author | haftmann |
Fri, 20 Jul 2007 14:28:01 +0200 | |
changeset 23879 | 4776af8be741 |
parent 23550 | d4f1d6ef119c |
child 24286 | 7619080e49f0 |
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 \<^loc>+ b) \<^loc>* c = a \<^loc>* c \<^loc>+ b \<^loc>* c" |
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assumes right_distrib: "a \<^loc>* (b \<^loc>+ c) = a \<^loc>* b \<^loc>+ a \<^loc>* c" |
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class mult_zero = times + zero + |
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assumes mult_zero_left [simp]: "\<^loc>0 \<^loc>* a = \<^loc>0" |
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assumes mult_zero_right [simp]: "a \<^loc>* \<^loc>0 = \<^loc>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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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 \<^loc>+ b) \<^loc>* c = a \<^loc>* c \<^loc>+ b \<^loc>* 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 |
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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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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]: "\<^loc>0 \<noteq> \<^loc>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> \<^loc>0 \<Longrightarrow> b \<noteq> \<^loc>0 \<Longrightarrow> a \<^loc>* b \<noteq> \<^loc>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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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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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> \<^loc>0 \<Longrightarrow> inverse a \<^loc>* a = \<^loc>1" |
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assumes right_inverse [simp]: "a \<noteq> \<^loc>0 \<Longrightarrow> a \<^loc>* inverse a = \<^loc>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 \<^loc>* a = \<^loc>1" |
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assumes divide_inverse: "a \<^loc>/ b = a \<^loc>* 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 |
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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 \<^loc>0 = \<^loc>0" |
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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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lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)" |
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apply (rule equals_zero_I) |
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187 |
apply (simp add: right_distrib [symmetric]) |
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|
188 |
done |
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189 |
|
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|
190 |
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 |
|
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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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replacing HOL/Real/PRat, PNat by the rational number development
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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]) |
|
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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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|
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 |
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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 + |
214 |
assumes mult_left_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b" |
|
215 |
assumes mult_right_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> a \<^loc>* c \<sqsubseteq> b \<^loc>* c" |
|
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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 |
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|
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
huffman
parents:
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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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|
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 + |
231 |
assumes mult_strict_left_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b" |
|
232 |
assumes mult_strict_right_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> a \<^loc>* c \<sqsubset> b \<^loc>* c" |
|
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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 + |
249 |
assumes mult_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* 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 |
|
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|
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 + |
260 |
assumes mult_strict_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b" |
|
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|
261 |
|
14738 | 262 |
instance pordered_comm_semiring \<subseteq> pordered_semiring |
21199
2d83f93c3580
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changeset
|
263 |
proof |
2d83f93c3580
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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
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|
268 |
qed |
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|
269 |
|
14738 | 270 |
instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring .. |
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|
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 |
|
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|
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 |
|
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|
288 |
|
22390 | 289 |
class pordered_ring = ring + pordered_cancel_semiring |
14270 | 290 |
|
14738 | 291 |
instance pordered_ring \<subseteq> pordered_ab_group_add .. |
14270 | 292 |
|
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22422
diff
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|
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 + |
22390 | 300 |
assumes abs_if: "abs a = (if a \<sqsubset> 0 then (uminus a) else a)" |
14270 | 301 |
|
23521 | 302 |
(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors. |
303 |
Basically, ordered_ring + no_zero_divisors = ordered_ring_strict. |
|
304 |
*) |
|
305 |
class ordered_ring = ring + ordered_semiring + lordered_ab_group + abs_if |
|
14270 | 306 |
|
23550 | 307 |
instance ordered_ring \<subseteq> lordered_ring |
308 |
proof |
|
309 |
fix x :: 'a |
|
310 |
show "\<bar>x\<bar> = sup x (- x)" |
|
311 |
by (simp only: abs_if sup_eq_if) |
|
312 |
qed |
|
23521 | 313 |
|
314 |
class ordered_ring_strict = ring + ordered_semiring_strict + lordered_ab_group + abs_if |
|
315 |
||
316 |
instance ordered_ring_strict \<subseteq> ordered_ring .. |
|
14270 | 317 |
|
22390 | 318 |
class pordered_comm_ring = comm_ring + pordered_comm_semiring |
14270 | 319 |
|
23527 | 320 |
instance pordered_comm_ring \<subseteq> pordered_ring .. |
321 |
||
23073 | 322 |
instance pordered_comm_ring \<subseteq> pordered_cancel_comm_semiring .. |
323 |
||
22390 | 324 |
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict + |
325 |
(*previously ordered_semiring*) |
|
326 |
assumes zero_less_one [simp]: "\<^loc>0 \<sqsubset> \<^loc>1" |
|
14270 | 327 |
|
23521 | 328 |
class ordered_idom = comm_ring_1 + ordered_comm_semiring_strict + lordered_ab_group + abs_if |
22390 | 329 |
(*previously ordered_ring*) |
14270 | 330 |
|
14738 | 331 |
instance ordered_idom \<subseteq> ordered_ring_strict .. |
14272
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|
332 |
|
23073 | 333 |
instance ordered_idom \<subseteq> pordered_comm_ring .. |
334 |
||
22390 | 335 |
class ordered_field = field + ordered_idom |
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|
336 |
|
15923 | 337 |
lemmas linorder_neqE_ordered_idom = |
338 |
linorder_neqE[where 'a = "?'b::ordered_idom"] |
|
339 |
||
14272
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|
340 |
lemma eq_add_iff1: |
23477
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tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
341 |
"(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
|
342 |
by (simp add: ring_simps) |
14272
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diff
changeset
|
343 |
|
5efbb548107d
Tidying of the integer development; towards removing the
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diff
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|
344 |
lemma eq_add_iff2: |
23477
f4b83f03cac9
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nipkow
parents:
23413
diff
changeset
|
345 |
"(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
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diff
changeset
|
346 |
by (simp add: ring_simps) |
14272
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Tidying of the integer development; towards removing the
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diff
changeset
|
347 |
|
5efbb548107d
Tidying of the integer development; towards removing the
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parents:
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diff
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|
348 |
lemma less_add_iff1: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
349 |
"(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
|
350 |
by (simp add: ring_simps) |
14272
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Tidying of the integer development; towards removing the
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parents:
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diff
changeset
|
351 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
352 |
lemma less_add_iff2: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
353 |
"(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
|
354 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
355 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
356 |
lemma le_add_iff1: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
357 |
"(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
|
358 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
359 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
360 |
lemma le_add_iff2: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
361 |
"(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
|
362 |
by (simp add: ring_simps) |
14272
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Tidying of the integer development; towards removing the
paulson
parents:
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diff
changeset
|
363 |
|
23389 | 364 |
|
14270 | 365 |
subsection {* Ordering Rules for Multiplication *} |
366 |
||
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
367 |
lemma mult_left_le_imp_le: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
368 |
"[|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
|
369 |
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
|
370 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
371 |
lemma mult_right_le_imp_le: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
372 |
"[|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
|
373 |
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
|
374 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
375 |
lemma mult_left_less_imp_less: |
23521 | 376 |
"[|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
|
377 |
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
|
378 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
379 |
lemma mult_right_less_imp_less: |
23521 | 380 |
"[|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
|
381 |
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
|
382 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
383 |
lemma mult_strict_left_mono_neg: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
384 |
"[|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
|
385 |
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
|
386 |
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
|
387 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
388 |
|
14738 | 389 |
lemma mult_left_mono_neg: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
390 |
"[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::pordered_ring)" |
14738 | 391 |
apply (drule mult_left_mono [of _ _ "-c"]) |
392 |
apply (simp_all add: minus_mult_left [symmetric]) |
|
393 |
done |
|
394 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
395 |
lemma mult_strict_right_mono_neg: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
396 |
"[|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
|
397 |
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
|
398 |
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
|
399 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
400 |
|
14738 | 401 |
lemma mult_right_mono_neg: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
402 |
"[|b \<le> a; c \<le> 0|] ==> a * c \<le> (b::'a::pordered_ring) * c" |
14738 | 403 |
apply (drule mult_right_mono [of _ _ "-c"]) |
404 |
apply (simp) |
|
405 |
apply (simp_all add: minus_mult_right [symmetric]) |
|
406 |
done |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
407 |
|
23389 | 408 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
409 |
subsection{* Products of Signs *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
410 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
411 |
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
|
412 |
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
|
413 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
414 |
lemma mult_nonneg_nonneg: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b" |
14738 | 415 |
by (drule mult_left_mono [of 0 b], auto) |
416 |
||
417 |
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
|
418 |
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
|
419 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
420 |
lemma mult_nonneg_nonpos: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0" |
14738 | 421 |
by (drule mult_left_mono [of b 0], auto) |
422 |
||
423 |
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" |
|
424 |
by (drule mult_strict_right_mono[of b 0], auto) |
|
425 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
426 |
lemma mult_nonneg_nonpos2: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" |
14738 | 427 |
by (drule mult_right_mono[of b 0], auto) |
428 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
429 |
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
|
430 |
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
|
431 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
432 |
lemma mult_nonpos_nonpos: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b" |
14738 | 433 |
by (drule mult_right_mono_neg[of a 0 b ], auto) |
434 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
435 |
lemma zero_less_mult_pos: |
14738 | 436 |
"[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)" |
21328 | 437 |
apply (cases "b\<le>0") |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
438 |
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
|
439 |
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
|
440 |
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
|
441 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
442 |
|
14738 | 443 |
lemma zero_less_mult_pos2: |
444 |
"[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)" |
|
21328 | 445 |
apply (cases "b\<le>0") |
14738 | 446 |
apply (auto simp add: order_le_less linorder_not_less) |
447 |
apply (drule_tac mult_pos_neg2 [of a b]) |
|
448 |
apply (auto dest: order_less_not_sym) |
|
449 |
done |
|
450 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
451 |
lemma zero_less_mult_iff: |
14738 | 452 |
"((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
|
453 |
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
|
454 |
mult_neg_neg) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
455 |
apply (blast dest: zero_less_mult_pos) |
14738 | 456 |
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
|
457 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
458 |
|
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
459 |
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
|
460 |
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
|
461 |
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
|
462 |
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
|
463 |
|
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
464 |
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
|
465 |
apply intro_classes |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
466 |
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
|
467 |
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
|
468 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
469 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
470 |
lemma zero_le_mult_iff: |
14738 | 471 |
"((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
|
472 |
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
|
473 |
zero_less_mult_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
474 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
475 |
lemma mult_less_0_iff: |
14738 | 476 |
"(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
|
477 |
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
|
478 |
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
|
479 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
480 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
481 |
lemma mult_le_0_iff: |
14738 | 482 |
"(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
|
483 |
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
|
484 |
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
|
485 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
486 |
|
14738 | 487 |
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
|
488 |
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos) |
14738 | 489 |
|
490 |
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
|
491 |
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2) |
14738 | 492 |
|
23095 | 493 |
lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a" |
494 |
by (simp add: zero_le_mult_iff linorder_linear) |
|
495 |
||
496 |
lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))" |
|
497 |
by (simp add: not_less) |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
498 |
|
14738 | 499 |
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom} |
500 |
theorems available to members of @{term ordered_idom} *} |
|
501 |
||
502 |
instance ordered_idom \<subseteq> ordered_semidom |
|
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
503 |
proof |
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
504 |
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
|
505 |
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
|
506 |
qed |
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
507 |
|
14738 | 508 |
instance ordered_idom \<subseteq> idom .. |
509 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
510 |
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
|
511 |
|
17085 | 512 |
lemmas one_neq_zero = zero_neq_one [THEN not_sym] |
513 |
declare one_neq_zero [simp] |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
514 |
|
14738 | 515 |
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
|
516 |
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
|
517 |
|
14738 | 518 |
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0" |
519 |
by (simp add: linorder_not_le) |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
520 |
|
14738 | 521 |
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0" |
522 |
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
|
523 |
|
23389 | 524 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
525 |
subsection{*More Monotonicity*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
526 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
527 |
text{*Strict monotonicity in both arguments*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
528 |
lemma mult_strict_mono: |
14738 | 529 |
"[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)" |
21328 | 530 |
apply (cases "c=0") |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
531 |
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
|
532 |
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
|
533 |
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
|
534 |
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
|
535 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
536 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
537 |
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
|
538 |
lemma mult_strict_mono': |
14738 | 539 |
"[| 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
|
540 |
apply (rule mult_strict_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
541 |
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
|
542 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
543 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
544 |
lemma mult_mono: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
545 |
"[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] |
14738 | 546 |
==> 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
|
547 |
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
|
548 |
apply (erule mult_left_mono, assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
549 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
550 |
|
21258 | 551 |
lemma mult_mono': |
552 |
"[|a \<le> b; c \<le> d; 0 \<le> a; 0 \<le> c|] |
|
553 |
==> a * c \<le> b * (d::'a::pordered_semiring)" |
|
554 |
apply (rule mult_mono) |
|
555 |
apply (fast intro: order_trans)+ |
|
556 |
done |
|
557 |
||
14738 | 558 |
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
|
559 |
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
|
560 |
apply (simp add: order_less_trans [OF zero_less_one]) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
561 |
done |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
562 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
563 |
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
|
564 |
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
|
565 |
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
|
566 |
apply (erule order_less_le_trans) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
567 |
apply (erule mult_left_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
568 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
569 |
apply (erule mult_strict_right_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
570 |
apply assumption |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
571 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
572 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
573 |
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
|
574 |
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
|
575 |
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
|
576 |
apply (erule order_le_less_trans) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
577 |
apply (erule mult_strict_left_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
578 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
579 |
apply (erule mult_right_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
580 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
581 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
582 |
|
23389 | 583 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
584 |
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
|
585 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
586 |
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
|
587 |
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
|
588 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
589 |
text{*These ``disjunction'' versions produce two cases when the comparison is |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
590 |
an assumption, but effectively four when the comparison is a goal.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
591 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
592 |
lemma mult_less_cancel_right_disj: |
14738 | 593 |
"(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))" |
21328 | 594 |
apply (cases "c = 0") |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
595 |
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
|
596 |
mult_strict_right_mono_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
597 |
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
|
598 |
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
|
599 |
linorder_not_le [symmetric, of a]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
600 |
apply (erule_tac [!] notE) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
601 |
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
|
602 |
mult_right_mono_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
603 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
604 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
605 |
lemma mult_less_cancel_left_disj: |
14738 | 606 |
"(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))" |
21328 | 607 |
apply (cases "c = 0") |
14738 | 608 |
apply (auto simp add: linorder_neq_iff mult_strict_left_mono |
609 |
mult_strict_left_mono_neg) |
|
610 |
apply (auto simp add: linorder_not_less |
|
611 |
linorder_not_le [symmetric, of "c*a"] |
|
612 |
linorder_not_le [symmetric, of a]) |
|
613 |
apply (erule_tac [!] notE) |
|
614 |
apply (auto simp add: order_less_imp_le mult_left_mono |
|
615 |
mult_left_mono_neg) |
|
616 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
617 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
618 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
619 |
text{*The ``conjunction of implication'' lemmas produce two cases when the |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
620 |
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
|
621 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
622 |
lemma mult_less_cancel_right: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
623 |
fixes c :: "'a :: ordered_ring_strict" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
624 |
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
|
625 |
by (insert mult_less_cancel_right_disj [of a c b], auto) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
626 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
627 |
lemma mult_less_cancel_left: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
628 |
fixes c :: "'a :: ordered_ring_strict" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
629 |
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
|
630 |
by (insert mult_less_cancel_left_disj [of c a b], auto) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
631 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
632 |
lemma mult_le_cancel_right: |
14738 | 633 |
"(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
|
634 |
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
|
635 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
636 |
lemma mult_le_cancel_left: |
14738 | 637 |
"(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
|
638 |
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
|
639 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
640 |
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
|
641 |
assumes less: "c*a < c*b" and nonneg: "0 \<le> c" |
14738 | 642 |
shows "a < (b::'a::ordered_semiring_strict)" |
14377 | 643 |
proof (rule ccontr) |
644 |
assume "~ a < b" |
|
645 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
23389 | 646 |
hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono) |
14377 | 647 |
with this and less show False |
648 |
by (simp add: linorder_not_less [symmetric]) |
|
649 |
qed |
|
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_less_imp_less_right: |
14738 | 652 |
assumes less: "a*c < b*c" and nonneg: "0 <= c" |
653 |
shows "a < (b::'a::ordered_semiring_strict)" |
|
654 |
proof (rule ccontr) |
|
655 |
assume "~ a < b" |
|
656 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
23389 | 657 |
hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono) |
14738 | 658 |
with this and less show False |
659 |
by (simp add: linorder_not_less [symmetric]) |
|
660 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
661 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
662 |
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
|
663 |
lemma mult_cancel_right [simp]: |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
664 |
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
|
665 |
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
|
666 |
proof - |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
667 |
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
|
668 |
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
|
669 |
thus ?thesis |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
670 |
by (simp add: disj_commute) |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
671 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
672 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
673 |
lemma mult_cancel_left [simp]: |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
674 |
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
|
675 |
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
|
676 |
proof - |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
677 |
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
|
678 |
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
|
679 |
thus ?thesis |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
680 |
by simp |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
681 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
682 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
683 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
684 |
subsubsection{*Special Cancellation Simprules for Multiplication*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
685 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
686 |
text{*These also produce two cases when the comparison is a goal.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
687 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
688 |
lemma mult_le_cancel_right1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
689 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
690 |
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
|
691 |
by (insert mult_le_cancel_right [of 1 c b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
692 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
693 |
lemma mult_le_cancel_right2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
694 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
695 |
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
|
696 |
by (insert mult_le_cancel_right [of a c 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
697 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
698 |
lemma mult_le_cancel_left1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
699 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
700 |
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
|
701 |
by (insert mult_le_cancel_left [of c 1 b], simp) |
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_left2: |
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*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
|
706 |
by (insert mult_le_cancel_left [of c a 1], 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_less_cancel_right1: |
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 "(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
|
711 |
by (insert mult_less_cancel_right [of 1 c b], 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_less_cancel_right2: |
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 "(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
|
716 |
by (insert mult_less_cancel_right [of a c 1], 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_less_cancel_left1: |
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 < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
721 |
by (insert mult_less_cancel_left [of c 1 b], 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_left2: |
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*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
726 |
by (insert mult_less_cancel_left [of c a 1], 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_cancel_right1 [simp]: |
23544 | 729 |
fixes c :: "'a :: ring_1_no_zero_divisors" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
730 |
shows "(c = b*c) = (c = 0 | b=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
731 |
by (insert mult_cancel_right [of 1 c b], force) |
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_cancel_right2 [simp]: |
23544 | 734 |
fixes c :: "'a :: ring_1_no_zero_divisors" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
735 |
shows "(a*c = c) = (c = 0 | a=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
736 |
by (insert mult_cancel_right [of a c 1], 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_cancel_left1 [simp]: |
23544 | 739 |
fixes c :: "'a :: ring_1_no_zero_divisors" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
740 |
shows "(c = c*b) = (c = 0 | b=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
741 |
by (insert mult_cancel_left [of c 1 b], force) |
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_left2 [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*a = c) = (c = 0 | a=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
746 |
by (insert mult_cancel_left [of c a 1], simp) |
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 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
749 |
text{*Simprules for comparisons where common factors can be cancelled.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
750 |
lemmas mult_compare_simps = |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
751 |
mult_le_cancel_right mult_le_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
752 |
mult_le_cancel_right1 mult_le_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
753 |
mult_le_cancel_left1 mult_le_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
754 |
mult_less_cancel_right mult_less_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
755 |
mult_less_cancel_right1 mult_less_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
756 |
mult_less_cancel_left1 mult_less_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
757 |
mult_cancel_right mult_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
758 |
mult_cancel_right1 mult_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
759 |
mult_cancel_left1 mult_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
760 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
761 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
762 |
subsection {* Fields *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
763 |
|
14288 | 764 |
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" |
765 |
proof |
|
766 |
assume neq: "b \<noteq> 0" |
|
767 |
{ |
|
768 |
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) |
|
769 |
also assume "a / b = 1" |
|
770 |
finally show "a = b" by simp |
|
771 |
next |
|
772 |
assume "a = b" |
|
773 |
with neq show "a / b = 1" by (simp add: divide_inverse) |
|
774 |
} |
|
775 |
qed |
|
776 |
||
777 |
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" |
|
778 |
by (simp add: divide_inverse) |
|
779 |
||
23398 | 780 |
lemma divide_self[simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1" |
14288 | 781 |
by (simp add: divide_inverse) |
782 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
783 |
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
|
784 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
785 |
|
15228 | 786 |
lemma divide_self_if [simp]: |
787 |
"a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)" |
|
788 |
by (simp add: divide_self) |
|
789 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
790 |
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
|
791 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
792 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
793 |
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
|
794 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
795 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
796 |
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
|
797 |
by (simp add: divide_inverse ring_distribs) |
14293 | 798 |
|
23482 | 799 |
(* what ordering?? this is a straight instance of mult_eq_0_iff |
14270 | 800 |
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement |
801 |
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
|
802 |
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
|
803 |
"(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
|
804 |
by simp |
23482 | 805 |
*) |
23496 | 806 |
(* 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
|
807 |
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
|
808 |
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
|
809 |
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
|
810 |
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
|
811 |
shows "a=b" |
14377 | 812 |
proof - |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
813 |
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
|
814 |
by (simp add: eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
815 |
thus "a=b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
816 |
by (simp add: mult_assoc cnz) |
14377 | 817 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
818 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
819 |
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
|
820 |
"(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
|
821 |
by simp |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
822 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
823 |
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
|
824 |
"(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
|
825 |
by simp |
23496 | 826 |
*) |
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
|
827 |
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
|
828 |
"a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)" |
14377 | 829 |
proof |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
830 |
assume ianz: "inverse a = 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
831 |
assume "a \<noteq> 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
832 |
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
|
833 |
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
|
834 |
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
|
835 |
thus False by (simp add: eq_commute) |
14377 | 836 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
837 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
838 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
839 |
subsection{*Basic Properties of @{term inverse}*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
840 |
|
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
|
841 |
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
|
842 |
apply (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
843 |
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
|
844 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
845 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
846 |
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
|
847 |
"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
|
848 |
apply (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
849 |
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
|
850 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
851 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
852 |
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
|
853 |
"(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
|
854 |
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
|
855 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
856 |
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
|
857 |
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
|
858 |
shows "inverse(-a) = -inverse(a::'a::division_ring)" |
14377 | 859 |
proof - |
860 |
have "-a * inverse (- a) = -a * - inverse a" |
|
861 |
by simp |
|
862 |
thus ?thesis |
|
23496 | 863 |
by (simp only: mult_cancel_left, simp) |
14377 | 864 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
865 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
866 |
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
|
867 |
"inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})" |
14377 | 868 |
proof cases |
869 |
assume "a=0" thus ?thesis by (simp add: inverse_zero) |
|
870 |
next |
|
871 |
assume "a\<noteq>0" |
|
872 |
thus ?thesis by (simp add: nonzero_inverse_minus_eq) |
|
873 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
874 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
875 |
lemma nonzero_inverse_eq_imp_eq: |
14269 | 876 |
assumes inveq: "inverse a = inverse b" |
877 |
and anz: "a \<noteq> 0" |
|
878 |
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
|
879 |
shows "a = (b::'a::division_ring)" |
14377 | 880 |
proof - |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
881 |
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
|
882 |
by (simp add: inveq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
883 |
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
|
884 |
by simp |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
885 |
thus "a = b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
886 |
by (simp add: mult_assoc anz bnz) |
14377 | 887 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
888 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
889 |
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
|
890 |
"inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})" |
21328 | 891 |
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
|
892 |
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
|
893 |
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
|
894 |
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
|
895 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
896 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
897 |
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
|
898 |
"(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
|
899 |
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
|
900 |
|
14270 | 901 |
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
|
902 |
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
|
903 |
shows "inverse(inverse (a::'a::division_ring)) = a" |
14270 | 904 |
proof - |
905 |
have "(inverse (inverse a) * inverse a) * a = a" |
|
906 |
by (simp add: nonzero_imp_inverse_nonzero) |
|
907 |
thus ?thesis |
|
908 |
by (simp add: mult_assoc) |
|
909 |
qed |
|
910 |
||
911 |
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
|
912 |
"inverse(inverse (a::'a::{division_ring,division_by_zero})) = a" |
14270 | 913 |
proof cases |
914 |
assume "a=0" thus ?thesis by simp |
|
915 |
next |
|
916 |
assume "a\<noteq>0" |
|
917 |
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) |
|
918 |
qed |
|
919 |
||
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
|
920 |
lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)" |
14270 | 921 |
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
|
922 |
have "inverse 1 * 1 = (1::'a::division_ring)" |
14270 | 923 |
by (rule left_inverse [OF zero_neq_one [symmetric]]) |
924 |
thus ?thesis by simp |
|
925 |
qed |
|
926 |
||
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
927 |
lemma inverse_unique: |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
928 |
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
|
929 |
shows "inverse a = (b::'a::division_ring)" |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
930 |
proof - |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
931 |
have "a \<noteq> 0" using ab by auto |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
932 |
moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
933 |
ultimately show ?thesis by (simp add: mult_assoc [symmetric]) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
934 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
935 |
|
14270 | 936 |
lemma nonzero_inverse_mult_distrib: |
937 |
assumes anz: "a \<noteq> 0" |
|
938 |
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
|
939 |
shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)" |
14270 | 940 |
proof - |
941 |
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" |
|
23482 | 942 |
by (simp add: anz bnz) |
14270 | 943 |
hence "inverse(a*b) * a = inverse(b)" |
944 |
by (simp add: mult_assoc bnz) |
|
945 |
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" |
|
946 |
by simp |
|
947 |
thus ?thesis |
|
948 |
by (simp add: mult_assoc anz) |
|
949 |
qed |
|
950 |
||
951 |
text{*This version builds in division by zero while also re-orienting |
|
952 |
the right-hand side.*} |
|
953 |
lemma inverse_mult_distrib [simp]: |
|
954 |
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" |
|
955 |
proof cases |
|
956 |
assume "a \<noteq> 0 & b \<noteq> 0" |
|
22993 | 957 |
thus ?thesis |
958 |
by (simp add: nonzero_inverse_mult_distrib mult_commute) |
|
14270 | 959 |
next |
960 |
assume "~ (a \<noteq> 0 & b \<noteq> 0)" |
|
22993 | 961 |
thus ?thesis |
962 |
by force |
|
14270 | 963 |
qed |
964 |
||
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
|
965 |
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
|
966 |
"[|(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
|
967 |
==> 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
|
968 |
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
|
969 |
|
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
|
970 |
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
|
971 |
"[|(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
|
972 |
==> 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
|
973 |
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
|
974 |
|
14270 | 975 |
text{*There is no slick version using division by zero.*} |
976 |
lemma inverse_add: |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
977 |
"[|a \<noteq> 0; b \<noteq> 0|] |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
978 |
==> 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
|
979 |
by (simp add: division_ring_inverse_add mult_ac) |
14270 | 980 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
981 |
lemma inverse_divide [simp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
982 |
"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
|
983 |
by (simp add: divide_inverse mult_commute) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
984 |
|
23389 | 985 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
986 |
subsection {* Calculations with fractions *} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
987 |
|
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
988 |
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
|
989 |
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
|
990 |
because the latter are covered by a simproc. *} |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
991 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
992 |
lemma nonzero_mult_divide_mult_cancel_left[simp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
993 |
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
|
994 |
proof - |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
995 |
have "(c*a)/(c*b) = c * a * (inverse b * inverse c)" |
23482 | 996 |
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
|
997 |
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
|
998 |
by (simp only: mult_ac) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
999 |
also have "... = a * inverse b" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1000 |
by simp |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1001 |
finally show ?thesis |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1002 |
by (simp add: divide_inverse) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1003 |
qed |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1004 |
|
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1005 |
lemma mult_divide_mult_cancel_left: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1006 |
"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})" |
21328 | 1007 |
apply (cases "b = 0") |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1008 |
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
|
1009 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1010 |
|
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1011 |
lemma nonzero_mult_divide_mult_cancel_right: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1012 |
"[|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
|
1013 |
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) |
14321 | 1014 |
|
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1015 |
lemma mult_divide_mult_cancel_right: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1016 |
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" |
21328 | 1017 |
apply (cases "b = 0") |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1018 |
apply (simp_all add: nonzero_mult_divide_mult_cancel_right) |
14321 | 1019 |
done |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1020 |
|
14284
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
1021 |
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
|
1022 |
by (simp add: divide_inverse) |
14284
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
1023 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1024 |
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
|
1025 |
by (simp add: divide_inverse mult_assoc) |
14288 | 1026 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1027 |
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
|
1028 |
by (simp add: divide_inverse mult_ac) |
14288 | 1029 |
|
23482 | 1030 |
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left |
1031 |
||
14288 | 1032 |
lemma divide_divide_eq_right [simp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1033 |
"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
|
1034 |
by (simp add: divide_inverse mult_ac) |
14288 | 1035 |
|
1036 |
lemma divide_divide_eq_left [simp]: |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1037 |
"(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
|
1038 |
by (simp add: divide_inverse mult_assoc) |
14288 | 1039 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1040 |
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
|
1041 |
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
|
1042 |
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
|
1043 |
apply (erule ssubst) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1044 |
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
|
1045 |
apply (erule ssubst) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1046 |
apply (rule add_divide_distrib [THEN sym]) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1047 |
apply (subst mult_commute) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1048 |
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
|
1049 |
apply assumption |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1050 |
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
|
1051 |
apply assumption |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1052 |
done |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1053 |
|
23389 | 1054 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1055 |
subsubsection{*Special Cancellation Simprules for Division*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1056 |
|
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1057 |
lemma mult_divide_mult_cancel_left_if[simp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1058 |
fixes c :: "'a :: {field,division_by_zero}" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1059 |
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
|
1060 |
by (simp add: mult_divide_mult_cancel_left) |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1061 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1062 |
lemma nonzero_mult_divide_cancel_right[simp]: |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1063 |
"b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)" |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1064 |
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
|
1065 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1066 |
lemma nonzero_mult_divide_cancel_left[simp]: |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1067 |
"a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)" |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1068 |
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
|
1069 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1070 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1071 |
lemma nonzero_divide_mult_cancel_right[simp]: |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1072 |
"\<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
|
1073 |
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
|
1074 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1075 |
lemma nonzero_divide_mult_cancel_left[simp]: |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1076 |
"\<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
|
1077 |
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
|
1078 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1079 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1080 |
lemma nonzero_mult_divide_mult_cancel_left2[simp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1081 |
"[|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
|
1082 |
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
|
1083 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1084 |
lemma nonzero_mult_divide_mult_cancel_right2[simp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1085 |
"[|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
|
1086 |
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
|
1087 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1088 |
|
14293 | 1089 |
subsection {* Division and Unary Minus *} |
1090 |
||
1091 |
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)" |
|
1092 |
by (simp add: divide_inverse minus_mult_left) |
|
1093 |
||
1094 |
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)" |
|
1095 |
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) |
|
1096 |
||
1097 |
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)" |
|
1098 |
by (simp add: divide_inverse nonzero_inverse_minus_eq) |
|
1099 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1100 |
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
|
1101 |
by (simp add: divide_inverse minus_mult_left [symmetric]) |
14293 | 1102 |
|
1103 |
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
|
1104 |
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
|
1105 |
|
14293 | 1106 |
|
1107 |
text{*The effect is to extract signs from divisions*} |
|
17085 | 1108 |
lemmas divide_minus_left = minus_divide_left [symmetric] |
1109 |
lemmas divide_minus_right = minus_divide_right [symmetric] |
|
1110 |
declare divide_minus_left [simp] divide_minus_right [simp] |
|
14293 | 1111 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1112 |
text{*Also, extract signs from products*} |
17085 | 1113 |
lemmas mult_minus_left = minus_mult_left [symmetric] |
1114 |
lemmas mult_minus_right = minus_mult_right [symmetric] |
|
1115 |
declare mult_minus_left [simp] mult_minus_right [simp] |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1116 |
|
14293 | 1117 |
lemma minus_divide_divide [simp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1118 |
"(-a)/(-b) = a / (b::'a::{field,division_by_zero})" |
21328 | 1119 |
apply (cases "b=0", simp) |
14293 | 1120 |
apply (simp add: nonzero_minus_divide_divide) |
1121 |
done |
|
1122 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1123 |
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
|
1124 |
by (simp add: diff_minus add_divide_distrib) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1125 |
|
23482 | 1126 |
lemma add_divide_eq_iff: |
1127 |
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z" |
|
1128 |
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left) |
|
1129 |
||
1130 |
lemma divide_add_eq_iff: |
|
1131 |
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z" |
|
1132 |
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left) |
|
1133 |
||
1134 |
lemma diff_divide_eq_iff: |
|
1135 |
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z" |
|
1136 |
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left) |
|
1137 |
||
1138 |
lemma divide_diff_eq_iff: |
|
1139 |
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z" |
|
1140 |
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left) |
|
1141 |
||
1142 |
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)" |
|
1143 |
proof - |
|
1144 |
assume [simp]: "c\<noteq>0" |
|
23496 | 1145 |
have "(a = b/c) = (a*c = (b/c)*c)" by simp |
1146 |
also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc) |
|
23482 | 1147 |
finally show ?thesis . |
1148 |
qed |
|
1149 |
||
1150 |
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)" |
|
1151 |
proof - |
|
1152 |
assume [simp]: "c\<noteq>0" |
|
23496 | 1153 |
have "(b/c = a) = ((b/c)*c = a*c)" by simp |
1154 |
also have "... = (b = a*c)" by (simp add: divide_inverse mult_assoc) |
|
23482 | 1155 |
finally show ?thesis . |
1156 |
qed |
|
1157 |
||
1158 |
lemma eq_divide_eq: |
|
1159 |
"((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)" |
|
1160 |
by (simp add: nonzero_eq_divide_eq) |
|
1161 |
||
1162 |
lemma divide_eq_eq: |
|
1163 |
"(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)" |
|
1164 |
by (force simp add: nonzero_divide_eq_eq) |
|
1165 |
||
1166 |
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==> |
|
1167 |
b = a * c ==> b / c = a" |
|
1168 |
by (subst divide_eq_eq, simp) |
|
1169 |
||
1170 |
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==> |
|
1171 |
a * c = b ==> a = b / c" |
|
1172 |
by (subst eq_divide_eq, simp) |
|
1173 |
||
1174 |
||
1175 |
lemmas field_eq_simps = ring_simps |
|
1176 |
(* pull / out*) |
|
1177 |
add_divide_eq_iff divide_add_eq_iff |
|
1178 |
diff_divide_eq_iff divide_diff_eq_iff |
|
1179 |
(* multiply eqn *) |
|
1180 |
nonzero_eq_divide_eq nonzero_divide_eq_eq |
|
1181 |
(* is added later: |
|
1182 |
times_divide_eq_left times_divide_eq_right |
|
1183 |
*) |
|
1184 |
||
1185 |
text{*An example:*} |
|
1186 |
lemma fixes a b c d e f :: "'a::field" |
|
1187 |
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" |
|
1188 |
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0") |
|
1189 |
apply(simp add:field_eq_simps) |
|
1190 |
apply(simp) |
|
1191 |
done |
|
1192 |
||
1193 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1194 |
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
|
1195 |
x / y - w / z = (x * z - w * y) / (y * z)" |
23482 | 1196 |
by (simp add:field_eq_simps times_divide_eq) |
1197 |
||
1198 |
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==> |
|
1199 |
(x / y = w / z) = (x * z = w * y)" |
|
1200 |
by (simp add:field_eq_simps times_divide_eq) |
|
14293 | 1201 |
|
23389 | 1202 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1203 |
subsection {* Ordered Fields *} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1204 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1205 |
lemma positive_imp_inverse_positive: |
23482 | 1206 |
assumes a_gt_0: "0 < a" shows "0 < inverse (a::'a::ordered_field)" |
1207 |
proof - |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1208 |
have "0 < a * inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1209 |
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
|
1210 |
thus "0 < inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1211 |
by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff) |
23482 | 1212 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1213 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1214 |
lemma negative_imp_inverse_negative: |
23482 | 1215 |
"a < 0 ==> inverse a < (0::'a::ordered_field)" |
1216 |
by (insert positive_imp_inverse_positive [of "-a"], |
|
1217 |
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
|
1218 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1219 |
lemma inverse_le_imp_le: |
23482 | 1220 |
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a" |
1221 |
shows "b \<le> (a::'a::ordered_field)" |
|
1222 |
proof (rule classical) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1223 |
assume "~ b \<le> a" |
23482 | 1224 |
hence "a < b" by (simp add: linorder_not_le) |
1225 |
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
|
1226 |
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
|
1227 |
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
|
1228 |
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
|
1229 |
by (simp add: bpos order_less_imp_le mult_right_mono) |
23482 | 1230 |
thus "b \<le> a" by (simp add: mult_assoc apos bpos order_less_imp_not_eq2) |
1231 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1232 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1233 |
lemma inverse_positive_imp_positive: |
23482 | 1234 |
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0" |
1235 |
shows "0 < (a::'a::ordered_field)" |
|
23389 | 1236 |
proof - |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1237 |
have "0 < inverse (inverse a)" |
23389 | 1238 |
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
|
1239 |
thus "0 < a" |
23389 | 1240 |
using nz by (simp add: nonzero_inverse_inverse_eq) |
1241 |
qed |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1242 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1243 |
lemma inverse_positive_iff_positive [simp]: |
23482 | 1244 |
"(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))" |
21328 | 1245 |
apply (cases "a = 0", simp) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1246 |
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
|
1247 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1248 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1249 |
lemma inverse_negative_imp_negative: |
23482 | 1250 |
assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0" |
1251 |
shows "a < (0::'a::ordered_field)" |
|
23389 | 1252 |
proof - |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1253 |
have "inverse (inverse a) < 0" |
23389 | 1254 |
using inv_less_0 by (rule negative_imp_inverse_negative) |
23482 | 1255 |
thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq) |
23389 | 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_negative_iff_negative [simp]: |
23482 | 1259 |
"(inverse a < 0) = (a < (0::'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_negative_imp_negative negative_imp_inverse_negative) |
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_nonnegative_iff_nonnegative [simp]: |
23482 | 1265 |
"(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
|
1266 |
by (simp add: linorder_not_less [symmetric]) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1267 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1268 |
lemma inverse_nonpositive_iff_nonpositive [simp]: |
23482 | 1269 |
"(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
|
1270 |
by (simp add: linorder_not_less [symmetric]) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1271 |
|
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
|
1272 |
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
|
1273 |
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
|
1274 |
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
|
1275 |
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
|
1276 |
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
|
1277 |
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
|
1278 |
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
|
1279 |
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
|
1280 |
|
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
|
1281 |
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
|
1282 |
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
|
1283 |
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
|
1284 |
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
|
1285 |
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
|
1286 |
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
|
1287 |
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
|
1288 |
qed |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1289 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1290 |
subsection{*Anti-Monotonicity of @{term inverse}*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1291 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1292 |
lemma less_imp_inverse_less: |
23482 | 1293 |
assumes less: "a < b" and apos: "0 < a" |
1294 |
shows "inverse b < inverse (a::'a::ordered_field)" |
|
1295 |
proof (rule ccontr) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1296 |
assume "~ inverse b < inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1297 |
hence "inverse a \<le> inverse b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1298 |
by (simp add: linorder_not_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1299 |
hence "~ (a < b)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1300 |
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
|
1301 |
thus False |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1302 |
by (rule notE [OF _ less]) |
23482 | 1303 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1304 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1305 |
lemma inverse_less_imp_less: |
23482 | 1306 |
"[|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
|
1307 |
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
|
1308 |
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
|
1309 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1310 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1311 |
text{*Both premises are essential. Consider -1 and 1.*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1312 |
lemma inverse_less_iff_less [simp]: |
23482 | 1313 |
"[|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
|
1314 |
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
|
1315 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1316 |
lemma le_imp_inverse_le: |
23482 | 1317 |
"[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)" |
1318 |
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
|
1319 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1320 |
lemma inverse_le_iff_le [simp]: |
23482 | 1321 |
"[|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
|
1322 |
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
|
1323 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1324 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1325 |
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
|
1326 |
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
|
1327 |
lemma inverse_le_imp_le_neg: |
23482 | 1328 |
"[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)" |
1329 |
apply (rule classical) |
|
1330 |
apply (subgoal_tac "a < 0") |
|
1331 |
prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) |
|
1332 |
apply (insert inverse_le_imp_le [of "-b" "-a"]) |
|
1333 |
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
|
1334 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1335 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1336 |
lemma less_imp_inverse_less_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1337 |
"[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)" |
23482 | 1338 |
apply (subgoal_tac "a < 0") |
1339 |
prefer 2 apply (blast intro: order_less_trans) |
|
1340 |
apply (insert less_imp_inverse_less [of "-b" "-a"]) |
|
1341 |
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
|
1342 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1343 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1344 |
lemma inverse_less_imp_less_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1345 |
"[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)" |
23482 | 1346 |
apply (rule classical) |
1347 |
apply (subgoal_tac "a < 0") |
|
1348 |
prefer 2 |
|
1349 |
apply (force simp add: linorder_not_less intro: order_le_less_trans) |
|
1350 |
apply (insert inverse_less_imp_less [of "-b" "-a"]) |
|
1351 |
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
|
1352 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1353 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1354 |
lemma inverse_less_iff_less_neg [simp]: |
23482 | 1355 |
"[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))" |
1356 |
apply (insert inverse_less_iff_less [of "-b" "-a"]) |
|
1357 |
apply (simp del: inverse_less_iff_less |
|
1358 |
add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
|
1359 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1360 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1361 |
lemma le_imp_inverse_le_neg: |
23482 | 1362 |
"[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)" |
1363 |
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
|
1364 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1365 |
lemma inverse_le_iff_le_neg [simp]: |
23482 | 1366 |
"[|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
|
1367 |
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
|
1368 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1369 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1370 |
subsection{*Inverses and the Number One*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1371 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1372 |
lemma one_less_inverse_iff: |
23482 | 1373 |
"(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))" |
1374 |
proof cases |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1375 |
assume "0 < x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1376 |
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
|
1377 |
show ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1378 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1379 |
assume notless: "~ (0 < x)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1380 |
have "~ (1 < inverse x)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1381 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1382 |
assume "1 < inverse x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1383 |
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
|
1384 |
also have "... < 1" by (rule zero_less_one) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1385 |
finally show False by auto |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1386 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1387 |
with notless show ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1388 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1389 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1390 |
lemma inverse_eq_1_iff [simp]: |
23482 | 1391 |
"(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
|
1392 |
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
|
1393 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1394 |
lemma one_le_inverse_iff: |
23482 | 1395 |
"(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
|
1396 |
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
|
1397 |
eq_commute [of 1]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1398 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1399 |
lemma inverse_less_1_iff: |
23482 | 1400 |
"(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
|
1401 |
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
|
1402 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1403 |
lemma inverse_le_1_iff: |
23482 | 1404 |
"(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
|
1405 |
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
|
1406 |
|
23389 | 1407 |
|
14288 | 1408 |
subsection{*Simplification of Inequalities Involving Literal Divisors*} |
1409 |
||
1410 |
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)" |
|
1411 |
proof - |
|
1412 |
assume less: "0<c" |
|
1413 |
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" |
|
1414 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1415 |
also have "... = (a*c \<le> b)" |
|
1416 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1417 |
finally show ?thesis . |
|
1418 |
qed |
|
1419 |
||
1420 |
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)" |
|
1421 |
proof - |
|
1422 |
assume less: "c<0" |
|
1423 |
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" |
|
1424 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1425 |
also have "... = (b \<le> a*c)" |
|
1426 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1427 |
finally show ?thesis . |
|
1428 |
qed |
|
1429 |
||
1430 |
lemma le_divide_eq: |
|
1431 |
"(a \<le> b/c) = |
|
1432 |
(if 0 < c then a*c \<le> b |
|
1433 |
else if c < 0 then b \<le> a*c |
|
1434 |
else a \<le> (0::'a::{ordered_field,division_by_zero}))" |
|
21328 | 1435 |
apply (cases "c=0", simp) |
14288 | 1436 |
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) |
1437 |
done |
|
1438 |
||
1439 |
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)" |
|
1440 |
proof - |
|
1441 |
assume less: "0<c" |
|
1442 |
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" |
|
1443 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1444 |
also have "... = (b \<le> a*c)" |
|
1445 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1446 |
finally show ?thesis . |
|
1447 |
qed |
|
1448 |
||
1449 |
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)" |
|
1450 |
proof - |
|
1451 |
assume less: "c<0" |
|
1452 |
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" |
|
1453 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1454 |
also have "... = (a*c \<le> b)" |
|
1455 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1456 |
finally show ?thesis . |
|
1457 |
qed |
|
1458 |
||
1459 |
lemma divide_le_eq: |
|
1460 |
"(b/c \<le> a) = |
|
1461 |
(if 0 < c then b \<le> a*c |
|
1462 |
else if c < 0 then a*c \<le> b |
|
1463 |
else 0 \<le> (a::'a::{ordered_field,division_by_zero}))" |
|
21328 | 1464 |
apply (cases "c=0", simp) |
14288 | 1465 |
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) |
1466 |
done |
|
1467 |
||
1468 |
lemma pos_less_divide_eq: |
|
1469 |
"0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)" |
|
1470 |
proof - |
|
1471 |
assume less: "0<c" |
|
1472 |
hence "(a < b/c) = (a*c < (b/c)*c)" |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1473 |
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) |
14288 | 1474 |
also have "... = (a*c < b)" |
1475 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1476 |
finally show ?thesis . |
|
1477 |
qed |
|
1478 |
||
1479 |
lemma neg_less_divide_eq: |
|
1480 |
"c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)" |
|
1481 |
proof - |
|
1482 |
assume less: "c<0" |
|
1483 |
hence "(a < b/c) = ((b/c)*c < a*c)" |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1484 |
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) |
14288 | 1485 |
also have "... = (b < a*c)" |
1486 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1487 |
finally show ?thesis . |
|
1488 |
qed |
|
1489 |
||
1490 |
lemma less_divide_eq: |
|
1491 |
"(a < b/c) = |
|
1492 |
(if 0 < c then a*c < b |
|
1493 |
else if c < 0 then b < a*c |
|
1494 |
else a < (0::'a::{ordered_field,division_by_zero}))" |
|
21328 | 1495 |
apply (cases "c=0", simp) |
14288 | 1496 |
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) |
1497 |
done |
|
1498 |
||
1499 |
lemma pos_divide_less_eq: |
|
1500 |
"0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)" |
|
1501 |
proof - |
|
1502 |
assume less: "0<c" |
|
1503 |
hence "(b/c < a) = ((b/c)*c < a*c)" |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1504 |
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) |
14288 | 1505 |
also have "... = (b < a*c)" |
1506 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1507 |
finally show ?thesis . |
|
1508 |
qed |
|
1509 |
||
1510 |
lemma neg_divide_less_eq: |
|
1511 |
"c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)" |
|
1512 |
proof - |
|
1513 |
assume less: "c<0" |
|
1514 |
hence "(b/c < a) = (a*c < (b/c)*c)" |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1515 |
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) |
14288 | 1516 |
also have "... = (a*c < b)" |
1517 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1518 |
finally show ?thesis . |
|
1519 |
qed |
|
1520 |
||
1521 |
lemma divide_less_eq: |
|
1522 |
"(b/c < a) = |
|
1523 |
(if 0 < c then b < a*c |
|
1524 |
else if c < 0 then a*c < b |
|
1525 |
else 0 < (a::'a::{ordered_field,division_by_zero}))" |
|
21328 | 1526 |
apply (cases "c=0", simp) |
14288 | 1527 |
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) |
1528 |
done |
|
1529 |
||
23482 | 1530 |
|
1531 |
subsection{*Field simplification*} |
|
1532 |
||
1533 |
text{* Lemmas @{text field_simps} multiply with denominators in |
|
1534 |
in(equations) if they can be proved to be non-zero (for equations) or |
|
1535 |
positive/negative (for inequations). *} |
|
14288 | 1536 |
|
23482 | 1537 |
lemmas field_simps = field_eq_simps |
1538 |
(* multiply ineqn *) |
|
1539 |
pos_divide_less_eq neg_divide_less_eq |
|
1540 |
pos_less_divide_eq neg_less_divide_eq |
|
1541 |
pos_divide_le_eq neg_divide_le_eq |
|
1542 |
pos_le_divide_eq neg_le_divide_eq |
|
14288 | 1543 |
|
23482 | 1544 |
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs |
23483 | 1545 |
of positivity/negativity needed for @{text field_simps}. Have not added @{text |
23482 | 1546 |
sign_simps} to @{text field_simps} because the former can lead to case |
1547 |
explosions. *} |
|
14288 | 1548 |
|
23482 | 1549 |
lemmas sign_simps = group_simps |
1550 |
zero_less_mult_iff mult_less_0_iff |
|
14288 | 1551 |
|
23482 | 1552 |
(* Only works once linear arithmetic is installed: |
1553 |
text{*An example:*} |
|
1554 |
lemma fixes a b c d e f :: "'a::ordered_field" |
|
1555 |
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow> |
|
1556 |
((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) < |
|
1557 |
((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u" |
|
1558 |
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0") |
|
1559 |
prefer 2 apply(simp add:sign_simps) |
|
1560 |
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0") |
|
1561 |
prefer 2 apply(simp add:sign_simps) |
|
1562 |
apply(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1563 |
done |
23482 | 1564 |
*) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1565 |
|
23389 | 1566 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1567 |
subsection{*Division and Signs*} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1568 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1569 |
lemma zero_less_divide_iff: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1570 |
"((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
|
1571 |
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
|
1572 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1573 |
lemma divide_less_0_iff: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1574 |
"(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
|
1575 |
(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
|
1576 |
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
|
1577 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1578 |
lemma zero_le_divide_iff: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1579 |
"((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
|
1580 |
(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
|
1581 |
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
|
1582 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1583 |
lemma divide_le_0_iff: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1584 |
"(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
|
1585 |
(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
|
1586 |
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
|
1587 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1588 |
lemma divide_eq_0_iff [simp]: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1589 |
"(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))" |
23482 | 1590 |
by (simp add: divide_inverse) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1591 |
|
23482 | 1592 |
lemma divide_pos_pos: |
1593 |
"0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y" |
|
1594 |
by(simp add:field_simps) |
|
1595 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1596 |
|
23482 | 1597 |
lemma divide_nonneg_pos: |
1598 |
"0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y" |
|
1599 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1600 |
|
23482 | 1601 |
lemma divide_neg_pos: |
1602 |
"(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0" |
|
1603 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1604 |
|
23482 | 1605 |
lemma divide_nonpos_pos: |
1606 |
"(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0" |
|
1607 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1608 |
|
23482 | 1609 |
lemma divide_pos_neg: |
1610 |
"0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0" |
|
1611 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1612 |
|
23482 | 1613 |
lemma divide_nonneg_neg: |
1614 |
"0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0" |
|
1615 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1616 |
|
23482 | 1617 |
lemma divide_neg_neg: |
1618 |
"(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y" |
|
1619 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1620 |
|
23482 | 1621 |
lemma divide_nonpos_neg: |
1622 |
"(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y" |
|
1623 |
by(simp add:field_simps) |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1624 |
|
23389 | 1625 |
|
14288 | 1626 |
subsection{*Cancellation Laws for Division*} |
1627 |
||
1628 |
lemma divide_cancel_right [simp]: |
|
1629 |
"(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))" |
|
23482 | 1630 |
apply (cases "c=0", simp) |
23496 | 1631 |
apply (simp add: divide_inverse) |
14288 | 1632 |
done |
1633 |
||
1634 |
lemma divide_cancel_left [simp]: |
|
1635 |
"(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" |
|
23482 | 1636 |
apply (cases "c=0", simp) |
23496 | 1637 |
apply (simp add: divide_inverse) |
14288 | 1638 |
done |
1639 |
||
23389 | 1640 |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1641 |
subsection {* Division and the Number One *} |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1642 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1643 |
text{*Simplify expressions equated with 1*} |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1644 |
lemma divide_eq_1_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1645 |
"(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" |
23482 | 1646 |
apply (cases "b=0", simp) |
1647 |
apply (simp add: right_inverse_eq) |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1648 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1649 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1650 |
lemma one_eq_divide_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1651 |
"(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" |
23482 | 1652 |
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
|
1653 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1654 |
lemma zero_eq_1_divide_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1655 |
"((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)" |
23482 | 1656 |
apply (cases "a=0", simp) |
1657 |
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
|
1658 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1659 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1660 |
lemma one_divide_eq_0_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1661 |
"(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)" |
23482 | 1662 |
apply (cases "a=0", simp) |
1663 |
apply (insert zero_neq_one [THEN not_sym]) |
|
1664 |
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
|
1665 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1666 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1667 |
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*} |
18623 | 1668 |
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified] |
1669 |
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified] |
|
1670 |
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified] |
|
1671 |
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified] |
|
17085 | 1672 |
|
1673 |
declare zero_less_divide_1_iff [simp] |
|
1674 |
declare divide_less_0_1_iff [simp] |
|
1675 |
declare zero_le_divide_1_iff [simp] |
|
1676 |
declare divide_le_0_1_iff [simp] |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1677 |
|
23389 | 1678 |
|
14293 | 1679 |
subsection {* Ordering Rules for Division *} |
1680 |
||
1681 |
lemma divide_strict_right_mono: |
|
1682 |
"[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)" |
|
1683 |
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono |
|
23482 | 1684 |
positive_imp_inverse_positive) |
14293 | 1685 |
|
1686 |
lemma divide_right_mono: |
|
1687 |
"[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})" |
|
23482 | 1688 |
by (force simp add: divide_strict_right_mono order_le_less) |
14293 | 1689 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1690 |
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
|
1691 |
==> c <= 0 ==> b / c <= a / c" |
23482 | 1692 |
apply (drule divide_right_mono [of _ _ "- c"]) |
1693 |
apply auto |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1694 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1695 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1696 |
lemma divide_strict_right_mono_neg: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1697 |
"[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)" |
23482 | 1698 |
apply (drule divide_strict_right_mono [of _ _ "-c"], simp) |
1699 |
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
|
1700 |
done |
14293 | 1701 |
|
1702 |
text{*The last premise ensures that @{term a} and @{term b} |
|
1703 |
have the same sign*} |
|
1704 |
lemma divide_strict_left_mono: |
|
23482 | 1705 |
"[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)" |
1706 |
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono) |
|
14293 | 1707 |
|
1708 |
lemma divide_left_mono: |
|
23482 | 1709 |
"[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)" |
1710 |
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono) |
|
14293 | 1711 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1712 |
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
|
1713 |
==> 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
|
1714 |
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
|
1715 |
apply (auto simp add: mult_commute) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1716 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1717 |
|
14293 | 1718 |
lemma divide_strict_left_mono_neg: |
23482 | 1719 |
"[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)" |
1720 |
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg) |
|
1721 |
||
14293 | 1722 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1723 |
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
|
1724 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1725 |
lemma le_divide_eq_1: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1726 |
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
|
1727 |
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
|
1728 |
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
|
1729 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1730 |
lemma divide_le_eq_1: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1731 |
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
|
1732 |
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
|
1733 |
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
|
1734 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1735 |
lemma less_divide_eq_1: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1736 |
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
|
1737 |
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
|
1738 |
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
|
1739 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1740 |
lemma divide_less_eq_1: |
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 "(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
|
1743 |
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
|
1744 |
|
23389 | 1745 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1746 |
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
|
1747 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1748 |
lemma le_divide_eq_1_pos [simp]: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1749 |
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
|
1750 |
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
|
1751 |
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
|
1752 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1753 |
lemma le_divide_eq_1_neg [simp]: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1754 |
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
|
1755 |
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
|
1756 |
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
|
1757 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1758 |
lemma divide_le_eq_1_pos [simp]: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1759 |
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
|
1760 |
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
|
1761 |
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
|
1762 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1763 |
lemma divide_le_eq_1_neg [simp]: |
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 "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
|
1766 |
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
|
1767 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1768 |
lemma less_divide_eq_1_pos [simp]: |
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 "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
|
1771 |
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
|
1772 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1773 |
lemma less_divide_eq_1_neg [simp]: |
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 "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
|
1776 |
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
|
1777 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1778 |
lemma divide_less_eq_1_pos [simp]: |
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 "0 < a \<Longrightarrow> (b/a < 1) = (b < a)" |
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1781 |
by (auto simp add: divide_less_eq) |
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1782 |
|
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1783 |
lemma divide_less_eq_1_neg [simp]: |
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1784 |
fixes a :: "'a :: {ordered_field,division_by_zero}" |
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset
|
1785 |
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
|
1786 |
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
|
1787 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1788 |
lemma eq_divide_eq_1 [simp]: |
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 "(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
|
1791 |
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
|
1792 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1793 |
lemma divide_eq_eq_1 [simp]: |
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 "(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
|
1796 |
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
|
1797 |
|
23389 | 1798 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1799 |
subsection {* Reasoning about inequalities with division *} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1800 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1801 |
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
|
1802 |
==> x * y <= x" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1803 |
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
|
1804 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1805 |
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
|
1806 |
==> y * x <= x" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1807 |
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
|
1808 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1809 |
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
|
1810 |
x / y <= z"; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1811 |
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
|
1812 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1813 |
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==> |
23482 | 1814 |
z <= x / y" |
1815 |
by(simp add:field_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1816 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1817 |
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
|
1818 |
x / y < z" |
23482 | 1819 |
by(simp add:field_simps) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1820 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1821 |
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
|
1822 |
z < x / y" |
23482 | 1823 |
by(simp add:field_simps) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1824 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1825 |
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
|
1826 |
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
|
1827 |
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
|
1828 |
apply simp; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1829 |
apply (subst times_divide_eq_left); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1830 |
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
|
1831 |
apply (rule mult_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1832 |
apply simp_all |
14293 | 1833 |
done |
1834 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1835 |
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
|
1836 |
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
|
1837 |
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
|
1838 |
apply simp; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1839 |
apply (subst times_divide_eq_left); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1840 |
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
|
1841 |
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
|
1842 |
apply simp_all |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1843 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1844 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1845 |
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
|
1846 |
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
|
1847 |
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
|
1848 |
apply simp_all |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1849 |
apply (subst times_divide_eq_left); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1850 |
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
|
1851 |
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
|
1852 |
apply simp_all |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1853 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1854 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1855 |
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
|
1856 |
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
|
1857 |
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
|
1858 |
seem to need them.*} |
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 |
declare times_divide_eq [simp] |
14293 | 1861 |
|
23389 | 1862 |
|
14293 | 1863 |
subsection {* Ordered Fields are Dense *} |
1864 |
||
14738 | 1865 |
lemma less_add_one: "a < (a+1::'a::ordered_semidom)" |
14293 | 1866 |
proof - |
14738 | 1867 |
have "a+0 < (a+1::'a::ordered_semidom)" |
23482 | 1868 |
by (blast intro: zero_less_one add_strict_left_mono) |
14293 | 1869 |
thus ?thesis by simp |
1870 |
qed |
|
1871 |
||
14738 | 1872 |
lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)" |
23482 | 1873 |
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
|
1874 |
|
14293 | 1875 |
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)" |
23482 | 1876 |
by (simp add: field_simps zero_less_two) |
14293 | 1877 |
|
1878 |
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b" |
|
23482 | 1879 |
by (simp add: field_simps zero_less_two) |
14293 | 1880 |
|
1881 |
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b" |
|
1882 |
by (blast intro!: less_half_sum gt_half_sum) |
|
1883 |
||
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1884 |
|
14293 | 1885 |
subsection {* Absolute Value *} |
1886 |
||
14738 | 1887 |
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
|
1888 |
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
|
1889 |
|
14738 | 1890 |
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" |
1891 |
proof - |
|
1892 |
let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b" |
|
1893 |
let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b" |
|
1894 |
have a: "(abs a) * (abs b) = ?x" |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1895 |
by (simp only: abs_prts[of a] abs_prts[of b] ring_simps) |
14738 | 1896 |
{ |
1897 |
fix u v :: 'a |
|
15481 | 1898 |
have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> |
1899 |
u * v = pprt a * pprt b + pprt a * nprt b + |
|
1900 |
nprt a * pprt b + nprt a * nprt b" |
|
14738 | 1901 |
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
|
1902 |
apply (simp add: ring_simps) |
14738 | 1903 |
done |
1904 |
} |
|
1905 |
note b = this[OF refl[of a] refl[of b]] |
|
1906 |
note addm = add_mono[of "0::'a" _ "0::'a", simplified] |
|
1907 |
note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified] |
|
1908 |
have xy: "- ?x <= ?y" |
|
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1909 |
apply (simp) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1910 |
apply (rule_tac y="0::'a" in order_trans) |
16568 | 1911 |
apply (rule addm2) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1912 |
apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos) |
16568 | 1913 |
apply (rule addm) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1914 |
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
|
1915 |
done |
14738 | 1916 |
have yx: "?y <= ?x" |
16568 | 1917 |
apply (simp add:diff_def) |
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1918 |
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
|
1919 |
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
|
1920 |
apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+) |
14738 | 1921 |
done |
1922 |
have i1: "a*b <= abs a * abs b" by (simp only: a b yx) |
|
1923 |
have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy) |
|
1924 |
show ?thesis |
|
1925 |
apply (rule abs_leI) |
|
1926 |
apply (simp add: i1) |
|
1927 |
apply (simp add: i2[simplified minus_le_iff]) |
|
1928 |
done |
|
1929 |
qed |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1930 |
|
14738 | 1931 |
lemma abs_eq_mult: |
1932 |
assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)" |
|
1933 |
shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)" |
|
1934 |
proof - |
|
1935 |
have s: "(0 <= a*b) | (a*b <= 0)" |
|
1936 |
apply (auto) |
|
1937 |
apply (rule_tac split_mult_pos_le) |
|
1938 |
apply (rule_tac contrapos_np[of "a*b <= 0"]) |
|
1939 |
apply (simp) |
|
1940 |
apply (rule_tac split_mult_neg_le) |
|
1941 |
apply (insert prems) |
|
1942 |
apply (blast) |
|
1943 |
done |
|
1944 |
have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)" |
|
1945 |
by (simp add: prts[symmetric]) |
|
1946 |
show ?thesis |
|
1947 |
proof cases |
|
1948 |
assume "0 <= a * b" |
|
1949 |
then show ?thesis |
|
1950 |
apply (simp_all add: mulprts abs_prts) |
|
1951 |
apply (insert prems) |
|
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1952 |
apply (auto simp add: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1953 |
ring_simps |
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1954 |
iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt] |
15197 | 1955 |
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
|
1956 |
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
|
1957 |
apply(drule (1) mult_nonneg_nonpos2[of b a], simp) |
14738 | 1958 |
done |
1959 |
next |
|
1960 |
assume "~(0 <= a*b)" |
|
1961 |
with s have "a*b <= 0" by simp |
|
1962 |
then show ?thesis |
|
1963 |
apply (simp_all add: mulprts abs_prts) |
|
1964 |
apply (insert prems) |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1965 |
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
|
1966 |
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
|
1967 |
apply(drule (1) mult_nonpos_nonpos[of a b],simp) |
14738 | 1968 |
done |
1969 |
qed |
|
1970 |
qed |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1971 |
|
14738 | 1972 |
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" |
1973 |
by (simp add: abs_eq_mult linorder_linear) |
|
14293 | 1974 |
|
14738 | 1975 |
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)" |
1976 |
by (simp add: abs_if) |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1977 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1978 |
lemma nonzero_abs_inverse: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1979 |
"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
|
1980 |
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
|
1981 |
negative_imp_inverse_negative) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1982 |
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
|
1983 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1984 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1985 |
lemma abs_inverse [simp]: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1986 |
"abs (inverse (a::'a::{ordered_field,division_by_zero})) = |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1987 |
inverse (abs a)" |
21328 | 1988 |
apply (cases "a=0", simp) |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1989 |
apply (simp add: nonzero_abs_inverse) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1990 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1991 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1992 |
lemma nonzero_abs_divide: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1993 |
"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
|
1994 |
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
|
1995 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1996 |
lemma abs_divide [simp]: |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1997 |
"abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b" |
21328 | 1998 |
apply (cases "b=0", simp) |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1999 |
apply (simp add: nonzero_abs_divide) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2000 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2001 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2002 |
lemma abs_mult_less: |
14738 | 2003 |
"[| 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
|
2004 |
proof - |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2005 |
assume ac: "abs a < c" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2006 |
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
|
2007 |
assume "abs b < d" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
2008 |
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
|
2009 |
qed |
14293 | 2010 |
|
14738 | 2011 |
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))" |
2012 |
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff) |
|
2013 |
||
2014 |
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))" |
|
2015 |
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff) |
|
2016 |
||
2017 |
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" |
|
2018 |
apply (simp add: order_less_le abs_le_iff) |
|
2019 |
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff) |
|
2020 |
apply (simp add: le_minus_self_iff linorder_neq_iff) |
|
2021 |
done |
|
2022 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2023 |
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
|
2024 |
(abs y) * x = abs (y * x)"; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2025 |
apply (subst abs_mult); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2026 |
apply simp; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2027 |
done; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2028 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2029 |
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
|
2030 |
abs x / y = abs (x / y)"; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2031 |
apply (subst abs_divide); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2032 |
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
|
2033 |
done; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
2034 |
|
23389 | 2035 |
|
19404 | 2036 |
subsection {* Bounds of products via negative and positive Part *} |
15178 | 2037 |
|
15580 | 2038 |
lemma mult_le_prts: |
2039 |
assumes |
|
2040 |
"a1 <= (a::'a::lordered_ring)" |
|
2041 |
"a <= a2" |
|
2042 |
"b1 <= b" |
|
2043 |
"b <= b2" |
|
2044 |
shows |
|
2045 |
"a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1" |
|
2046 |
proof - |
|
2047 |
have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" |
|
2048 |
apply (subst prts[symmetric])+ |
|
2049 |
apply simp |
|
2050 |
done |
|
2051 |
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
|
2052 |
by (simp add: ring_simps) |
15580 | 2053 |
moreover have "pprt a * pprt b <= pprt a2 * pprt b2" |
2054 |
by (simp_all add: prems mult_mono) |
|
2055 |
moreover have "pprt a * nprt b <= pprt a1 * nprt b2" |
|
2056 |
proof - |
|
2057 |
have "pprt a * nprt b <= pprt a * nprt b2" |
|
2058 |
by (simp add: mult_left_mono prems) |
|
2059 |
moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2" |
|
2060 |
by (simp add: mult_right_mono_neg prems) |
|
2061 |
ultimately show ?thesis |
|
2062 |
by simp |
|
2063 |
qed |
|
2064 |
moreover have "nprt a * pprt b <= nprt a2 * pprt b1" |
|
2065 |
proof - |
|
2066 |
have "nprt a * pprt b <= nprt a2 * pprt b" |
|
2067 |
by (simp add: mult_right_mono prems) |
|
2068 |
moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1" |
|
2069 |
by (simp add: mult_left_mono_neg prems) |
|
2070 |
ultimately show ?thesis |
|
2071 |
by simp |
|
2072 |
qed |
|
2073 |
moreover have "nprt a * nprt b <= nprt a1 * nprt b1" |
|
2074 |
proof - |
|
2075 |
have "nprt a * nprt b <= nprt a * nprt b1" |
|
2076 |
by (simp add: mult_left_mono_neg prems) |
|
2077 |
moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1" |
|
2078 |
by (simp add: mult_right_mono_neg prems) |
|
2079 |
ultimately show ?thesis |
|
2080 |
by simp |
|
2081 |
qed |
|
2082 |
ultimately show ?thesis |
|
2083 |
by - (rule add_mono | simp)+ |
|
2084 |
qed |
|
19404 | 2085 |
|
2086 |
lemma mult_ge_prts: |
|
15178 | 2087 |
assumes |
19404 | 2088 |
"a1 <= (a::'a::lordered_ring)" |
2089 |
"a <= a2" |
|
2090 |
"b1 <= b" |
|
2091 |
"b <= b2" |
|
15178 | 2092 |
shows |
19404 | 2093 |
"a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1" |
2094 |
proof - |
|
2095 |
from prems have a1:"- a2 <= -a" by auto |
|
2096 |
from prems have a2: "-a <= -a1" by auto |
|
2097 |
from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 prems(3) prems(4), simplified nprt_neg pprt_neg] |
|
2098 |
have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp |
|
2099 |
then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b" |
|
2100 |
by (simp only: minus_le_iff) |
|
2101 |
then show ?thesis by simp |
|
15178 | 2102 |
qed |
2103 |
||
23389 | 2104 |
|
22842 | 2105 |
subsection {* Theorems for proof tools *} |
2106 |
||
2107 |
lemma add_mono_thms_ordered_semiring: |
|
2108 |
fixes i j k :: "'a\<Colon>pordered_ab_semigroup_add" |
|
2109 |
shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l" |
|
2110 |
and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l" |
|
2111 |
and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l" |
|
2112 |
and "i = j \<and> k = l \<Longrightarrow> i + k = j + l" |
|
2113 |
by (rule add_mono, clarify+)+ |
|
2114 |
||
2115 |
lemma add_mono_thms_ordered_field: |
|
2116 |
fixes i j k :: "'a\<Colon>pordered_cancel_ab_semigroup_add" |
|
2117 |
shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l" |
|
2118 |
and "i = j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
2119 |
and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l" |
|
2120 |
and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
2121 |
and "i < j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
2122 |
by (auto intro: add_strict_right_mono add_strict_left_mono |
|
2123 |
add_less_le_mono add_le_less_mono add_strict_mono) |
|
2124 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
2125 |
end |