1103 |
|
14293 | 1104 |
subsection {* Division and Unary Minus *} |
1105 |
||
1106 |
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)" |
|
1107 |
by (simp add: divide_inverse minus_mult_left) |
|
1108 |
||
1109 |
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)" |
|
1110 |
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) |
|
1111 |
||
1112 |
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)" |
|
1113 |
by (simp add: divide_inverse nonzero_inverse_minus_eq) |
|
1114 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1115 |
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1116 |
by (simp add: divide_inverse minus_mult_left [symmetric]) |
14293 | 1117 |
|
1118 |
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})" |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1119 |
by (simp add: divide_inverse minus_mult_right [symmetric]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1120 |
|
14293 | 1121 |
|
1122 |
text{*The effect is to extract signs from divisions*} |
|
17085 | 1123 |
lemmas divide_minus_left = minus_divide_left [symmetric] |
1124 |
lemmas divide_minus_right = minus_divide_right [symmetric] |
|
1125 |
declare divide_minus_left [simp] divide_minus_right [simp] |
|
14293 | 1126 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1127 |
text{*Also, extract signs from products*} |
17085 | 1128 |
lemmas mult_minus_left = minus_mult_left [symmetric] |
1129 |
lemmas mult_minus_right = minus_mult_right [symmetric] |
|
1130 |
declare mult_minus_left [simp] mult_minus_right [simp] |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1131 |
|
14293 | 1132 |
lemma minus_divide_divide [simp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1133 |
"(-a)/(-b) = a / (b::'a::{field,division_by_zero})" |
21328 | 1134 |
apply (cases "b=0", simp) |
14293 | 1135 |
apply (simp add: nonzero_minus_divide_divide) |
1136 |
done |
|
1137 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1138 |
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1139 |
by (simp add: diff_minus add_divide_distrib) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1140 |
|
23482 | 1141 |
lemma add_divide_eq_iff: |
1142 |
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z" |
|
1143 |
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left) |
|
1144 |
||
1145 |
lemma divide_add_eq_iff: |
|
1146 |
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z" |
|
1147 |
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left) |
|
1148 |
||
1149 |
lemma diff_divide_eq_iff: |
|
1150 |
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z" |
|
1151 |
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left) |
|
1152 |
||
1153 |
lemma divide_diff_eq_iff: |
|
1154 |
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z" |
|
1155 |
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left) |
|
1156 |
||
1157 |
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)" |
|
1158 |
proof - |
|
1159 |
assume [simp]: "c\<noteq>0" |
|
23496 | 1160 |
have "(a = b/c) = (a*c = (b/c)*c)" by simp |
1161 |
also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc) |
|
23482 | 1162 |
finally show ?thesis . |
1163 |
qed |
|
1164 |
||
1165 |
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)" |
|
1166 |
proof - |
|
1167 |
assume [simp]: "c\<noteq>0" |
|
23496 | 1168 |
have "(b/c = a) = ((b/c)*c = a*c)" by simp |
1169 |
also have "... = (b = a*c)" by (simp add: divide_inverse mult_assoc) |
|
23482 | 1170 |
finally show ?thesis . |
1171 |
qed |
|
1172 |
||
1173 |
lemma eq_divide_eq: |
|
1174 |
"((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)" |
|
1175 |
by (simp add: nonzero_eq_divide_eq) |
|
1176 |
||
1177 |
lemma divide_eq_eq: |
|
1178 |
"(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)" |
|
1179 |
by (force simp add: nonzero_divide_eq_eq) |
|
1180 |
||
1181 |
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==> |
|
1182 |
b = a * c ==> b / c = a" |
|
1183 |
by (subst divide_eq_eq, simp) |
|
1184 |
||
1185 |
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==> |
|
1186 |
a * c = b ==> a = b / c" |
|
1187 |
by (subst eq_divide_eq, simp) |
|
1188 |
||
1189 |
||
1190 |
lemmas field_eq_simps = ring_simps |
|
1191 |
(* pull / out*) |
|
1192 |
add_divide_eq_iff divide_add_eq_iff |
|
1193 |
diff_divide_eq_iff divide_diff_eq_iff |
|
1194 |
(* multiply eqn *) |
|
1195 |
nonzero_eq_divide_eq nonzero_divide_eq_eq |
|
1196 |
(* is added later: |
|
1197 |
times_divide_eq_left times_divide_eq_right |
|
1198 |
*) |
|
1199 |
||
1200 |
text{*An example:*} |
|
1201 |
lemma fixes a b c d e f :: "'a::field" |
|
1202 |
shows "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f \<rbrakk> \<Longrightarrow> ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) = 1" |
|
1203 |
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0") |
|
1204 |
apply(simp add:field_eq_simps) |
|
1205 |
apply(simp) |
|
1206 |
done |
|
1207 |
||
1208 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1209 |
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1210 |
x / y - w / z = (x * z - w * y) / (y * z)" |
23482 | 1211 |
by (simp add:field_eq_simps times_divide_eq) |
1212 |
||
1213 |
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==> |
|
1214 |
(x / y = w / z) = (x * z = w * y)" |
|
1215 |
by (simp add:field_eq_simps times_divide_eq) |
|
14293 | 1216 |
|
23389 | 1217 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1218 |
subsection {* Ordered Fields *} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1219 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1220 |
lemma positive_imp_inverse_positive: |
23482 | 1221 |
assumes a_gt_0: "0 < a" shows "0 < inverse (a::'a::ordered_field)" |
1222 |
proof - |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1223 |
have "0 < a * inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1224 |
by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1225 |
thus "0 < inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1226 |
by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff) |
23482 | 1227 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1228 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1229 |
lemma negative_imp_inverse_negative: |
23482 | 1230 |
"a < 0 ==> inverse a < (0::'a::ordered_field)" |
1231 |
by (insert positive_imp_inverse_positive [of "-a"], |
|
1232 |
simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1233 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1234 |
lemma inverse_le_imp_le: |
23482 | 1235 |
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a" |
1236 |
shows "b \<le> (a::'a::ordered_field)" |
|
1237 |
proof (rule classical) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1238 |
assume "~ b \<le> a" |
23482 | 1239 |
hence "a < b" by (simp add: linorder_not_le) |
1240 |
hence bpos: "0 < b" by (blast intro: apos order_less_trans) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1241 |
hence "a * inverse a \<le> a * inverse b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1242 |
by (simp add: apos invle order_less_imp_le mult_left_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1243 |
hence "(a * inverse a) * b \<le> (a * inverse b) * b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1244 |
by (simp add: bpos order_less_imp_le mult_right_mono) |
23482 | 1245 |
thus "b \<le> a" by (simp add: mult_assoc apos bpos order_less_imp_not_eq2) |
1246 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1247 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1248 |
lemma inverse_positive_imp_positive: |
23482 | 1249 |
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0" |
1250 |
shows "0 < (a::'a::ordered_field)" |
|
23389 | 1251 |
proof - |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1252 |
have "0 < inverse (inverse a)" |
23389 | 1253 |
using inv_gt_0 by (rule positive_imp_inverse_positive) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1254 |
thus "0 < a" |
23389 | 1255 |
using nz by (simp add: nonzero_inverse_inverse_eq) |
1256 |
qed |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1257 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1258 |
lemma inverse_positive_iff_positive [simp]: |
23482 | 1259 |
"(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))" |
21328 | 1260 |
apply (cases "a = 0", simp) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1261 |
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1262 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1263 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1264 |
lemma inverse_negative_imp_negative: |
23482 | 1265 |
assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0" |
1266 |
shows "a < (0::'a::ordered_field)" |
|
23389 | 1267 |
proof - |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1268 |
have "inverse (inverse a) < 0" |
23389 | 1269 |
using inv_less_0 by (rule negative_imp_inverse_negative) |
23482 | 1270 |
thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq) |
23389 | 1271 |
qed |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1272 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1273 |
lemma inverse_negative_iff_negative [simp]: |
23482 | 1274 |
"(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))" |
21328 | 1275 |
apply (cases "a = 0", simp) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1276 |
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1277 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1278 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1279 |
lemma inverse_nonnegative_iff_nonnegative [simp]: |
23482 | 1280 |
"(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))" |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1281 |
by (simp add: linorder_not_less [symmetric]) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1282 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1283 |
lemma inverse_nonpositive_iff_nonpositive [simp]: |
23482 | 1284 |
"(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))" |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1285 |
by (simp add: linorder_not_less [symmetric]) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1286 |
|
23406
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1287 |
lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)" |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1288 |
proof |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1289 |
fix x::'a |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1290 |
have m1: "- (1::'a) < 0" by simp |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1291 |
from add_strict_right_mono[OF m1, where c=x] |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1292 |
have "(- 1) + x < x" by simp |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1293 |
thus "\<exists>y. y < x" by blast |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1294 |
qed |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1295 |
|
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1296 |
lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)" |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1297 |
proof |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1298 |
fix x::'a |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1299 |
have m1: " (1::'a) > 0" by simp |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1300 |
from add_strict_right_mono[OF m1, where c=x] |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1301 |
have "1 + x > x" by simp |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1302 |
thus "\<exists>y. y > x" by blast |
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset
|
1303 |
qed |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1304 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1305 |
subsection{*Anti-Monotonicity of @{term inverse}*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1306 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1307 |
lemma less_imp_inverse_less: |
23482 | 1308 |
assumes less: "a < b" and apos: "0 < a" |
1309 |
shows "inverse b < inverse (a::'a::ordered_field)" |
|
1310 |
proof (rule ccontr) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1311 |
assume "~ inverse b < inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1312 |
hence "inverse a \<le> inverse b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1313 |
by (simp add: linorder_not_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1314 |
hence "~ (a < b)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1315 |
by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1316 |
thus False |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1317 |
by (rule notE [OF _ less]) |
23482 | 1318 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1319 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1320 |
lemma inverse_less_imp_less: |
23482 | 1321 |
"[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1322 |
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1323 |
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1324 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1325 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1326 |
text{*Both premises are essential. Consider -1 and 1.*} |
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1327 |
lemma inverse_less_iff_less [simp,noatp]: |
23482 | 1328 |
"[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1329 |
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1330 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1331 |
lemma le_imp_inverse_le: |
23482 | 1332 |
"[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)" |
1333 |
by (force simp add: order_le_less less_imp_inverse_less) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1334 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1335 |
lemma inverse_le_iff_le [simp,noatp]: |
23482 | 1336 |
"[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1337 |
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1338 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1339 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1340 |
text{*These results refer to both operands being negative. The opposite-sign |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1341 |
case is trivial, since inverse preserves signs.*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1342 |
lemma inverse_le_imp_le_neg: |
23482 | 1343 |
"[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)" |
1344 |
apply (rule classical) |
|
1345 |
apply (subgoal_tac "a < 0") |
|
1346 |
prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) |
|
1347 |
apply (insert inverse_le_imp_le [of "-b" "-a"]) |
|
1348 |
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
|
1349 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1350 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1351 |
lemma less_imp_inverse_less_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1352 |
"[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)" |
23482 | 1353 |
apply (subgoal_tac "a < 0") |
1354 |
prefer 2 apply (blast intro: order_less_trans) |
|
1355 |
apply (insert less_imp_inverse_less [of "-b" "-a"]) |
|
1356 |
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
|
1357 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1358 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1359 |
lemma inverse_less_imp_less_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1360 |
"[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)" |
23482 | 1361 |
apply (rule classical) |
1362 |
apply (subgoal_tac "a < 0") |
|
1363 |
prefer 2 |
|
1364 |
apply (force simp add: linorder_not_less intro: order_le_less_trans) |
|
1365 |
apply (insert inverse_less_imp_less [of "-b" "-a"]) |
|
1366 |
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
|
1367 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1368 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1369 |
lemma inverse_less_iff_less_neg [simp,noatp]: |
23482 | 1370 |
"[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))" |
1371 |
apply (insert inverse_less_iff_less [of "-b" "-a"]) |
|
1372 |
apply (simp del: inverse_less_iff_less |
|
1373 |
add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
|
1374 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1375 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1376 |
lemma le_imp_inverse_le_neg: |
23482 | 1377 |
"[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)" |
1378 |
by (force simp add: order_le_less less_imp_inverse_less_neg) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1379 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1380 |
lemma inverse_le_iff_le_neg [simp,noatp]: |
23482 | 1381 |
"[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1382 |
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1383 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1384 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1385 |
subsection{*Inverses and the Number One*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1386 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1387 |
lemma one_less_inverse_iff: |
23482 | 1388 |
"(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))" |
1389 |
proof cases |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1390 |
assume "0 < x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1391 |
with inverse_less_iff_less [OF zero_less_one, of x] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1392 |
show ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1393 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1394 |
assume notless: "~ (0 < x)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1395 |
have "~ (1 < inverse x)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1396 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1397 |
assume "1 < inverse x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1398 |
also with notless have "... \<le> 0" by (simp add: linorder_not_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1399 |
also have "... < 1" by (rule zero_less_one) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1400 |
finally show False by auto |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1401 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1402 |
with notless show ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1403 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1404 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1405 |
lemma inverse_eq_1_iff [simp]: |
23482 | 1406 |
"(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1407 |
by (insert inverse_eq_iff_eq [of x 1], simp) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1408 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1409 |
lemma one_le_inverse_iff: |
23482 | 1410 |
"(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1411 |
by (force simp add: order_le_less one_less_inverse_iff zero_less_one |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1412 |
eq_commute [of 1]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1413 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1414 |
lemma inverse_less_1_iff: |
23482 | 1415 |
"(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1416 |
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1417 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1418 |
lemma inverse_le_1_iff: |
23482 | 1419 |
"(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1420 |
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1421 |
|
23389 | 1422 |
|
14288 | 1423 |
subsection{*Simplification of Inequalities Involving Literal Divisors*} |
1424 |
||
1425 |
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)" |
|
1426 |
proof - |
|
1427 |
assume less: "0<c" |
|
1428 |
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" |
|
1429 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1430 |
also have "... = (a*c \<le> b)" |
|
1431 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1432 |
finally show ?thesis . |
|
1433 |
qed |
|
1434 |
||
1435 |
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)" |
|
1436 |
proof - |
|
1437 |
assume less: "c<0" |
|
1438 |
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" |
|
1439 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1440 |
also have "... = (b \<le> a*c)" |
|
1441 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1442 |
finally show ?thesis . |
|
1443 |
qed |
|
1444 |
||
1445 |
lemma le_divide_eq: |
|
1446 |
"(a \<le> b/c) = |
|
1447 |
(if 0 < c then a*c \<le> b |
|
1448 |
else if c < 0 then b \<le> a*c |
|
1449 |
else a \<le> (0::'a::{ordered_field,division_by_zero}))" |
|
21328 | 1450 |
apply (cases "c=0", simp) |
14288 | 1451 |
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) |
1452 |
done |
|
1453 |
||
1454 |
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)" |
|
1455 |
proof - |
|
1456 |
assume less: "0<c" |
|
1457 |
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" |
|
1458 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1459 |
also have "... = (b \<le> a*c)" |
|
1460 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1461 |
finally show ?thesis . |
|
1462 |
qed |
|
1463 |
||
1464 |
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)" |
|
1465 |
proof - |
|
1466 |
assume less: "c<0" |
|
1467 |
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" |
|
1468 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1469 |
also have "... = (a*c \<le> b)" |
|
1470 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1471 |
finally show ?thesis . |
|
1472 |
qed |
|
1473 |
||
1474 |
lemma divide_le_eq: |
|
1475 |
"(b/c \<le> a) = |
|
1476 |
(if 0 < c then b \<le> a*c |
|
1477 |
else if c < 0 then a*c \<le> b |
|
1478 |
else 0 \<le> (a::'a::{ordered_field,division_by_zero}))" |
|
21328 | 1479 |
apply (cases "c=0", simp) |
14288 | 1480 |
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) |
1481 |
done |
|
1482 |
||
1483 |
lemma pos_less_divide_eq: |
|
1484 |
"0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)" |
|
1485 |
proof - |
|
1486 |
assume less: "0<c" |
|
1487 |
hence "(a < b/c) = (a*c < (b/c)*c)" |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1488 |
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) |
14288 | 1489 |
also have "... = (a*c < b)" |
1490 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1491 |
finally show ?thesis . |
|
1492 |
qed |
|
1493 |
||
1494 |
lemma neg_less_divide_eq: |
|
1495 |
"c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)" |
|
1496 |
proof - |
|
1497 |
assume less: "c<0" |
|
1498 |
hence "(a < b/c) = ((b/c)*c < a*c)" |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1499 |
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) |
14288 | 1500 |
also have "... = (b < a*c)" |
1501 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1502 |
finally show ?thesis . |
|
1503 |
qed |
|
1504 |
||
1505 |
lemma less_divide_eq: |
|
1506 |
"(a < b/c) = |
|
1507 |
(if 0 < c then a*c < b |
|
1508 |
else if c < 0 then b < a*c |
|
1509 |
else a < (0::'a::{ordered_field,division_by_zero}))" |
|
21328 | 1510 |
apply (cases "c=0", simp) |
14288 | 1511 |
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) |
1512 |
done |
|
1513 |
||
1514 |
lemma pos_divide_less_eq: |
|
1515 |
"0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)" |
|
1516 |
proof - |
|
1517 |
assume less: "0<c" |
|
1518 |
hence "(b/c < a) = ((b/c)*c < a*c)" |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1519 |
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) |
14288 | 1520 |
also have "... = (b < a*c)" |
1521 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1522 |
finally show ?thesis . |
|
1523 |
qed |
|
1524 |
||
1525 |
lemma neg_divide_less_eq: |
|
1526 |
"c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)" |
|
1527 |
proof - |
|
1528 |
assume less: "c<0" |
|
1529 |
hence "(b/c < a) = (a*c < (b/c)*c)" |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1530 |
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) |
14288 | 1531 |
also have "... = (a*c < b)" |
1532 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1533 |
finally show ?thesis . |
|
1534 |
qed |
|
1535 |
||
1536 |
lemma divide_less_eq: |
|
1537 |
"(b/c < a) = |
|
1538 |
(if 0 < c then b < a*c |
|
1539 |
else if c < 0 then a*c < b |
|
1540 |
else 0 < (a::'a::{ordered_field,division_by_zero}))" |
|
21328 | 1541 |
apply (cases "c=0", simp) |
14288 | 1542 |
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) |
1543 |
done |
|
1544 |
||
23482 | 1545 |
|
1546 |
subsection{*Field simplification*} |
|
1547 |
||
1548 |
text{* Lemmas @{text field_simps} multiply with denominators in |
|
1549 |
in(equations) if they can be proved to be non-zero (for equations) or |
|
1550 |
positive/negative (for inequations). *} |
|
14288 | 1551 |
|
23482 | 1552 |
lemmas field_simps = field_eq_simps |
1553 |
(* multiply ineqn *) |
|
1554 |
pos_divide_less_eq neg_divide_less_eq |
|
1555 |
pos_less_divide_eq neg_less_divide_eq |
|
1556 |
pos_divide_le_eq neg_divide_le_eq |
|
1557 |
pos_le_divide_eq neg_le_divide_eq |
|
14288 | 1558 |
|
23482 | 1559 |
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs |
23483 | 1560 |
of positivity/negativity needed for @{text field_simps}. Have not added @{text |
23482 | 1561 |
sign_simps} to @{text field_simps} because the former can lead to case |
1562 |
explosions. *} |
|
14288 | 1563 |
|
23482 | 1564 |
lemmas sign_simps = group_simps |
1565 |
zero_less_mult_iff mult_less_0_iff |
|
14288 | 1566 |
|
23482 | 1567 |
(* Only works once linear arithmetic is installed: |
1568 |
text{*An example:*} |
|
1569 |
lemma fixes a b c d e f :: "'a::ordered_field" |
|
1570 |
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow> |
|
1571 |
((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) < |
|
1572 |
((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u" |
|
1573 |
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0") |
|
1574 |
prefer 2 apply(simp add:sign_simps) |
|
1575 |
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0") |
|
1576 |
prefer 2 apply(simp add:sign_simps) |
|
1577 |
apply(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1578 |
done |
23482 | 1579 |
*) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1580 |
|
23389 | 1581 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1582 |
subsection{*Division and Signs*} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1583 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1584 |
lemma zero_less_divide_iff: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1585 |
"((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1586 |
by (simp add: divide_inverse zero_less_mult_iff) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1587 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1588 |
lemma divide_less_0_iff: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1589 |
"(a/b < (0::'a::{ordered_field,division_by_zero})) = |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1590 |
(0 < a & b < 0 | a < 0 & 0 < b)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1591 |
by (simp add: divide_inverse mult_less_0_iff) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1592 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1593 |
lemma zero_le_divide_iff: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1594 |
"((0::'a::{ordered_field,division_by_zero}) \<le> a/b) = |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1595 |
(0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1596 |
by (simp add: divide_inverse zero_le_mult_iff) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1597 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1598 |
lemma divide_le_0_iff: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1599 |
"(a/b \<le> (0::'a::{ordered_field,division_by_zero})) = |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1600 |
(0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1601 |
by (simp add: divide_inverse mult_le_0_iff) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1602 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1603 |
lemma divide_eq_0_iff [simp,noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1604 |
"(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))" |
23482 | 1605 |
by (simp add: divide_inverse) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1606 |
|
23482 | 1607 |
lemma divide_pos_pos: |
1608 |
"0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y" |
|
1609 |
by(simp add:field_simps) |
|
1610 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1611 |
|
23482 | 1612 |
lemma divide_nonneg_pos: |
1613 |
"0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y" |
|
1614 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1615 |
|
23482 | 1616 |
lemma divide_neg_pos: |
1617 |
"(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0" |
|
1618 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1619 |
|
23482 | 1620 |
lemma divide_nonpos_pos: |
1621 |
"(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0" |
|
1622 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1623 |
|
23482 | 1624 |
lemma divide_pos_neg: |
1625 |
"0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0" |
|
1626 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1627 |
|
23482 | 1628 |
lemma divide_nonneg_neg: |
1629 |
"0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0" |
|
1630 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1631 |
|
23482 | 1632 |
lemma divide_neg_neg: |
1633 |
"(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y" |
|
1634 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1635 |
|
23482 | 1636 |
lemma divide_nonpos_neg: |
1637 |
"(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y" |
|
1638 |
by(simp add:field_simps) |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1639 |
|
23389 | 1640 |
|
14288 | 1641 |
subsection{*Cancellation Laws for Division*} |
1642 |
||
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1643 |
lemma divide_cancel_right [simp,noatp]: |
14288 | 1644 |
"(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))" |
23482 | 1645 |
apply (cases "c=0", simp) |
23496 | 1646 |
apply (simp add: divide_inverse) |
14288 | 1647 |
done |
1648 |
||
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1649 |
lemma divide_cancel_left [simp,noatp]: |
14288 | 1650 |
"(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" |
23482 | 1651 |
apply (cases "c=0", simp) |
23496 | 1652 |
apply (simp add: divide_inverse) |
14288 | 1653 |
done |
1654 |
||
23389 | 1655 |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1656 |
subsection {* Division and the Number One *} |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1657 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1658 |
text{*Simplify expressions equated with 1*} |
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1659 |
lemma divide_eq_1_iff [simp,noatp]: |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1660 |
"(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" |
23482 | 1661 |
apply (cases "b=0", simp) |
1662 |
apply (simp add: right_inverse_eq) |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1663 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1664 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1665 |
lemma one_eq_divide_iff [simp,noatp]: |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1666 |
"(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" |
23482 | 1667 |
by (simp add: eq_commute [of 1]) |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1668 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1669 |
lemma zero_eq_1_divide_iff [simp,noatp]: |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1670 |
"((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)" |
23482 | 1671 |
apply (cases "a=0", simp) |
1672 |
apply (auto simp add: nonzero_eq_divide_eq) |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1673 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1674 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1675 |
lemma one_divide_eq_0_iff [simp,noatp]: |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1676 |
"(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)" |
23482 | 1677 |
apply (cases "a=0", simp) |
1678 |
apply (insert zero_neq_one [THEN not_sym]) |
|
1679 |
apply (auto simp add: nonzero_divide_eq_eq) |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1680 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1681 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1682 |
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*} |
18623 | 1683 |
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified] |
1684 |
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified] |
|
1685 |
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified] |
|
1686 |
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified] |
|
17085 | 1687 |
|
1688 |
declare zero_less_divide_1_iff [simp] |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1689 |
declare divide_less_0_1_iff [simp,noatp] |
17085 | 1690 |
declare zero_le_divide_1_iff [simp] |
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1691 |
declare divide_le_0_1_iff [simp,noatp] |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1692 |
|
23389 | 1693 |
|
14293 | 1694 |
subsection {* Ordering Rules for Division *} |
1695 |
||
1696 |
lemma divide_strict_right_mono: |
|
1697 |
"[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)" |
|
1698 |
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono |
|
23482 | 1699 |
positive_imp_inverse_positive) |
14293 | 1700 |
|
1701 |
lemma divide_right_mono: |
|
1702 |
"[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})" |
|
23482 | 1703 |
by (force simp add: divide_strict_right_mono order_le_less) |
14293 | 1704 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1705 |
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1706 |
==> c <= 0 ==> b / c <= a / c" |
23482 | 1707 |
apply (drule divide_right_mono [of _ _ "- c"]) |
1708 |
apply auto |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1709 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1710 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1711 |
lemma divide_strict_right_mono_neg: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1712 |
"[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)" |
23482 | 1713 |
apply (drule divide_strict_right_mono [of _ _ "-c"], simp) |
1714 |
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1715 |
done |
14293 | 1716 |
|
1717 |
text{*The last premise ensures that @{term a} and @{term b} |
|
1718 |
have the same sign*} |
|
1719 |
lemma divide_strict_left_mono: |
|
23482 | 1720 |
"[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)" |
1721 |
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono) |
|
14293 | 1722 |
|
1723 |
lemma divide_left_mono: |
|
23482 | 1724 |
"[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)" |
1725 |
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono) |
|
14293 | 1726 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1727 |
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1728 |
==> c <= 0 ==> 0 < a * b ==> c / a <= c / b" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1729 |
apply (drule divide_left_mono [of _ _ "- c"]) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1730 |
apply (auto simp add: mult_commute) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1731 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1732 |
|
14293 | 1733 |
lemma divide_strict_left_mono_neg: |
23482 | 1734 |
"[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)" |
1735 |
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg) |
|
1736 |
||
14293 | 1737 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1738 |
text{*Simplify quotients that are compared with the value 1.*} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1739 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1740 |
lemma le_divide_eq_1 [noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1741 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1742 |
shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1743 |
by (auto simp add: le_divide_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1744 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1745 |
lemma divide_le_eq_1 [noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1746 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1747 |
shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1748 |
by (auto simp add: divide_le_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1749 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1750 |
lemma less_divide_eq_1 [noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1751 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1752 |
shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1753 |
by (auto simp add: less_divide_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1754 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1755 |
lemma divide_less_eq_1 [noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1756 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1757 |
shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1758 |
by (auto simp add: divide_less_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1759 |
|
23389 | 1760 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1761 |
subsection{*Conditional Simplification Rules: No Case Splits*} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1762 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1763 |
lemma le_divide_eq_1_pos [simp,noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1764 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1765 |
shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1766 |
by (auto simp add: le_divide_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1767 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1768 |
lemma le_divide_eq_1_neg [simp,noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1769 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1770 |
shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1771 |
by (auto simp add: le_divide_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1772 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1773 |
lemma divide_le_eq_1_pos [simp,noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1774 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1775 |
shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1776 |
by (auto simp add: divide_le_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1777 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1778 |
lemma divide_le_eq_1_neg [simp,noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1779 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1780 |
shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1781 |
by (auto simp add: divide_le_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1782 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1783 |
lemma less_divide_eq_1_pos [simp,noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1784 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1785 |
shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1786 |
by (auto simp add: less_divide_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1787 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1788 |
lemma less_divide_eq_1_neg [simp,noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1789 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1790 |
shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1791 |
by (auto simp add: less_divide_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1792 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1793 |
lemma divide_less_eq_1_pos [simp,noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1794 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1795 |
shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)" |
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1796 |
by (auto simp add: divide_less_eq) |
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1797 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1798 |
lemma divide_less_eq_1_neg [simp,noatp]: |
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1799 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1800 |
shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1801 |
by (auto simp add: divide_less_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1802 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1803 |
lemma eq_divide_eq_1 [simp,noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1804 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1805 |
shows "(1 = b/a) = ((a \<noteq> 0 & a = b))" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1806 |
by (auto simp add: eq_divide_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1807 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1808 |
lemma divide_eq_eq_1 [simp,noatp]: |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1809 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1810 |
shows "(b/a = 1) = ((a \<noteq> 0 & a = b))" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1811 |
by (auto simp add: divide_eq_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1812 |
|
23389 | 1813 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1814 |
subsection {* Reasoning about inequalities with division *} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1815 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1816 |
lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1 |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1817 |
==> x * y <= x" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1818 |
by (auto simp add: mult_compare_simps); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1819 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1820 |
lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1 |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1821 |
==> y * x <= x" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1822 |
by (auto simp add: mult_compare_simps); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1823 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1824 |
lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1825 |
x / y <= z"; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1826 |
by (subst pos_divide_le_eq, assumption+); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1827 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1828 |
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==> |
23482 | 1829 |
z <= x / y" |
1830 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1831 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1832 |
lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1833 |
x / y < z" |
23482 | 1834 |
by(simp add:field_simps) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1835 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1836 |
lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1837 |
z < x / y" |
23482 | 1838 |
by(simp add:field_simps) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1839 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1840 |
lemma frac_le: "(0::'a::ordered_field) <= x ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1841 |
x <= y ==> 0 < w ==> w <= z ==> x / z <= y / w" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1842 |
apply (rule mult_imp_div_pos_le) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1843 |
apply simp; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1844 |
apply (subst times_divide_eq_left); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1845 |
apply (rule mult_imp_le_div_pos, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1846 |
apply (rule mult_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1847 |
apply simp_all |
14293 | 1848 |
done |
1849 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1850 |
lemma frac_less: "(0::'a::ordered_field) <= x ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1851 |
x < y ==> 0 < w ==> w <= z ==> x / z < y / w" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1852 |
apply (rule mult_imp_div_pos_less) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1853 |
apply simp; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1854 |
apply (subst times_divide_eq_left); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1855 |
apply (rule mult_imp_less_div_pos, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1856 |
apply (erule mult_less_le_imp_less) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1857 |
apply simp_all |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1858 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1859 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1860 |
lemma frac_less2: "(0::'a::ordered_field) < x ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1861 |
x <= y ==> 0 < w ==> w < z ==> x / z < y / w" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1862 |
apply (rule mult_imp_div_pos_less) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1863 |
apply simp_all |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1864 |
apply (subst times_divide_eq_left); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1865 |
apply (rule mult_imp_less_div_pos, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1866 |
apply (erule mult_le_less_imp_less) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1867 |
apply simp_all |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1868 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1869 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1870 |
text{*It's not obvious whether these should be simprules or not. |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1871 |
Their effect is to gather terms into one big fraction, like |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1872 |
a*b*c / x*y*z. The rationale for that is unclear, but many proofs |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1873 |
seem to need them.*} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1874 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1875 |
declare times_divide_eq [simp] |
14293 | 1876 |
|
23389 | 1877 |
|
14293 | 1878 |
subsection {* Ordered Fields are Dense *} |
1879 |
||
14738 | 1880 |
lemma less_add_one: "a < (a+1::'a::ordered_semidom)" |
14293 | 1881 |
proof - |
14738 | 1882 |
have "a+0 < (a+1::'a::ordered_semidom)" |
23482 | 1883 |
by (blast intro: zero_less_one add_strict_left_mono) |
14293 | 1884 |
thus ?thesis by simp |
1885 |
qed |
|
1886 |
||
14738 | 1887 |
lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)" |
23482 | 1888 |
by (blast intro: order_less_trans zero_less_one less_add_one) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1889 |
|
14293 | 1890 |
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)" |
23482 | 1891 |
by (simp add: field_simps zero_less_two) |
14293 | 1892 |
|
1893 |
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b" |
|
23482 | 1894 |
by (simp add: field_simps zero_less_two) |
14293 | 1895 |
|
24422 | 1896 |
instance ordered_field < dense_linear_order |
1897 |
proof |
|
1898 |
fix x y :: 'a |
|
1899 |
have "x < x + 1" by simp |
|
1900 |
then show "\<exists>y. x < y" .. |
|
1901 |
have "x - 1 < x" by simp |
|
1902 |
then show "\<exists>y. y < x" .. |
|
1903 |
show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum) |
|
1904 |
qed |
|
14293 | 1905 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1906 |
|
14293 | 1907 |
subsection {* Absolute Value *} |
1908 |
||
24491 | 1909 |
lemma mult_sgn_abs: "sgn x * abs x = (x::'a::{ordered_idom,linorder})" |
1910 |
using less_linear[of x 0] |
|
24506 | 1911 |
by(auto simp: sgn_if abs_if) |
24491 | 1912 |
|
14738 | 1913 |
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1914 |
by (simp add: abs_if zero_less_one [THEN order_less_not_sym]) |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1915 |
|
14738 | 1916 |
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" |
1917 |
proof - |
|
1918 |
let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b" |
|
1919 |
let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b" |
|
1920 |
have a: "(abs a) * (abs b) = ?x" |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1921 |
by (simp only: abs_prts[of a] abs_prts[of b] ring_simps) |
14738 | 1922 |
{ |
1923 |
fix u v :: 'a |
|
15481 | 1924 |
have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> |
1925 |
u * v = pprt a * pprt b + pprt a * nprt b + |
|
1926 |
nprt a * pprt b + nprt a * nprt b" |
|
14738 | 1927 |
apply (subst prts[of u], subst prts[of v]) |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1928 |
apply (simp add: ring_simps) |
14738 | 1929 |
done |
1930 |
} |
|
1931 |
note b = this[OF refl[of a] refl[of b]] |
|
1932 |
note addm = add_mono[of "0::'a" _ "0::'a", simplified] |
|
1933 |
note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified] |
|
1934 |
have xy: "- ?x <= ?y" |
|
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1935 |
apply (simp) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1936 |
apply (rule_tac y="0::'a" in order_trans) |
16568 | 1937 |
apply (rule addm2) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1938 |
apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos) |
16568 | 1939 |
apply (rule addm) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1940 |
apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos) |
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1941 |
done |
14738 | 1942 |
have yx: "?y <= ?x" |
16568 | 1943 |
apply (simp add:diff_def) |
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1944 |
apply (rule_tac y=0 in order_trans) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1945 |
apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1946 |
apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+) |
14738 | 1947 |
done |
1948 |
have i1: "a*b <= abs a * abs b" by (simp only: a b yx) |
|
1949 |
have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy) |
|
1950 |
show ?thesis |
|
1951 |
apply (rule abs_leI) |
|
1952 |
apply (simp add: i1) |
|
1953 |
apply (simp add: i2[simplified minus_le_iff]) |
|
1954 |
done |
|
1955 |
qed |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1956 |
|
14738 | 1957 |
lemma abs_eq_mult: |
1958 |
assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)" |
|
1959 |
shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)" |
|
1960 |
proof - |
|
1961 |
have s: "(0 <= a*b) | (a*b <= 0)" |
|
1962 |
apply (auto) |
|
1963 |
apply (rule_tac split_mult_pos_le) |
|
1964 |
apply (rule_tac contrapos_np[of "a*b <= 0"]) |
|
1965 |
apply (simp) |
|
1966 |
apply (rule_tac split_mult_neg_le) |
|
1967 |
apply (insert prems) |
|
1968 |
apply (blast) |
|
1969 |
done |
|
1970 |
have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)" |
|
1971 |
by (simp add: prts[symmetric]) |
|
1972 |
show ?thesis |
|
1973 |
proof cases |
|
1974 |
assume "0 <= a * b" |
|
1975 |
then show ?thesis |
|
1976 |
apply (simp_all add: mulprts abs_prts) |
|
1977 |
apply (insert prems) |
|
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1978 |
apply (auto simp add: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1979 |
ring_simps |
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1980 |
iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt] |
15197 | 1981 |
iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id]) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1982 |
apply(drule (1) mult_nonneg_nonpos[of a b], simp) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1983 |
apply(drule (1) mult_nonneg_nonpos2[of b a], simp) |
14738 | 1984 |
done |
1985 |
next |
|
1986 |
assume "~(0 <= a*b)" |
|
1987 |
with s have "a*b <= 0" by simp |
|
1988 |
then show ?thesis |
|
1989 |
apply (simp_all add: mulprts abs_prts) |
|
1990 |
apply (insert prems) |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1991 |
apply (auto simp add: ring_simps) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1992 |
apply(drule (1) mult_nonneg_nonneg[of a b],simp) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1993 |
apply(drule (1) mult_nonpos_nonpos[of a b],simp) |
14738 | 1994 |
done |
1995 |
qed |
|
1996 |
qed |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1997 |
|
14738 | 1998 |
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" |
1999 |
by (simp add: abs_eq_mult linorder_linear) |
|
14293 | 2000 |
|
14738 | 2001 |
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)" |
2002 |
by (simp add: abs_if) |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2003 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2004 |
lemma nonzero_abs_inverse: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2005 |
"a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2006 |
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2007 |
negative_imp_inverse_negative) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2008 |
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2009 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2010 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2011 |
lemma abs_inverse [simp]: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2012 |
"abs (inverse (a::'a::{ordered_field,division_by_zero})) = |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2013 |
inverse (abs a)" |
21328 | 2014 |
apply (cases "a=0", simp) |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2015 |
apply (simp add: nonzero_abs_inverse) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2016 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2017 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2018 |
lemma nonzero_abs_divide: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2019 |
"b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2020 |
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2021 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
2022 |
lemma abs_divide [simp]: |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2023 |
"abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b" |
21328 | 2024 |
apply (cases "b=0", simp) |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2025 |
apply (simp add: nonzero_abs_divide) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2026 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2027 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2028 |
lemma abs_mult_less: |
14738 | 2029 |
"[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)" |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2030 |
proof - |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2031 |
assume ac: "abs a < c" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2032 |
hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2033 |
assume "abs b < d" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2034 |
thus ?thesis by (simp add: ac cpos mult_strict_mono) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2035 |
qed |
14293 | 2036 |
|
14738 | 2037 |
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))" |
2038 |
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff) |
|
2039 |
||
2040 |
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))" |
|
2041 |
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff) |
|
2042 |
||
2043 |
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" |
|
2044 |
apply (simp add: order_less_le abs_le_iff) |
|
2045 |
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff) |
|
2046 |
apply (simp add: le_minus_self_iff linorder_neq_iff) |
|
2047 |
done |
|
2048 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2049 |
lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2050 |
(abs y) * x = abs (y * x)"; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2051 |
apply (subst abs_mult); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2052 |
apply simp; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2053 |
done; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2054 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2055 |
lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2056 |
abs x / y = abs (x / y)"; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2057 |
apply (subst abs_divide); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2058 |
apply (simp add: order_less_imp_le); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2059 |
done; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2060 |
|
23389 | 2061 |
|
19404 | 2062 |
subsection {* Bounds of products via negative and positive Part *} |
15178 | 2063 |
|
15580 | 2064 |
lemma mult_le_prts: |
2065 |
assumes |
|
2066 |
"a1 <= (a::'a::lordered_ring)" |
|
2067 |
"a <= a2" |
|
2068 |
"b1 <= b" |
|
2069 |
"b <= b2" |
|
2070 |
shows |
|
2071 |
"a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1" |
|
2072 |
proof - |
|
2073 |
have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" |
|
2074 |
apply (subst prts[symmetric])+ |
|
2075 |
apply simp |
|
2076 |
done |
|
2077 |
then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b" |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
2078 |
by (simp add: ring_simps) |
15580 | 2079 |
moreover have "pprt a * pprt b <= pprt a2 * pprt b2" |
2080 |
by (simp_all add: prems mult_mono) |
|
2081 |
moreover have "pprt a * nprt b <= pprt a1 * nprt b2" |
|
2082 |
proof - |
|
2083 |
have "pprt a * nprt b <= pprt a * nprt b2" |
|
2084 |
by (simp add: mult_left_mono prems) |
|
2085 |
moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2" |
|
2086 |
by (simp add: mult_right_mono_neg prems) |
|
2087 |
ultimately show ?thesis |
|
2088 |
by simp |
|
2089 |
qed |
|
2090 |
moreover have "nprt a * pprt b <= nprt a2 * pprt b1" |
|
2091 |
proof - |
|
2092 |
have "nprt a * pprt b <= nprt a2 * pprt b" |
|
2093 |
by (simp add: mult_right_mono prems) |
|
2094 |
moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1" |
|
2095 |
by (simp add: mult_left_mono_neg prems) |
|
2096 |
ultimately show ?thesis |
|
2097 |
by simp |
|
2098 |
qed |
|
2099 |
moreover have "nprt a * nprt b <= nprt a1 * nprt b1" |
|
2100 |
proof - |
|
2101 |
have "nprt a * nprt b <= nprt a * nprt b1" |
|
2102 |
by (simp add: mult_left_mono_neg prems) |
|
2103 |
moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1" |
|
2104 |
by (simp add: mult_right_mono_neg prems) |
|
2105 |
ultimately show ?thesis |
|
2106 |
by simp |
|
2107 |
qed |
|
2108 |
ultimately show ?thesis |
|
2109 |
by - (rule add_mono | simp)+ |
|
2110 |
qed |
|
19404 | 2111 |
|
2112 |
lemma mult_ge_prts: |
|
15178 | 2113 |
assumes |
19404 | 2114 |
"a1 <= (a::'a::lordered_ring)" |
2115 |
"a <= a2" |
|
2116 |
"b1 <= b" |
|
2117 |
"b <= b2" |
|
15178 | 2118 |
shows |
19404 | 2119 |
"a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1" |
2120 |
proof - |
|
2121 |
from prems have a1:"- a2 <= -a" by auto |
|
2122 |
from prems have a2: "-a <= -a1" by auto |
|
2123 |
from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 prems(3) prems(4), simplified nprt_neg pprt_neg] |
|
2124 |
have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp |
|
2125 |
then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b" |
|
2126 |
by (simp only: minus_le_iff) |
|
2127 |
then show ?thesis by simp |
|
15178 | 2128 |
qed |
2129 |
||
23389 | 2130 |
|
22842 | 2131 |
subsection {* Theorems for proof tools *} |
2132 |
||
24427 | 2133 |
lemma add_mono_thms_ordered_semiring [noatp]: |
22842 | 2134 |
fixes i j k :: "'a\<Colon>pordered_ab_semigroup_add" |
2135 |
shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l" |
|
2136 |
and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l" |
|
2137 |
and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l" |
|
2138 |
and "i = j \<and> k = l \<Longrightarrow> i + k = j + l" |
|
2139 |
by (rule add_mono, clarify+)+ |
|
2140 |
||
24427 | 2141 |
lemma add_mono_thms_ordered_field [noatp]: |
22842 | 2142 |
fixes i j k :: "'a\<Colon>pordered_cancel_ab_semigroup_add" |
2143 |
shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l" |
|
2144 |
and "i = j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
2145 |
and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l" |
|
2146 |
and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
2147 |
and "i < j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
2148 |
by (auto intro: add_strict_right_mono add_strict_left_mono |
|
2149 |
add_less_le_mono add_le_less_mono add_strict_mono) |
|
2150 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
2151 |
end |