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permissions  rwrr 
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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, Lawrence C Paulson, and Markus Wenzel, 
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with contributions by Jeremy Avigad 
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*) 
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14738  7 
header {* (Ordered) Rings and Fields *} 
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15229  9 
theory Ring_and_Field 
15140  10 
imports OrderedGroup 
15131  11 
begin 
14504  12 

14738  13 
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} 

19 
Most of the used notions can also be looked up in 

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\begin{itemize} 

14770  21 
\item \url{http://www.mathworld.com} by Eric Weisstein et. al. 
14738  22 
\item \emph{Algebra I} by van der Waerden, Springer. 
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\end{itemize} 

24 
*} 

14504  25 

22390  26 
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" 

14504  29 

22390  30 
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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22390  34 
class semiring_0 = semiring + comm_monoid_add + mult_zero 
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22390  36 
class semiring_0_cancel = semiring + comm_monoid_add + cancel_ab_semigroup_add 
14504  37 

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instance semiring_0_cancel \<subseteq> semiring_0 
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proof 
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fix a :: 'a 
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have "0 * a + 0 * a = 0 * a + 0" 
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by (simp add: left_distrib [symmetric]) 
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thus "0 * a = 0" 
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by (simp only: add_left_cancel) 
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have "a * 0 + a * 0 = a * 0 + 0" 
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by (simp add: right_distrib [symmetric]) 
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thus "a * 0 = 0" 
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by (simp only: add_left_cancel) 
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qed 
14940  51 

22390  52 
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" 

14504  54 

14738  55 
instance comm_semiring \<subseteq> semiring 
56 
proof 

57 
fix a b c :: 'a 

58 
show "(a + b) * c = a * c + b * c" by (simp add: distrib) 

59 
have "a * (b + c) = (b + c) * a" by (simp add: mult_ac) 

60 
also have "... = b * a + c * a" by (simp only: distrib) 

61 
also have "... = a * b + a * c" by (simp add: mult_ac) 

62 
finally show "a * (b + c) = a * b + a * c" by blast 

14504  63 
qed 
64 

22390  65 
class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero 
14504  66 

14738  67 
instance comm_semiring_0 \<subseteq> semiring_0 .. 
14504  68 

22390  69 
class comm_semiring_0_cancel = comm_semiring + comm_monoid_add + cancel_ab_semigroup_add 
14940  70 

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instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel .. 

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instance comm_semiring_0_cancel \<subseteq> comm_semiring_0 .. 
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22390  75 
class zero_neq_one = zero + one + 
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assumes zero_neq_one [simp]: "\<^loc>0 \<noteq> \<^loc>1" 

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22390  78 
class semiring_1 = zero_neq_one + semiring_0 + monoid_mult 
14504  79 

22390  80 
class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult 
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(*previously almost_semiring*) 

14738  82 

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instance comm_semiring_1 \<subseteq> semiring_1 .. 

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22390  85 
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" 

14504  87 

22390  88 
class semiring_1_cancel = semiring + comm_monoid_add + zero_neq_one 
89 
+ cancel_ab_semigroup_add + monoid_mult 

14940  90 

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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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22390  95 
class comm_semiring_1_cancel = comm_semiring + comm_monoid_add + comm_monoid_mult 
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+ zero_neq_one + cancel_ab_semigroup_add 

14738  97 

14940  98 
instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel .. 
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instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel .. 

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instance comm_semiring_1_cancel \<subseteq> comm_semiring_1 .. 
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22390  104 
class ring = semiring + ab_group_add 
14738  105 

14940  106 
instance ring \<subseteq> semiring_0_cancel .. 
14504  107 

22390  108 
class comm_ring = comm_semiring + ab_group_add 
14738  109 

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instance comm_ring \<subseteq> ring .. 

14504  111 

14940  112 
instance comm_ring \<subseteq> comm_semiring_0_cancel .. 
14738  113 

22390  114 
class ring_1 = ring + zero_neq_one + monoid_mult 
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14940  116 
instance ring_1 \<subseteq> semiring_1_cancel .. 
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22390  118 
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult 
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(*previously ring*) 

14738  120 

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instance comm_ring_1 \<subseteq> ring_1 .. 

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14738  123 
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 dom = ring_1 + ring_no_zero_divisors 
23326  128 
hide const dom 
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22390  130 
class idom = comm_ring_1 + no_zero_divisors 
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instance idom \<subseteq> dom .. 
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22390  134 
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> dom 
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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) 
14738  166 
qed 
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instance field \<subseteq> idom .. 
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class division_by_zero = zero + inverse + 
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assumes inverse_zero [simp]: "inverse \<^loc>0 = \<^loc>0" 

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23389  173 

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subsection {* Distribution rules *} 
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theorems ring_distrib = right_distrib left_distrib 
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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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apply (simp add: right_distrib [symmetric]) 
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done 
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lemma minus_mult_minus [simp]: "( a) * ( b) = a * (b::'a::ring)" 
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by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma minus_mult_commute: "( a) * b = a * ( b::'a::ring)" 
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by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma right_diff_distrib: "a * (b  c) = a * b  a * (c::'a::ring)" 
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by (simp add: right_distrib diff_minus 
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minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma left_diff_distrib: "(a  b) * c = a * c  b * (c::'a::ring)" 
14738  204 
by (simp add: left_distrib diff_minus 
205 
minus_mult_left [symmetric] minus_mult_right [symmetric]) 

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22390  207 
class mult_mono = times + zero + ord + 
208 
assumes mult_left_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b" 

209 
assumes mult_right_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> a \<^loc>* c \<sqsubseteq> b \<^loc>* c" 

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22390  211 
class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add 
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22390  213 
class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add 
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+ semiring + comm_monoid_add + cancel_ab_semigroup_add 
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14940  216 
instance pordered_cancel_semiring \<subseteq> semiring_0_cancel .. 
217 

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instance pordered_cancel_semiring \<subseteq> pordered_semiring .. 
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22390  220 
class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + 
221 
assumes mult_strict_left_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b" 

222 
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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223 

14940  224 
instance ordered_semiring_strict \<subseteq> semiring_0_cancel .. 
225 

14738  226 
instance ordered_semiring_strict \<subseteq> pordered_cancel_semiring 
227 
apply intro_classes 

21328  228 
apply (cases "a < b & 0 < c") 
14738  229 
apply (auto simp add: mult_strict_left_mono order_less_le) 
230 
apply (auto simp add: mult_strict_left_mono order_le_less) 

231 
apply (simp add: mult_strict_right_mono) 

14270  232 
done 
233 

22390  234 
class mult_mono1 = times + zero + ord + 
235 
assumes mult_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b" 

14270  236 

22390  237 
class pordered_comm_semiring = comm_semiring_0 
238 
+ pordered_ab_semigroup_add + mult_mono1 

14270  239 

22390  240 
class pordered_cancel_comm_semiring = comm_semiring_0_cancel 
241 
+ pordered_ab_semigroup_add + mult_mono1 

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14738  243 
instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring .. 
14270  244 

22390  245 
class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add + 
246 
assumes mult_strict_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b" 

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14738  248 
instance pordered_comm_semiring \<subseteq> pordered_semiring 
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249 
proof 
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250 
fix a b c :: 'a 
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assume A: "a <= b" "0 <= c" 
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with mult_mono show "c * a <= c * b" . 
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253 

2d83f93c3580
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254 
from mult_commute have "a * c = c * a" .. 
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255 
also from mult_mono A have "\<dots> <= c * b" . 
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256 
also from mult_commute have "c * b = b * c" .. 
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finally show "a * c <= b * c" . 
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258 
qed 
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14738  260 
instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring .. 
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14738  262 
instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict 
263 
by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+) 

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14738  265 
instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring 
266 
apply (intro_classes) 

21328  267 
apply (cases "a < b & 0 < c") 
14738  268 
apply (auto simp add: mult_strict_left_mono order_less_le) 
269 
apply (auto simp add: mult_strict_left_mono order_le_less) 

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done 
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22390  272 
class pordered_ring = ring + pordered_cancel_semiring 
14270  273 

14738  274 
instance pordered_ring \<subseteq> pordered_ab_group_add .. 
14270  275 

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class lordered_ring = pordered_ring + lordered_ab_group_abs 
14270  277 

14940  278 
instance lordered_ring \<subseteq> lordered_ab_group_meet .. 
279 

280 
instance lordered_ring \<subseteq> lordered_ab_group_join .. 

281 

22390  282 
class abs_if = minus + ord + zero + 
283 
assumes abs_if: "abs a = (if a \<sqsubset> 0 then (uminus a) else a)" 

14270  284 

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class ordered_ring_strict = ring + ordered_semiring_strict + abs_if + lordered_ab_group 
14270  286 

14738  287 
instance ordered_ring_strict \<subseteq> lordered_ring 
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by intro_classes (simp add: abs_if sup_eq_if) 
14270  289 

22390  290 
class pordered_comm_ring = comm_ring + pordered_comm_semiring 
14270  291 

23073  292 
instance pordered_comm_ring \<subseteq> pordered_cancel_comm_semiring .. 
293 

22390  294 
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict + 
295 
(*previously ordered_semiring*) 

296 
assumes zero_less_one [simp]: "\<^loc>0 \<sqsubset> \<^loc>1" 

14270  297 

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class ordered_idom = comm_ring_1 + ordered_comm_semiring_strict + abs_if + lordered_ab_group 
22390  299 
(*previously ordered_ring*) 
14270  300 

14738  301 
instance ordered_idom \<subseteq> ordered_ring_strict .. 
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23073  303 
instance ordered_idom \<subseteq> pordered_comm_ring .. 
304 

22390  305 
class ordered_field = field + ordered_idom 
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15923  307 
lemmas linorder_neqE_ordered_idom = 
308 
linorder_neqE[where 'a = "?'b::ordered_idom"] 

309 

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lemma eq_add_iff1: 
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"(a*e + c = b*e + d) = ((ab)*e + c = (d::'a::ring))" 
14738  312 
apply (simp add: diff_minus left_distrib) 
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apply (simp add: diff_minus left_distrib add_ac) 
14738  314 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done 
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316 

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lemma eq_add_iff2: 
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"(a*e + c = b*e + d) = (c = (ba)*e + (d::'a::ring))" 
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apply (simp add: diff_minus left_distrib add_ac) 
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done 
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lemma less_add_iff1: 
14738  324 
"(a*e + c < b*e + d) = ((ab)*e + c < (d::'a::pordered_ring))" 
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apply (simp add: diff_minus left_distrib add_ac) 
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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327 
done 
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328 

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329 
lemma less_add_iff2: 
14738  330 
"(a*e + c < b*e + d) = (c < (ba)*e + (d::'a::pordered_ring))" 
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apply (simp add: diff_minus left_distrib add_ac) 
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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333 
done 
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334 

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335 
lemma le_add_iff1: 
14738  336 
"(a*e + c \<le> b*e + d) = ((ab)*e + c \<le> (d::'a::pordered_ring))" 
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apply (simp add: diff_minus left_distrib add_ac) 
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338 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
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339 
done 
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340 

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341 
lemma le_add_iff2: 
14738  342 
"(a*e + c \<le> b*e + d) = (c \<le> (ba)*e + (d::'a::pordered_ring))" 
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apply (simp add: diff_minus left_distrib add_ac) 
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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345 
done 
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346 

23389  347 

14270  348 
subsection {* Ordering Rules for Multiplication *} 
349 

14348
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350 
lemma mult_left_le_imp_le: 
14738  351 
"[c*a \<le> c*b; 0 < c] ==> a \<le> (b::'a::ordered_semiring_strict)" 
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by (force simp add: mult_strict_left_mono linorder_not_less [symmetric]) 
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353 

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354 
lemma mult_right_le_imp_le: 
14738  355 
"[a*c \<le> b*c; 0 < c] ==> a \<le> (b::'a::ordered_semiring_strict)" 
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356 
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric]) 
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357 

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358 
lemma mult_left_less_imp_less: 
14738  359 
"[c*a < c*b; 0 \<le> c] ==> a < (b::'a::ordered_semiring_strict)" 
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360 
by (force simp add: mult_left_mono linorder_not_le [symmetric]) 
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361 

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362 
lemma mult_right_less_imp_less: 
14738  363 
"[a*c < b*c; 0 \<le> c] ==> a < (b::'a::ordered_semiring_strict)" 
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364 
by (force simp add: mult_right_mono linorder_not_le [symmetric]) 
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365 

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366 
lemma mult_strict_left_mono_neg: 
14738  367 
"[b < a; c < 0] ==> c * a < c * (b::'a::ordered_ring_strict)" 
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368 
apply (drule mult_strict_left_mono [of _ _ "c"]) 
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369 
apply (simp_all add: minus_mult_left [symmetric]) 
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370 
done 
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371 

14738  372 
lemma mult_left_mono_neg: 
373 
"[b \<le> a; c \<le> 0] ==> c * a \<le> c * (b::'a::pordered_ring)" 

374 
apply (drule mult_left_mono [of _ _ "c"]) 

375 
apply (simp_all add: minus_mult_left [symmetric]) 

376 
done 

377 

14265
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378 
lemma mult_strict_right_mono_neg: 
14738  379 
"[b < a; c < 0] ==> a * c < b * (c::'a::ordered_ring_strict)" 
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380 
apply (drule mult_strict_right_mono [of _ _ "c"]) 
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381 
apply (simp_all add: minus_mult_right [symmetric]) 
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382 
done 
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383 

14738  384 
lemma mult_right_mono_neg: 
385 
"[b \<le> a; c \<le> 0] ==> a * c \<le> (b::'a::pordered_ring) * c" 

386 
apply (drule mult_right_mono [of _ _ "c"]) 

387 
apply (simp) 

388 
apply (simp_all add: minus_mult_right [symmetric]) 

389 
done 

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390 

23389  391 

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392 
subsection{* Products of Signs *} 
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393 

16775
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394 
lemma mult_pos_pos: "[ (0::'a::ordered_semiring_strict) < a; 0 < b ] ==> 0 < a*b" 
14265
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395 
by (drule mult_strict_left_mono [of 0 b], auto) 
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396 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

397 
lemma mult_nonneg_nonneg: "[ (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b ] ==> 0 \<le> a*b" 
14738  398 
by (drule mult_left_mono [of 0 b], auto) 
399 

400 
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

401 
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

402 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

403 
lemma mult_nonneg_nonpos: "[ (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 ] ==> a*b \<le> 0" 
14738  404 
by (drule mult_left_mono [of b 0], auto) 
405 

406 
lemma mult_pos_neg2: "[ (0::'a::ordered_semiring_strict) < a; b < 0 ] ==> b*a < 0" 

407 
by (drule mult_strict_right_mono[of b 0], auto) 

408 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

409 
lemma mult_nonneg_nonpos2: "[ (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 ] ==> b*a \<le> 0" 
14738  410 
by (drule mult_right_mono[of b 0], auto) 
411 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

412 
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

413 
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

414 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

415 
lemma mult_nonpos_nonpos: "[ a \<le> (0::'a::pordered_ring); b \<le> 0 ] ==> 0 \<le> a*b" 
14738  416 
by (drule mult_right_mono_neg[of a 0 b ], auto) 
417 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

418 
lemma zero_less_mult_pos: 
14738  419 
"[ 0 < a*b; 0 < a] ==> 0 < (b::'a::ordered_semiring_strict)" 
21328  420 
apply (cases "b\<le>0") 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

421 
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

422 
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

423 
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

424 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

425 

14738  426 
lemma zero_less_mult_pos2: 
427 
"[ 0 < b*a; 0 < a] ==> 0 < (b::'a::ordered_semiring_strict)" 

21328  428 
apply (cases "b\<le>0") 
14738  429 
apply (auto simp add: order_le_less linorder_not_less) 
430 
apply (drule_tac mult_pos_neg2 [of a b]) 

431 
apply (auto dest: order_less_not_sym) 

432 
done 

433 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

434 
lemma zero_less_mult_iff: 
14738  435 
"((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

436 
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

437 
mult_neg_neg) 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

438 
apply (blast dest: zero_less_mult_pos) 
14738  439 
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

440 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

441 

22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

442 
lemma mult_eq_0_iff [simp]: 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

443 
fixes a b :: "'a::ring_no_zero_divisors" 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

444 
shows "(a * b = 0) = (a = 0 \<or> b = 0)" 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

445 
by (cases "a = 0 \<or> b = 0", auto dest: no_zero_divisors) 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

446 

775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

447 
instance ordered_ring_strict \<subseteq> ring_no_zero_divisors 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

448 
apply intro_classes 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

449 
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

450 
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

451 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

452 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

453 
lemma zero_le_mult_iff: 
14738  454 
"((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

455 
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

456 
zero_less_mult_iff) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

457 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

458 
lemma mult_less_0_iff: 
14738  459 
"(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

460 
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

461 
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

462 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

463 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

464 
lemma mult_le_0_iff: 
14738  465 
"(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

466 
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

467 
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

468 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

469 

14738  470 
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

471 
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos) 
14738  472 

473 
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

474 
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2) 
14738  475 

23095  476 
lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a" 
477 
by (simp add: zero_le_mult_iff linorder_linear) 

478 

479 
lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))" 

480 
by (simp add: not_less) 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

481 

14738  482 
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom} 
483 
theorems available to members of @{term ordered_idom} *} 

484 

485 
instance ordered_idom \<subseteq> ordered_semidom 

14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

486 
proof 
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

487 
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

488 
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

489 
qed 
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

490 

14738  491 
instance ordered_idom \<subseteq> idom .. 
492 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

493 
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

494 

17085  495 
lemmas one_neq_zero = zero_neq_one [THEN not_sym] 
496 
declare one_neq_zero [simp] 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

497 

14738  498 
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

499 
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

500 

14738  501 
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0" 
502 
by (simp add: linorder_not_le) 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

503 

14738  504 
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0" 
505 
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

506 

23389  507 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

508 
subsection{*More Monotonicity*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

509 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

510 
text{*Strict monotonicity in both arguments*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

511 
lemma mult_strict_mono: 
14738  512 
"[a<b; c<d; 0<b; 0\<le>c] ==> a * c < b * (d::'a::ordered_semiring_strict)" 
21328  513 
apply (cases "c=0") 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

514 
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

515 
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

516 
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

517 
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

518 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

519 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

520 
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

521 
lemma mult_strict_mono': 
14738  522 
"[ 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

523 
apply (rule mult_strict_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

524 
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

525 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

526 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

527 
lemma mult_mono: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

528 
"[a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c] 
14738  529 
==> 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

530 
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

531 
apply (erule mult_left_mono, assumption) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

532 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

533 

21258  534 
lemma mult_mono': 
535 
"[a \<le> b; c \<le> d; 0 \<le> a; 0 \<le> c] 

536 
==> a * c \<le> b * (d::'a::pordered_semiring)" 

537 
apply (rule mult_mono) 

538 
apply (fast intro: order_trans)+ 

539 
done 

540 

14738  541 
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

542 
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

543 
apply (simp add: order_less_trans [OF zero_less_one]) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

544 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

545 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

546 
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

547 
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

548 
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

549 
apply (erule order_less_le_trans) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

550 
apply (erule mult_left_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

551 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

552 
apply (erule mult_strict_right_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

553 
apply assumption 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

554 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

555 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

556 
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

557 
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

558 
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

559 
apply (erule order_le_less_trans) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

560 
apply (erule mult_strict_left_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

561 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

562 
apply (erule mult_right_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

563 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

564 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

565 

23389  566 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

567 
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

568 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

569 
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

570 
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

571 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

572 
text{*These ``disjunction'' versions produce two cases when the comparison is 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

573 
an assumption, but effectively four when the comparison is a goal.*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

574 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

575 
lemma mult_less_cancel_right_disj: 
14738  576 
"(a*c < b*c) = ((0 < c & a < b)  (c < 0 & b < (a::'a::ordered_ring_strict)))" 
21328  577 
apply (cases "c = 0") 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

578 
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

579 
mult_strict_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

580 
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

581 
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

582 
linorder_not_le [symmetric, of a]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

583 
apply (erule_tac [!] notE) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

584 
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

585 
mult_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

586 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

587 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

588 
lemma mult_less_cancel_left_disj: 
14738  589 
"(c*a < c*b) = ((0 < c & a < b)  (c < 0 & b < (a::'a::ordered_ring_strict)))" 
21328  590 
apply (cases "c = 0") 
14738  591 
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
592 
mult_strict_left_mono_neg) 

593 
apply (auto simp add: linorder_not_less 

594 
linorder_not_le [symmetric, of "c*a"] 

595 
linorder_not_le [symmetric, of a]) 

596 
apply (erule_tac [!] notE) 

597 
apply (auto simp add: order_less_imp_le mult_left_mono 

598 
mult_left_mono_neg) 

599 
done 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

600 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

601 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

602 
text{*The ``conjunction of implication'' lemmas produce two cases when the 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

603 
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

604 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

605 
lemma mult_less_cancel_right: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

606 
fixes c :: "'a :: ordered_ring_strict" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

607 
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

608 
by (insert mult_less_cancel_right_disj [of a c b], auto) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

609 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

610 
lemma mult_less_cancel_left: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

611 
fixes c :: "'a :: ordered_ring_strict" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

612 
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

613 
by (insert mult_less_cancel_left_disj [of c a b], auto) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

614 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

615 
lemma mult_le_cancel_right: 
14738  616 
"(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

617 
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

618 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

619 
lemma mult_le_cancel_left: 
14738  620 
"(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

621 
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

622 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

623 
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

624 
assumes less: "c*a < c*b" and nonneg: "0 \<le> c" 
14738  625 
shows "a < (b::'a::ordered_semiring_strict)" 
14377  626 
proof (rule ccontr) 
627 
assume "~ a < b" 

628 
hence "b \<le> a" by (simp add: linorder_not_less) 

23389  629 
hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono) 
14377  630 
with this and less show False 
631 
by (simp add: linorder_not_less [symmetric]) 

632 
qed 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

633 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

634 
lemma mult_less_imp_less_right: 
14738  635 
assumes less: "a*c < b*c" and nonneg: "0 <= c" 
636 
shows "a < (b::'a::ordered_semiring_strict)" 

637 
proof (rule ccontr) 

638 
assume "~ a < b" 

639 
hence "b \<le> a" by (simp add: linorder_not_less) 

23389  640 
hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono) 
14738  641 
with this and less show False 
642 
by (simp add: linorder_not_less [symmetric]) 

643 
qed 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

644 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

645 
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

646 
lemma mult_cancel_right [simp]: 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

647 
fixes a b c :: "'a::ring_no_zero_divisors" 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

648 
shows "(a * c = b * c) = (c = 0 \<or> a = b)" 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

649 
proof  
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

650 
have "(a * c = b * c) = ((a  b) * c = 0)" 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

651 
by (simp add: left_diff_distrib) 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

652 
thus ?thesis 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

653 
by (simp add: disj_commute) 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

654 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

655 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

656 
lemma mult_cancel_left [simp]: 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

657 
fixes a b c :: "'a::ring_no_zero_divisors" 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

658 
shows "(c * a = c * b) = (c = 0 \<or> a = b)" 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

659 
proof  
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

660 
have "(c * a = c * b) = (c * (a  b) = 0)" 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

661 
by (simp add: right_diff_distrib) 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

662 
thus ?thesis 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

663 
by simp 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

664 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

665 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

666 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

667 
subsubsection{*Special Cancellation Simprules for Multiplication*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

668 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

669 
text{*These also produce two cases when the comparison is a goal.*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

670 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

671 
lemma mult_le_cancel_right1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

672 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

673 
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

674 
by (insert mult_le_cancel_right [of 1 c b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

675 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

676 
lemma mult_le_cancel_right2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

677 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

678 
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

679 
by (insert mult_le_cancel_right [of a c 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

680 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

681 
lemma mult_le_cancel_left1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

682 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

683 
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

684 
by (insert mult_le_cancel_left [of c 1 b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

685 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

686 
lemma mult_le_cancel_left2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

687 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

688 
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

689 
by (insert mult_le_cancel_left [of c a 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

690 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

691 
lemma mult_less_cancel_right1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

692 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

693 
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

694 
by (insert mult_less_cancel_right [of 1 c b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

695 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

696 
lemma mult_less_cancel_right2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

697 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

698 
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

699 
by (insert mult_less_cancel_right [of a c 1], simp) 
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 
lemma mult_less_cancel_left1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

702 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

703 
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

704 
by (insert mult_less_cancel_left [of c 1 b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

705 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

706 
lemma mult_less_cancel_left2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

707 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

708 
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

709 
by (insert mult_less_cancel_left [of c a 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

710 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

711 
lemma mult_cancel_right1 [simp]: 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

712 
fixes c :: "'a :: dom" 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

713 
shows "(c = b*c) = (c = 0  b=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

714 
by (insert mult_cancel_right [of 1 c b], force) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

715 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

716 
lemma mult_cancel_right2 [simp]: 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

717 
fixes c :: "'a :: dom" 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

718 
shows "(a*c = c) = (c = 0  a=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

719 
by (insert mult_cancel_right [of a c 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

720 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

721 
lemma mult_cancel_left1 [simp]: 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

722 
fixes c :: "'a :: dom" 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

723 
shows "(c = c*b) = (c = 0  b=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

724 
by (insert mult_cancel_left [of c 1 b], force) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

725 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

726 
lemma mult_cancel_left2 [simp]: 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

727 
fixes c :: "'a :: dom" 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

728 
shows "(c*a = c) = (c = 0  a=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

729 
by (insert mult_cancel_left [of c a 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

730 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

731 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

732 
text{*Simprules for comparisons where common factors can be cancelled.*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

733 
lemmas mult_compare_simps = 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

734 
mult_le_cancel_right mult_le_cancel_left 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

735 
mult_le_cancel_right1 mult_le_cancel_right2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

736 
mult_le_cancel_left1 mult_le_cancel_left2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

737 
mult_less_cancel_right mult_less_cancel_left 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

738 
mult_less_cancel_right1 mult_less_cancel_right2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

739 
mult_less_cancel_left1 mult_less_cancel_left2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

740 
mult_cancel_right mult_cancel_left 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

741 
mult_cancel_right1 mult_cancel_right2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

742 
mult_cancel_left1 mult_cancel_left2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

743 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

744 

14738  745 
text{*This list of rewrites decides ring equalities by ordered rewriting.*} 
15178  746 
lemmas ring_eq_simps = 
747 
(* mult_ac*) 

14738  748 
left_distrib right_distrib left_diff_distrib right_diff_distrib 
15178  749 
group_eq_simps 
750 
(* add_ac 

14738  751 
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 
15178  752 
diff_eq_eq eq_diff_eq *) 
14738  753 

23389  754 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

755 
subsection {* Fields *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

756 

14288  757 
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" 
758 
proof 

759 
assume neq: "b \<noteq> 0" 

760 
{ 

761 
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) 

762 
also assume "a / b = 1" 

763 
finally show "a = b" by simp 

764 
next 

765 
assume "a = b" 

766 
with neq show "a / b = 1" by (simp add: divide_inverse) 

767 
} 

768 
qed 

769 

770 
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" 

771 
by (simp add: divide_inverse) 

772 

23398  773 
lemma divide_self[simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1" 
14288  774 
by (simp add: divide_inverse) 
775 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

776 
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

777 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

778 

15228  779 
lemma divide_self_if [simp]: 
780 
"a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)" 

781 
by (simp add: divide_self) 

782 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

783 
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

784 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

785 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

786 
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

787 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

788 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

789 
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c" 
14293  790 
by (simp add: divide_inverse left_distrib) 
791 

792 

14270  793 
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement 
794 
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

795 
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

796 
"(a*b = (0::'a::division_ring)) = (a = 0  b = 0)" 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

797 
by simp 
14270  798 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

799 
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

800 
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

801 
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

802 
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

803 
shows "a=b" 
14377  804 
proof  
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

805 
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

806 
by (simp add: eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

807 
thus "a=b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

808 
by (simp add: mult_assoc cnz) 
14377  809 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

810 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

811 
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

812 
"(a*c = b*c) = (c = (0::'a::division_ring)  a=b)" 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

813 
by simp 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

814 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

815 
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

816 
"(c*a = c*b) = (c = (0::'a::division_ring)  a=b)" 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

817 
by simp 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

818 

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

819 
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

820 
"a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)" 
14377  821 
proof 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

822 
assume ianz: "inverse a = 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

823 
assume "a \<noteq> 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

824 
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

825 
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

826 
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

827 
thus False by (simp add: eq_commute) 
14377  828 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

829 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

830 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

831 
subsection{*Basic Properties of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

832 

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

833 
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

834 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

835 
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

836 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

837 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

838 
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

839 
"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

840 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

841 
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

842 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

843 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

844 
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

845 
"(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

846 
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

847 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

848 
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

849 
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

850 
shows "inverse(a) = inverse(a::'a::division_ring)" 
14377  851 
proof  
852 
have "a * inverse ( a) = a *  inverse a" 

853 
by simp 

854 
thus ?thesis 

855 
by (simp only: field_mult_cancel_left, simp) 

856 
qed 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

857 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

858 
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

859 
"inverse(a) = inverse(a::'a::{division_ring,division_by_zero})" 
14377  860 
proof cases 
861 
assume "a=0" thus ?thesis by (simp add: inverse_zero) 

862 
next 

863 
assume "a\<noteq>0" 

864 
thus ?thesis by (simp add: nonzero_inverse_minus_eq) 

865 
qed 

14268
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 nonzero_inverse_eq_imp_eq: 
14269  868 
assumes inveq: "inverse a = inverse b" 
869 
and anz: "a \<noteq> 0" 

870 
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

871 
shows "a = (b::'a::division_ring)" 
14377  872 
proof  
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

873 
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

874 
by (simp add: inveq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

875 
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

876 
by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

877 
thus "a = b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

878 
by (simp add: mult_assoc anz bnz) 
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_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

882 
"inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})" 
21328  883 
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

884 
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

885 
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

886 
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

887 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

888 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

889 
lemma inverse_eq_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

890 
"(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

891 
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

892 

14270  893 
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

894 
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

895 
shows "inverse(inverse (a::'a::division_ring)) = a" 
14270  896 
proof  
897 
have "(inverse (inverse a) * inverse a) * a = a" 

898 
by (simp add: nonzero_imp_inverse_nonzero) 

899 
thus ?thesis 

900 
by (simp add: mult_assoc) 

901 
qed 

902 

903 
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

904 
"inverse(inverse (a::'a::{division_ring,division_by_zero})) = a" 
14270  905 
proof cases 
906 
assume "a=0" thus ?thesis by simp 

907 
next 

908 
assume "a\<noteq>0" 

909 
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) 

910 
qed 

911 

20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset

912 
lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)" 
14270  913 
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

914 
have "inverse 1 * 1 = (1::'a::division_ring)" 
14270  915 
by (rule left_inverse [OF zero_neq_one [symmetric]]) 
916 
thus ?thesis by simp 

917 
qed 

918 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

919 
lemma inverse_unique: 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

920 
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

921 
shows "inverse a = (b::'a::division_ring)" 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

922 
proof  
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

923 
have "a \<noteq> 0" using ab by auto 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

924 
moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

925 
ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

926 
qed 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

927 

14270  928 
lemma nonzero_inverse_mult_distrib: 
929 
assumes anz: "a \<noteq> 0" 

930 
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

931 
shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)" 
14270  932 
proof  
933 
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 

934 
by (simp add: field_mult_eq_0_iff anz bnz) 

935 
hence "inverse(a*b) * a = inverse(b)" 

936 
by (simp add: mult_assoc bnz) 

937 
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 

938 
by simp 

939 
thus ?thesis 

940 
by (simp add: mult_assoc anz) 

941 
qed 

942 

943 
text{*This version builds in division by zero while also reorienting 

944 
the righthand side.*} 

945 
lemma inverse_mult_distrib [simp]: 

946 
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" 

947 
proof cases 

948 
assume "a \<noteq> 0 & b \<noteq> 0" 

22993  949 
thus ?thesis 
950 
by (simp add: nonzero_inverse_mult_distrib mult_commute) 

14270  951 
next 
952 
assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

22993  953 
thus ?thesis 
954 
by force 

14270  955 
qed 
956 

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

957 
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

958 
"[(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

959 
==> inverse a + inverse b = inverse a * (a+b) * inverse b" 
22993  960 
by (simp add: right_distrib left_distrib mult_assoc) 
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

961 

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

962 
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

963 
"[(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

964 
==> inverse a  inverse b = inverse a * (ba) * inverse b" 
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset

965 
by (simp add: right_diff_distrib left_diff_distrib mult_assoc) 
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset

966 

14270  967 
text{*There is no slick version using division by zero.*} 
968 
lemma inverse_add: 

969 
"[a \<noteq> 0; b \<noteq> 0] 

970 
==> 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

971 
by (simp add: division_ring_inverse_add mult_ac) 
14270  972 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

973 
lemma inverse_divide [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

974 
"inverse (a/b) = b / (a::'a::{field,division_by_zero})" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

975 
by (simp add: divide_inverse mult_commute) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

976 

23389  977 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

978 
subsection {* Calculations with fractions *} 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

979 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

980 
lemma nonzero_mult_divide_cancel_left: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

981 
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

982 
shows "(c*a)/(c*b) = a/(b::'a::field)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

983 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

984 
have "(c*a)/(c*b) = c * a * (inverse b * inverse c)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

985 
by (simp add: field_mult_eq_0_iff divide_inverse 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

986 
nonzero_inverse_mult_distrib) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

987 
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

988 
by (simp only: mult_ac) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

989 
also have "... = a * inverse b" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

990 
by simp 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

991 
finally show ?thesis 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

992 
by (simp add: divide_inverse) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

993 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

994 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

995 
lemma mult_divide_cancel_left: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

996 
"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})" 
21328  997 
apply (cases "b = 0") 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

998 
apply (simp_all add: nonzero_mult_divide_cancel_left) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

999 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1000 

14321  1001 
lemma nonzero_mult_divide_cancel_right: 
1002 
"[b\<noteq>0; c\<noteq>0] ==> (a*c) / (b*c) = a/(b::'a::field)" 

1003 
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 

1004 

1005 
lemma mult_divide_cancel_right: 

1006 
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" 

21328  1007 
apply (cases "b = 0") 
14321  1008 
apply (simp_all add: nonzero_mult_divide_cancel_right) 
1009 
done 

14284
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

1010 
lemma divide_1 [simp]: "a/1 = (a::'a::field)" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

1011 
by (simp add: divide_inverse) 
14284
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

1012 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1013 
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

1014 
by (simp add: divide_inverse mult_assoc) 
14288  1015 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

1016 
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

1017 
by (simp add: divide_inverse mult_ac) 
14288  1018 

1019 
lemma divide_divide_eq_right [simp]: 

1020 
"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

1021 
by (simp add: divide_inverse mult_ac) 
14288  1022 

1023 
lemma divide_divide_eq_left [simp]: 

1024 
"(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

1025 
by (simp add: divide_inverse mult_assoc) 
14288  1026 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1027 
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

1028 
x / y + w / z = (x * z + w * y) / (y * z)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1029 
apply (subgoal_tac "x / y = (x * z) / (y * z)") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1030 
apply (erule ssubst) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1031 
apply (subgoal_tac "w / z = (w * y) / (y * z)") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1032 
apply (erule ssubst) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1033 
apply (rule add_divide_distrib [THEN sym]) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1034 
apply (subst mult_commute) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1035 
apply (erule nonzero_mult_divide_cancel_left [THEN sym]) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1036 
apply assumption 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1037 
apply (erule nonzero_mult_divide_cancel_right [THEN sym]) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1038 
apply assumption 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1039 
done 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1040 

23389  1041 

23406
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1042 
lemma nonzero_mult_divide_cancel_right': 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1043 
"b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)" 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1044 
proof  
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1045 
assume b: "b \<noteq> 0" 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1046 
have "a * b / b = a * (b / b)" by (simp add: times_divide_eq_right) 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1047 
also have "\<dots> = a" by (simp add: divide_self b) 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1048 
finally show "a * b / b = a" . 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1049 
qed 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1050 

167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1051 
lemma nonzero_mult_divide_cancel_left': 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1052 
"a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)" 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1053 
proof  
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1054 
assume b: "a \<noteq> 0" 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1055 
have "a * b / a = b * a / a" by (simp add: mult_commute) 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1056 
also have "\<dots> = b * (a / a)" by (simp add: times_divide_eq_right) 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1057 
also have "\<dots> = b" by (simp add: divide_self b) 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1058 
finally show "a * b / a = b" . 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1059 
qed 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1060 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1061 
subsubsection{*Special Cancellation Simprules for Division*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1062 

23400
a64b39e5809b
The simprocs "divide_cancel_factor" and "ring_eq_cancel_factor" no
nipkow
parents:
23398
diff
changeset

1063 
(* FIXME need not be a simprule once "divide_cancel_factor" has been fixed *) 
a64b39e5809b
The simprocs "divide_cancel_factor" and "ring_eq_cancel_factor" no
nipkow
parents:
23398
diff
changeset

1064 
lemma mult_divide_cancel_left_if[simp]: 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1065 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1066 
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1067 
by (simp add: mult_divide_cancel_left) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1068 

23400
a64b39e5809b
The simprocs "divide_cancel_factor" and "ring_eq_cancel_factor" no
nipkow
parents:
23398
diff
changeset

1069 
(* Not needed anymore because now subsumed by simproc "divide_cancel_factor" 
a64b39e5809b
The simprocs "divide_cancel_factor" and "ring_eq_cancel_factor" no
nipkow
parents:
23398
diff
changeset

1070 
lemma mult_divide_cancel_right_if: 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1071 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1072 
shows "(a*c) / (b*c) = (if c=0 then 0 else a/b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1073 
by (simp add: mult_divide_cancel_right) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1074 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1075 
lemma mult_divide_cancel_left_if1 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1076 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1077 
shows "c / (c*b) = (if c=0 then 0 else 1/b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1078 
apply (insert mult_divide_cancel_left_if [of c 1 b]) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1079 
apply (simp del: mult_divide_cancel_left_if) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1080 
done 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1081 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1082 
lemma mult_divide_cancel_left_if2 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1083 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1084 
shows "(c*a) / c = (if c=0 then 0 else a)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1085 
apply (insert mult_divide_cancel_left_if [of c a 1]) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1086 
apply (simp del: mult_divide_cancel_left_if) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1087 
done 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1088 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1089 
lemma mult_divide_cancel_right_if1 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1090 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1091 
shows "c / (b*c) = (if c=0 then 0 else 1/b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1092 
apply (insert mult_divide_cancel_right_if [of 1 c b]) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1093 
apply (simp del: mult_divide_cancel_right_if) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1094 
done 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1095 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1096 
lemma mult_divide_cancel_right_if2 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1097 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1098 
shows "(a*c) / c = (if c=0 then 0 else a)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1099 
apply (insert mult_divide_cancel_right_if [of a c 1]) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1100 
apply (simp del: mult_divide_cancel_right_if) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1101 
done 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1102 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1103 
text{*Two lemmas for cancelling the denominator*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1104 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1105 
lemma times_divide_self_right [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1106 
fixes a :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1107 
shows "a * (b/a) = (if a=0 then 0 else b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1108 
by (simp add: times_divide_eq_right) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1109 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1110 
lemma times_divide_self_left [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1111 
fixes a :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1112 
shows "(b/a) * a = (if a=0 then 0 else b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1113 
by (simp add: times_divide_eq_left) 
23400
a64b39e5809b
The simprocs "divide_cancel_factor" and "ring_eq_cancel_factor" no
nipkow
parents:
23398
diff
changeset

1114 
*) 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1115 

14293  1116 
subsection {* Division and Unary Minus *} 
1117 

1118 
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==>  (a/b) = (a) / (b::'a::field)" 

1119 
by (simp add: divide_inverse minus_mult_left) 

1120 

1121 
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==>  (a/b) = a / (b::'a::field)" 

1122 
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) 

1123 

1124 
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (a)/(b) = a / (b::'a::field)" 

1125 
by (simp add: divide_inverse nonzero_inverse_minus_eq) 

1126 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

1127 
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

1128 
by (simp add: divide_inverse minus_mult_left [symmetric]) 
14293  1129 

1130 
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

1131 
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

1132 

14293  1133 

1134 
text{*The effect is to extract signs from divisions*} 

17085  1135 
lemmas divide_minus_left = minus_divide_left [symmetric] 
1136 
lemmas divide_minus_right = minus_divide_right [symmetric] 

1137 
declare divide_minus_left [simp] divide_minus_right [simp] 

14293  1138 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1139 
text{*Also, extract signs from products*} 
17085  1140 
lemmas mult_minus_left = minus_mult_left [symmetric] 
1141 
lemmas mult_minus_right = minus_mult_right [symmetric] 

1142 
declare mult_minus_left [simp] mult_minus_right [simp] 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1143 

14293  1144 
lemma minus_divide_divide [simp]: 
1145 
"(a)/(b) = a / (b::'a::{field,division_by_zero})" 

21328  1146 
apply (cases "b=0", simp) 
14293  1147 
apply (simp add: nonzero_minus_divide_divide) 
1148 
done 

1149 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

1150 
lemma diff_divide_distrib: "(ab)/(c::'a::field) = a/c  b/c" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1151 
by (simp add: diff_minus add_divide_distrib) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1152 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1153 
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

1154 
x / y  w / z = (x * z  w * y) / (y * z)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1155 
apply (subst diff_def)+ 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1156 
apply (subst minus_divide_left) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1157 
apply (subst add_frac_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1158 
apply simp_all 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1159 
done 
14293  1160 

23389  1161 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1162 
subsection {* Ordered Fields *} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1163 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1164 
lemma positive_imp_inverse_positive: 
14269  1165 
assumes a_gt_0: "0 < a" shows "0 < inverse (a::'a::ordered_field)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1166 
proof  
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1167 
have "0 < a * inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1168 
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

1169 
thus "0 < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1170 
by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1171 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1172 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1173 
lemma negative_imp_inverse_negative: 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1174 
"a < 0 ==> inverse a < (0::'a::ordered_field)" 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1175 
by (insert positive_imp_inverse_positive [of "a"], 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1176 
simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1177 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1178 
lemma inverse_le_imp_le: 
14269  1179 
assumes invle: "inverse a \<le> inverse b" 
1180 
and apos: "0 < a" 

1181 
shows "b \<le> (a::'a::ordered_field)" 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1182 
proof (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1183 
assume "~ b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1184 
hence "a < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1185 
by (simp add: linorder_not_le) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1186 
hence bpos: "0 < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1187 
by (blast intro: apos order_less_trans) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1188 
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

1189 
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

1190 
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

1191 
by (simp add: bpos 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

1192 
thus "b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1193 
by (simp add: mult_assoc apos bpos order_less_imp_not_eq2) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1194 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1195 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1196 
lemma inverse_positive_imp_positive: 
23389  1197 
assumes inv_gt_0: "0 < inverse a" 
1198 
and nz: "a \<noteq> 0" 

1199 
shows "0 < (a::'a::ordered_field)" 

1200 
proof  

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1201 
have "0 < inverse (inverse a)" 
23389  1202 
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

1203 
thus "0 < a" 
23389  1204 
using nz by (simp add: nonzero_inverse_inverse_eq) 
1205 
qed 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1206 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1207 
lemma inverse_positive_iff_positive [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1208 
"(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))" 
21328  1209 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1210 
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1211 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1212 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1213 
lemma inverse_negative_imp_negative: 
23389  1214 
assumes inv_less_0: "inverse a < 0" 
1215 
and nz: "a \<noteq> 0" 

1216 
shows "a < (0::'a::ordered_field)" 

1217 
proof  

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1218 
have "inverse (inverse a) < 0" 
23389  1219 
using inv_less_0 by (rule negative_imp_inverse_negative) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1220 
thus "a < 0" 
23389  1221 
using nz by (simp add: nonzero_inverse_inverse_eq) 
1222 
qed 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1223 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1224 
lemma inverse_negative_iff_negative [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1225 
"(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))" 
21328  1226 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1227 
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1228 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1229 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1230 
lemma inverse_nonnegative_iff_nonnegative [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1231 
"(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1232 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1233 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1234 
lemma inverse_nonpositive_iff_nonpositive [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1235 
"(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1236 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1237 

23406
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1238 
lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)" 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1239 
proof 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1240 
fix x::'a 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1241 
have m1: " (1::'a) < 0" by simp 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1242 
from add_strict_right_mono[OF m1, where c=x] 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1243 
have "( 1) + x < x" by simp 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1244 
thus "\<exists>y. y < x" by blast 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1245 
qed 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1246 

167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1247 
lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)" 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1248 
proof 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1249 
fix x::'a 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1250 
have m1: " (1::'a) > 0" by simp 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1251 
from add_strict_right_mono[OF m1, where c=x] 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1252 
have "1 + x > x" by simp 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1253 
thus "\<exists>y. y > x" by blast 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

1254 
qed 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1255 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1256 
subsection{*AntiMonotonicity of @{term inverse}*} 