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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, Tobias Nipkow, Lawrence C Paulson, and Markus Wenzel, 
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with contributions by Jeremy Avigad 
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*) 
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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 

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fix a b c :: 'a 

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

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

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

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

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

14504  63 
qed 
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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 
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+ 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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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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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  202 
by (simp add: left_distrib diff_minus 
203 
minus_mult_left [symmetric] minus_mult_right [symmetric]) 

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lemmas ring_distribs = 
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right_distrib left_distrib left_diff_distrib right_diff_distrib 
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text{*This list of rewrites simplifies ring terms by multiplying 
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everything out and bringing sums and products into a canonical form 
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(by ordered rewriting). As a result it decides ring equalities but 
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211 
also helps with inequalities. *} 
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lemmas ring_simps = group_simps ring_distribs 
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22390  214 
class mult_mono = times + zero + ord + 
215 
assumes mult_left_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b" 

216 
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  218 
class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add 
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22390  220 
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  223 
instance pordered_cancel_semiring \<subseteq> semiring_0_cancel .. 
224 

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instance pordered_cancel_semiring \<subseteq> pordered_semiring .. 
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22390  227 
class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + 
228 
assumes mult_strict_left_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b" 

229 
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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14940  231 
instance ordered_semiring_strict \<subseteq> semiring_0_cancel .. 
232 

14738  233 
instance ordered_semiring_strict \<subseteq> pordered_cancel_semiring 
234 
apply intro_classes 

21328  235 
apply (cases "a < b & 0 < c") 
14738  236 
apply (auto simp add: mult_strict_left_mono order_less_le) 
237 
apply (auto simp add: mult_strict_left_mono order_le_less) 

238 
apply (simp add: mult_strict_right_mono) 

14270  239 
done 
240 

22390  241 
class mult_mono1 = times + zero + ord + 
242 
assumes mult_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b" 

14270  243 

22390  244 
class pordered_comm_semiring = comm_semiring_0 
245 
+ pordered_ab_semigroup_add + mult_mono1 

14270  246 

22390  247 
class pordered_cancel_comm_semiring = comm_semiring_0_cancel 
248 
+ pordered_ab_semigroup_add + mult_mono1 

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14738  250 
instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring .. 
14270  251 

22390  252 
class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add + 
253 
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  255 
instance pordered_comm_semiring \<subseteq> pordered_semiring 
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256 
proof 
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257 
fix a b c :: 'a 
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258 
assume A: "a <= b" "0 <= c" 
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259 
with mult_mono show "c * a <= c * b" . 
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260 

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from mult_commute have "a * c = c * a" .. 
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also from mult_mono A have "\<dots> <= c * b" . 
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also from mult_commute have "c * b = b * c" .. 
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264 
finally show "a * c <= b * c" . 
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265 
qed 
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14738  267 
instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring .. 
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14738  269 
instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict 
270 
by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+) 

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14738  272 
instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring 
273 
apply (intro_classes) 

21328  274 
apply (cases "a < b & 0 < c") 
14738  275 
apply (auto simp add: mult_strict_left_mono order_less_le) 
276 
apply (auto simp add: mult_strict_left_mono order_le_less) 

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done 
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22390  279 
class pordered_ring = ring + pordered_cancel_semiring 
14270  280 

14738  281 
instance pordered_ring \<subseteq> pordered_ab_group_add .. 
14270  282 

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class lordered_ring = pordered_ring + lordered_ab_group_abs 
14270  284 

14940  285 
instance lordered_ring \<subseteq> lordered_ab_group_meet .. 
286 

287 
instance lordered_ring \<subseteq> lordered_ab_group_join .. 

288 

22390  289 
class abs_if = minus + ord + zero + 
290 
assumes abs_if: "abs a = (if a \<sqsubset> 0 then (uminus a) else a)" 

14270  291 

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class ordered_ring_strict = ring + ordered_semiring_strict + abs_if + lordered_ab_group 
14270  293 

14738  294 
instance ordered_ring_strict \<subseteq> lordered_ring 
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by intro_classes (simp add: abs_if sup_eq_if) 
14270  296 

22390  297 
class pordered_comm_ring = comm_ring + pordered_comm_semiring 
14270  298 

23073  299 
instance pordered_comm_ring \<subseteq> pordered_cancel_comm_semiring .. 
300 

22390  301 
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict + 
302 
(*previously ordered_semiring*) 

303 
assumes zero_less_one [simp]: "\<^loc>0 \<sqsubset> \<^loc>1" 

14270  304 

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class ordered_idom = comm_ring_1 + ordered_comm_semiring_strict + abs_if + lordered_ab_group 
22390  306 
(*previously ordered_ring*) 
14270  307 

14738  308 
instance ordered_idom \<subseteq> ordered_ring_strict .. 
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23073  310 
instance ordered_idom \<subseteq> pordered_comm_ring .. 
311 

22390  312 
class ordered_field = field + ordered_idom 
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15923  314 
lemmas linorder_neqE_ordered_idom = 
315 
linorder_neqE[where 'a = "?'b::ordered_idom"] 

316 

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lemma eq_add_iff1: 
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"(a*e + c = b*e + d) = ((ab)*e + c = (d::'a::ring))" 
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by (simp add: ring_simps) 
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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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by (simp add: ring_simps) 
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lemma less_add_iff1: 
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"(a*e + c < b*e + d) = ((ab)*e + c < (d::'a::pordered_ring))" 
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by (simp add: ring_simps) 
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328 

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lemma less_add_iff2: 
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330 
"(a*e + c < b*e + d) = (c < (ba)*e + (d::'a::pordered_ring))" 
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331 
by (simp add: ring_simps) 
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332 

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lemma le_add_iff1: 
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334 
"(a*e + c \<le> b*e + d) = ((ab)*e + c \<le> (d::'a::pordered_ring))" 
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335 
by (simp add: ring_simps) 
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336 

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337 
lemma le_add_iff2: 
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338 
"(a*e + c \<le> b*e + d) = (c \<le> (ba)*e + (d::'a::pordered_ring))" 
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by (simp add: ring_simps) 
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340 

23389  341 

14270  342 
subsection {* Ordering Rules for Multiplication *} 
343 

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lemma mult_left_le_imp_le: 
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"[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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347 

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lemma mult_right_le_imp_le: 
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"[a*c \<le> b*c; 0 < c] ==> a \<le> (b::'a::ordered_semiring_strict)" 
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by (force simp add: mult_strict_right_mono linorder_not_less [symmetric]) 
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351 

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lemma mult_left_less_imp_less: 
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353 
"[c*a < c*b; 0 \<le> c] ==> a < (b::'a::ordered_semiring_strict)" 
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354 
by (force simp add: mult_left_mono linorder_not_le [symmetric]) 
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355 

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356 
lemma mult_right_less_imp_less: 
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357 
"[a*c < b*c; 0 \<le> c] ==> a < (b::'a::ordered_semiring_strict)" 
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358 
by (force simp add: mult_right_mono linorder_not_le [symmetric]) 
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359 

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lemma mult_strict_left_mono_neg: 
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361 
"[b < a; c < 0] ==> c * a < c * (b::'a::ordered_ring_strict)" 
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apply (drule mult_strict_left_mono [of _ _ "c"]) 
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apply (simp_all add: minus_mult_left [symmetric]) 
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364 
done 
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14738  366 
lemma mult_left_mono_neg: 
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367 
"[b \<le> a; c \<le> 0] ==> c * a \<le> c * (b::'a::pordered_ring)" 
14738  368 
apply (drule mult_left_mono [of _ _ "c"]) 
369 
apply (simp_all add: minus_mult_left [symmetric]) 

370 
done 

371 

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lemma mult_strict_right_mono_neg: 
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"[b < a; c < 0] ==> a * c < b * (c::'a::ordered_ring_strict)" 
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apply (drule mult_strict_right_mono [of _ _ "c"]) 
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apply (simp_all add: minus_mult_right [symmetric]) 
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376 
done 
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377 

14738  378 
lemma mult_right_mono_neg: 
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379 
"[b \<le> a; c \<le> 0] ==> a * c \<le> (b::'a::pordered_ring) * c" 
14738  380 
apply (drule mult_right_mono [of _ _ "c"]) 
381 
apply (simp) 

382 
apply (simp_all add: minus_mult_right [symmetric]) 

383 
done 

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

384 

23389  385 

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

386 
subsection{* Products of Signs *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

387 

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

388 
lemma mult_pos_pos: "[ (0::'a::ordered_semiring_strict) < a; 0 < b ] ==> 0 < a*b" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

389 
by (drule mult_strict_left_mono [of 0 b], auto) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

390 

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

391 
lemma mult_nonneg_nonneg: "[ (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b ] ==> 0 \<le> a*b" 
14738  392 
by (drule mult_left_mono [of 0 b], auto) 
393 

394 
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

395 
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

396 

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

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

400 
lemma mult_pos_neg2: "[ (0::'a::ordered_semiring_strict) < a; b < 0 ] ==> b*a < 0" 

401 
by (drule mult_strict_right_mono[of b 0], auto) 

402 

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

403 
lemma mult_nonneg_nonpos2: "[ (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 ] ==> b*a \<le> 0" 
14738  404 
by (drule mult_right_mono[of b 0], auto) 
405 

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

406 
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

407 
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

408 

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

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

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

412 
lemma zero_less_mult_pos: 
14738  413 
"[ 0 < a*b; 0 < a] ==> 0 < (b::'a::ordered_semiring_strict)" 
21328  414 
apply (cases "b\<le>0") 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

415 
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

416 
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

417 
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

418 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

419 

14738  420 
lemma zero_less_mult_pos2: 
421 
"[ 0 < b*a; 0 < a] ==> 0 < (b::'a::ordered_semiring_strict)" 

21328  422 
apply (cases "b\<le>0") 
14738  423 
apply (auto simp add: order_le_less linorder_not_less) 
424 
apply (drule_tac mult_pos_neg2 [of a b]) 

425 
apply (auto dest: order_less_not_sym) 

426 
done 

427 

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

428 
lemma zero_less_mult_iff: 
14738  429 
"((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

430 
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

431 
mult_neg_neg) 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

432 
apply (blast dest: zero_less_mult_pos) 
14738  433 
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

434 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

435 

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

436 
lemma mult_eq_0_iff [simp]: 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

437 
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

438 
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

439 
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

440 

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

441 
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

442 
apply intro_classes 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

443 
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

444 
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

445 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

446 

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

447 
lemma zero_le_mult_iff: 
14738  448 
"((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

449 
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

450 
zero_less_mult_iff) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

451 

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

452 
lemma mult_less_0_iff: 
14738  453 
"(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

454 
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

455 
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

456 
done 
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_le_0_iff: 
14738  459 
"(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

460 
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

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 

14738  464 
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

465 
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos) 
14738  466 

467 
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

468 
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2) 
14738  469 

23095  470 
lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a" 
471 
by (simp add: zero_le_mult_iff linorder_linear) 

472 

473 
lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))" 

474 
by (simp add: not_less) 

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

475 

14738  476 
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom} 
477 
theorems available to members of @{term ordered_idom} *} 

478 

479 
instance ordered_idom \<subseteq> ordered_semidom 

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

480 
proof 
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

481 
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

482 
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

483 
qed 
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

484 

14738  485 
instance ordered_idom \<subseteq> idom .. 
486 

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

487 
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

488 

17085  489 
lemmas one_neq_zero = zero_neq_one [THEN not_sym] 
490 
declare one_neq_zero [simp] 

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

491 

14738  492 
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

493 
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

494 

14738  495 
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0" 
496 
by (simp add: linorder_not_le) 

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

497 

14738  498 
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0" 
499 
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

500 

23389  501 

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

502 
subsection{*More Monotonicity*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

503 

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

504 
text{*Strict monotonicity in both arguments*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

505 
lemma mult_strict_mono: 
14738  506 
"[a<b; c<d; 0<b; 0\<le>c] ==> a * c < b * (d::'a::ordered_semiring_strict)" 
21328  507 
apply (cases "c=0") 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

508 
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

509 
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

510 
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

511 
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

512 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

513 

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

514 
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

515 
lemma mult_strict_mono': 
14738  516 
"[ 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

517 
apply (rule mult_strict_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

518 
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

519 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

520 

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

521 
lemma mult_mono: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

522 
"[a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c] 
14738  523 
==> 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

524 
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

525 
apply (erule mult_left_mono, assumption) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

526 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

527 

21258  528 
lemma mult_mono': 
529 
"[a \<le> b; c \<le> d; 0 \<le> a; 0 \<le> c] 

530 
==> a * c \<le> b * (d::'a::pordered_semiring)" 

531 
apply (rule mult_mono) 

532 
apply (fast intro: order_trans)+ 

533 
done 

534 

14738  535 
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

536 
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

537 
apply (simp add: order_less_trans [OF zero_less_one]) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

538 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

539 

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

540 
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

541 
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

542 
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

543 
apply (erule order_less_le_trans) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

544 
apply (erule mult_left_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

545 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

546 
apply (erule mult_strict_right_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

547 
apply assumption 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

548 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

549 

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

550 
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

551 
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

552 
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

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

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

555 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

556 
apply (erule mult_right_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

557 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

558 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

559 

23389  560 

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

561 
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

562 

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

563 
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

564 
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

565 

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

566 
text{*These ``disjunction'' versions produce two cases when the comparison is 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

567 
an assumption, but effectively four when the comparison is a goal.*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

568 

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

569 
lemma mult_less_cancel_right_disj: 
14738  570 
"(a*c < b*c) = ((0 < c & a < b)  (c < 0 & b < (a::'a::ordered_ring_strict)))" 
21328  571 
apply (cases "c = 0") 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

572 
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

573 
mult_strict_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

574 
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

575 
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

576 
linorder_not_le [symmetric, of a]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

577 
apply (erule_tac [!] notE) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

578 
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

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

580 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

581 

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

582 
lemma mult_less_cancel_left_disj: 
14738  583 
"(c*a < c*b) = ((0 < c & a < b)  (c < 0 & b < (a::'a::ordered_ring_strict)))" 
21328  584 
apply (cases "c = 0") 
14738  585 
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
586 
mult_strict_left_mono_neg) 

587 
apply (auto simp add: linorder_not_less 

588 
linorder_not_le [symmetric, of "c*a"] 

589 
linorder_not_le [symmetric, of a]) 

590 
apply (erule_tac [!] notE) 

591 
apply (auto simp add: order_less_imp_le mult_left_mono 

592 
mult_left_mono_neg) 

593 
done 

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

594 

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

595 

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

596 
text{*The ``conjunction of implication'' lemmas produce two cases when the 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

597 
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

598 

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

599 
lemma mult_less_cancel_right: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

600 
fixes c :: "'a :: ordered_ring_strict" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

601 
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

602 
by (insert mult_less_cancel_right_disj [of a c b], auto) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

603 

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

604 
lemma mult_less_cancel_left: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

605 
fixes c :: "'a :: ordered_ring_strict" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

606 
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

607 
by (insert mult_less_cancel_left_disj [of c a b], auto) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

608 

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

609 
lemma mult_le_cancel_right: 
14738  610 
"(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

611 
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

612 

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

613 
lemma mult_le_cancel_left: 
14738  614 
"(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

615 
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

616 

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

617 
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

618 
assumes less: "c*a < c*b" and nonneg: "0 \<le> c" 
14738  619 
shows "a < (b::'a::ordered_semiring_strict)" 
14377  620 
proof (rule ccontr) 
621 
assume "~ a < b" 

622 
hence "b \<le> a" by (simp add: linorder_not_less) 

23389  623 
hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono) 
14377  624 
with this and less show False 
625 
by (simp add: linorder_not_less [symmetric]) 

626 
qed 

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

627 

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

628 
lemma mult_less_imp_less_right: 
14738  629 
assumes less: "a*c < b*c" and nonneg: "0 <= c" 
630 
shows "a < (b::'a::ordered_semiring_strict)" 

631 
proof (rule ccontr) 

632 
assume "~ a < b" 

633 
hence "b \<le> a" by (simp add: linorder_not_less) 

23389  634 
hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono) 
14738  635 
with this and less show False 
636 
by (simp add: linorder_not_less [symmetric]) 

637 
qed 

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

638 

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

639 
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

640 
lemma mult_cancel_right [simp]: 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

641 
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

642 
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

643 
proof  
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

644 
have "(a * c = b * c) = ((a  b) * c = 0)" 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

645 
by (simp add: ring_distribs) 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

646 
thus ?thesis 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

647 
by (simp add: disj_commute) 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

648 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

649 

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

650 
lemma mult_cancel_left [simp]: 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

651 
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

652 
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

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

654 
have "(c * a = c * b) = (c * (a  b) = 0)" 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

655 
by (simp add: ring_distribs) 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

656 
thus ?thesis 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

657 
by simp 
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

658 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

659 

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

660 

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

661 
subsubsection{*Special Cancellation Simprules for Multiplication*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

662 

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

663 
text{*These also produce two cases when the comparison is a goal.*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

664 

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

665 
lemma mult_le_cancel_right1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

666 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

667 
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

668 
by (insert mult_le_cancel_right [of 1 c b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

669 

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

670 
lemma mult_le_cancel_right2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

671 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

672 
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

673 
by (insert mult_le_cancel_right [of a c 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

674 

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

675 
lemma mult_le_cancel_left1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

676 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

677 
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

678 
by (insert mult_le_cancel_left [of c 1 b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

679 

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

680 
lemma mult_le_cancel_left2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

681 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

682 
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

683 
by (insert mult_le_cancel_left [of c a 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

684 

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

685 
lemma mult_less_cancel_right1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

686 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

687 
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

688 
by (insert mult_less_cancel_right [of 1 c b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

689 

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

690 
lemma mult_less_cancel_right2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

691 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

692 
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

693 
by (insert mult_less_cancel_right [of a c 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

694 

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

695 
lemma mult_less_cancel_left1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

696 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

697 
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

698 
by (insert mult_less_cancel_left [of c 1 b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

699 

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

700 
lemma mult_less_cancel_left2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

701 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

702 
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

703 
by (insert mult_less_cancel_left [of c a 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

704 

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

705 
lemma mult_cancel_right1 [simp]: 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

706 
fixes c :: "'a :: dom" 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

707 
shows "(c = b*c) = (c = 0  b=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

708 
by (insert mult_cancel_right [of 1 c b], force) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

709 

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

710 
lemma mult_cancel_right2 [simp]: 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

711 
fixes c :: "'a :: dom" 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

712 
shows "(a*c = c) = (c = 0  a=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

713 
by (insert mult_cancel_right [of a c 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

714 

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

715 
lemma mult_cancel_left1 [simp]: 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

716 
fixes c :: "'a :: dom" 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

717 
shows "(c = c*b) = (c = 0  b=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

718 
by (insert mult_cancel_left [of c 1 b], force) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

719 

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

720 
lemma mult_cancel_left2 [simp]: 
22990
775e9de3db48
added classes ring_no_zero_divisors and dom (noncommutative version of idom);
huffman
parents:
22987
diff
changeset

721 
fixes c :: "'a :: dom" 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

722 
shows "(c*a = c) = (c = 0  a=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

723 
by (insert mult_cancel_left [of c a 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

724 

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

727 
lemmas mult_compare_simps = 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

728 
mult_le_cancel_right mult_le_cancel_left 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

729 
mult_le_cancel_right1 mult_le_cancel_right2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

730 
mult_le_cancel_left1 mult_le_cancel_left2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

731 
mult_less_cancel_right mult_less_cancel_left 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

732 
mult_less_cancel_right1 mult_less_cancel_right2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

733 
mult_less_cancel_left1 mult_less_cancel_left2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

734 
mult_cancel_right mult_cancel_left 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

735 
mult_cancel_right1 mult_cancel_right2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

736 
mult_cancel_left1 mult_cancel_left2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

737 

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

738 

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

739 
subsection {* Fields *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

740 

14288  741 
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" 
742 
proof 

743 
assume neq: "b \<noteq> 0" 

744 
{ 

745 
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) 

746 
also assume "a / b = 1" 

747 
finally show "a = b" by simp 

748 
next 

749 
assume "a = b" 

750 
with neq show "a / b = 1" by (simp add: divide_inverse) 

751 
} 

752 
qed 

753 

754 
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" 

755 
by (simp add: divide_inverse) 

756 

23398  757 
lemma divide_self[simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1" 
14288  758 
by (simp add: divide_inverse) 
759 

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

760 
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

761 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

762 

15228  763 
lemma divide_self_if [simp]: 
764 
"a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)" 

765 
by (simp add: divide_self) 

766 

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

767 
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

768 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

769 

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

770 
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

771 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

772 

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

773 
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c" 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

774 
by (simp add: divide_inverse ring_distribs) 
14293  775 

23482  776 
(* what ordering?? this is a straight instance of mult_eq_0_iff 
14270  777 
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement 
778 
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

779 
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

780 
"(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

781 
by simp 
23482  782 
*) 
14270  783 

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

784 
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

785 
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

786 
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

787 
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

788 
shows "a=b" 
14377  789 
proof  
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

790 
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

791 
by (simp add: eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

792 
thus "a=b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

793 
by (simp add: mult_assoc cnz) 
14377  794 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

795 

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

796 
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

797 
"(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

798 
by simp 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

799 

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

800 
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

801 
"(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

802 
by simp 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

803 

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

804 
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

805 
"a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)" 
14377  806 
proof 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

807 
assume ianz: "inverse a = 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

808 
assume "a \<noteq> 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

809 
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

810 
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

811 
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

812 
thus False by (simp add: eq_commute) 
14377  813 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

814 

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

815 

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

816 
subsection{*Basic Properties of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

817 

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

818 
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

819 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

820 
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

821 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

822 

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

823 
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

824 
"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

825 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

826 
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

827 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

828 

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

829 
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

830 
"(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

831 
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

832 

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

833 
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

834 
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

835 
shows "inverse(a) = inverse(a::'a::division_ring)" 
14377  836 
proof  
837 
have "a * inverse ( a) = a *  inverse a" 

838 
by simp 

839 
thus ?thesis 

840 
by (simp only: field_mult_cancel_left, simp) 

841 
qed 

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

842 

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

843 
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

844 
"inverse(a) = inverse(a::'a::{division_ring,division_by_zero})" 
14377  845 
proof cases 
846 
assume "a=0" thus ?thesis by (simp add: inverse_zero) 

847 
next 

848 
assume "a\<noteq>0" 

849 
thus ?thesis by (simp add: nonzero_inverse_minus_eq) 

850 
qed 

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

851 

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

852 
lemma nonzero_inverse_eq_imp_eq: 
14269  853 
assumes inveq: "inverse a = inverse b" 
854 
and anz: "a \<noteq> 0" 

855 
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

856 
shows "a = (b::'a::division_ring)" 
14377  857 
proof  
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

858 
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

859 
by (simp add: inveq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

860 
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

861 
by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

862 
thus "a = b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

863 
by (simp add: mult_assoc anz bnz) 
14377  864 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

865 

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

866 
lemma inverse_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

867 
"inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})" 
21328  868 
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

869 
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

870 
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

871 
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

872 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

873 

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

874 
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

875 
"(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

876 
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

877 

14270  878 
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

879 
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

880 
shows "inverse(inverse (a::'a::division_ring)) = a" 
14270  881 
proof  
882 
have "(inverse (inverse a) * inverse a) * a = a" 

883 
by (simp add: nonzero_imp_inverse_nonzero) 

884 
thus ?thesis 

885 
by (simp add: mult_assoc) 

886 
qed 

887 

888 
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

889 
"inverse(inverse (a::'a::{division_ring,division_by_zero})) = a" 
14270  890 
proof cases 
891 
assume "a=0" thus ?thesis by simp 

892 
next 

893 
assume "a\<noteq>0" 

894 
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) 

895 
qed 

896 

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

897 
lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)" 
14270  898 
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

899 
have "inverse 1 * 1 = (1::'a::division_ring)" 
14270  900 
by (rule left_inverse [OF zero_neq_one [symmetric]]) 
901 
thus ?thesis by simp 

902 
qed 

903 

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

904 
lemma inverse_unique: 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

905 
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

906 
shows "inverse a = (b::'a::division_ring)" 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

907 
proof  
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

908 
have "a \<noteq> 0" using ab by auto 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

909 
moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

910 
ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

911 
qed 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

912 

14270  913 
lemma nonzero_inverse_mult_distrib: 
914 
assumes anz: "a \<noteq> 0" 

915 
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

916 
shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)" 
14270  917 
proof  
918 
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 

23482  919 
by (simp add: anz bnz) 
14270  920 
hence "inverse(a*b) * a = inverse(b)" 
921 
by (simp add: mult_assoc bnz) 

922 
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 

923 
by simp 

924 
thus ?thesis 

925 
by (simp add: mult_assoc anz) 

926 
qed 

927 

928 
text{*This version builds in division by zero while also reorienting 

929 
the righthand side.*} 

930 
lemma inverse_mult_distrib [simp]: 

931 
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" 

932 
proof cases 

933 
assume "a \<noteq> 0 & b \<noteq> 0" 

22993  934 
thus ?thesis 
935 
by (simp add: nonzero_inverse_mult_distrib mult_commute) 

14270  936 
next 
937 
assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

22993  938 
thus ?thesis 
939 
by force 

14270  940 
qed 
941 

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

942 
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

943 
"[(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

944 
==> inverse a + inverse b = inverse a * (a+b) * inverse b" 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

945 
by (simp add: ring_simps) 
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset

946 

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

947 
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

948 
"[(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

949 
==> inverse a  inverse b = inverse a * (ba) * inverse b" 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

950 
by (simp add: ring_simps) 
20496
23eb6034c06d
added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents:
19404
diff
changeset

951 

14270  952 
text{*There is no slick version using division by zero.*} 
953 
lemma inverse_add: 

23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

954 
"[a \<noteq> 0; b \<noteq> 0] 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

955 
==> 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

956 
by (simp add: division_ring_inverse_add mult_ac) 
14270  957 

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

958 
lemma inverse_divide [simp]: 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

959 
"inverse (a/b) = b / (a::'a::{field,division_by_zero})" 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

960 
by (simp add: divide_inverse mult_commute) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

961 

23389  962 

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

963 
subsection {* Calculations with fractions *} 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

964 

23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

965 
text{* There is a whole bunch of simprules just for class @{text 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

966 
field} but none for class @{text field} and @{text nonzero_divides} 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

967 
because the latter are covered by a simproc. *} 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

968 

5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

969 
lemma nonzero_mult_divide_mult_cancel_left[simp]: 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

970 
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)" 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

971 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

972 
have "(c*a)/(c*b) = c * a * (inverse b * inverse c)" 
23482  973 
by (simp add: divide_inverse nonzero_inverse_mult_distrib) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

974 
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

975 
by (simp only: mult_ac) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

976 
also have "... = a * inverse b" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

977 
by simp 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

978 
finally show ?thesis 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

979 
by (simp add: divide_inverse) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

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

981 

23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

982 
lemma mult_divide_mult_cancel_left: 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

983 
"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})" 
21328  984 
apply (cases "b = 0") 
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

985 
apply (simp_all add: nonzero_mult_divide_mult_cancel_left) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

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

987 

23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

988 
lemma nonzero_mult_divide_mult_cancel_right: 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

989 
"[b\<noteq>0; c\<noteq>0] ==> (a*c) / (b*c) = a/(b::'a::field)" 
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

990 
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
14321  991 

23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

992 
lemma mult_divide_mult_cancel_right: 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

993 
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" 
21328  994 
apply (cases "b = 0") 
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

995 
apply (simp_all add: nonzero_mult_divide_mult_cancel_right) 
14321  996 
done 
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

997 

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

998 
lemma divide_1 [simp]: "a/1 = (a::'a::field)" 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

999 
by (simp add: divide_inverse) 
14284
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

1000 

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

1001 
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

1002 
by (simp add: divide_inverse mult_assoc) 
14288  1003 

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

1004 
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

1005 
by (simp add: divide_inverse mult_ac) 
14288  1006 

23482  1007 
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left 
1008 

14288  1009 
lemma divide_divide_eq_right [simp]: 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1010 
"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

1011 
by (simp add: divide_inverse mult_ac) 
14288  1012 

1013 
lemma divide_divide_eq_left [simp]: 

23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1014 
"(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

1015 
by (simp add: divide_inverse mult_assoc) 
14288  1016 

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

1017 
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

1018 
x / y + w / z = (x * z + w * y) / (y * z)" 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1019 
apply (subgoal_tac "x / y = (x * z) / (y * z)") 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1020 
apply (erule ssubst) 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1021 
apply (subgoal_tac "w / z = (w * y) / (y * z)") 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1022 
apply (erule ssubst) 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1023 
apply (rule add_divide_distrib [THEN sym]) 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1024 
apply (subst mult_commute) 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1025 
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym]) 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1026 
apply assumption 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1027 
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym]) 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1028 
apply assumption 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1029 
done 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1030 

23389  1031 

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

1032 
subsubsection{*Special Cancellation Simprules for Division*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1033 

23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1034 
lemma mult_divide_mult_cancel_left_if[simp]: 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1035 
fixes c :: "'a :: {field,division_by_zero}" 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1036 
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)" 
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1037 
by (simp add: mult_divide_mult_cancel_left) 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1038 

5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1039 
lemma nonzero_mult_divide_cancel_right[simp]: 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1040 
"b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)" 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1041 
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1042 

5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1043 
lemma nonzero_mult_divide_cancel_left[simp]: 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1044 
"a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)" 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1045 
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1046 

5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1047 

5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1048 
lemma nonzero_divide_mult_cancel_right[simp]: 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1049 
"\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)" 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1050 
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1051 

5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1052 
lemma nonzero_divide_mult_cancel_left[simp]: 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1053 
"\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)" 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1054 
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1055 

5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1056 

5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1057 
lemma nonzero_mult_divide_mult_cancel_left2[simp]: 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1058 
"[b\<noteq>0; c\<noteq>0] ==> (c*a) / (b*c) = a/(b::'a::field)" 
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1059 
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac) 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1060 

5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1061 
lemma nonzero_mult_divide_mult_cancel_right2[simp]: 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1062 
"[b\<noteq>0; c\<noteq>0] ==> (a*c) / (c*b) = a/(b::'a::field)" 
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1063 
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac) 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

1064 

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

1065 

14293  1066 
subsection {* Division and Unary Minus *} 
1067 

1068 
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==>  (a/b) = (a) / (b::'a::field)" 

1069 
by (simp add: divide_inverse minus_mult_left) 

1070 

1071 
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==>  (a/b) = a / (b::'a::field)" 

1072 
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) 

1073 

1074 
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (a)/(b) = a / (b::'a::field)" 

1075 
by (simp add: divide_inverse nonzero_inverse_minus_eq) 

1076 

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

1077 
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

1078 
by (simp add: divide_inverse minus_mult_left [symmetric]) 
14293  1079 

1080 
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

1081 
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

1082 

14293  1083 

1084 
text{*The effect is to extract signs from divisions*} 

17085  1085 
lemmas divide_minus_left = minus_divide_left [symmetric] 
1086 
lemmas divide_minus_right = minus_divide_right [symmetric] 

1087 
declare divide_minus_left [simp] divide_minus_right [simp] 

14293  1088 

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

1089 
text{*Also, extract signs from products*} 
17085  1090 
lemmas mult_minus_left = minus_mult_left [symmetric] 
1091 
lemmas mult_minus_right = minus_mult_right [symmetric] 

1092 
declare mult_minus_left [simp] mult_minus_right [simp] 

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

1093 

14293  1094 
lemma minus_divide_divide [simp]: 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

1095 
"(a)/(b) = a / (b::'a::{field,division_by_zero})" 
21328  1096 
apply (cases "b=0", simp) 
14293  1097 
apply (simp add: nonzero_minus_divide_divide) 
1098 
done 

1099 

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

1100 
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

1101 
by (simp add: diff_minus add_divide_distrib) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1102 

23482  1103 
lemma add_divide_eq_iff: 
1104 
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z" 

1105 
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left) 

1106 

1107 
lemma divide_add_eq_iff: 

1108 
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z" 

1109 
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left) 

1110 

1111 
lemma diff_divide_eq_iff: 

1112 
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x  y/z = (z*x  y)/z" 

1113 
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left) 

1114 

1115 
lemma divide_diff_eq_iff: 

1116 
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z  y = (x  z*y)/z" 

1117 
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left) 

1118 

1119 
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)" 

1120 
proof  

1121 
assume [simp]: "c\<noteq>0" 

1122 
have "(a = b/c) = (a*c = (b/c)*c)" 

1123 
by (simp add: field_mult_cancel_right) 

1124 
also have "... = (a*c = b)" 

1125 
by (simp add: divide_inverse mult_assoc) 

1126 
finally show ?thesis . 

1127 
qed 

1128 

1129 
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)" 

1130 
proof  

1131 
assume [simp]: "c\<noteq>0" 

1132 
have "(b/c = a) = ((b/c)*c = a*c)" 

1133 
by (simp add: field_mult_cancel_right) 

1134 
also have "... = (b = a*c)" 

1135 
by (simp add: divide_inverse mult_assoc) 

1136 
finally show ?thesis . 

1137 
qed 

1138 

1139 
lemma eq_divide_eq: 

1140 
"((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)" 

1141 
by (simp add: nonzero_eq_divide_eq) 

1142 

1143 
lemma divide_eq_eq: 

1144 
"(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)" 

1145 
by (force simp add: nonzero_divide_eq_eq) 

1146 

1147 
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==> 

1148 
b = a * c ==> b / c = a" 

1149 
by (subst divide_eq_eq, simp) 

1150 

1151 
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==> 

1152 
a * c = b ==> a = b / c" 

1153 
by (subst eq_divide_eq, simp) 

1154 

1155 

1156 
lemmas field_eq_simps = ring_simps 

1157 
(* pull / out*) 

1158 
add_divide_eq_iff divide_add_eq_iff 

1159 
diff_divide_eq_iff divide_diff_eq_iff 

1160 
(* multiply eqn *) 

1161 
nonzero_eq_divide_eq nonzero_divide_eq_eq 

1162 
(* is added later: 

1163 
times_divide_eq_left times_divide_eq_right 

1164 
*) 

1165 

1166 
text{*An example:*} 

1167 
lemma fixes a b c d e f :: "'a::field" 

1168 
shows "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f \<rbrakk> \<Longrightarrow> ((ab)*(cd)*(ef))/((cd)*(ef)*(ab)) = 1" 

1169 
apply(subgoal_tac "(cd)*(ef)*(ab) \<noteq> 0") 

1170 
apply(simp add:field_eq_simps) 

1171 
apply(simp) 

1172 
done 

1173 

1174 

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

1175 
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

1176 
x / y  w / z = (x * z  w * y) / (y * z)" 
23482  1177 
by (simp add:field_eq_simps times_divide_eq) 
1178 

1179 
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==> 

1180 
(x / y = w / z) = (x * z = w * y)" 

1181 
by (simp add:field_eq_simps times_divide_eq) 

14293  1182 

23389  1183 

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

1184 
subsection {* Ordered Fields *} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1185 

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

1186 
lemma positive_imp_inverse_positive: 
23482  1187 
assumes a_gt_0: "0 < a" shows "0 < inverse (a::'a::ordered_field)" 
1188 
proof  

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

1189 
have "0 < a * inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1190 
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

1191 
thus "0 < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1192 
by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff) 
23482  1193 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1194 

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

1195 
lemma negative_imp_inverse_negative: 
23482  1196 
"a < 0 ==> inverse a < (0::'a::ordered_field)" 
1197 
by (insert positive_imp_inverse_positive [of "a"], 

1198 
simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 

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

1199 

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

1200 
lemma inverse_le_imp_le: 
23482  1201 
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a" 
1202 
shows "b \<le> (a::'a::ordered_field)" 

1203 
proof (rule classical) 

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

1204 
assume "~ b \<le> a" 
23482  1205 
hence "a < b" by (simp add: linorder_not_le) 
1206 
hence bpos: "0 < b" by (blast intro: apos order_less_trans) 

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

1207 
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

1208 
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

1209 
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

1210 
by (simp add: bpos order_less_imp_le mult_right_mono) 
23482  1211 
thus "b \<le> a" by (simp add: mult_assoc apos bpos order_less_imp_not_eq2) 
1212 
qed 

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

1213 

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

1214 
lemma inverse_positive_imp_positive: 
23482  1215 
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0" 
1216 
shows "0 < (a::'a::ordered_field)" 

23389  1217 
proof  
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1218 
have "0 < inverse (inverse a)" 
23389  1219 
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

1220 
thus "0 < a" 
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_positive_iff_positive [simp]: 
23482  1225 
"(0 < inverse a) = (0 < (a::'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_positive_imp_positive positive_imp_inverse_positive) 
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_negative_imp_negative: 
23482  1231 
assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0" 
1232 
shows "a < (0::'a::ordered_field)" 

23389  1233 
proof  
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1234 
have "inverse (inverse a) < 0" 
23389  1235 
using inv_less_0 by (rule negative_imp_inverse_negative) 
23482  1236 
thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq) 
23389  1237 
qed 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1238 

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

1239 
lemma inverse_negative_iff_negative [simp]: 
23482  1240 
"(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))" 
21328  1241 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1242 
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

1243 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1244 

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

1245 
lemma inverse_nonnegative_iff_nonnegative [simp]: 
23482  1246 
"(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))" 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1247 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1248 

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

1249 
lemma inverse_nonpositive_iff_nonpositive [simp]: 
23482  1250 
"(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))" 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1251 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1252 

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

1253 
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

1254 
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

1255 
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

1256 
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

1257 
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

1258 
have "( 1) + x < x" by simp 
