(* Title: HOL/Ring_and_Field.thy
ID: $Id$
Author: Gertrud Bauer, Steven Obua, Tobias Nipkow, Lawrence C Paulson, and Markus Wenzel,
with contributions by Jeremy Avigad
*)
header {* (Ordered) Rings and Fields *}
theory Ring_and_Field
imports OrderedGroup
begin
text {*
The theory of partially ordered rings is taken from the books:
\begin{itemize}
\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
\end{itemize}
Most of the used notions can also be looked up in
\begin{itemize}
\item \url{http://www.mathworld.com} by Eric Weisstein et. al.
\item \emph{Algebra I} by van der Waerden, Springer.
\end{itemize}
*}
class semiring = ab_semigroup_add + semigroup_mult +
assumes left_distrib: "(a \<^loc>+ b) \<^loc>* c = a \<^loc>* c \<^loc>+ b \<^loc>* c"
assumes right_distrib: "a \<^loc>* (b \<^loc>+ c) = a \<^loc>* b \<^loc>+ a \<^loc>* c"
class mult_zero = times + zero +
assumes mult_zero_left [simp]: "\<^loc>0 \<^loc>* a = \<^loc>0"
assumes mult_zero_right [simp]: "a \<^loc>* \<^loc>0 = \<^loc>0"
class semiring_0 = semiring + comm_monoid_add + mult_zero
class semiring_0_cancel = semiring + comm_monoid_add + cancel_ab_semigroup_add
instance semiring_0_cancel \<subseteq> semiring_0
proof
fix a :: 'a
have "0 * a + 0 * a = 0 * a + 0"
by (simp add: left_distrib [symmetric])
thus "0 * a = 0"
by (simp only: add_left_cancel)
have "a * 0 + a * 0 = a * 0 + 0"
by (simp add: right_distrib [symmetric])
thus "a * 0 = 0"
by (simp only: add_left_cancel)
qed
class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
assumes distrib: "(a \<^loc>+ b) \<^loc>* c = a \<^loc>* c \<^loc>+ b \<^loc>* c"
instance comm_semiring \<subseteq> semiring
proof
fix a b c :: 'a
show "(a + b) * c = a * c + b * c" by (simp add: distrib)
have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
also have "... = b * a + c * a" by (simp only: distrib)
also have "... = a * b + a * c" by (simp add: mult_ac)
finally show "a * (b + c) = a * b + a * c" by blast
qed
class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
instance comm_semiring_0 \<subseteq> semiring_0 ..
class comm_semiring_0_cancel = comm_semiring + comm_monoid_add + cancel_ab_semigroup_add
instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel ..
instance comm_semiring_0_cancel \<subseteq> comm_semiring_0 ..
class zero_neq_one = zero + one +
assumes zero_neq_one [simp]: "\<^loc>0 \<noteq> \<^loc>1"
class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
(*previously almost_semiring*)
instance comm_semiring_1 \<subseteq> semiring_1 ..
class no_zero_divisors = zero + times +
assumes no_zero_divisors: "a \<noteq> \<^loc>0 \<Longrightarrow> b \<noteq> \<^loc>0 \<Longrightarrow> a \<^loc>* b \<noteq> \<^loc>0"
class semiring_1_cancel = semiring + comm_monoid_add + zero_neq_one
+ cancel_ab_semigroup_add + monoid_mult
instance semiring_1_cancel \<subseteq> semiring_0_cancel ..
instance semiring_1_cancel \<subseteq> semiring_1 ..
class comm_semiring_1_cancel = comm_semiring + comm_monoid_add + comm_monoid_mult
+ zero_neq_one + cancel_ab_semigroup_add
instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel ..
instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel ..
instance comm_semiring_1_cancel \<subseteq> comm_semiring_1 ..
class ring = semiring + ab_group_add
instance ring \<subseteq> semiring_0_cancel ..
class comm_ring = comm_semiring + ab_group_add
instance comm_ring \<subseteq> ring ..
instance comm_ring \<subseteq> comm_semiring_0_cancel ..
class ring_1 = ring + zero_neq_one + monoid_mult
instance ring_1 \<subseteq> semiring_1_cancel ..
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
(*previously ring*)
instance comm_ring_1 \<subseteq> ring_1 ..
instance comm_ring_1 \<subseteq> comm_semiring_1_cancel ..
class ring_no_zero_divisors = ring + no_zero_divisors
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
class idom = comm_ring_1 + no_zero_divisors
instance idom \<subseteq> ring_1_no_zero_divisors ..
class division_ring = ring_1 + inverse +
assumes left_inverse [simp]: "a \<noteq> \<^loc>0 \<Longrightarrow> inverse a \<^loc>* a = \<^loc>1"
assumes right_inverse [simp]: "a \<noteq> \<^loc>0 \<Longrightarrow> a \<^loc>* inverse a = \<^loc>1"
instance division_ring \<subseteq> ring_1_no_zero_divisors
proof
fix a b :: 'a
assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
show "a * b \<noteq> 0"
proof
assume ab: "a * b = 0"
hence "0 = inverse a * (a * b) * inverse b"
by simp
also have "\<dots> = (inverse a * a) * (b * inverse b)"
by (simp only: mult_assoc)
also have "\<dots> = 1"
using a b by simp
finally show False
by simp
qed
qed
class field = comm_ring_1 + inverse +
assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a \<^loc>* a = \<^loc>1"
assumes divide_inverse: "a \<^loc>/ b = a \<^loc>* inverse b"
instance field \<subseteq> division_ring
proof
fix a :: 'a
assume "a \<noteq> 0"
thus "inverse a * a = 1" by (rule field_inverse)
thus "a * inverse a = 1" by (simp only: mult_commute)
qed
instance field \<subseteq> idom ..
class division_by_zero = zero + inverse +
assumes inverse_zero [simp]: "inverse \<^loc>0 = \<^loc>0"
subsection {* Distribution rules *}
text{*For the @{text combine_numerals} simproc*}
lemma combine_common_factor:
"a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)"
by (simp add: left_distrib add_ac)
lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)"
apply (rule equals_zero_I)
apply (simp add: left_distrib [symmetric])
done
lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"
apply (rule equals_zero_I)
apply (simp add: right_distrib [symmetric])
done
lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)"
by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)"
by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"
by (simp add: right_distrib diff_minus
minus_mult_left [symmetric] minus_mult_right [symmetric])
lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)"
by (simp add: left_distrib diff_minus
minus_mult_left [symmetric] minus_mult_right [symmetric])
lemmas ring_distribs =
right_distrib left_distrib left_diff_distrib right_diff_distrib
text{*This list of rewrites simplifies ring terms by multiplying
everything out and bringing sums and products into a canonical form
(by ordered rewriting). As a result it decides ring equalities but
also helps with inequalities. *}
lemmas ring_simps = group_simps ring_distribs
class mult_mono = times + zero + ord +
assumes mult_left_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b"
assumes mult_right_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> a \<^loc>* c \<sqsubseteq> b \<^loc>* c"
class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add
class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
+ semiring + comm_monoid_add + cancel_ab_semigroup_add
instance pordered_cancel_semiring \<subseteq> semiring_0_cancel ..
instance pordered_cancel_semiring \<subseteq> pordered_semiring ..
class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono
instance ordered_semiring \<subseteq> pordered_cancel_semiring ..
class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
assumes mult_strict_left_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b"
assumes mult_strict_right_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> a \<^loc>* c \<sqsubset> b \<^loc>* c"
instance ordered_semiring_strict \<subseteq> semiring_0_cancel ..
instance ordered_semiring_strict \<subseteq> ordered_semiring
proof
fix a b c :: 'a
assume A: "a \<le> b" "0 \<le> c"
from A show "c * a \<le> c * b"
unfolding order_le_less
using mult_strict_left_mono by auto
from A show "a * c \<le> b * c"
unfolding order_le_less
using mult_strict_right_mono by auto
qed
class mult_mono1 = times + zero + ord +
assumes mult_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b"
class pordered_comm_semiring = comm_semiring_0
+ pordered_ab_semigroup_add + mult_mono1
class pordered_cancel_comm_semiring = comm_semiring_0_cancel
+ pordered_ab_semigroup_add + mult_mono1
instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring ..
class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
assumes mult_strict_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b"
instance pordered_comm_semiring \<subseteq> pordered_semiring
proof
fix a b c :: 'a
assume "a \<le> b" "0 \<le> c"
thus "c * a \<le> c * b" by (rule mult_mono)
thus "a * c \<le> b * c" by (simp only: mult_commute)
qed
instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring ..
instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict
proof
fix a b c :: 'a
assume "a < b" "0 < c"
thus "c * a < c * b" by (rule mult_strict_mono)
thus "a * c < b * c" by (simp only: mult_commute)
qed
instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring
proof
fix a b c :: 'a
assume "a \<le> b" "0 \<le> c"
thus "c * a \<le> c * b"
unfolding order_le_less
using mult_strict_mono by auto
qed
class pordered_ring = ring + pordered_cancel_semiring
instance pordered_ring \<subseteq> pordered_ab_group_add ..
class lordered_ring = pordered_ring + lordered_ab_group_abs
instance lordered_ring \<subseteq> lordered_ab_group_meet ..
instance lordered_ring \<subseteq> lordered_ab_group_join ..
class abs_if = minus + ord + zero + abs +
assumes abs_if: "abs a = (if a \<sqsubset> 0 then (uminus a) else a)"
(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
*)
class ordered_ring = ring + ordered_semiring + lordered_ab_group + abs_if
instance ordered_ring \<subseteq> lordered_ring
proof
fix x :: 'a
show "\<bar>x\<bar> = sup x (- x)"
by (simp only: abs_if sup_eq_if)
qed
class ordered_ring_strict = ring + ordered_semiring_strict + lordered_ab_group + abs_if
instance ordered_ring_strict \<subseteq> ordered_ring ..
class pordered_comm_ring = comm_ring + pordered_comm_semiring
instance pordered_comm_ring \<subseteq> pordered_ring ..
instance pordered_comm_ring \<subseteq> pordered_cancel_comm_semiring ..
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
(*previously ordered_semiring*)
assumes zero_less_one [simp]: "\<^loc>0 \<sqsubset> \<^loc>1"
class ordered_idom = comm_ring_1 + ordered_comm_semiring_strict + lordered_ab_group + abs_if
(*previously ordered_ring*)
instance ordered_idom \<subseteq> ordered_ring_strict ..
instance ordered_idom \<subseteq> pordered_comm_ring ..
class ordered_field = field + ordered_idom
lemmas linorder_neqE_ordered_idom =
linorder_neqE[where 'a = "?'b::ordered_idom"]
lemma eq_add_iff1:
"(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
by (simp add: ring_simps)
lemma eq_add_iff2:
"(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
by (simp add: ring_simps)
lemma less_add_iff1:
"(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"
by (simp add: ring_simps)
lemma less_add_iff2:
"(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"
by (simp add: ring_simps)
lemma le_add_iff1:
"(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"
by (simp add: ring_simps)
lemma le_add_iff2:
"(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"
by (simp add: ring_simps)
subsection {* Ordering Rules for Multiplication *}
lemma mult_left_le_imp_le:
"[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
lemma mult_right_le_imp_le:
"[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
lemma mult_left_less_imp_less:
"[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
by (force simp add: mult_left_mono linorder_not_le [symmetric])
lemma mult_right_less_imp_less:
"[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
by (force simp add: mult_right_mono linorder_not_le [symmetric])
lemma mult_strict_left_mono_neg:
"[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"
apply (drule mult_strict_left_mono [of _ _ "-c"])
apply (simp_all add: minus_mult_left [symmetric])
done
lemma mult_left_mono_neg:
"[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::pordered_ring)"
apply (drule mult_left_mono [of _ _ "-c"])
apply (simp_all add: minus_mult_left [symmetric])
done
lemma mult_strict_right_mono_neg:
"[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"
apply (drule mult_strict_right_mono [of _ _ "-c"])
apply (simp_all add: minus_mult_right [symmetric])
done
lemma mult_right_mono_neg:
"[|b \<le> a; c \<le> 0|] ==> a * c \<le> (b::'a::pordered_ring) * c"
apply (drule mult_right_mono [of _ _ "-c"])
apply (simp)
apply (simp_all add: minus_mult_right [symmetric])
done
subsection{* Products of Signs *}
lemma mult_pos_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
by (drule mult_strict_left_mono [of 0 b], auto)
lemma mult_nonneg_nonneg: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
by (drule mult_left_mono [of 0 b], auto)
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
by (drule mult_strict_left_mono [of b 0], auto)
lemma mult_nonneg_nonpos: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
by (drule mult_left_mono [of b 0], auto)
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0"
by (drule mult_strict_right_mono[of b 0], auto)
lemma mult_nonneg_nonpos2: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0"
by (drule mult_right_mono[of b 0], auto)
lemma mult_neg_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
by (drule mult_strict_right_mono_neg, auto)
lemma mult_nonpos_nonpos: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
by (drule mult_right_mono_neg[of a 0 b ], auto)
lemma zero_less_mult_pos:
"[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
apply (cases "b\<le>0")
apply (auto simp add: order_le_less linorder_not_less)
apply (drule_tac mult_pos_neg [of a b])
apply (auto dest: order_less_not_sym)
done
lemma zero_less_mult_pos2:
"[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
apply (cases "b\<le>0")
apply (auto simp add: order_le_less linorder_not_less)
apply (drule_tac mult_pos_neg2 [of a b])
apply (auto dest: order_less_not_sym)
done
lemma zero_less_mult_iff:
"((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
apply (auto simp add: order_le_less linorder_not_less mult_pos_pos
mult_neg_neg)
apply (blast dest: zero_less_mult_pos)
apply (blast dest: zero_less_mult_pos2)
done
lemma mult_eq_0_iff [simp]:
fixes a b :: "'a::ring_no_zero_divisors"
shows "(a * b = 0) = (a = 0 \<or> b = 0)"
by (cases "a = 0 \<or> b = 0", auto dest: no_zero_divisors)
instance ordered_ring_strict \<subseteq> ring_no_zero_divisors
apply intro_classes
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
done
lemma zero_le_mult_iff:
"((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
zero_less_mult_iff)
lemma mult_less_0_iff:
"(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
apply (insert zero_less_mult_iff [of "-a" b])
apply (force simp add: minus_mult_left[symmetric])
done
lemma mult_le_0_iff:
"(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
apply (insert zero_le_mult_iff [of "-a" b])
apply (force simp add: minus_mult_left[symmetric])
done
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)"
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
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)"
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a"
by (simp add: zero_le_mult_iff linorder_linear)
lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))"
by (simp add: not_less)
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
theorems available to members of @{term ordered_idom} *}
instance ordered_idom \<subseteq> ordered_semidom
proof
have "(0::'a) \<le> 1*1" by (rule zero_le_square)
thus "(0::'a) < 1" by (simp add: order_le_less)
qed
instance ordered_idom \<subseteq> idom ..
text{*All three types of comparision involving 0 and 1 are covered.*}
lemmas one_neq_zero = zero_neq_one [THEN not_sym]
declare one_neq_zero [simp]
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
by (rule zero_less_one [THEN order_less_imp_le])
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
by (simp add: linorder_not_le)
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
by (simp add: linorder_not_less)
subsection{*More Monotonicity*}
text{*Strict monotonicity in both arguments*}
lemma mult_strict_mono:
"[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
apply (cases "c=0")
apply (simp add: mult_pos_pos)
apply (erule mult_strict_right_mono [THEN order_less_trans])
apply (force simp add: order_le_less)
apply (erule mult_strict_left_mono, assumption)
done
text{*This weaker variant has more natural premises*}
lemma mult_strict_mono':
"[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
apply (rule mult_strict_mono)
apply (blast intro: order_le_less_trans)+
done
lemma mult_mono:
"[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|]
==> a * c \<le> b * (d::'a::pordered_semiring)"
apply (erule mult_right_mono [THEN order_trans], assumption)
apply (erule mult_left_mono, assumption)
done
lemma mult_mono':
"[|a \<le> b; c \<le> d; 0 \<le> a; 0 \<le> c|]
==> a * c \<le> b * (d::'a::pordered_semiring)"
apply (rule mult_mono)
apply (fast intro: order_trans)+
done
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
apply (insert mult_strict_mono [of 1 m 1 n])
apply (simp add: order_less_trans [OF zero_less_one])
done
lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==>
c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d"
apply (subgoal_tac "a * c < b * c")
apply (erule order_less_le_trans)
apply (erule mult_left_mono)
apply simp
apply (erule mult_strict_right_mono)
apply assumption
done
lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==>
c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d"
apply (subgoal_tac "a * c <= b * c")
apply (erule order_le_less_trans)
apply (erule mult_strict_left_mono)
apply simp
apply (erule mult_right_mono)
apply simp
done
subsection{*Cancellation Laws for Relationships With a Common Factor*}
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
also with the relations @{text "\<le>"} and equality.*}
text{*These ``disjunction'' versions produce two cases when the comparison is
an assumption, but effectively four when the comparison is a goal.*}
lemma mult_less_cancel_right_disj:
"(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
apply (cases "c = 0")
apply (auto simp add: linorder_neq_iff mult_strict_right_mono
mult_strict_right_mono_neg)
apply (auto simp add: linorder_not_less
linorder_not_le [symmetric, of "a*c"]
linorder_not_le [symmetric, of a])
apply (erule_tac [!] notE)
apply (auto simp add: order_less_imp_le mult_right_mono
mult_right_mono_neg)
done
lemma mult_less_cancel_left_disj:
"(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
apply (cases "c = 0")
apply (auto simp add: linorder_neq_iff mult_strict_left_mono
mult_strict_left_mono_neg)
apply (auto simp add: linorder_not_less
linorder_not_le [symmetric, of "c*a"]
linorder_not_le [symmetric, of a])
apply (erule_tac [!] notE)
apply (auto simp add: order_less_imp_le mult_left_mono
mult_left_mono_neg)
done
text{*The ``conjunction of implication'' lemmas produce two cases when the
comparison is a goal, but give four when the comparison is an assumption.*}
lemma mult_less_cancel_right:
fixes c :: "'a :: ordered_ring_strict"
shows "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
by (insert mult_less_cancel_right_disj [of a c b], auto)
lemma mult_less_cancel_left:
fixes c :: "'a :: ordered_ring_strict"
shows "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
by (insert mult_less_cancel_left_disj [of c a b], auto)
lemma mult_le_cancel_right:
"(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)
lemma mult_le_cancel_left:
"(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)
lemma mult_less_imp_less_left:
assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
shows "a < (b::'a::ordered_semiring_strict)"
proof (rule ccontr)
assume "~ a < b"
hence "b \<le> a" by (simp add: linorder_not_less)
hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono)
with this and less show False
by (simp add: linorder_not_less [symmetric])
qed
lemma mult_less_imp_less_right:
assumes less: "a*c < b*c" and nonneg: "0 <= c"
shows "a < (b::'a::ordered_semiring_strict)"
proof (rule ccontr)
assume "~ a < b"
hence "b \<le> a" by (simp add: linorder_not_less)
hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono)
with this and less show False
by (simp add: linorder_not_less [symmetric])
qed
text{*Cancellation of equalities with a common factor*}
lemma mult_cancel_right [simp]:
fixes a b c :: "'a::ring_no_zero_divisors"
shows "(a * c = b * c) = (c = 0 \<or> a = b)"
proof -
have "(a * c = b * c) = ((a - b) * c = 0)"
by (simp add: ring_distribs)
thus ?thesis
by (simp add: disj_commute)
qed
lemma mult_cancel_left [simp]:
fixes a b c :: "'a::ring_no_zero_divisors"
shows "(c * a = c * b) = (c = 0 \<or> a = b)"
proof -
have "(c * a = c * b) = (c * (a - b) = 0)"
by (simp add: ring_distribs)
thus ?thesis
by simp
qed
subsubsection{*Special Cancellation Simprules for Multiplication*}
text{*These also produce two cases when the comparison is a goal.*}
lemma mult_le_cancel_right1:
fixes c :: "'a :: ordered_idom"
shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
by (insert mult_le_cancel_right [of 1 c b], simp)
lemma mult_le_cancel_right2:
fixes c :: "'a :: ordered_idom"
shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
by (insert mult_le_cancel_right [of a c 1], simp)
lemma mult_le_cancel_left1:
fixes c :: "'a :: ordered_idom"
shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
by (insert mult_le_cancel_left [of c 1 b], simp)
lemma mult_le_cancel_left2:
fixes c :: "'a :: ordered_idom"
shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
by (insert mult_le_cancel_left [of c a 1], simp)
lemma mult_less_cancel_right1:
fixes c :: "'a :: ordered_idom"
shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
by (insert mult_less_cancel_right [of 1 c b], simp)
lemma mult_less_cancel_right2:
fixes c :: "'a :: ordered_idom"
shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
by (insert mult_less_cancel_right [of a c 1], simp)
lemma mult_less_cancel_left1:
fixes c :: "'a :: ordered_idom"
shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
by (insert mult_less_cancel_left [of c 1 b], simp)
lemma mult_less_cancel_left2:
fixes c :: "'a :: ordered_idom"
shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
by (insert mult_less_cancel_left [of c a 1], simp)
lemma mult_cancel_right1 [simp]:
fixes c :: "'a :: ring_1_no_zero_divisors"
shows "(c = b*c) = (c = 0 | b=1)"
by (insert mult_cancel_right [of 1 c b], force)
lemma mult_cancel_right2 [simp]:
fixes c :: "'a :: ring_1_no_zero_divisors"
shows "(a*c = c) = (c = 0 | a=1)"
by (insert mult_cancel_right [of a c 1], simp)
lemma mult_cancel_left1 [simp]:
fixes c :: "'a :: ring_1_no_zero_divisors"
shows "(c = c*b) = (c = 0 | b=1)"
by (insert mult_cancel_left [of c 1 b], force)
lemma mult_cancel_left2 [simp]:
fixes c :: "'a :: ring_1_no_zero_divisors"
shows "(c*a = c) = (c = 0 | a=1)"
by (insert mult_cancel_left [of c a 1], simp)
text{*Simprules for comparisons where common factors can be cancelled.*}
lemmas mult_compare_simps =
mult_le_cancel_right mult_le_cancel_left
mult_le_cancel_right1 mult_le_cancel_right2
mult_le_cancel_left1 mult_le_cancel_left2
mult_less_cancel_right mult_less_cancel_left
mult_less_cancel_right1 mult_less_cancel_right2
mult_less_cancel_left1 mult_less_cancel_left2
mult_cancel_right mult_cancel_left
mult_cancel_right1 mult_cancel_right2
mult_cancel_left1 mult_cancel_left2
subsection {* Fields *}
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
proof
assume neq: "b \<noteq> 0"
{
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
also assume "a / b = 1"
finally show "a = b" by simp
next
assume "a = b"
with neq show "a / b = 1" by (simp add: divide_inverse)
}
qed
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
by (simp add: divide_inverse)
lemma divide_self[simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
by (simp add: divide_inverse)
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
by (simp add: divide_inverse)
lemma divide_self_if [simp]:
"a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
by (simp add: divide_self)
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
by (simp add: divide_inverse)
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
by (simp add: divide_inverse)
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
by (simp add: divide_inverse ring_distribs)
(* what ordering?? this is a straight instance of mult_eq_0_iff
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
of an ordering.*}
lemma field_mult_eq_0_iff [simp]:
"(a*b = (0::'a::division_ring)) = (a = 0 | b = 0)"
by simp
*)
(* subsumed by mult_cancel lemmas on ring_no_zero_divisors
text{*Cancellation of equalities with a common factor*}
lemma field_mult_cancel_right_lemma:
assumes cnz: "c \<noteq> (0::'a::division_ring)"
and eq: "a*c = b*c"
shows "a=b"
proof -
have "(a * c) * inverse c = (b * c) * inverse c"
by (simp add: eq)
thus "a=b"
by (simp add: mult_assoc cnz)
qed
lemma field_mult_cancel_right [simp]:
"(a*c = b*c) = (c = (0::'a::division_ring) | a=b)"
by simp
lemma field_mult_cancel_left [simp]:
"(c*a = c*b) = (c = (0::'a::division_ring) | a=b)"
by simp
*)
lemma nonzero_imp_inverse_nonzero:
"a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)"
proof
assume ianz: "inverse a = 0"
assume "a \<noteq> 0"
hence "1 = a * inverse a" by simp
also have "... = 0" by (simp add: ianz)
finally have "1 = (0::'a::division_ring)" .
thus False by (simp add: eq_commute)
qed
subsection{*Basic Properties of @{term inverse}*}
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::division_ring)"
apply (rule ccontr)
apply (blast dest: nonzero_imp_inverse_nonzero)
done
lemma inverse_nonzero_imp_nonzero:
"inverse a = 0 ==> a = (0::'a::division_ring)"
apply (rule ccontr)
apply (blast dest: nonzero_imp_inverse_nonzero)
done
lemma inverse_nonzero_iff_nonzero [simp]:
"(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
by (force dest: inverse_nonzero_imp_nonzero)
lemma nonzero_inverse_minus_eq:
assumes [simp]: "a\<noteq>0"
shows "inverse(-a) = -inverse(a::'a::division_ring)"
proof -
have "-a * inverse (- a) = -a * - inverse a"
by simp
thus ?thesis
by (simp only: mult_cancel_left, simp)
qed
lemma inverse_minus_eq [simp]:
"inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
proof cases
assume "a=0" thus ?thesis by (simp add: inverse_zero)
next
assume "a\<noteq>0"
thus ?thesis by (simp add: nonzero_inverse_minus_eq)
qed
lemma nonzero_inverse_eq_imp_eq:
assumes inveq: "inverse a = inverse b"
and anz: "a \<noteq> 0"
and bnz: "b \<noteq> 0"
shows "a = (b::'a::division_ring)"
proof -
have "a * inverse b = a * inverse a"
by (simp add: inveq)
hence "(a * inverse b) * b = (a * inverse a) * b"
by simp
thus "a = b"
by (simp add: mult_assoc anz bnz)
qed
lemma inverse_eq_imp_eq:
"inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
apply (cases "a=0 | b=0")
apply (force dest!: inverse_zero_imp_zero
simp add: eq_commute [of "0::'a"])
apply (force dest!: nonzero_inverse_eq_imp_eq)
done
lemma inverse_eq_iff_eq [simp]:
"(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
by (force dest!: inverse_eq_imp_eq)
lemma nonzero_inverse_inverse_eq:
assumes [simp]: "a \<noteq> 0"
shows "inverse(inverse (a::'a::division_ring)) = a"
proof -
have "(inverse (inverse a) * inverse a) * a = a"
by (simp add: nonzero_imp_inverse_nonzero)
thus ?thesis
by (simp add: mult_assoc)
qed
lemma inverse_inverse_eq [simp]:
"inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
proof cases
assume "a=0" thus ?thesis by simp
next
assume "a\<noteq>0"
thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
qed
lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)"
proof -
have "inverse 1 * 1 = (1::'a::division_ring)"
by (rule left_inverse [OF zero_neq_one [symmetric]])
thus ?thesis by simp
qed
lemma inverse_unique:
assumes ab: "a*b = 1"
shows "inverse a = (b::'a::division_ring)"
proof -
have "a \<noteq> 0" using ab by auto
moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
ultimately show ?thesis by (simp add: mult_assoc [symmetric])
qed
lemma nonzero_inverse_mult_distrib:
assumes anz: "a \<noteq> 0"
and bnz: "b \<noteq> 0"
shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)"
proof -
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)"
by (simp add: anz bnz)
hence "inverse(a*b) * a = inverse(b)"
by (simp add: mult_assoc bnz)
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)"
by simp
thus ?thesis
by (simp add: mult_assoc anz)
qed
text{*This version builds in division by zero while also re-orienting
the right-hand side.*}
lemma inverse_mult_distrib [simp]:
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
proof cases
assume "a \<noteq> 0 & b \<noteq> 0"
thus ?thesis
by (simp add: nonzero_inverse_mult_distrib mult_commute)
next
assume "~ (a \<noteq> 0 & b \<noteq> 0)"
thus ?thesis
by force
qed
lemma division_ring_inverse_add:
"[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
==> inverse a + inverse b = inverse a * (a+b) * inverse b"
by (simp add: ring_simps)
lemma division_ring_inverse_diff:
"[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
==> inverse a - inverse b = inverse a * (b-a) * inverse b"
by (simp add: ring_simps)
text{*There is no slick version using division by zero.*}
lemma inverse_add:
"[|a \<noteq> 0; b \<noteq> 0|]
==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
by (simp add: division_ring_inverse_add mult_ac)
lemma inverse_divide [simp]:
"inverse (a/b) = b / (a::'a::{field,division_by_zero})"
by (simp add: divide_inverse mult_commute)
subsection {* Calculations with fractions *}
text{* There is a whole bunch of simp-rules just for class @{text
field} but none for class @{text field} and @{text nonzero_divides}
because the latter are covered by a simproc. *}
lemma nonzero_mult_divide_mult_cancel_left[simp]:
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)"
proof -
have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
by (simp add: divide_inverse nonzero_inverse_mult_distrib)
also have "... = a * inverse b * (inverse c * c)"
by (simp only: mult_ac)
also have "... = a * inverse b"
by simp
finally show ?thesis
by (simp add: divide_inverse)
qed
lemma mult_divide_mult_cancel_left:
"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
apply (cases "b = 0")
apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
done
lemma nonzero_mult_divide_mult_cancel_right:
"[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left)
lemma mult_divide_mult_cancel_right:
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
apply (cases "b = 0")
apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
done
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
by (simp add: divide_inverse)
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
by (simp add: divide_inverse mult_assoc)
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
by (simp add: divide_inverse mult_ac)
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
lemma divide_divide_eq_right [simp]:
"a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
by (simp add: divide_inverse mult_ac)
lemma divide_divide_eq_left [simp]:
"(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
by (simp add: divide_inverse mult_assoc)
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
x / y + w / z = (x * z + w * y) / (y * z)"
apply (subgoal_tac "x / y = (x * z) / (y * z)")
apply (erule ssubst)
apply (subgoal_tac "w / z = (w * y) / (y * z)")
apply (erule ssubst)
apply (rule add_divide_distrib [THEN sym])
apply (subst mult_commute)
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym])
apply assumption
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
apply assumption
done
subsubsection{*Special Cancellation Simprules for Division*}
lemma mult_divide_mult_cancel_left_if[simp]:
fixes c :: "'a :: {field,division_by_zero}"
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
by (simp add: mult_divide_mult_cancel_left)
lemma nonzero_mult_divide_cancel_right[simp]:
"b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp
lemma nonzero_mult_divide_cancel_left[simp]:
"a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp
lemma nonzero_divide_mult_cancel_right[simp]:
"\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)"
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp
lemma nonzero_divide_mult_cancel_left[simp]:
"\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)"
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp
lemma nonzero_mult_divide_mult_cancel_left2[simp]:
"[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)"
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac)
lemma nonzero_mult_divide_mult_cancel_right2[simp]:
"[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)"
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac)
subsection {* Division and Unary Minus *}
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
by (simp add: divide_inverse minus_mult_left)
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
by (simp add: divide_inverse nonzero_inverse_minus_eq)
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
by (simp add: divide_inverse minus_mult_left [symmetric])
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
by (simp add: divide_inverse minus_mult_right [symmetric])
text{*The effect is to extract signs from divisions*}
lemmas divide_minus_left = minus_divide_left [symmetric]
lemmas divide_minus_right = minus_divide_right [symmetric]
declare divide_minus_left [simp] divide_minus_right [simp]
text{*Also, extract signs from products*}
lemmas mult_minus_left = minus_mult_left [symmetric]
lemmas mult_minus_right = minus_mult_right [symmetric]
declare mult_minus_left [simp] mult_minus_right [simp]
lemma minus_divide_divide [simp]:
"(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
apply (cases "b=0", simp)
apply (simp add: nonzero_minus_divide_divide)
done
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
by (simp add: diff_minus add_divide_distrib)
lemma add_divide_eq_iff:
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z"
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
lemma divide_add_eq_iff:
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z"
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
lemma diff_divide_eq_iff:
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z"
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
lemma divide_diff_eq_iff:
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z"
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
proof -
assume [simp]: "c\<noteq>0"
have "(a = b/c) = (a*c = (b/c)*c)" by simp
also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc)
finally show ?thesis .
qed
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
proof -
assume [simp]: "c\<noteq>0"
have "(b/c = a) = ((b/c)*c = a*c)" by simp
also have "... = (b = a*c)" by (simp add: divide_inverse mult_assoc)
finally show ?thesis .
qed
lemma eq_divide_eq:
"((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
by (simp add: nonzero_eq_divide_eq)
lemma divide_eq_eq:
"(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
by (force simp add: nonzero_divide_eq_eq)
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
b = a * c ==> b / c = a"
by (subst divide_eq_eq, simp)
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
a * c = b ==> a = b / c"
by (subst eq_divide_eq, simp)
lemmas field_eq_simps = ring_simps
(* pull / out*)
add_divide_eq_iff divide_add_eq_iff
diff_divide_eq_iff divide_diff_eq_iff
(* multiply eqn *)
nonzero_eq_divide_eq nonzero_divide_eq_eq
(* is added later:
times_divide_eq_left times_divide_eq_right
*)
text{*An example:*}
lemma fixes a b c d e f :: "'a::field"
shows "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f \<rbrakk> \<Longrightarrow> ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) = 1"
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
apply(simp add:field_eq_simps)
apply(simp)
done
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
x / y - w / z = (x * z - w * y) / (y * z)"
by (simp add:field_eq_simps times_divide_eq)
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
(x / y = w / z) = (x * z = w * y)"
by (simp add:field_eq_simps times_divide_eq)
subsection {* Ordered Fields *}
lemma positive_imp_inverse_positive:
assumes a_gt_0: "0 < a" shows "0 < inverse (a::'a::ordered_field)"
proof -
have "0 < a * inverse a"
by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
thus "0 < inverse a"
by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
qed
lemma negative_imp_inverse_negative:
"a < 0 ==> inverse a < (0::'a::ordered_field)"
by (insert positive_imp_inverse_positive [of "-a"],
simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
lemma inverse_le_imp_le:
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
shows "b \<le> (a::'a::ordered_field)"
proof (rule classical)
assume "~ b \<le> a"
hence "a < b" by (simp add: linorder_not_le)
hence bpos: "0 < b" by (blast intro: apos order_less_trans)
hence "a * inverse a \<le> a * inverse b"
by (simp add: apos invle order_less_imp_le mult_left_mono)
hence "(a * inverse a) * b \<le> (a * inverse b) * b"
by (simp add: bpos order_less_imp_le mult_right_mono)
thus "b \<le> a" by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
qed
lemma inverse_positive_imp_positive:
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
shows "0 < (a::'a::ordered_field)"
proof -
have "0 < inverse (inverse a)"
using inv_gt_0 by (rule positive_imp_inverse_positive)
thus "0 < a"
using nz by (simp add: nonzero_inverse_inverse_eq)
qed
lemma inverse_positive_iff_positive [simp]:
"(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
apply (cases "a = 0", simp)
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
done
lemma inverse_negative_imp_negative:
assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
shows "a < (0::'a::ordered_field)"
proof -
have "inverse (inverse a) < 0"
using inv_less_0 by (rule negative_imp_inverse_negative)
thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
qed
lemma inverse_negative_iff_negative [simp]:
"(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
apply (cases "a = 0", simp)
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
done
lemma inverse_nonnegative_iff_nonnegative [simp]:
"(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
by (simp add: linorder_not_less [symmetric])
lemma inverse_nonpositive_iff_nonpositive [simp]:
"(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
by (simp add: linorder_not_less [symmetric])
lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
proof
fix x::'a
have m1: "- (1::'a) < 0" by simp
from add_strict_right_mono[OF m1, where c=x]
have "(- 1) + x < x" by simp
thus "\<exists>y. y < x" by blast
qed
lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
proof
fix x::'a
have m1: " (1::'a) > 0" by simp
from add_strict_right_mono[OF m1, where c=x]
have "1 + x > x" by simp
thus "\<exists>y. y > x" by blast
qed
subsection{*Anti-Monotonicity of @{term inverse}*}
lemma less_imp_inverse_less:
assumes less: "a < b" and apos: "0 < a"
shows "inverse b < inverse (a::'a::ordered_field)"
proof (rule ccontr)
assume "~ inverse b < inverse a"
hence "inverse a \<le> inverse b"
by (simp add: linorder_not_less)
hence "~ (a < b)"
by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
thus False
by (rule notE [OF _ less])
qed
lemma inverse_less_imp_less:
"[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq)
done
text{*Both premises are essential. Consider -1 and 1.*}
lemma inverse_less_iff_less [simp]:
"[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)
lemma le_imp_inverse_le:
"[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
by (force simp add: order_le_less less_imp_inverse_less)
lemma inverse_le_iff_le [simp]:
"[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)
text{*These results refer to both operands being negative. The opposite-sign
case is trivial, since inverse preserves signs.*}
lemma inverse_le_imp_le_neg:
"[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
apply (rule classical)
apply (subgoal_tac "a < 0")
prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans)
apply (insert inverse_le_imp_le [of "-b" "-a"])
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)
done
lemma less_imp_inverse_less_neg:
"[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
apply (subgoal_tac "a < 0")
prefer 2 apply (blast intro: order_less_trans)
apply (insert less_imp_inverse_less [of "-b" "-a"])
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)
done
lemma inverse_less_imp_less_neg:
"[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
apply (rule classical)
apply (subgoal_tac "a < 0")
prefer 2
apply (force simp add: linorder_not_less intro: order_le_less_trans)
apply (insert inverse_less_imp_less [of "-b" "-a"])
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)
done
lemma inverse_less_iff_less_neg [simp]:
"[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
apply (insert inverse_less_iff_less [of "-b" "-a"])
apply (simp del: inverse_less_iff_less
add: order_less_imp_not_eq nonzero_inverse_minus_eq)
done
lemma le_imp_inverse_le_neg:
"[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
by (force simp add: order_le_less less_imp_inverse_less_neg)
lemma inverse_le_iff_le_neg [simp]:
"[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)
subsection{*Inverses and the Number One*}
lemma one_less_inverse_iff:
"(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"
proof cases
assume "0 < x"
with inverse_less_iff_less [OF zero_less_one, of x]
show ?thesis by simp
next
assume notless: "~ (0 < x)"
have "~ (1 < inverse x)"
proof
assume "1 < inverse x"
also with notless have "... \<le> 0" by (simp add: linorder_not_less)
also have "... < 1" by (rule zero_less_one)
finally show False by auto
qed
with notless show ?thesis by simp
qed
lemma inverse_eq_1_iff [simp]:
"(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
by (insert inverse_eq_iff_eq [of x 1], simp)
lemma one_le_inverse_iff:
"(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
by (force simp add: order_le_less one_less_inverse_iff zero_less_one
eq_commute [of 1])
lemma inverse_less_1_iff:
"(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff)
lemma inverse_le_1_iff:
"(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff)
subsection{*Simplification of Inequalities Involving Literal Divisors*}
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
proof -
assume less: "0<c"
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
also have "... = (a*c \<le> b)"
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)
finally show ?thesis .
qed
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
proof -
assume less: "c<0"
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
also have "... = (b \<le> a*c)"
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)
finally show ?thesis .
qed
lemma le_divide_eq:
"(a \<le> b/c) =
(if 0 < c then a*c \<le> b
else if c < 0 then b \<le> a*c
else a \<le> (0::'a::{ordered_field,division_by_zero}))"
apply (cases "c=0", simp)
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff)
done
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
proof -
assume less: "0<c"
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
also have "... = (b \<le> a*c)"
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)
finally show ?thesis .
qed
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
proof -
assume less: "c<0"
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
also have "... = (a*c \<le> b)"
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)
finally show ?thesis .
qed
lemma divide_le_eq:
"(b/c \<le> a) =
(if 0 < c then b \<le> a*c
else if c < 0 then a*c \<le> b
else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
apply (cases "c=0", simp)
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff)
done
lemma pos_less_divide_eq:
"0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
proof -
assume less: "0<c"
hence "(a < b/c) = (a*c < (b/c)*c)"
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
also have "... = (a*c < b)"
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)
finally show ?thesis .
qed
lemma neg_less_divide_eq:
"c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
proof -
assume less: "c<0"
hence "(a < b/c) = ((b/c)*c < a*c)"
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
also have "... = (b < a*c)"
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)
finally show ?thesis .
qed
lemma less_divide_eq:
"(a < b/c) =
(if 0 < c then a*c < b
else if c < 0 then b < a*c
else a < (0::'a::{ordered_field,division_by_zero}))"
apply (cases "c=0", simp)
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff)
done
lemma pos_divide_less_eq:
"0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
proof -
assume less: "0<c"
hence "(b/c < a) = ((b/c)*c < a*c)"
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
also have "... = (b < a*c)"
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)
finally show ?thesis .
qed
lemma neg_divide_less_eq:
"c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
proof -
assume less: "c<0"
hence "(b/c < a) = (a*c < (b/c)*c)"
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
also have "... = (a*c < b)"
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)
finally show ?thesis .
qed
lemma divide_less_eq:
"(b/c < a) =
(if 0 < c then b < a*c
else if c < 0 then a*c < b
else 0 < (a::'a::{ordered_field,division_by_zero}))"
apply (cases "c=0", simp)
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff)
done
subsection{*Field simplification*}
text{* Lemmas @{text field_simps} multiply with denominators in
in(equations) if they can be proved to be non-zero (for equations) or
positive/negative (for inequations). *}
lemmas field_simps = field_eq_simps
(* multiply ineqn *)
pos_divide_less_eq neg_divide_less_eq
pos_less_divide_eq neg_less_divide_eq
pos_divide_le_eq neg_divide_le_eq
pos_le_divide_eq neg_le_divide_eq
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
of positivity/negativity needed for @{text field_simps}. Have not added @{text
sign_simps} to @{text field_simps} because the former can lead to case
explosions. *}
lemmas sign_simps = group_simps
zero_less_mult_iff mult_less_0_iff
(* Only works once linear arithmetic is installed:
text{*An example:*}
lemma fixes a b c d e f :: "'a::ordered_field"
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
prefer 2 apply(simp add:sign_simps)
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
prefer 2 apply(simp add:sign_simps)
apply(simp add:field_simps)
done
*)
subsection{*Division and Signs*}
lemma zero_less_divide_iff:
"((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
by (simp add: divide_inverse zero_less_mult_iff)
lemma divide_less_0_iff:
"(a/b < (0::'a::{ordered_field,division_by_zero})) =
(0 < a & b < 0 | a < 0 & 0 < b)"
by (simp add: divide_inverse mult_less_0_iff)
lemma zero_le_divide_iff:
"((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
(0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
by (simp add: divide_inverse zero_le_mult_iff)
lemma divide_le_0_iff:
"(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
(0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
by (simp add: divide_inverse mult_le_0_iff)
lemma divide_eq_0_iff [simp]:
"(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
by (simp add: divide_inverse)
lemma divide_pos_pos:
"0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y"
by(simp add:field_simps)
lemma divide_nonneg_pos:
"0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y"
by(simp add:field_simps)
lemma divide_neg_pos:
"(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
by(simp add:field_simps)
lemma divide_nonpos_pos:
"(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
by(simp add:field_simps)
lemma divide_pos_neg:
"0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
by(simp add:field_simps)
lemma divide_nonneg_neg:
"0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0"
by(simp add:field_simps)
lemma divide_neg_neg:
"(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
by(simp add:field_simps)
lemma divide_nonpos_neg:
"(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
by(simp add:field_simps)
subsection{*Cancellation Laws for Division*}
lemma divide_cancel_right [simp]:
"(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
apply (cases "c=0", simp)
apply (simp add: divide_inverse)
done
lemma divide_cancel_left [simp]:
"(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
apply (cases "c=0", simp)
apply (simp add: divide_inverse)
done
subsection {* Division and the Number One *}
text{*Simplify expressions equated with 1*}
lemma divide_eq_1_iff [simp]:
"(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
apply (cases "b=0", simp)
apply (simp add: right_inverse_eq)
done
lemma one_eq_divide_iff [simp]:
"(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
by (simp add: eq_commute [of 1])
lemma zero_eq_1_divide_iff [simp]:
"((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
apply (cases "a=0", simp)
apply (auto simp add: nonzero_eq_divide_eq)
done
lemma one_divide_eq_0_iff [simp]:
"(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
apply (cases "a=0", simp)
apply (insert zero_neq_one [THEN not_sym])
apply (auto simp add: nonzero_divide_eq_eq)
done
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
declare zero_less_divide_1_iff [simp]
declare divide_less_0_1_iff [simp]
declare zero_le_divide_1_iff [simp]
declare divide_le_0_1_iff [simp]
subsection {* Ordering Rules for Division *}
lemma divide_strict_right_mono:
"[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono
positive_imp_inverse_positive)
lemma divide_right_mono:
"[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
by (force simp add: divide_strict_right_mono order_le_less)
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b
==> c <= 0 ==> b / c <= a / c"
apply (drule divide_right_mono [of _ _ "- c"])
apply auto
done
lemma divide_strict_right_mono_neg:
"[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
done
text{*The last premise ensures that @{term a} and @{term b}
have the same sign*}
lemma divide_strict_left_mono:
"[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
lemma divide_left_mono:
"[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b
==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
apply (drule divide_left_mono [of _ _ "- c"])
apply (auto simp add: mult_commute)
done
lemma divide_strict_left_mono_neg:
"[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
text{*Simplify quotients that are compared with the value 1.*}
lemma le_divide_eq_1:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
by (auto simp add: le_divide_eq)
lemma divide_le_eq_1:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
by (auto simp add: divide_le_eq)
lemma less_divide_eq_1:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
by (auto simp add: less_divide_eq)
lemma divide_less_eq_1:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
by (auto simp add: divide_less_eq)
subsection{*Conditional Simplification Rules: No Case Splits*}
lemma le_divide_eq_1_pos [simp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
by (auto simp add: le_divide_eq)
lemma le_divide_eq_1_neg [simp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
by (auto simp add: le_divide_eq)
lemma divide_le_eq_1_pos [simp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
by (auto simp add: divide_le_eq)
lemma divide_le_eq_1_neg [simp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
by (auto simp add: divide_le_eq)
lemma less_divide_eq_1_pos [simp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
by (auto simp add: less_divide_eq)
lemma less_divide_eq_1_neg [simp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
by (auto simp add: less_divide_eq)
lemma divide_less_eq_1_pos [simp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
by (auto simp add: divide_less_eq)
lemma divide_less_eq_1_neg [simp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
by (auto simp add: divide_less_eq)
lemma eq_divide_eq_1 [simp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
by (auto simp add: eq_divide_eq)
lemma divide_eq_eq_1 [simp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
by (auto simp add: divide_eq_eq)
subsection {* Reasoning about inequalities with division *}
lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
==> x * y <= x"
by (auto simp add: mult_compare_simps);
lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
==> y * x <= x"
by (auto simp add: mult_compare_simps);
lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
x / y <= z";
by (subst pos_divide_le_eq, assumption+);
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
z <= x / y"
by(simp add:field_simps)
lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
x / y < z"
by(simp add:field_simps)
lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
z < x / y"
by(simp add:field_simps)
lemma frac_le: "(0::'a::ordered_field) <= x ==>
x <= y ==> 0 < w ==> w <= z ==> x / z <= y / w"
apply (rule mult_imp_div_pos_le)
apply simp;
apply (subst times_divide_eq_left);
apply (rule mult_imp_le_div_pos, assumption)
apply (rule mult_mono)
apply simp_all
done
lemma frac_less: "(0::'a::ordered_field) <= x ==>
x < y ==> 0 < w ==> w <= z ==> x / z < y / w"
apply (rule mult_imp_div_pos_less)
apply simp;
apply (subst times_divide_eq_left);
apply (rule mult_imp_less_div_pos, assumption)
apply (erule mult_less_le_imp_less)
apply simp_all
done
lemma frac_less2: "(0::'a::ordered_field) < x ==>
x <= y ==> 0 < w ==> w < z ==> x / z < y / w"
apply (rule mult_imp_div_pos_less)
apply simp_all
apply (subst times_divide_eq_left);
apply (rule mult_imp_less_div_pos, assumption)
apply (erule mult_le_less_imp_less)
apply simp_all
done
text{*It's not obvious whether these should be simprules or not.
Their effect is to gather terms into one big fraction, like
a*b*c / x*y*z. The rationale for that is unclear, but many proofs
seem to need them.*}
declare times_divide_eq [simp]
subsection {* Ordered Fields are Dense *}
lemma less_add_one: "a < (a+1::'a::ordered_semidom)"
proof -
have "a+0 < (a+1::'a::ordered_semidom)"
by (blast intro: zero_less_one add_strict_left_mono)
thus ?thesis by simp
qed
lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"
by (blast intro: order_less_trans zero_less_one less_add_one)
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
by (simp add: field_simps zero_less_two)
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
by (simp add: field_simps zero_less_two)
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"
by (blast intro!: less_half_sum gt_half_sum)
subsection {* Absolute Value *}
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))"
proof -
let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
have a: "(abs a) * (abs b) = ?x"
by (simp only: abs_prts[of a] abs_prts[of b] ring_simps)
{
fix u v :: 'a
have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow>
u * v = pprt a * pprt b + pprt a * nprt b +
nprt a * pprt b + nprt a * nprt b"
apply (subst prts[of u], subst prts[of v])
apply (simp add: ring_simps)
done
}
note b = this[OF refl[of a] refl[of b]]
note addm = add_mono[of "0::'a" _ "0::'a", simplified]
note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
have xy: "- ?x <= ?y"
apply (simp)
apply (rule_tac y="0::'a" in order_trans)
apply (rule addm2)
apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
apply (rule addm)
apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
done
have yx: "?y <= ?x"
apply (simp add:diff_def)
apply (rule_tac y=0 in order_trans)
apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
done
have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
show ?thesis
apply (rule abs_leI)
apply (simp add: i1)
apply (simp add: i2[simplified minus_le_iff])
done
qed
lemma abs_eq_mult:
assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"
proof -
have s: "(0 <= a*b) | (a*b <= 0)"
apply (auto)
apply (rule_tac split_mult_pos_le)
apply (rule_tac contrapos_np[of "a*b <= 0"])
apply (simp)
apply (rule_tac split_mult_neg_le)
apply (insert prems)
apply (blast)
done
have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
by (simp add: prts[symmetric])
show ?thesis
proof cases
assume "0 <= a * b"
then show ?thesis
apply (simp_all add: mulprts abs_prts)
apply (insert prems)
apply (auto simp add:
ring_simps
iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id])
apply(drule (1) mult_nonneg_nonpos[of a b], simp)
apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
done
next
assume "~(0 <= a*b)"
with s have "a*b <= 0" by simp
then show ?thesis
apply (simp_all add: mulprts abs_prts)
apply (insert prems)
apply (auto simp add: ring_simps)
apply(drule (1) mult_nonneg_nonneg[of a b],simp)
apply(drule (1) mult_nonpos_nonpos[of a b],simp)
done
qed
qed
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)"
by (simp add: abs_eq_mult linorder_linear)
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
by (simp add: abs_if)
lemma nonzero_abs_inverse:
"a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq
negative_imp_inverse_negative)
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym)
done
lemma abs_inverse [simp]:
"abs (inverse (a::'a::{ordered_field,division_by_zero})) =
inverse (abs a)"
apply (cases "a=0", simp)
apply (simp add: nonzero_abs_inverse)
done
lemma nonzero_abs_divide:
"b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
by (simp add: divide_inverse abs_mult nonzero_abs_inverse)
lemma abs_divide [simp]:
"abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
apply (cases "b=0", simp)
apply (simp add: nonzero_abs_divide)
done
lemma abs_mult_less:
"[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
proof -
assume ac: "abs a < c"
hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
assume "abs b < d"
thus ?thesis by (simp add: ac cpos mult_strict_mono)
qed
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))"
apply (simp add: order_less_le abs_le_iff)
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)
apply (simp add: le_minus_self_iff linorder_neq_iff)
done
lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==>
(abs y) * x = abs (y * x)";
apply (subst abs_mult);
apply simp;
done;
lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==>
abs x / y = abs (x / y)";
apply (subst abs_divide);
apply (simp add: order_less_imp_le);
done;
subsection {* Bounds of products via negative and positive Part *}
lemma mult_le_prts:
assumes
"a1 <= (a::'a::lordered_ring)"
"a <= a2"
"b1 <= b"
"b <= b2"
shows
"a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
proof -
have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
apply (subst prts[symmetric])+
apply simp
done
then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
by (simp add: ring_simps)
moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
by (simp_all add: prems mult_mono)
moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
proof -
have "pprt a * nprt b <= pprt a * nprt b2"
by (simp add: mult_left_mono prems)
moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
by (simp add: mult_right_mono_neg prems)
ultimately show ?thesis
by simp
qed
moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
proof -
have "nprt a * pprt b <= nprt a2 * pprt b"
by (simp add: mult_right_mono prems)
moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
by (simp add: mult_left_mono_neg prems)
ultimately show ?thesis
by simp
qed
moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
proof -
have "nprt a * nprt b <= nprt a * nprt b1"
by (simp add: mult_left_mono_neg prems)
moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
by (simp add: mult_right_mono_neg prems)
ultimately show ?thesis
by simp
qed
ultimately show ?thesis
by - (rule add_mono | simp)+
qed
lemma mult_ge_prts:
assumes
"a1 <= (a::'a::lordered_ring)"
"a <= a2"
"b1 <= b"
"b <= b2"
shows
"a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
proof -
from prems have a1:"- a2 <= -a" by auto
from prems have a2: "-a <= -a1" by auto
from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 prems(3) prems(4), simplified nprt_neg pprt_neg]
have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp
then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
by (simp only: minus_le_iff)
then show ?thesis by simp
qed
subsection {* Theorems for proof tools *}
lemma add_mono_thms_ordered_semiring:
fixes i j k :: "'a\<Colon>pordered_ab_semigroup_add"
shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
by (rule add_mono, clarify+)+
lemma add_mono_thms_ordered_field:
fixes i j k :: "'a\<Colon>pordered_cancel_ab_semigroup_add"
shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
by (auto intro: add_strict_right_mono add_strict_left_mono
add_less_le_mono add_le_less_mono add_strict_mono)
end