author | wenzelm |
Sat, 01 Sep 2007 18:17:38 +0200 | |
changeset 24515 | d4dc5dc2db98 |
parent 24506 | 020db6ec334a |
child 24748 | ee0a0eb6b738 |
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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|
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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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||
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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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||
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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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||
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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 |
|
14265
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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 |
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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 |
21199
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|
218 |
|
22390 | 219 |
class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add |
22987
550709aa8e66
instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents:
22842
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|
220 |
+ semiring + comm_monoid_add + cancel_ab_semigroup_add |
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221 |
|
14940 | 222 |
instance pordered_cancel_semiring \<subseteq> semiring_0_cancel .. |
223 |
||
21199
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|
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 |
|
21199
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diff
changeset
|
256 |
|
14738 | 257 |
instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring .. |
14270 | 258 |
|
22390 | 259 |
class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add + |
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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diff
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|
261 |
|
14738 | 262 |
instance pordered_comm_semiring \<subseteq> pordered_semiring |
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
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diff
changeset
|
263 |
proof |
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
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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
2d83f93c3580
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20633
diff
changeset
|
268 |
qed |
14265
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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 |
|
14272
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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 |
|
14272
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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 |
|
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
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 |
|
24506 | 302 |
class sgn_if = sgn + zero + one + minus + ord + |
303 |
assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 \<sqsubset> x then 1 else uminus 1)" |
|
304 |
||
23521 | 305 |
(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors. |
306 |
Basically, ordered_ring + no_zero_divisors = ordered_ring_strict. |
|
307 |
*) |
|
308 |
class ordered_ring = ring + ordered_semiring + lordered_ab_group + abs_if |
|
14270 | 309 |
|
23550 | 310 |
instance ordered_ring \<subseteq> lordered_ring |
311 |
proof |
|
312 |
fix x :: 'a |
|
313 |
show "\<bar>x\<bar> = sup x (- x)" |
|
314 |
by (simp only: abs_if sup_eq_if) |
|
315 |
qed |
|
23521 | 316 |
|
24506 | 317 |
class ordered_ring_strict = |
318 |
ring + ordered_semiring_strict + lordered_ab_group + abs_if |
|
23521 | 319 |
|
320 |
instance ordered_ring_strict \<subseteq> ordered_ring .. |
|
14270 | 321 |
|
22390 | 322 |
class pordered_comm_ring = comm_ring + pordered_comm_semiring |
14270 | 323 |
|
23527 | 324 |
instance pordered_comm_ring \<subseteq> pordered_ring .. |
325 |
||
23073 | 326 |
instance pordered_comm_ring \<subseteq> pordered_cancel_comm_semiring .. |
327 |
||
22390 | 328 |
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict + |
329 |
(*previously ordered_semiring*) |
|
330 |
assumes zero_less_one [simp]: "\<^loc>0 \<sqsubset> \<^loc>1" |
|
14270 | 331 |
|
24422 | 332 |
lemma pos_add_strict: |
333 |
fixes a b c :: "'a\<Colon>ordered_semidom" |
|
334 |
shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c" |
|
335 |
using add_strict_mono [of 0 a b c] by simp |
|
336 |
||
24506 | 337 |
class ordered_idom = |
338 |
comm_ring_1 + |
|
339 |
ordered_comm_semiring_strict + |
|
340 |
lordered_ab_group + |
|
341 |
abs_if + sgn_if |
|
22390 | 342 |
(*previously ordered_ring*) |
14270 | 343 |
|
14738 | 344 |
instance ordered_idom \<subseteq> ordered_ring_strict .. |
14272
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diff
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|
345 |
|
23073 | 346 |
instance ordered_idom \<subseteq> pordered_comm_ring .. |
347 |
||
22390 | 348 |
class ordered_field = field + ordered_idom |
14272
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Tidying of the integer development; towards removing the
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parents:
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diff
changeset
|
349 |
|
24515
d4dc5dc2db98
linorder_neqE_ordered_idom: proper proof, avoid illegal schematic type vars;
wenzelm
parents:
24506
diff
changeset
|
350 |
lemma linorder_neqE_ordered_idom: |
d4dc5dc2db98
linorder_neqE_ordered_idom: proper proof, avoid illegal schematic type vars;
wenzelm
parents:
24506
diff
changeset
|
351 |
fixes x y :: "'a :: ordered_idom" |
d4dc5dc2db98
linorder_neqE_ordered_idom: proper proof, avoid illegal schematic type vars;
wenzelm
parents:
24506
diff
changeset
|
352 |
assumes "x \<noteq> y" obtains "x < y" | "y < x" |
d4dc5dc2db98
linorder_neqE_ordered_idom: proper proof, avoid illegal schematic type vars;
wenzelm
parents:
24506
diff
changeset
|
353 |
using assms by (rule linorder_neqE) |
15923 | 354 |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
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diff
changeset
|
355 |
lemma eq_add_iff1: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
356 |
"(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
357 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
358 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
359 |
lemma eq_add_iff2: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
360 |
"(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
361 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
362 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
363 |
lemma less_add_iff1: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
364 |
"(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
365 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
366 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
367 |
lemma less_add_iff2: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
368 |
"(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
369 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
370 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
371 |
lemma le_add_iff1: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
372 |
"(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
373 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
374 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
375 |
lemma le_add_iff2: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
376 |
"(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
377 |
by (simp add: ring_simps) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
378 |
|
23389 | 379 |
|
14270 | 380 |
subsection {* Ordering Rules for Multiplication *} |
381 |
||
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
382 |
lemma mult_left_le_imp_le: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
383 |
"[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
384 |
by (force simp add: mult_strict_left_mono linorder_not_less [symmetric]) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
385 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
386 |
lemma mult_right_le_imp_le: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
387 |
"[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
388 |
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric]) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
389 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
390 |
lemma mult_left_less_imp_less: |
23521 | 391 |
"[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
392 |
by (force simp add: mult_left_mono linorder_not_le [symmetric]) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
393 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
394 |
lemma mult_right_less_imp_less: |
23521 | 395 |
"[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
396 |
by (force simp add: mult_right_mono linorder_not_le [symmetric]) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
397 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
398 |
lemma mult_strict_left_mono_neg: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
399 |
"[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
400 |
apply (drule mult_strict_left_mono [of _ _ "-c"]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
401 |
apply (simp_all add: minus_mult_left [symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
402 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
403 |
|
14738 | 404 |
lemma mult_left_mono_neg: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
405 |
"[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::pordered_ring)" |
14738 | 406 |
apply (drule mult_left_mono [of _ _ "-c"]) |
407 |
apply (simp_all add: minus_mult_left [symmetric]) |
|
408 |
done |
|
409 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
410 |
lemma mult_strict_right_mono_neg: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
411 |
"[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
412 |
apply (drule mult_strict_right_mono [of _ _ "-c"]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
413 |
apply (simp_all add: minus_mult_right [symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
414 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
415 |
|
14738 | 416 |
lemma mult_right_mono_neg: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
417 |
"[|b \<le> a; c \<le> 0|] ==> a * c \<le> (b::'a::pordered_ring) * c" |
14738 | 418 |
apply (drule mult_right_mono [of _ _ "-c"]) |
419 |
apply (simp) |
|
420 |
apply (simp_all add: minus_mult_right [symmetric]) |
|
421 |
done |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
422 |
|
23389 | 423 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
424 |
subsection{* Products of Signs *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
425 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
426 |
lemma mult_pos_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
427 |
by (drule mult_strict_left_mono [of 0 b], auto) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
428 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
429 |
lemma mult_nonneg_nonneg: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b" |
14738 | 430 |
by (drule mult_left_mono [of 0 b], auto) |
431 |
||
432 |
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0" |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
433 |
by (drule mult_strict_left_mono [of b 0], auto) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
434 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
435 |
lemma mult_nonneg_nonpos: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0" |
14738 | 436 |
by (drule mult_left_mono [of b 0], auto) |
437 |
||
438 |
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" |
|
439 |
by (drule mult_strict_right_mono[of b 0], auto) |
|
440 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
441 |
lemma mult_nonneg_nonpos2: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" |
14738 | 442 |
by (drule mult_right_mono[of b 0], auto) |
443 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
444 |
lemma mult_neg_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
445 |
by (drule mult_strict_right_mono_neg, auto) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
446 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
447 |
lemma mult_nonpos_nonpos: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b" |
14738 | 448 |
by (drule mult_right_mono_neg[of a 0 b ], auto) |
449 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
450 |
lemma zero_less_mult_pos: |
14738 | 451 |
"[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)" |
21328 | 452 |
apply (cases "b\<le>0") |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
453 |
apply (auto simp add: order_le_less linorder_not_less) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
454 |
apply (drule_tac mult_pos_neg [of a b]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
455 |
apply (auto dest: order_less_not_sym) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
456 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
457 |
|
14738 | 458 |
lemma zero_less_mult_pos2: |
459 |
"[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)" |
|
21328 | 460 |
apply (cases "b\<le>0") |
14738 | 461 |
apply (auto simp add: order_le_less linorder_not_less) |
462 |
apply (drule_tac mult_pos_neg2 [of a b]) |
|
463 |
apply (auto dest: order_less_not_sym) |
|
464 |
done |
|
465 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
466 |
lemma zero_less_mult_iff: |
14738 | 467 |
"((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
468 |
apply (auto simp add: order_le_less linorder_not_less mult_pos_pos |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
469 |
mult_neg_neg) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
470 |
apply (blast dest: zero_less_mult_pos) |
14738 | 471 |
apply (blast dest: zero_less_mult_pos2) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
472 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
473 |
|
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
474 |
lemma mult_eq_0_iff [simp]: |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
475 |
fixes a b :: "'a::ring_no_zero_divisors" |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
476 |
shows "(a * b = 0) = (a = 0 \<or> b = 0)" |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
477 |
by (cases "a = 0 \<or> b = 0", auto dest: no_zero_divisors) |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
478 |
|
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
479 |
instance ordered_ring_strict \<subseteq> ring_no_zero_divisors |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
480 |
apply intro_classes |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
481 |
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
482 |
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+ |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
483 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
484 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
485 |
lemma zero_le_mult_iff: |
14738 | 486 |
"((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
487 |
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
488 |
zero_less_mult_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
489 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
490 |
lemma mult_less_0_iff: |
14738 | 491 |
"(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
492 |
apply (insert zero_less_mult_iff [of "-a" b]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
493 |
apply (force simp add: minus_mult_left[symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
494 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
495 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
496 |
lemma mult_le_0_iff: |
14738 | 497 |
"(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
498 |
apply (insert zero_le_mult_iff [of "-a" b]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
499 |
apply (force simp add: minus_mult_left[symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
500 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
501 |
|
14738 | 502 |
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
503 |
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos) |
14738 | 504 |
|
505 |
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> (0::_::pordered_cancel_semiring)" |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
506 |
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2) |
14738 | 507 |
|
23095 | 508 |
lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a" |
509 |
by (simp add: zero_le_mult_iff linorder_linear) |
|
510 |
||
511 |
lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))" |
|
512 |
by (simp add: not_less) |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
513 |
|
14738 | 514 |
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom} |
515 |
theorems available to members of @{term ordered_idom} *} |
|
516 |
||
517 |
instance ordered_idom \<subseteq> ordered_semidom |
|
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
518 |
proof |
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
519 |
have "(0::'a) \<le> 1*1" by (rule zero_le_square) |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
520 |
thus "(0::'a) < 1" by (simp add: order_le_less) |
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
521 |
qed |
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
522 |
|
14738 | 523 |
instance ordered_idom \<subseteq> idom .. |
524 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
525 |
text{*All three types of comparision involving 0 and 1 are covered.*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
526 |
|
17085 | 527 |
lemmas one_neq_zero = zero_neq_one [THEN not_sym] |
528 |
declare one_neq_zero [simp] |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
529 |
|
14738 | 530 |
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
531 |
by (rule zero_less_one [THEN order_less_imp_le]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
532 |
|
14738 | 533 |
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0" |
534 |
by (simp add: linorder_not_le) |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
535 |
|
14738 | 536 |
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0" |
537 |
by (simp add: linorder_not_less) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
538 |
|
23389 | 539 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
540 |
subsection{*More Monotonicity*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
541 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
542 |
text{*Strict monotonicity in both arguments*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
543 |
lemma mult_strict_mono: |
14738 | 544 |
"[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)" |
21328 | 545 |
apply (cases "c=0") |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
546 |
apply (simp add: mult_pos_pos) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
547 |
apply (erule mult_strict_right_mono [THEN order_less_trans]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
548 |
apply (force simp add: order_le_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
549 |
apply (erule mult_strict_left_mono, assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
550 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
551 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
552 |
text{*This weaker variant has more natural premises*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
553 |
lemma mult_strict_mono': |
14738 | 554 |
"[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
555 |
apply (rule mult_strict_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
556 |
apply (blast intro: order_le_less_trans)+ |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
557 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
558 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
559 |
lemma mult_mono: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
560 |
"[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] |
14738 | 561 |
==> a * c \<le> b * (d::'a::pordered_semiring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
562 |
apply (erule mult_right_mono [THEN order_trans], assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
563 |
apply (erule mult_left_mono, assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
564 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
565 |
|
21258 | 566 |
lemma mult_mono': |
567 |
"[|a \<le> b; c \<le> d; 0 \<le> a; 0 \<le> c|] |
|
568 |
==> a * c \<le> b * (d::'a::pordered_semiring)" |
|
569 |
apply (rule mult_mono) |
|
570 |
apply (fast intro: order_trans)+ |
|
571 |
done |
|
572 |
||
14738 | 573 |
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
574 |
apply (insert mult_strict_mono [of 1 m 1 n]) |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
575 |
apply (simp add: order_less_trans [OF zero_less_one]) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
576 |
done |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
577 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
578 |
lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
579 |
c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
580 |
apply (subgoal_tac "a * c < b * c") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
581 |
apply (erule order_less_le_trans) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
582 |
apply (erule mult_left_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
583 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
584 |
apply (erule mult_strict_right_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
585 |
apply assumption |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
586 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
587 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
588 |
lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
589 |
c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
590 |
apply (subgoal_tac "a * c <= b * c") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
591 |
apply (erule order_le_less_trans) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
592 |
apply (erule mult_strict_left_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
593 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
594 |
apply (erule mult_right_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
595 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
596 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
597 |
|
23389 | 598 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
599 |
subsection{*Cancellation Laws for Relationships With a Common Factor*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
600 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
601 |
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"}, |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
602 |
also with the relations @{text "\<le>"} and equality.*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
603 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
604 |
text{*These ``disjunction'' versions produce two cases when the comparison is |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
605 |
an assumption, but effectively four when the comparison is a goal.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
606 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
607 |
lemma mult_less_cancel_right_disj: |
14738 | 608 |
"(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))" |
21328 | 609 |
apply (cases "c = 0") |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
610 |
apply (auto simp add: linorder_neq_iff mult_strict_right_mono |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
611 |
mult_strict_right_mono_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
612 |
apply (auto simp add: linorder_not_less |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
613 |
linorder_not_le [symmetric, of "a*c"] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
614 |
linorder_not_le [symmetric, of a]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
615 |
apply (erule_tac [!] notE) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
616 |
apply (auto simp add: order_less_imp_le mult_right_mono |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
617 |
mult_right_mono_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
618 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
619 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
620 |
lemma mult_less_cancel_left_disj: |
14738 | 621 |
"(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))" |
21328 | 622 |
apply (cases "c = 0") |
14738 | 623 |
apply (auto simp add: linorder_neq_iff mult_strict_left_mono |
624 |
mult_strict_left_mono_neg) |
|
625 |
apply (auto simp add: linorder_not_less |
|
626 |
linorder_not_le [symmetric, of "c*a"] |
|
627 |
linorder_not_le [symmetric, of a]) |
|
628 |
apply (erule_tac [!] notE) |
|
629 |
apply (auto simp add: order_less_imp_le mult_left_mono |
|
630 |
mult_left_mono_neg) |
|
631 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
632 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
633 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
634 |
text{*The ``conjunction of implication'' lemmas produce two cases when the |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
635 |
comparison is a goal, but give four when the comparison is an assumption.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
636 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
637 |
lemma mult_less_cancel_right: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
638 |
fixes c :: "'a :: ordered_ring_strict" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
639 |
shows "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
640 |
by (insert mult_less_cancel_right_disj [of a c b], auto) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
641 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
642 |
lemma mult_less_cancel_left: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
643 |
fixes c :: "'a :: ordered_ring_strict" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
644 |
shows "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
645 |
by (insert mult_less_cancel_left_disj [of c a b], auto) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
646 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
647 |
lemma mult_le_cancel_right: |
14738 | 648 |
"(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
649 |
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
650 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
651 |
lemma mult_le_cancel_left: |
14738 | 652 |
"(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
653 |
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
654 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
655 |
lemma mult_less_imp_less_left: |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
656 |
assumes less: "c*a < c*b" and nonneg: "0 \<le> c" |
14738 | 657 |
shows "a < (b::'a::ordered_semiring_strict)" |
14377 | 658 |
proof (rule ccontr) |
659 |
assume "~ a < b" |
|
660 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
23389 | 661 |
hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono) |
14377 | 662 |
with this and less show False |
663 |
by (simp add: linorder_not_less [symmetric]) |
|
664 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
665 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
666 |
lemma mult_less_imp_less_right: |
14738 | 667 |
assumes less: "a*c < b*c" and nonneg: "0 <= c" |
668 |
shows "a < (b::'a::ordered_semiring_strict)" |
|
669 |
proof (rule ccontr) |
|
670 |
assume "~ a < b" |
|
671 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
23389 | 672 |
hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono) |
14738 | 673 |
with this and less show False |
674 |
by (simp add: linorder_not_less [symmetric]) |
|
675 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
676 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
677 |
text{*Cancellation of equalities with a common factor*} |
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
678 |
lemma mult_cancel_right [simp,noatp]: |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
679 |
fixes a b c :: "'a::ring_no_zero_divisors" |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
680 |
shows "(a * c = b * c) = (c = 0 \<or> a = b)" |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
681 |
proof - |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
682 |
have "(a * c = b * c) = ((a - b) * c = 0)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
683 |
by (simp add: ring_distribs) |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
684 |
thus ?thesis |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
685 |
by (simp add: disj_commute) |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
686 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
687 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
688 |
lemma mult_cancel_left [simp,noatp]: |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
689 |
fixes a b c :: "'a::ring_no_zero_divisors" |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
690 |
shows "(c * a = c * b) = (c = 0 \<or> a = b)" |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
691 |
proof - |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
692 |
have "(c * a = c * b) = (c * (a - b) = 0)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
693 |
by (simp add: ring_distribs) |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
694 |
thus ?thesis |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
695 |
by simp |
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
696 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
697 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
698 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
699 |
subsubsection{*Special Cancellation Simprules for Multiplication*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
700 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
701 |
text{*These also produce two cases when the comparison is a goal.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
702 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
703 |
lemma mult_le_cancel_right1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
704 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
705 |
shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
706 |
by (insert mult_le_cancel_right [of 1 c b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
707 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
708 |
lemma mult_le_cancel_right2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
709 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
710 |
shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
711 |
by (insert mult_le_cancel_right [of a c 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
712 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
713 |
lemma mult_le_cancel_left1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
714 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
715 |
shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
716 |
by (insert mult_le_cancel_left [of c 1 b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
717 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
718 |
lemma mult_le_cancel_left2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
719 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
720 |
shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
721 |
by (insert mult_le_cancel_left [of c a 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
722 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
723 |
lemma mult_less_cancel_right1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
724 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
725 |
shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
726 |
by (insert mult_less_cancel_right [of 1 c b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
727 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
728 |
lemma mult_less_cancel_right2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
729 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
730 |
shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
731 |
by (insert mult_less_cancel_right [of a c 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
732 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
733 |
lemma mult_less_cancel_left1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
734 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
735 |
shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
736 |
by (insert mult_less_cancel_left [of c 1 b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
737 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
738 |
lemma mult_less_cancel_left2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
739 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
740 |
shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
741 |
by (insert mult_less_cancel_left [of c a 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
742 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
743 |
lemma mult_cancel_right1 [simp]: |
23544 | 744 |
fixes c :: "'a :: ring_1_no_zero_divisors" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
745 |
shows "(c = b*c) = (c = 0 | b=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
746 |
by (insert mult_cancel_right [of 1 c b], force) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
747 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
748 |
lemma mult_cancel_right2 [simp]: |
23544 | 749 |
fixes c :: "'a :: ring_1_no_zero_divisors" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
750 |
shows "(a*c = c) = (c = 0 | a=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
751 |
by (insert mult_cancel_right [of a c 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
752 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
753 |
lemma mult_cancel_left1 [simp]: |
23544 | 754 |
fixes c :: "'a :: ring_1_no_zero_divisors" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
755 |
shows "(c = c*b) = (c = 0 | b=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
756 |
by (insert mult_cancel_left [of c 1 b], force) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
757 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
758 |
lemma mult_cancel_left2 [simp]: |
23544 | 759 |
fixes c :: "'a :: ring_1_no_zero_divisors" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
760 |
shows "(c*a = c) = (c = 0 | a=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
761 |
by (insert mult_cancel_left [of c a 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
762 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
763 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
764 |
text{*Simprules for comparisons where common factors can be cancelled.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
765 |
lemmas mult_compare_simps = |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
766 |
mult_le_cancel_right mult_le_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
767 |
mult_le_cancel_right1 mult_le_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
768 |
mult_le_cancel_left1 mult_le_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
769 |
mult_less_cancel_right mult_less_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
770 |
mult_less_cancel_right1 mult_less_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
771 |
mult_less_cancel_left1 mult_less_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
772 |
mult_cancel_right mult_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
773 |
mult_cancel_right1 mult_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
774 |
mult_cancel_left1 mult_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
775 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
776 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
777 |
subsection {* Fields *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
778 |
|
14288 | 779 |
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" |
780 |
proof |
|
781 |
assume neq: "b \<noteq> 0" |
|
782 |
{ |
|
783 |
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) |
|
784 |
also assume "a / b = 1" |
|
785 |
finally show "a = b" by simp |
|
786 |
next |
|
787 |
assume "a = b" |
|
788 |
with neq show "a / b = 1" by (simp add: divide_inverse) |
|
789 |
} |
|
790 |
qed |
|
791 |
||
792 |
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" |
|
793 |
by (simp add: divide_inverse) |
|
794 |
||
23398 | 795 |
lemma divide_self[simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1" |
14288 | 796 |
by (simp add: divide_inverse) |
797 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
798 |
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
799 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
800 |
|
15228 | 801 |
lemma divide_self_if [simp]: |
802 |
"a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)" |
|
803 |
by (simp add: divide_self) |
|
804 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
805 |
lemma divide_zero_left [simp]: "0/a = (0::'a::field)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
806 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
807 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
808 |
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
809 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
810 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
811 |
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
812 |
by (simp add: divide_inverse ring_distribs) |
14293 | 813 |
|
23482 | 814 |
(* what ordering?? this is a straight instance of mult_eq_0_iff |
14270 | 815 |
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement |
816 |
of an ordering.*} |
|
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
817 |
lemma field_mult_eq_0_iff [simp]: |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
818 |
"(a*b = (0::'a::division_ring)) = (a = 0 | b = 0)" |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
819 |
by simp |
23482 | 820 |
*) |
23496 | 821 |
(* subsumed by mult_cancel lemmas on ring_no_zero_divisors |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
822 |
text{*Cancellation of equalities with a common factor*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
823 |
lemma field_mult_cancel_right_lemma: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
824 |
assumes cnz: "c \<noteq> (0::'a::division_ring)" |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
825 |
and eq: "a*c = b*c" |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
826 |
shows "a=b" |
14377 | 827 |
proof - |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
828 |
have "(a * c) * inverse c = (b * c) * inverse c" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
829 |
by (simp add: eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
830 |
thus "a=b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
831 |
by (simp add: mult_assoc cnz) |
14377 | 832 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
833 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
834 |
lemma field_mult_cancel_right [simp]: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
835 |
"(a*c = b*c) = (c = (0::'a::division_ring) | a=b)" |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
836 |
by simp |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
837 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
838 |
lemma field_mult_cancel_left [simp]: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
839 |
"(c*a = c*b) = (c = (0::'a::division_ring) | a=b)" |
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents:
22987
diff
changeset
|
840 |
by simp |
23496 | 841 |
*) |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
842 |
lemma nonzero_imp_inverse_nonzero: |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
843 |
"a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)" |
14377 | 844 |
proof |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
845 |
assume ianz: "inverse a = 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
846 |
assume "a \<noteq> 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
847 |
hence "1 = a * inverse a" by simp |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
848 |
also have "... = 0" by (simp add: ianz) |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
849 |
finally have "1 = (0::'a::division_ring)" . |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
850 |
thus False by (simp add: eq_commute) |
14377 | 851 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
852 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
853 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
854 |
subsection{*Basic Properties of @{term inverse}*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
855 |
|
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
856 |
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::division_ring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
857 |
apply (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
858 |
apply (blast dest: nonzero_imp_inverse_nonzero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
859 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
860 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
861 |
lemma inverse_nonzero_imp_nonzero: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
862 |
"inverse a = 0 ==> a = (0::'a::division_ring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
863 |
apply (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
864 |
apply (blast dest: nonzero_imp_inverse_nonzero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
865 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
866 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
867 |
lemma inverse_nonzero_iff_nonzero [simp]: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
868 |
"(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
869 |
by (force dest: inverse_nonzero_imp_nonzero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
870 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
871 |
lemma nonzero_inverse_minus_eq: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
872 |
assumes [simp]: "a\<noteq>0" |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
873 |
shows "inverse(-a) = -inverse(a::'a::division_ring)" |
14377 | 874 |
proof - |
875 |
have "-a * inverse (- a) = -a * - inverse a" |
|
876 |
by simp |
|
877 |
thus ?thesis |
|
23496 | 878 |
by (simp only: mult_cancel_left, simp) |
14377 | 879 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
880 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
881 |
lemma inverse_minus_eq [simp]: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
882 |
"inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})" |
14377 | 883 |
proof cases |
884 |
assume "a=0" thus ?thesis by (simp add: inverse_zero) |
|
885 |
next |
|
886 |
assume "a\<noteq>0" |
|
887 |
thus ?thesis by (simp add: nonzero_inverse_minus_eq) |
|
888 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
889 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
890 |
lemma nonzero_inverse_eq_imp_eq: |
14269 | 891 |
assumes inveq: "inverse a = inverse b" |
892 |
and anz: "a \<noteq> 0" |
|
893 |
and bnz: "b \<noteq> 0" |
|
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
894 |
shows "a = (b::'a::division_ring)" |
14377 | 895 |
proof - |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
896 |
have "a * inverse b = a * inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
897 |
by (simp add: inveq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
898 |
hence "(a * inverse b) * b = (a * inverse a) * b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
899 |
by simp |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
900 |
thus "a = b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
901 |
by (simp add: mult_assoc anz bnz) |
14377 | 902 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
903 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
904 |
lemma inverse_eq_imp_eq: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
905 |
"inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})" |
21328 | 906 |
apply (cases "a=0 | b=0") |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
907 |
apply (force dest!: inverse_zero_imp_zero |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
908 |
simp add: eq_commute [of "0::'a"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
909 |
apply (force dest!: nonzero_inverse_eq_imp_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
910 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
911 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
912 |
lemma inverse_eq_iff_eq [simp]: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
913 |
"(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))" |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
914 |
by (force dest!: inverse_eq_imp_eq) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
915 |
|
14270 | 916 |
lemma nonzero_inverse_inverse_eq: |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
917 |
assumes [simp]: "a \<noteq> 0" |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
918 |
shows "inverse(inverse (a::'a::division_ring)) = a" |
14270 | 919 |
proof - |
920 |
have "(inverse (inverse a) * inverse a) * a = a" |
|
921 |
by (simp add: nonzero_imp_inverse_nonzero) |
|
922 |
thus ?thesis |
|
923 |
by (simp add: mult_assoc) |
|
924 |
qed |
|
925 |
||
926 |
lemma inverse_inverse_eq [simp]: |
|
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
927 |
"inverse(inverse (a::'a::{division_ring,division_by_zero})) = a" |
14270 | 928 |
proof cases |
929 |
assume "a=0" thus ?thesis by simp |
|
930 |
next |
|
931 |
assume "a\<noteq>0" |
|
932 |
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) |
|
933 |
qed |
|
934 |
||
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
935 |
lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)" |
14270 | 936 |
proof - |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
937 |
have "inverse 1 * 1 = (1::'a::division_ring)" |
14270 | 938 |
by (rule left_inverse [OF zero_neq_one [symmetric]]) |
939 |
thus ?thesis by simp |
|
940 |
qed |
|
941 |
||
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
942 |
lemma inverse_unique: |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
943 |
assumes ab: "a*b = 1" |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
944 |
shows "inverse a = (b::'a::division_ring)" |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
945 |
proof - |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
946 |
have "a \<noteq> 0" using ab by auto |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
947 |
moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
948 |
ultimately show ?thesis by (simp add: mult_assoc [symmetric]) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
949 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
950 |
|
14270 | 951 |
lemma nonzero_inverse_mult_distrib: |
952 |
assumes anz: "a \<noteq> 0" |
|
953 |
and bnz: "b \<noteq> 0" |
|
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
954 |
shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)" |
14270 | 955 |
proof - |
956 |
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" |
|
23482 | 957 |
by (simp add: anz bnz) |
14270 | 958 |
hence "inverse(a*b) * a = inverse(b)" |
959 |
by (simp add: mult_assoc bnz) |
|
960 |
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" |
|
961 |
by simp |
|
962 |
thus ?thesis |
|
963 |
by (simp add: mult_assoc anz) |
|
964 |
qed |
|
965 |
||
966 |
text{*This version builds in division by zero while also re-orienting |
|
967 |
the right-hand side.*} |
|
968 |
lemma inverse_mult_distrib [simp]: |
|
969 |
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" |
|
970 |
proof cases |
|
971 |
assume "a \<noteq> 0 & b \<noteq> 0" |
|
22993 | 972 |
thus ?thesis |
973 |
by (simp add: nonzero_inverse_mult_distrib mult_commute) |
|
14270 | 974 |
next |
975 |
assume "~ (a \<noteq> 0 & b \<noteq> 0)" |
|
22993 | 976 |
thus ?thesis |
977 |
by force |
|
14270 | 978 |
qed |
979 |
||
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
980 |
lemma division_ring_inverse_add: |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
981 |
"[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|] |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
982 |
==> inverse a + inverse b = inverse a * (a+b) * inverse b" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
983 |
by (simp add: ring_simps) |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
984 |
|
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
985 |
lemma division_ring_inverse_diff: |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
986 |
"[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|] |
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
987 |
==> inverse a - inverse b = inverse a * (b-a) * inverse b" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
988 |
by (simp add: ring_simps) |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
989 |
|
14270 | 990 |
text{*There is no slick version using division by zero.*} |
991 |
lemma inverse_add: |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
992 |
"[|a \<noteq> 0; b \<noteq> 0|] |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
993 |
==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)" |
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset
|
994 |
by (simp add: division_ring_inverse_add mult_ac) |
14270 | 995 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
996 |
lemma inverse_divide [simp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
997 |
"inverse (a/b) = b / (a::'a::{field,division_by_zero})" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
998 |
by (simp add: divide_inverse mult_commute) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
999 |
|
23389 | 1000 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1001 |
subsection {* Calculations with fractions *} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1002 |
|
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1003 |
text{* There is a whole bunch of simp-rules just for class @{text |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1004 |
field} but none for class @{text field} and @{text nonzero_divides} |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1005 |
because the latter are covered by a simproc. *} |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1006 |
|
24427 | 1007 |
lemma nonzero_mult_divide_mult_cancel_left[simp,noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1008 |
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)" |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1009 |
proof - |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1010 |
have "(c*a)/(c*b) = c * a * (inverse b * inverse c)" |
23482 | 1011 |
by (simp add: divide_inverse nonzero_inverse_mult_distrib) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1012 |
also have "... = a * inverse b * (inverse c * c)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1013 |
by (simp only: mult_ac) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1014 |
also have "... = a * inverse b" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1015 |
by simp |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1016 |
finally show ?thesis |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1017 |
by (simp add: divide_inverse) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1018 |
qed |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1019 |
|
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1020 |
lemma mult_divide_mult_cancel_left: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1021 |
"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})" |
21328 | 1022 |
apply (cases "b = 0") |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1023 |
apply (simp_all add: nonzero_mult_divide_mult_cancel_left) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1024 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1025 |
|
24427 | 1026 |
lemma nonzero_mult_divide_mult_cancel_right [noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1027 |
"[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)" |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1028 |
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) |
14321 | 1029 |
|
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1030 |
lemma mult_divide_mult_cancel_right: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1031 |
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" |
21328 | 1032 |
apply (cases "b = 0") |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1033 |
apply (simp_all add: nonzero_mult_divide_mult_cancel_right) |
14321 | 1034 |
done |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1035 |
|
14284
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
1036 |
lemma divide_1 [simp]: "a/1 = (a::'a::field)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1037 |
by (simp add: divide_inverse) |
14284
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
1038 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1039 |
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1040 |
by (simp add: divide_inverse mult_assoc) |
14288 | 1041 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1042 |
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1043 |
by (simp add: divide_inverse mult_ac) |
14288 | 1044 |
|
23482 | 1045 |
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left |
1046 |
||
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1047 |
lemma divide_divide_eq_right [simp,noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1048 |
"a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1049 |
by (simp add: divide_inverse mult_ac) |
14288 | 1050 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset
|
1051 |
lemma divide_divide_eq_left [simp,noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1052 |
"(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1053 |
by (simp add: divide_inverse mult_assoc) |
14288 | 1054 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1055 |
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1056 |
x / y + w / z = (x * z + w * y) / (y * z)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1057 |
apply (subgoal_tac "x / y = (x * z) / (y * z)") |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1058 |
apply (erule ssubst) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1059 |
apply (subgoal_tac "w / z = (w * y) / (y * z)") |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1060 |
apply (erule ssubst) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1061 |
apply (rule add_divide_distrib [THEN sym]) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1062 |
apply (subst mult_commute) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1063 |
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym]) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1064 |
apply assumption |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1065 |
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym]) |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1066 |
apply assumption |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1067 |
done |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1068 |
|
23389 | 1069 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1070 |
subsubsection{*Special Cancellation Simprules for Division*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1071 |
|
24427 | 1072 |
lemma mult_divide_mult_cancel_left_if[simp,noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1073 |
fixes c :: "'a :: {field,division_by_zero}" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1074 |
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)" |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1075 |
by (simp add: mult_divide_mult_cancel_left) |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1076 |
|
24427 | 1077 |
lemma nonzero_mult_divide_cancel_right[simp,noatp]: |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1078 |
"b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)" |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1079 |
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1080 |
|
24427 | 1081 |
lemma nonzero_mult_divide_cancel_left[simp,noatp]: |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1082 |
"a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)" |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1083 |
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1084 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1085 |
|
24427 | 1086 |
lemma nonzero_divide_mult_cancel_right[simp,noatp]: |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1087 |
"\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)" |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1088 |
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1089 |
|
24427 | 1090 |
lemma nonzero_divide_mult_cancel_left[simp,noatp]: |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1091 |
"\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)" |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1092 |
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1093 |
|
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1094 |
|
24427 | 1095 |
lemma nonzero_mult_divide_mult_cancel_left2[simp,noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1096 |
"[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)" |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1097 |
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac) |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1098 |
|
24427 | 1099 |
lemma nonzero_mult_divide_mult_cancel_right2[simp,noatp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1100 |
"[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)" |
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1101 |
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac) |
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset
|
1102 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1103 |
|
14293 | 1104 |
subsection {* Division and Unary Minus *} |
1105 |
||
1106 |
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)" |
|
1107 |
by (simp add: divide_inverse minus_mult_left) |
|
1108 |
||
1109 |
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)" |
|
1110 |
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) |
|
1111 |
||
1112 |
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)" |
|
1113 |
by (simp add: divide_inverse nonzero_inverse_minus_eq) |
|
1114 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1115 |
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1116 |
by (simp add: divide_inverse minus_mult_left [symmetric]) |
14293 | 1117 |
|
1118 |
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})" |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1119 |
by (simp add: divide_inverse minus_mult_right [symmetric]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1120 |
|
14293 | 1121 |
|
1122 |
text{*The effect is to extract signs from divisions*} |
|
17085 | 1123 |
lemmas divide_minus_left = minus_divide_left [symmetric] |
1124 |
lemmas divide_minus_right = minus_divide_right [symmetric] |
|
1125 |
declare divide_minus_left [simp] divide_minus_right [simp] |
|
14293 | 1126 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1127 |
text{*Also, extract signs from products*} |
17085 | 1128 |
lemmas mult_minus_left = minus_mult_left [symmetric] |
1129 |
lemmas mult_minus_right = minus_mult_right [symmetric] |
|
1130 |
declare mult_minus_left [simp] mult_minus_right [simp] |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1131 |
|
14293 | 1132 |
lemma minus_divide_divide [simp]: |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
1133 |
"(-a)/(-b) = a / (b::'a::{field,division_by_zero})" |
21328 | 1134 |
apply (cases "b=0", simp) |
14293 | 1135 |
apply (simp add: nonzero_minus_divide_divide) |
1136 |
done |
|
1137 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1138 |
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1139 |
by (simp add: diff_minus add_divide_distrib) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1140 |
|
23482 | 1141 |
lemma add_divide_eq_iff: |
1142 |
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z" |
|
1143 |
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left) |
|
1144 |
||
1145 |
lemma divide_add_eq_iff: |
|
1146 |
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z" |
|
1147 |
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left) |
|
1148 |
||
1149 |
lemma diff_divide_eq_iff: |
|
1150 |
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z" |
|
1151 |
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left) |
|
1152 |
||
1153 |
lemma divide_diff_eq_iff: |
|
1154 |
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z" |
|
1155 |
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left) |
|
1156 |
||
1157 |
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)" |
|
1158 |
proof - |
|
1159 |
assume [simp]: "c\<noteq>0" |
|
23496 | 1160 |
have "(a = b/c) = (a*c = (b/c)*c)" by simp |
1161 |
also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc) |
|
23482 | 1162 |
finally show ?thesis . |
1163 |
qed |
|
1164 |
||
1165 |
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)" |
|
1166 |
proof - |
|
1167 |
assume [simp]: "c\<noteq>0" |
|
23496 | 1168 |
have "(b/c = a) = ((b/c)*c = a*c)" by simp |
1169 |
also have "... = (b = a*c)" by (simp add: divide_inverse mult_assoc) |
|
23482 | 1170 |
finally show ?thesis . |
1171 |
qed |
|
1172 |
||
1173 |
lemma eq_divide_eq: |
|
1174 |
"((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)" |
|
1175 |
by (simp add: nonzero_eq_divide_eq) |
|
1176 |
||
1177 |
lemma divide_eq_eq: |
|
1178 |
"(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)" |
|
1179 |
by (force simp add: nonzero_divide_eq_eq) |
|
1180 |
||
1181 |
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==> |
|
1182 |
b = a * c ==> b / c = a" |
|
1183 |
by (subst divide_eq_eq, simp) |
|
1184 |
||
1185 |
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==> |
|
1186 |
a * c = b ==> a = b / c" |
|
1187 |
by (subst eq_divide_eq, simp) |
|
1188 |
||
1189 |
||
1190 |
lemmas field_eq_simps = ring_simps |
|
1191 |
(* pull / out*) |
|
1192 |
add_divide_eq_iff divide_add_eq_iff |
|
1193 |
diff_divide_eq_iff divide_diff_eq_iff |
|
1194 |
(* multiply eqn *) |
|
1195 |
nonzero_eq_divide_eq nonzero_divide_eq_eq |
|
1196 |
(* is added later: |
|
1197 |
times_divide_eq_left times_divide_eq_right |
|
1198 |
*) |
|
1199 |
||
1200 |
text{*An example:*} |
|
1201 |
lemma fixes a b c d e f :: "'a::field" |
|
1202 |
shows "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f \<rbrakk> \<Longrightarrow> ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) = 1" |
|
1203 |
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0") |
|
1204 |
apply(simp add:field_eq_simps) |
|
1205 |
apply(simp) |
|
1206 |
done |
|
1207 |
||
1208 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1209 |
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset
|
1210 |
x / y - w / z = (x * z - w * y) / (y * z)" |
23482 | 1211 |
by (simp add:field_eq_simps times_divide_eq) |
1212 |
||
1213 |
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==> |
|
1214 |
(x / y = w / z) = (x * z = w * y)" |
|
1215 |
by (simp add:field_eq_simps times_divide_eq) |
|
14293 | 1216 |
|
23389 | 1217 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1218 |
subsection {* Ordered Fields *} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1219 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1220 |
lemma positive_imp_inverse_positive: |
23482 | 1221 |
assumes a_gt_0: "0 < a" shows "0 < inverse (a::'a::ordered_field)" |
1222 |
proof - |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1223 |
have "0 < a * inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1224 |
by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1225 |
thus "0 < inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1226 |
by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff) |
23482 | 1227 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1228 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1229 |
lemma negative_imp_inverse_negative: |
23482 | 1230 |
"a < 0 ==> inverse a < (0::'a::ordered_field)" |
1231 |
by (insert positive_imp_inverse_positive [of "-a"], |
|
1232 |
simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1233 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1234 |
lemma inverse_le_imp_le: |
23482 | 1235 |
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a" |
1236 |
shows "b \<le> (a::'a::ordered_field)" |
|
1237 |
proof (rule classical) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1238 |
assume "~ b \<le> a" |
23482 | 1239 |
hence "a < b" by (simp add: linorder_not_le) |
1240 |
hence bpos: "0 < b" by (blast intro: apos order_less_trans) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1241 |
hence "a * inverse a \<le> a * inverse b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1242 |
by (simp add: apos invle order_less_imp_le mult_left_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1243 |
hence "(a * inverse a) * b \<le> (a * inverse b) * b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1244 |
by (simp add: bpos order_less_imp_le mult_right_mono) |
23482 | 1245 |
thus "b \<le> a" by (simp add: mult_assoc apos bpos order_less_imp_not_eq2) |
1246 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1247 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1248 |
lemma inverse_positive_imp_positive: |
23482 | 1249 |
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0" |
1250 |
shows "0 < (a::'a::ordered_field)" |
|
23389 | 1251 |
proof - |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1252 |
have "0 < inverse (inverse a)" |
23389 | 1253 |
using inv_gt_0 by (rule positive_imp_inverse_positive) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1254 |
thus "0 < a" |
23389 | 1255 |
using nz by (simp add: nonzero_inverse_inverse_eq) |
1256 |
qed |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1257 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1258 |
lemma inverse_positive_iff_positive [simp]: |
23482
2f4be6844f7c
tuned and used field_simps
nip
|