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parent 14603 | 985eb6708207 |
child 14754 | a080eeeaec14 |
permissions | -rw-r--r-- |
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(* Title: HOL/Ring_and_Field.thy |
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ID: $Id$ |
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Author: Gertrud Bauer and Markus Wenzel, TU Muenchen |
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Lawrence C Paulson, University of Cambridge |
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Revised and splitted into Ring_and_Field.thy and Group.thy |
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by Steven Obua, TU Muenchen, in May 2004 |
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License: GPL (GNU GENERAL PUBLIC LICENSE) |
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*) |
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header {* (Ordered) Rings and Fields *} |
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theory Ring_and_Field = OrderedGroup: |
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text {* |
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The theory of partially ordered rings is taken from the books: |
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\begin{itemize} |
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\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 |
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\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 |
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\end{itemize} |
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Most of the used notions can also be looked up in |
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\begin{itemize} |
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\item \emph{www.mathworld.com} by Eric Weisstein et. al. |
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\item \emph{Algebra I} by van der Waerden, Springer. |
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\end{itemize} |
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*} |
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axclass semiring \<subseteq> ab_semigroup_add, semigroup_mult |
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left_distrib: "(a + b) * c = a * c + b * c" |
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right_distrib: "a * (b + c) = a * b + a * c" |
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axclass semiring_0 \<subseteq> semiring, comm_monoid_add |
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axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult |
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mult_commute: "a * b = b * a" |
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distrib: "(a + b) * c = a * c + b * c" |
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instance comm_semiring \<subseteq> semiring |
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proof |
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fix a b c :: 'a |
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show "(a + b) * c = a * c + b * c" by (simp add: distrib) |
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have "a * (b + c) = (b + c) * a" by (simp add: mult_ac) |
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also have "... = b * a + c * a" by (simp only: distrib) |
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also have "... = a * b + a * c" by (simp add: mult_ac) |
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finally show "a * (b + c) = a * b + a * c" by blast |
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qed |
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axclass comm_semiring_0 \<subseteq> comm_semiring, comm_monoid_add |
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instance comm_semiring_0 \<subseteq> semiring_0 .. |
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axclass axclass_0_neq_1 \<subseteq> zero, one |
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zero_neq_one [simp]: "0 \<noteq> 1" |
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axclass semiring_1 \<subseteq> axclass_0_neq_1, semiring_0, monoid_mult |
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axclass comm_semiring_1 \<subseteq> axclass_0_neq_1, comm_semiring_0, comm_monoid_mult (* previously almost_semiring *) |
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instance comm_semiring_1 \<subseteq> semiring_1 .. |
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axclass axclass_no_zero_divisors \<subseteq> zero, times |
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no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0" |
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axclass comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *) |
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axclass ring \<subseteq> semiring, ab_group_add |
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instance ring \<subseteq> semiring_0 .. |
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axclass comm_ring \<subseteq> comm_semiring_0, ab_group_add |
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instance comm_ring \<subseteq> ring .. |
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instance comm_ring \<subseteq> comm_semiring_0 .. |
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axclass ring_1 \<subseteq> ring, semiring_1 |
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axclass comm_ring_1 \<subseteq> comm_ring, comm_semiring_1 (* previously ring *) |
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instance comm_ring_1 \<subseteq> ring_1 .. |
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instance comm_ring_1 \<subseteq> comm_semiring_1_cancel .. |
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axclass idom \<subseteq> comm_ring_1, axclass_no_zero_divisors |
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axclass field \<subseteq> comm_ring_1, inverse |
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left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1" |
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divide_inverse: "a / b = a * inverse b" |
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lemma mult_zero_left [simp]: "0 * a = (0::'a::{semiring_0, cancel_semigroup_add})" |
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proof - |
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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 ?thesis |
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by (simp only: add_left_cancel) |
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qed |
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lemma mult_zero_right [simp]: "a * 0 = (0::'a::{semiring_0, cancel_semigroup_add})" |
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proof - |
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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 ?thesis |
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by (simp only: add_left_cancel) |
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qed |
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lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)" |
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proof cases |
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assume "a=0" thus ?thesis by simp |
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next |
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assume anz [simp]: "a\<noteq>0" |
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{ assume "a * b = 0" |
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hence "inverse a * (a * b) = 0" by simp |
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hence "b = 0" by (simp (no_asm_use) add: mult_assoc [symmetric])} |
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thus ?thesis by force |
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qed |
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instance field \<subseteq> idom |
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by (intro_classes, simp) |
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axclass division_by_zero \<subseteq> zero, inverse |
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inverse_zero [simp]: "inverse 0 = 0" |
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subsection {* Distribution rules *} |
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theorems ring_distrib = right_distrib left_distrib |
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text{*For the @{text combine_numerals} simproc*} |
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lemma combine_common_factor: |
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"a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)" |
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by (simp add: left_distrib add_ac) |
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lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)" |
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apply (rule equals_zero_I) |
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apply (simp add: left_distrib [symmetric]) |
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done |
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lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)" |
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apply (rule equals_zero_I) |
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apply (simp add: right_distrib [symmetric]) |
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done |
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lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)" |
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by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) |
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lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)" |
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by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) |
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lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)" |
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by (simp add: right_distrib diff_minus |
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minus_mult_left [symmetric] minus_mult_right [symmetric]) |
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lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)" |
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by (simp add: left_distrib diff_minus |
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minus_mult_left [symmetric] minus_mult_right [symmetric]) |
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axclass pordered_semiring \<subseteq> semiring_0, pordered_ab_semigroup_add |
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mult_left_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b" |
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mult_right_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> a * c <= b * c" |
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axclass pordered_cancel_semiring \<subseteq> pordered_semiring, cancel_ab_semigroup_add |
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axclass ordered_semiring_strict \<subseteq> semiring_0, ordered_cancel_ab_semigroup_add |
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mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
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mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c" |
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instance ordered_semiring_strict \<subseteq> pordered_cancel_semiring |
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apply intro_classes |
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apply (case_tac "a < b & 0 < c") |
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apply (auto simp add: mult_strict_left_mono order_less_le) |
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apply (auto simp add: mult_strict_left_mono order_le_less) |
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apply (simp add: mult_strict_right_mono) |
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done |
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axclass pordered_comm_semiring \<subseteq> comm_semiring_0, pordered_ab_semigroup_add |
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mult_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b" |
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axclass pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring, cancel_ab_semigroup_add |
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instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring .. |
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axclass ordered_comm_semiring_strict \<subseteq> comm_semiring_0, ordered_cancel_ab_semigroup_add |
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mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
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instance pordered_comm_semiring \<subseteq> pordered_semiring |
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by (intro_classes, insert mult_mono, simp_all add: mult_commute, blast+) |
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instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring .. |
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instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict |
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by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+) |
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instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring |
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apply (intro_classes) |
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apply (case_tac "a < b & 0 < c") |
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apply (auto simp add: mult_strict_left_mono order_less_le) |
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apply (auto simp add: mult_strict_left_mono order_le_less) |
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done |
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axclass pordered_ring \<subseteq> ring, pordered_semiring |
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instance pordered_ring \<subseteq> pordered_ab_group_add .. |
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instance pordered_ring \<subseteq> pordered_cancel_semiring .. |
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axclass lordered_ring \<subseteq> pordered_ring, lordered_ab_group_abs |
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axclass axclass_abs_if \<subseteq> minus, ord, zero |
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abs_if: "abs a = (if (a < 0) then (-a) else a)" |
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axclass ordered_ring_strict \<subseteq> ring, ordered_semiring_strict, axclass_abs_if |
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instance ordered_ring_strict \<subseteq> lordered_ab_group .. |
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instance ordered_ring_strict \<subseteq> lordered_ring |
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by (intro_classes, simp add: abs_if join_eq_if) |
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axclass pordered_comm_ring \<subseteq> comm_ring, pordered_comm_semiring |
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axclass ordered_semidom \<subseteq> comm_semiring_1_cancel, ordered_comm_semiring_strict (* previously ordered_semiring *) |
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zero_less_one [simp]: "0 < 1" |
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axclass ordered_idom \<subseteq> comm_ring_1, ordered_comm_semiring_strict, axclass_abs_if (* previously ordered_ring *) |
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instance ordered_idom \<subseteq> ordered_ring_strict .. |
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axclass ordered_field \<subseteq> field, ordered_idom |
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lemma eq_add_iff1: |
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"(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))" |
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apply (simp add: diff_minus left_distrib) |
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Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
230 |
apply (simp add: diff_minus left_distrib add_ac) |
14738 | 231 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
232 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
233 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
234 |
lemma eq_add_iff2: |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
235 |
"(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
236 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
237 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
238 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
239 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
240 |
lemma less_add_iff1: |
14738 | 241 |
"(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))" |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
242 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
243 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
244 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
245 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
246 |
lemma less_add_iff2: |
14738 | 247 |
"(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))" |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
248 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
249 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
250 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
251 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
252 |
lemma le_add_iff1: |
14738 | 253 |
"(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))" |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
254 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
255 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
256 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
257 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
258 |
lemma le_add_iff2: |
14738 | 259 |
"(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))" |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
260 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
261 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
262 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
263 |
|
14270 | 264 |
subsection {* Ordering Rules for Multiplication *} |
265 |
||
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
266 |
lemma mult_left_le_imp_le: |
14738 | 267 |
"[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
268 |
by (force simp add: mult_strict_left_mono linorder_not_less [symmetric]) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
269 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
270 |
lemma mult_right_le_imp_le: |
14738 | 271 |
"[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
272 |
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric]) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
273 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
274 |
lemma mult_left_less_imp_less: |
14738 | 275 |
"[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
276 |
by (force simp add: mult_left_mono linorder_not_le [symmetric]) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
277 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
278 |
lemma mult_right_less_imp_less: |
14738 | 279 |
"[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
280 |
by (force simp add: mult_right_mono linorder_not_le [symmetric]) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
281 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
282 |
lemma mult_strict_left_mono_neg: |
14738 | 283 |
"[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
284 |
apply (drule mult_strict_left_mono [of _ _ "-c"]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
285 |
apply (simp_all add: minus_mult_left [symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
286 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
287 |
|
14738 | 288 |
lemma mult_left_mono_neg: |
289 |
"[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::pordered_ring)" |
|
290 |
apply (drule mult_left_mono [of _ _ "-c"]) |
|
291 |
apply (simp_all add: minus_mult_left [symmetric]) |
|
292 |
done |
|
293 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
294 |
lemma mult_strict_right_mono_neg: |
14738 | 295 |
"[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
296 |
apply (drule mult_strict_right_mono [of _ _ "-c"]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
297 |
apply (simp_all add: minus_mult_right [symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
298 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
299 |
|
14738 | 300 |
lemma mult_right_mono_neg: |
301 |
"[|b \<le> a; c \<le> 0|] ==> a * c \<le> (b::'a::pordered_ring) * c" |
|
302 |
apply (drule mult_right_mono [of _ _ "-c"]) |
|
303 |
apply (simp) |
|
304 |
apply (simp_all add: minus_mult_right [symmetric]) |
|
305 |
done |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
306 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
307 |
subsection{* Products of Signs *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
308 |
|
14738 | 309 |
lemma mult_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
|
310 |
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
|
311 |
|
14738 | 312 |
lemma mult_pos_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b" |
313 |
by (drule mult_left_mono [of 0 b], auto) |
|
314 |
||
315 |
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
|
316 |
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
|
317 |
|
14738 | 318 |
lemma mult_pos_neg_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0" |
319 |
by (drule mult_left_mono [of b 0], auto) |
|
320 |
||
321 |
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" |
|
322 |
by (drule mult_strict_right_mono[of b 0], auto) |
|
323 |
||
324 |
lemma mult_pos_neg2_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" |
|
325 |
by (drule mult_right_mono[of b 0], auto) |
|
326 |
||
327 |
lemma mult_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
|
328 |
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
|
329 |
|
14738 | 330 |
lemma mult_neg_le: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b" |
331 |
by (drule mult_right_mono_neg[of a 0 b ], auto) |
|
332 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
333 |
lemma zero_less_mult_pos: |
14738 | 334 |
"[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
335 |
apply (case_tac "b\<le>0") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
336 |
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
|
337 |
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
|
338 |
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
|
339 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
340 |
|
14738 | 341 |
lemma zero_less_mult_pos2: |
342 |
"[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)" |
|
343 |
apply (case_tac "b\<le>0") |
|
344 |
apply (auto simp add: order_le_less linorder_not_less) |
|
345 |
apply (drule_tac mult_pos_neg2 [of a b]) |
|
346 |
apply (auto dest: order_less_not_sym) |
|
347 |
done |
|
348 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
349 |
lemma zero_less_mult_iff: |
14738 | 350 |
"((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
351 |
apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
352 |
apply (blast dest: zero_less_mult_pos) |
14738 | 353 |
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
|
354 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
355 |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
356 |
text{*A field has no "zero divisors", and this theorem holds without the |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
357 |
assumption of an ordering. See @{text field_mult_eq_0_iff} below.*} |
14738 | 358 |
lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring_strict)) = (a = 0 | b = 0)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
359 |
apply (case_tac "a < 0") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
360 |
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
|
361 |
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
|
362 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
363 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
364 |
lemma zero_le_mult_iff: |
14738 | 365 |
"((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
|
366 |
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
|
367 |
zero_less_mult_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
368 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
369 |
lemma mult_less_0_iff: |
14738 | 370 |
"(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
|
371 |
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
|
372 |
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
|
373 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
374 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
375 |
lemma mult_le_0_iff: |
14738 | 376 |
"(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
|
377 |
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
|
378 |
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
|
379 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
380 |
|
14738 | 381 |
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)" |
382 |
by (auto simp add: mult_pos_le mult_neg_le) |
|
383 |
||
384 |
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)" |
|
385 |
by (auto simp add: mult_pos_neg_le mult_pos_neg2_le) |
|
386 |
||
387 |
lemma zero_le_square: "(0::'a::ordered_ring_strict) \<le> a*a" |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
388 |
by (simp add: zero_le_mult_iff linorder_linear) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
389 |
|
14738 | 390 |
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom} |
391 |
theorems available to members of @{term ordered_idom} *} |
|
392 |
||
393 |
instance ordered_idom \<subseteq> ordered_semidom |
|
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
394 |
proof |
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
395 |
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
|
396 |
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
|
397 |
qed |
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
398 |
|
14738 | 399 |
instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors |
400 |
by (intro_classes, simp) |
|
401 |
||
402 |
instance ordered_idom \<subseteq> idom .. |
|
403 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
404 |
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
|
405 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
406 |
declare zero_neq_one [THEN not_sym, simp] |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
407 |
|
14738 | 408 |
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
|
409 |
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
|
410 |
|
14738 | 411 |
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0" |
412 |
by (simp add: linorder_not_le) |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
413 |
|
14738 | 414 |
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0" |
415 |
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
|
416 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
417 |
subsection{*More Monotonicity*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
418 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
419 |
lemma mult_left_mono_neg: |
14738 | 420 |
"[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::pordered_ring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
421 |
apply (drule mult_left_mono [of _ _ "-c"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
422 |
apply (simp_all add: minus_mult_left [symmetric]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
423 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
424 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
425 |
lemma mult_right_mono_neg: |
14738 | 426 |
"[|b \<le> a; c \<le> 0|] ==> a * c \<le> b * (c::'a::pordered_ring)" |
427 |
apply (drule mult_right_mono [of _ _ "-c"]) |
|
428 |
apply (simp_all add: minus_mult_right [symmetric]) |
|
429 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
430 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
431 |
text{*Strict monotonicity in both arguments*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
432 |
lemma mult_strict_mono: |
14738 | 433 |
"[|a<b; c<d; 0<b; 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
|
434 |
apply (case_tac "c=0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
435 |
apply (simp add: mult_pos) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
436 |
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
|
437 |
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
|
438 |
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
|
439 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
440 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
441 |
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
|
442 |
lemma mult_strict_mono': |
14738 | 443 |
"[| 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
|
444 |
apply (rule mult_strict_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
445 |
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
|
446 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
447 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
448 |
lemma mult_mono: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
449 |
"[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] |
14738 | 450 |
==> 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
|
451 |
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
|
452 |
apply (erule mult_left_mono, assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
453 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
454 |
|
14738 | 455 |
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
|
456 |
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
|
457 |
apply (simp add: order_less_trans [OF zero_less_one]) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
458 |
done |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
459 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
460 |
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
|
461 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
462 |
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
|
463 |
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
|
464 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
465 |
lemma mult_less_cancel_right: |
14738 | 466 |
"(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
467 |
apply (case_tac "c = 0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
468 |
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
|
469 |
mult_strict_right_mono_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
470 |
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
|
471 |
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
|
472 |
linorder_not_le [symmetric, of a]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
473 |
apply (erule_tac [!] notE) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
474 |
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
|
475 |
mult_right_mono_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
476 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
477 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
478 |
lemma mult_less_cancel_left: |
14738 | 479 |
"(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))" |
480 |
apply (case_tac "c = 0") |
|
481 |
apply (auto simp add: linorder_neq_iff mult_strict_left_mono |
|
482 |
mult_strict_left_mono_neg) |
|
483 |
apply (auto simp add: linorder_not_less |
|
484 |
linorder_not_le [symmetric, of "c*a"] |
|
485 |
linorder_not_le [symmetric, of a]) |
|
486 |
apply (erule_tac [!] notE) |
|
487 |
apply (auto simp add: order_less_imp_le mult_left_mono |
|
488 |
mult_left_mono_neg) |
|
489 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
490 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
491 |
lemma mult_le_cancel_right: |
14738 | 492 |
"(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
493 |
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
494 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
495 |
lemma mult_le_cancel_left: |
14738 | 496 |
"(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))" |
497 |
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
498 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
499 |
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
|
500 |
assumes less: "c*a < c*b" and nonneg: "0 \<le> c" |
14738 | 501 |
shows "a < (b::'a::ordered_semiring_strict)" |
14377 | 502 |
proof (rule ccontr) |
503 |
assume "~ a < b" |
|
504 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
505 |
hence "c*b \<le> c*a" by (rule mult_left_mono) |
|
506 |
with this and less show False |
|
507 |
by (simp add: linorder_not_less [symmetric]) |
|
508 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
509 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
510 |
lemma mult_less_imp_less_right: |
14738 | 511 |
assumes less: "a*c < b*c" and nonneg: "0 <= c" |
512 |
shows "a < (b::'a::ordered_semiring_strict)" |
|
513 |
proof (rule ccontr) |
|
514 |
assume "~ a < b" |
|
515 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
516 |
hence "b*c \<le> a*c" by (rule mult_right_mono) |
|
517 |
with this and less show False |
|
518 |
by (simp add: linorder_not_less [symmetric]) |
|
519 |
qed |
|
14268
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 |
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
|
522 |
lemma mult_cancel_right [simp]: |
14738 | 523 |
"(a*c = b*c) = (c = (0::'a::ordered_ring_strict) | a=b)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
524 |
apply (cut_tac linorder_less_linear [of 0 c]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
525 |
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
526 |
simp add: linorder_neq_iff) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
527 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
528 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
529 |
text{*These cancellation theorems require an ordering. Versions are proved |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
530 |
below that work for fields without an ordering.*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
531 |
lemma mult_cancel_left [simp]: |
14738 | 532 |
"(c*a = c*b) = (c = (0::'a::ordered_ring_strict) | a=b)" |
533 |
apply (cut_tac linorder_less_linear [of 0 c]) |
|
534 |
apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono |
|
535 |
simp add: linorder_neq_iff) |
|
536 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
537 |
|
14738 | 538 |
text{*This list of rewrites decides ring equalities by ordered rewriting.*} |
539 |
lemmas ring_eq_simps = |
|
540 |
mult_ac |
|
541 |
left_distrib right_distrib left_diff_distrib right_diff_distrib |
|
542 |
add_ac |
|
543 |
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 |
|
544 |
diff_eq_eq eq_diff_eq |
|
545 |
||
546 |
thm ring_eq_simps |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
547 |
subsection {* Fields *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
548 |
|
14288 | 549 |
lemma right_inverse [simp]: |
550 |
assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1" |
|
551 |
proof - |
|
552 |
have "a * inverse a = inverse a * a" by (simp add: mult_ac) |
|
553 |
also have "... = 1" using not0 by simp |
|
554 |
finally show ?thesis . |
|
555 |
qed |
|
556 |
||
557 |
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" |
|
558 |
proof |
|
559 |
assume neq: "b \<noteq> 0" |
|
560 |
{ |
|
561 |
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) |
|
562 |
also assume "a / b = 1" |
|
563 |
finally show "a = b" by simp |
|
564 |
next |
|
565 |
assume "a = b" |
|
566 |
with neq show "a / b = 1" by (simp add: divide_inverse) |
|
567 |
} |
|
568 |
qed |
|
569 |
||
570 |
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" |
|
571 |
by (simp add: divide_inverse) |
|
572 |
||
573 |
lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1" |
|
574 |
by (simp add: divide_inverse) |
|
575 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
576 |
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
|
577 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
578 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
579 |
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
|
580 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
581 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
582 |
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
|
583 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
584 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
585 |
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c" |
14293 | 586 |
by (simp add: divide_inverse left_distrib) |
587 |
||
588 |
||
14270 | 589 |
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement |
590 |
of an ordering.*} |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
591 |
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)" |
14377 | 592 |
proof cases |
593 |
assume "a=0" thus ?thesis by simp |
|
594 |
next |
|
595 |
assume anz [simp]: "a\<noteq>0" |
|
596 |
{ assume "a * b = 0" |
|
597 |
hence "inverse a * (a * b) = 0" by simp |
|
598 |
hence "b = 0" by (simp (no_asm_use) add: mult_assoc [symmetric])} |
|
599 |
thus ?thesis by force |
|
600 |
qed |
|
14270 | 601 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
602 |
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
|
603 |
lemma field_mult_cancel_right_lemma: |
14269 | 604 |
assumes cnz: "c \<noteq> (0::'a::field)" |
605 |
and eq: "a*c = b*c" |
|
606 |
shows "a=b" |
|
14377 | 607 |
proof - |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
608 |
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
|
609 |
by (simp add: eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
610 |
thus "a=b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
611 |
by (simp add: mult_assoc cnz) |
14377 | 612 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
613 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
614 |
lemma field_mult_cancel_right [simp]: |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
615 |
"(a*c = b*c) = (c = (0::'a::field) | a=b)" |
14377 | 616 |
proof cases |
617 |
assume "c=0" thus ?thesis by simp |
|
618 |
next |
|
619 |
assume "c\<noteq>0" |
|
620 |
thus ?thesis by (force dest: field_mult_cancel_right_lemma) |
|
621 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
622 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
623 |
lemma field_mult_cancel_left [simp]: |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
624 |
"(c*a = c*b) = (c = (0::'a::field) | a=b)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
625 |
by (simp add: mult_commute [of c] field_mult_cancel_right) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
626 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
627 |
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)" |
14377 | 628 |
proof |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
629 |
assume ianz: "inverse a = 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
630 |
assume "a \<noteq> 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
631 |
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
|
632 |
also have "... = 0" by (simp add: ianz) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
633 |
finally have "1 = (0::'a::field)" . |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
634 |
thus False by (simp add: eq_commute) |
14377 | 635 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
636 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
637 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
638 |
subsection{*Basic Properties of @{term inverse}*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
639 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
640 |
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
641 |
apply (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
642 |
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
|
643 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
644 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
645 |
lemma inverse_nonzero_imp_nonzero: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
646 |
"inverse a = 0 ==> a = (0::'a::field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
647 |
apply (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
648 |
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
|
649 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
650 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
651 |
lemma inverse_nonzero_iff_nonzero [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
652 |
"(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
653 |
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
|
654 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
655 |
lemma nonzero_inverse_minus_eq: |
14269 | 656 |
assumes [simp]: "a\<noteq>0" shows "inverse(-a) = -inverse(a::'a::field)" |
14377 | 657 |
proof - |
658 |
have "-a * inverse (- a) = -a * - inverse a" |
|
659 |
by simp |
|
660 |
thus ?thesis |
|
661 |
by (simp only: field_mult_cancel_left, simp) |
|
662 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
663 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
664 |
lemma inverse_minus_eq [simp]: |
14377 | 665 |
"inverse(-a) = -inverse(a::'a::{field,division_by_zero})"; |
666 |
proof cases |
|
667 |
assume "a=0" thus ?thesis by (simp add: inverse_zero) |
|
668 |
next |
|
669 |
assume "a\<noteq>0" |
|
670 |
thus ?thesis by (simp add: nonzero_inverse_minus_eq) |
|
671 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
672 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
673 |
lemma nonzero_inverse_eq_imp_eq: |
14269 | 674 |
assumes inveq: "inverse a = inverse b" |
675 |
and anz: "a \<noteq> 0" |
|
676 |
and bnz: "b \<noteq> 0" |
|
677 |
shows "a = (b::'a::field)" |
|
14377 | 678 |
proof - |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
679 |
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
|
680 |
by (simp add: inveq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
681 |
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
|
682 |
by simp |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
683 |
thus "a = b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
684 |
by (simp add: mult_assoc anz bnz) |
14377 | 685 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
686 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
687 |
lemma inverse_eq_imp_eq: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
688 |
"inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
689 |
apply (case_tac "a=0 | b=0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
690 |
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
|
691 |
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
|
692 |
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
|
693 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
694 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
695 |
lemma inverse_eq_iff_eq [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
696 |
"(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
697 |
by (force dest!: inverse_eq_imp_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
698 |
|
14270 | 699 |
lemma nonzero_inverse_inverse_eq: |
700 |
assumes [simp]: "a \<noteq> 0" shows "inverse(inverse (a::'a::field)) = a" |
|
701 |
proof - |
|
702 |
have "(inverse (inverse a) * inverse a) * a = a" |
|
703 |
by (simp add: nonzero_imp_inverse_nonzero) |
|
704 |
thus ?thesis |
|
705 |
by (simp add: mult_assoc) |
|
706 |
qed |
|
707 |
||
708 |
lemma inverse_inverse_eq [simp]: |
|
709 |
"inverse(inverse (a::'a::{field,division_by_zero})) = a" |
|
710 |
proof cases |
|
711 |
assume "a=0" thus ?thesis by simp |
|
712 |
next |
|
713 |
assume "a\<noteq>0" |
|
714 |
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) |
|
715 |
qed |
|
716 |
||
717 |
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)" |
|
718 |
proof - |
|
719 |
have "inverse 1 * 1 = (1::'a::field)" |
|
720 |
by (rule left_inverse [OF zero_neq_one [symmetric]]) |
|
721 |
thus ?thesis by simp |
|
722 |
qed |
|
723 |
||
724 |
lemma nonzero_inverse_mult_distrib: |
|
725 |
assumes anz: "a \<noteq> 0" |
|
726 |
and bnz: "b \<noteq> 0" |
|
727 |
shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)" |
|
728 |
proof - |
|
729 |
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" |
|
730 |
by (simp add: field_mult_eq_0_iff anz bnz) |
|
731 |
hence "inverse(a*b) * a = inverse(b)" |
|
732 |
by (simp add: mult_assoc bnz) |
|
733 |
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" |
|
734 |
by simp |
|
735 |
thus ?thesis |
|
736 |
by (simp add: mult_assoc anz) |
|
737 |
qed |
|
738 |
||
739 |
text{*This version builds in division by zero while also re-orienting |
|
740 |
the right-hand side.*} |
|
741 |
lemma inverse_mult_distrib [simp]: |
|
742 |
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" |
|
743 |
proof cases |
|
744 |
assume "a \<noteq> 0 & b \<noteq> 0" |
|
745 |
thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute) |
|
746 |
next |
|
747 |
assume "~ (a \<noteq> 0 & b \<noteq> 0)" |
|
748 |
thus ?thesis by force |
|
749 |
qed |
|
750 |
||
751 |
text{*There is no slick version using division by zero.*} |
|
752 |
lemma inverse_add: |
|
753 |
"[|a \<noteq> 0; b \<noteq> 0|] |
|
754 |
==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)" |
|
755 |
apply (simp add: left_distrib mult_assoc) |
|
756 |
apply (simp add: mult_commute [of "inverse a"]) |
|
757 |
apply (simp add: mult_assoc [symmetric] add_commute) |
|
758 |
done |
|
759 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
760 |
lemma inverse_divide [simp]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
761 |
"inverse (a/b) = b / (a::'a::{field,division_by_zero})" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
762 |
by (simp add: divide_inverse mult_commute) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
763 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
764 |
lemma nonzero_mult_divide_cancel_left: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
765 |
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
766 |
shows "(c*a)/(c*b) = a/(b::'a::field)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
767 |
proof - |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
768 |
have "(c*a)/(c*b) = c * a * (inverse b * inverse c)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
769 |
by (simp add: field_mult_eq_0_iff divide_inverse |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
770 |
nonzero_inverse_mult_distrib) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
771 |
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
|
772 |
by (simp only: mult_ac) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
773 |
also have "... = a * inverse b" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
774 |
by simp |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
775 |
finally show ?thesis |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
776 |
by (simp add: divide_inverse) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
777 |
qed |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
778 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
779 |
lemma mult_divide_cancel_left: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
780 |
"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
781 |
apply (case_tac "b = 0") |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
782 |
apply (simp_all add: nonzero_mult_divide_cancel_left) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
783 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
784 |
|
14321 | 785 |
lemma nonzero_mult_divide_cancel_right: |
786 |
"[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)" |
|
787 |
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) |
|
788 |
||
789 |
lemma mult_divide_cancel_right: |
|
790 |
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" |
|
791 |
apply (case_tac "b = 0") |
|
792 |
apply (simp_all add: nonzero_mult_divide_cancel_right) |
|
793 |
done |
|
794 |
||
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
795 |
(*For ExtractCommonTerm*) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
796 |
lemma mult_divide_cancel_eq_if: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
797 |
"(c*a) / (c*b) = |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
798 |
(if c=0 then 0 else a / (b::'a::{field,division_by_zero}))" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
799 |
by (simp add: mult_divide_cancel_left) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
800 |
|
14284
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
801 |
lemma divide_1 [simp]: "a/1 = (a::'a::field)" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
802 |
by (simp add: divide_inverse) |
14284
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
803 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
804 |
lemma times_divide_eq_right [simp]: "a * (b/c) = (a*b) / (c::'a::field)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
805 |
by (simp add: divide_inverse mult_assoc) |
14288 | 806 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
807 |
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
|
808 |
by (simp add: divide_inverse mult_ac) |
14288 | 809 |
|
810 |
lemma divide_divide_eq_right [simp]: |
|
811 |
"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
|
812 |
by (simp add: divide_inverse mult_ac) |
14288 | 813 |
|
814 |
lemma divide_divide_eq_left [simp]: |
|
815 |
"(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
|
816 |
by (simp add: divide_inverse mult_assoc) |
14288 | 817 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
818 |
|
14293 | 819 |
subsection {* Division and Unary Minus *} |
820 |
||
821 |
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)" |
|
822 |
by (simp add: divide_inverse minus_mult_left) |
|
823 |
||
824 |
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)" |
|
825 |
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) |
|
826 |
||
827 |
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)" |
|
828 |
by (simp add: divide_inverse nonzero_inverse_minus_eq) |
|
829 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
830 |
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
|
831 |
by (simp add: divide_inverse minus_mult_left [symmetric]) |
14293 | 832 |
|
833 |
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
|
834 |
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
|
835 |
|
14293 | 836 |
|
837 |
text{*The effect is to extract signs from divisions*} |
|
838 |
declare minus_divide_left [symmetric, simp] |
|
839 |
declare minus_divide_right [symmetric, simp] |
|
840 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
841 |
text{*Also, extract signs from products*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
842 |
declare minus_mult_left [symmetric, simp] |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
843 |
declare minus_mult_right [symmetric, simp] |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
844 |
|
14293 | 845 |
lemma minus_divide_divide [simp]: |
846 |
"(-a)/(-b) = a / (b::'a::{field,division_by_zero})" |
|
847 |
apply (case_tac "b=0", simp) |
|
848 |
apply (simp add: nonzero_minus_divide_divide) |
|
849 |
done |
|
850 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
851 |
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
852 |
by (simp add: diff_minus add_divide_distrib) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
853 |
|
14293 | 854 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
855 |
subsection {* Ordered Fields *} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
856 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
857 |
lemma positive_imp_inverse_positive: |
14269 | 858 |
assumes a_gt_0: "0 < a" shows "0 < inverse (a::'a::ordered_field)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
859 |
proof - |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
860 |
have "0 < a * inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
861 |
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
|
862 |
thus "0 < inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
863 |
by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
864 |
qed |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
865 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
866 |
lemma negative_imp_inverse_negative: |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
867 |
"a < 0 ==> inverse a < (0::'a::ordered_field)" |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
868 |
by (insert positive_imp_inverse_positive [of "-a"], |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
869 |
simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
870 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
871 |
lemma inverse_le_imp_le: |
14269 | 872 |
assumes invle: "inverse a \<le> inverse b" |
873 |
and apos: "0 < a" |
|
874 |
shows "b \<le> (a::'a::ordered_field)" |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
875 |
proof (rule classical) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
876 |
assume "~ b \<le> a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
877 |
hence "a < b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
878 |
by (simp add: linorder_not_le) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
879 |
hence bpos: "0 < b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
880 |
by (blast intro: apos order_less_trans) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
881 |
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
|
882 |
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
|
883 |
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
|
884 |
by (simp add: bpos order_less_imp_le mult_right_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
885 |
thus "b \<le> a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
886 |
by (simp add: mult_assoc apos bpos order_less_imp_not_eq2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
887 |
qed |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
888 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
889 |
lemma inverse_positive_imp_positive: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
890 |
assumes inv_gt_0: "0 < inverse a" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
891 |
and [simp]: "a \<noteq> 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
892 |
shows "0 < (a::'a::ordered_field)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
893 |
proof - |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
894 |
have "0 < inverse (inverse a)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
895 |
by (rule positive_imp_inverse_positive) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
896 |
thus "0 < a" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
897 |
by (simp add: nonzero_inverse_inverse_eq) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
898 |
qed |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
899 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
900 |
lemma inverse_positive_iff_positive [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
901 |
"(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
902 |
apply (case_tac "a = 0", simp) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
903 |
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
|
904 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
905 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
906 |
lemma inverse_negative_imp_negative: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
907 |
assumes inv_less_0: "inverse a < 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
908 |
and [simp]: "a \<noteq> 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
909 |
shows "a < (0::'a::ordered_field)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
910 |
proof - |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
911 |
have "inverse (inverse a) < 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
912 |
by (rule negative_imp_inverse_negative) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
913 |
thus "a < 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
914 |
by (simp add: nonzero_inverse_inverse_eq) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
915 |
qed |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
916 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
917 |
lemma inverse_negative_iff_negative [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
918 |
"(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
919 |
apply (case_tac "a = 0", simp) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
920 |
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
|
921 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
922 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
923 |
lemma inverse_nonnegative_iff_nonnegative [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
924 |
"(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
925 |
by (simp add: linorder_not_less [symmetric]) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
926 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
927 |
lemma inverse_nonpositive_iff_nonpositive [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
928 |
"(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
929 |
by (simp add: linorder_not_less [symmetric]) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
930 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
931 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
932 |
subsection{*Anti-Monotonicity of @{term inverse}*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
933 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
934 |
lemma less_imp_inverse_less: |
14269 | 935 |
assumes less: "a < b" |
936 |
and apos: "0 < a" |
|
937 |
shows "inverse b < inverse (a::'a::ordered_field)" |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
938 |
proof (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
939 |
assume "~ inverse b < inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
940 |
hence "inverse a \<le> inverse b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
941 |
by (simp add: linorder_not_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
942 |
hence "~ (a < b)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
943 |
by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
944 |
thus False |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
945 |
by (rule notE [OF _ less]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
946 |
qed |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
947 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
948 |
lemma inverse_less_imp_less: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
949 |
"[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
950 |
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
951 |
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
952 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
953 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
954 |
text{*Both premises are essential. Consider -1 and 1.*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
955 |
lemma inverse_less_iff_less [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
956 |
"[|0 < a; 0 < b|] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
957 |
==> (inverse a < inverse b) = (b < (a::'a::ordered_field))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
958 |
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
959 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
960 |
lemma le_imp_inverse_le: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
961 |
"[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
962 |
by (force simp add: order_le_less less_imp_inverse_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
963 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
964 |
lemma inverse_le_iff_le [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
965 |
"[|0 < a; 0 < b|] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
966 |
==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
967 |
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
968 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
969 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
970 |
text{*These results refer to both operands being negative. The opposite-sign |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
971 |
case is trivial, since inverse preserves signs.*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
972 |
lemma inverse_le_imp_le_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
973 |
"[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
974 |
apply (rule classical) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
975 |
apply (subgoal_tac "a < 0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
976 |
prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
977 |
apply (insert inverse_le_imp_le [of "-b" "-a"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
978 |
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
979 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
980 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
981 |
lemma less_imp_inverse_less_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
982 |
"[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
983 |
apply (subgoal_tac "a < 0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
984 |
prefer 2 apply (blast intro: order_less_trans) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
985 |
apply (insert less_imp_inverse_less [of "-b" "-a"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
986 |
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
987 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
988 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
989 |
lemma inverse_less_imp_less_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
990 |
"[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
991 |
apply (rule classical) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
992 |
apply (subgoal_tac "a < 0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
993 |
prefer 2 |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
994 |
apply (force simp add: linorder_not_less intro: order_le_less_trans) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
995 |
apply (insert inverse_less_imp_less [of "-b" "-a"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
996 |
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
997 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
998 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
999 |
lemma inverse_less_iff_less_neg [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1000 |
"[|a < 0; b < 0|] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1001 |
==> (inverse a < inverse b) = (b < (a::'a::ordered_field))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1002 |
apply (insert inverse_less_iff_less [of "-b" "-a"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1003 |
apply (simp del: inverse_less_iff_less |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1004 |
add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1005 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1006 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1007 |
lemma le_imp_inverse_le_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1008 |
"[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1009 |
by (force simp add: order_le_less less_imp_inverse_less_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1010 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1011 |
lemma inverse_le_iff_le_neg [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1012 |
"[|a < 0; b < 0|] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1013 |
==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1014 |
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1015 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1016 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1017 |
subsection{*Inverses and the Number One*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1018 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1019 |
lemma one_less_inverse_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1020 |
"(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1021 |
assume "0 < x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1022 |
with inverse_less_iff_less [OF zero_less_one, of x] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1023 |
show ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1024 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1025 |
assume notless: "~ (0 < x)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1026 |
have "~ (1 < inverse x)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1027 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1028 |
assume "1 < inverse x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1029 |
also with notless have "... \<le> 0" by (simp add: linorder_not_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1030 |
also have "... < 1" by (rule zero_less_one) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1031 |
finally show False by auto |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1032 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1033 |
with notless show ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1034 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1035 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1036 |
lemma inverse_eq_1_iff [simp]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1037 |
"(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1038 |
by (insert inverse_eq_iff_eq [of x 1], simp) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1039 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1040 |
lemma one_le_inverse_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1041 |
"(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1042 |
by (force simp add: order_le_less one_less_inverse_iff zero_less_one |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1043 |
eq_commute [of 1]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1044 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1045 |
lemma inverse_less_1_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1046 |
"(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1047 |
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1048 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1049 |
lemma inverse_le_1_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1050 |
"(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1051 |
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1052 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1053 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1054 |
subsection{*Division and Signs*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1055 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1056 |
lemma zero_less_divide_iff: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1057 |
"((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1058 |
by (simp add: divide_inverse zero_less_mult_iff) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1059 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1060 |
lemma divide_less_0_iff: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1061 |
"(a/b < (0::'a::{ordered_field,division_by_zero})) = |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1062 |
(0 < a & b < 0 | a < 0 & 0 < b)" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1063 |
by (simp add: divide_inverse mult_less_0_iff) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1064 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1065 |
lemma zero_le_divide_iff: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1066 |
"((0::'a::{ordered_field,division_by_zero}) \<le> a/b) = |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1067 |
(0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1068 |
by (simp add: divide_inverse zero_le_mult_iff) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1069 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1070 |
lemma divide_le_0_iff: |
14288 | 1071 |
"(a/b \<le> (0::'a::{ordered_field,division_by_zero})) = |
1072 |
(0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)" |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1073 |
by (simp add: divide_inverse mult_le_0_iff) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1074 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1075 |
lemma divide_eq_0_iff [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1076 |
"(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1077 |
by (simp add: divide_inverse field_mult_eq_0_iff) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1078 |
|
14288 | 1079 |
|
1080 |
subsection{*Simplification of Inequalities Involving Literal Divisors*} |
|
1081 |
||
1082 |
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)" |
|
1083 |
proof - |
|
1084 |
assume less: "0<c" |
|
1085 |
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" |
|
1086 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1087 |
also have "... = (a*c \<le> b)" |
|
1088 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1089 |
finally show ?thesis . |
|
1090 |
qed |
|
1091 |
||
1092 |
||
1093 |
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)" |
|
1094 |
proof - |
|
1095 |
assume less: "c<0" |
|
1096 |
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" |
|
1097 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1098 |
also have "... = (b \<le> a*c)" |
|
1099 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1100 |
finally show ?thesis . |
|
1101 |
qed |
|
1102 |
||
1103 |
lemma le_divide_eq: |
|
1104 |
"(a \<le> b/c) = |
|
1105 |
(if 0 < c then a*c \<le> b |
|
1106 |
else if c < 0 then b \<le> a*c |
|
1107 |
else a \<le> (0::'a::{ordered_field,division_by_zero}))" |
|
1108 |
apply (case_tac "c=0", simp) |
|
1109 |
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) |
|
1110 |
done |
|
1111 |
||
1112 |
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)" |
|
1113 |
proof - |
|
1114 |
assume less: "0<c" |
|
1115 |
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" |
|
1116 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1117 |
also have "... = (b \<le> a*c)" |
|
1118 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1119 |
finally show ?thesis . |
|
1120 |
qed |
|
1121 |
||
1122 |
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)" |
|
1123 |
proof - |
|
1124 |
assume less: "c<0" |
|
1125 |
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" |
|
1126 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1127 |
also have "... = (a*c \<le> b)" |
|
1128 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1129 |
finally show ?thesis . |
|
1130 |
qed |
|
1131 |
||
1132 |
lemma divide_le_eq: |
|
1133 |
"(b/c \<le> a) = |
|
1134 |
(if 0 < c then b \<le> a*c |
|
1135 |
else if c < 0 then a*c \<le> b |
|
1136 |
else 0 \<le> (a::'a::{ordered_field,division_by_zero}))" |
|
1137 |
apply (case_tac "c=0", simp) |
|
1138 |
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) |
|
1139 |
done |
|
1140 |
||
1141 |
||
1142 |
lemma pos_less_divide_eq: |
|
1143 |
"0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)" |
|
1144 |
proof - |
|
1145 |
assume less: "0<c" |
|
1146 |
hence "(a < b/c) = (a*c < (b/c)*c)" |
|
1147 |
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) |
|
1148 |
also have "... = (a*c < b)" |
|
1149 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1150 |
finally show ?thesis . |
|
1151 |
qed |
|
1152 |
||
1153 |
lemma neg_less_divide_eq: |
|
1154 |
"c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)" |
|
1155 |
proof - |
|
1156 |
assume less: "c<0" |
|
1157 |
hence "(a < b/c) = ((b/c)*c < a*c)" |
|
1158 |
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) |
|
1159 |
also have "... = (b < a*c)" |
|
1160 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1161 |
finally show ?thesis . |
|
1162 |
qed |
|
1163 |
||
1164 |
lemma less_divide_eq: |
|
1165 |
"(a < b/c) = |
|
1166 |
(if 0 < c then a*c < b |
|
1167 |
else if c < 0 then b < a*c |
|
1168 |
else a < (0::'a::{ordered_field,division_by_zero}))" |
|
1169 |
apply (case_tac "c=0", simp) |
|
1170 |
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) |
|
1171 |
done |
|
1172 |
||
1173 |
lemma pos_divide_less_eq: |
|
1174 |
"0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)" |
|
1175 |
proof - |
|
1176 |
assume less: "0<c" |
|
1177 |
hence "(b/c < a) = ((b/c)*c < a*c)" |
|
1178 |
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) |
|
1179 |
also have "... = (b < a*c)" |
|
1180 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1181 |
finally show ?thesis . |
|
1182 |
qed |
|
1183 |
||
1184 |
lemma neg_divide_less_eq: |
|
1185 |
"c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)" |
|
1186 |
proof - |
|
1187 |
assume less: "c<0" |
|
1188 |
hence "(b/c < a) = (a*c < (b/c)*c)" |
|
1189 |
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) |
|
1190 |
also have "... = (a*c < b)" |
|
1191 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1192 |
finally show ?thesis . |
|
1193 |
qed |
|
1194 |
||
1195 |
lemma divide_less_eq: |
|
1196 |
"(b/c < a) = |
|
1197 |
(if 0 < c then b < a*c |
|
1198 |
else if c < 0 then a*c < b |
|
1199 |
else 0 < (a::'a::{ordered_field,division_by_zero}))" |
|
1200 |
apply (case_tac "c=0", simp) |
|
1201 |
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) |
|
1202 |
done |
|
1203 |
||
1204 |
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)" |
|
1205 |
proof - |
|
1206 |
assume [simp]: "c\<noteq>0" |
|
1207 |
have "(a = b/c) = (a*c = (b/c)*c)" |
|
1208 |
by (simp add: field_mult_cancel_right) |
|
1209 |
also have "... = (a*c = b)" |
|
1210 |
by (simp add: divide_inverse mult_assoc) |
|
1211 |
finally show ?thesis . |
|
1212 |
qed |
|
1213 |
||
1214 |
lemma eq_divide_eq: |
|
1215 |
"((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)" |
|
1216 |
by (simp add: nonzero_eq_divide_eq) |
|
1217 |
||
1218 |
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)" |
|
1219 |
proof - |
|
1220 |
assume [simp]: "c\<noteq>0" |
|
1221 |
have "(b/c = a) = ((b/c)*c = a*c)" |
|
1222 |
by (simp add: field_mult_cancel_right) |
|
1223 |
also have "... = (b = a*c)" |
|
1224 |
by (simp add: divide_inverse mult_assoc) |
|
1225 |
finally show ?thesis . |
|
1226 |
qed |
|
1227 |
||
1228 |
lemma divide_eq_eq: |
|
1229 |
"(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)" |
|
1230 |
by (force simp add: nonzero_divide_eq_eq) |
|
1231 |
||
1232 |
subsection{*Cancellation Laws for Division*} |
|
1233 |
||
1234 |
lemma divide_cancel_right [simp]: |
|
1235 |
"(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))" |
|
1236 |
apply (case_tac "c=0", simp) |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1237 |
apply (simp add: divide_inverse field_mult_cancel_right) |
14288 | 1238 |
done |
1239 |
||
1240 |
lemma divide_cancel_left [simp]: |
|
1241 |
"(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" |
|
1242 |
apply (case_tac "c=0", simp) |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1243 |
apply (simp add: divide_inverse field_mult_cancel_left) |
14288 | 1244 |
done |
1245 |
||
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1246 |
subsection {* Division and the Number One *} |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1247 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1248 |
text{*Simplify expressions equated with 1*} |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1249 |
lemma divide_eq_1_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1250 |
"(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1251 |
apply (case_tac "b=0", simp) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1252 |
apply (simp add: right_inverse_eq) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1253 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1254 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1255 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1256 |
lemma one_eq_divide_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1257 |
"(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1258 |
by (simp add: eq_commute [of 1]) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1259 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1260 |
lemma zero_eq_1_divide_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1261 |
"((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)" |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1262 |
apply (case_tac "a=0", simp) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1263 |
apply (auto simp add: nonzero_eq_divide_eq) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1264 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1265 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1266 |
lemma one_divide_eq_0_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1267 |
"(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)" |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1268 |
apply (case_tac "a=0", simp) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1269 |
apply (insert zero_neq_one [THEN not_sym]) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1270 |
apply (auto simp add: nonzero_divide_eq_eq) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1271 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1272 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1273 |
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*} |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1274 |
declare zero_less_divide_iff [of "1", simp] |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1275 |
declare divide_less_0_iff [of "1", simp] |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1276 |
declare zero_le_divide_iff [of "1", simp] |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1277 |
declare divide_le_0_iff [of "1", simp] |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1278 |
|
14288 | 1279 |
|
14293 | 1280 |
subsection {* Ordering Rules for Division *} |
1281 |
||
1282 |
lemma divide_strict_right_mono: |
|
1283 |
"[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)" |
|
1284 |
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono |
|
1285 |
positive_imp_inverse_positive) |
|
1286 |
||
1287 |
lemma divide_right_mono: |
|
1288 |
"[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})" |
|
1289 |
by (force simp add: divide_strict_right_mono order_le_less) |
|
1290 |
||
1291 |
||
1292 |
text{*The last premise ensures that @{term a} and @{term b} |
|
1293 |
have the same sign*} |
|
1294 |
lemma divide_strict_left_mono: |
|
1295 |
"[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)" |
|
1296 |
by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono |
|
1297 |
order_less_imp_not_eq order_less_imp_not_eq2 |
|
1298 |
less_imp_inverse_less less_imp_inverse_less_neg) |
|
1299 |
||
1300 |
lemma divide_left_mono: |
|
1301 |
"[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)" |
|
1302 |
apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") |
|
1303 |
prefer 2 |
|
1304 |
apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) |
|
1305 |
apply (case_tac "c=0", simp add: divide_inverse) |
|
1306 |
apply (force simp add: divide_strict_left_mono order_le_less) |
|
1307 |
done |
|
1308 |
||
1309 |
lemma divide_strict_left_mono_neg: |
|
1310 |
"[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)" |
|
1311 |
apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") |
|
1312 |
prefer 2 |
|
1313 |
apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) |
|
1314 |
apply (drule divide_strict_left_mono [of _ _ "-c"]) |
|
1315 |
apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric]) |
|
1316 |
done |
|
1317 |
||
1318 |
lemma divide_strict_right_mono_neg: |
|
1319 |
"[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)" |
|
1320 |
apply (drule divide_strict_right_mono [of _ _ "-c"], simp) |
|
1321 |
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) |
|
1322 |
done |
|
1323 |
||
1324 |
||
1325 |
subsection {* Ordered Fields are Dense *} |
|
1326 |
||
14738 | 1327 |
lemma less_add_one: "a < (a+1::'a::ordered_semidom)" |
14293 | 1328 |
proof - |
14738 | 1329 |
have "a+0 < (a+1::'a::ordered_semidom)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1330 |
by (blast intro: zero_less_one add_strict_left_mono) |
14293 | 1331 |
thus ?thesis by simp |
1332 |
qed |
|
1333 |
||
14738 | 1334 |
lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1335 |
by (blast intro: order_less_trans zero_less_one less_add_one) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1336 |
|
14293 | 1337 |
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)" |
1338 |
by (simp add: zero_less_two pos_less_divide_eq right_distrib) |
|
1339 |
||
1340 |
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b" |
|
1341 |
by (simp add: zero_less_two pos_divide_less_eq right_distrib) |
|
1342 |
||
1343 |
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b" |
|
1344 |
by (blast intro!: less_half_sum gt_half_sum) |
|
1345 |
||
1346 |
subsection {* Absolute Value *} |
|
1347 |
||
14738 | 1348 |
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)" |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1349 |
by (simp add: abs_if zero_less_one [THEN order_less_not_sym]) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1350 |
|
14738 | 1351 |
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" |
1352 |
proof - |
|
1353 |
let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b" |
|
1354 |
let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b" |
|
1355 |
have a: "(abs a) * (abs b) = ?x" |
|
1356 |
by (simp only: abs_prts[of a] abs_prts[of b] ring_eq_simps) |
|
1357 |
{ |
|
1358 |
fix u v :: 'a |
|
1359 |
have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> u * v = ?y" |
|
1360 |
apply (subst prts[of u], subst prts[of v]) |
|
1361 |
apply (simp add: left_distrib right_distrib add_ac) |
|
1362 |
done |
|
1363 |
} |
|
1364 |
note b = this[OF refl[of a] refl[of b]] |
|
1365 |
note addm = add_mono[of "0::'a" _ "0::'a", simplified] |
|
1366 |
note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified] |
|
1367 |
have xy: "- ?x <= ?y" |
|
1368 |
apply (simp add: compare_rls) |
|
1369 |
apply (rule add_le_imp_le_left[of "-(pprt a * nprt b + nprt a * pprt b)"]) |
|
1370 |
apply (simp add: add_ac) |
|
1371 |
proof - |
|
1372 |
let ?r = "nprt a * nprt b +(nprt a * nprt b + (nprt a * pprt b + (pprt a * nprt b + (pprt a * pprt b + (pprt a * pprt b + |
|
1373 |
(- (nprt a * pprt b) + - (pprt a * nprt b)))))))" |
|
1374 |
let ?rr = "nprt a * nprt b + nprt a * nprt b + ((nprt a * pprt b) + (- (nprt a * pprt b))) + ((pprt a * nprt b) + - (pprt a * nprt b)) |
|
1375 |
+ pprt a * pprt b + pprt a * pprt b" |
|
1376 |
have a:"?r = ?rr" by (simp only: add_ac) |
|
1377 |
have "0 <= ?rr" |
|
1378 |
apply (simp) |
|
1379 |
apply (rule addm)+ |
|
1380 |
apply (simp_all add: mult_neg_le mult_pos_le) |
|
1381 |
done |
|
1382 |
with a show "0 <= ?r" by simp |
|
1383 |
qed |
|
1384 |
have yx: "?y <= ?x" |
|
1385 |
apply (simp add: add_ac) |
|
1386 |
apply (simp add: compare_rls) |
|
1387 |
apply (rule add_le_imp_le_right[of _ "-(pprt a * pprt b)"]) |
|
1388 |
apply (simp add: add_ac) |
|
1389 |
apply (rule addm2, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)+ |
|
1390 |
done |
|
1391 |
have i1: "a*b <= abs a * abs b" by (simp only: a b yx) |
|
1392 |
have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy) |
|
1393 |
show ?thesis |
|
1394 |
apply (rule abs_leI) |
|
1395 |
apply (simp add: i1) |
|
1396 |
apply (simp add: i2[simplified minus_le_iff]) |
|
1397 |
done |
|
1398 |
qed |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1399 |
|
14738 | 1400 |
lemma abs_eq_mult: |
1401 |
assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)" |
|
1402 |
shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)" |
|
1403 |
proof - |
|
1404 |
have s: "(0 <= a*b) | (a*b <= 0)" |
|
1405 |
apply (auto) |
|
1406 |
apply (rule_tac split_mult_pos_le) |
|
1407 |
apply (rule_tac contrapos_np[of "a*b <= 0"]) |
|
1408 |
apply (simp) |
|
1409 |
apply (rule_tac split_mult_neg_le) |
|
1410 |
apply (insert prems) |
|
1411 |
apply (blast) |
|
1412 |
done |
|
1413 |
have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)" |
|
1414 |
by (simp add: prts[symmetric]) |
|
1415 |
show ?thesis |
|
1416 |
proof cases |
|
1417 |
assume "0 <= a * b" |
|
1418 |
then show ?thesis |
|
1419 |
apply (simp_all add: mulprts abs_prts) |
|
1420 |
apply (insert prems) |
|
1421 |
apply (auto simp add: ring_eq_simps iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt] |
|
1422 |
iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id] order_antisym mult_pos_neg_le[of a b] mult_pos_neg2_le[of b a]) |
|
1423 |
done |
|
1424 |
next |
|
1425 |
assume "~(0 <= a*b)" |
|
1426 |
with s have "a*b <= 0" by simp |
|
1427 |
then show ?thesis |
|
1428 |
apply (simp_all add: mulprts abs_prts) |
|
1429 |
apply (insert prems) |
|
1430 |
apply (auto simp add: ring_eq_simps iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt] |
|
1431 |
iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id] order_antisym mult_pos_le[of a b] mult_neg_le[of a b]) |
|
1432 |
done |
|
1433 |
qed |
|
1434 |
qed |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1435 |
|
14738 | 1436 |
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" |
1437 |
by (simp add: abs_eq_mult linorder_linear) |
|
14293 | 1438 |
|
14738 | 1439 |
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)" |
1440 |
by (simp add: abs_if) |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1441 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1442 |
lemma nonzero_abs_inverse: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1443 |
"a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1444 |
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1445 |
negative_imp_inverse_negative) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1446 |
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1447 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1448 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1449 |
lemma abs_inverse [simp]: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1450 |
"abs (inverse (a::'a::{ordered_field,division_by_zero})) = |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1451 |
inverse (abs a)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1452 |
apply (case_tac "a=0", simp) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1453 |
apply (simp add: nonzero_abs_inverse) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1454 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1455 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1456 |
lemma nonzero_abs_divide: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1457 |
"b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1458 |
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1459 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1460 |
lemma abs_divide: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1461 |
"abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1462 |
apply (case_tac "b=0", simp) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1463 |
apply (simp add: nonzero_abs_divide) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1464 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1465 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1466 |
lemma abs_mult_less: |
14738 | 1467 |
"[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)" |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1468 |
proof - |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1469 |
assume ac: "abs a < c" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1470 |
hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1471 |
assume "abs b < d" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1472 |
thus ?thesis by (simp add: ac cpos mult_strict_mono) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1473 |
qed |
14293 | 1474 |
|
14738 | 1475 |
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))" |
1476 |
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff) |
|
1477 |
||
1478 |
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))" |
|
1479 |
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff) |
|
1480 |
||
1481 |
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" |
|
1482 |
apply (simp add: order_less_le abs_le_iff) |
|
1483 |
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff) |
|
1484 |
apply (simp add: le_minus_self_iff linorder_neq_iff) |
|
1485 |
done |
|
1486 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1487 |
text{*Moving this up spoils many proofs using @{text mult_le_cancel_right}*} |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1488 |
declare times_divide_eq_left [simp] |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1489 |
|
14738 | 1490 |
ML {* |
14334 | 1491 |
val left_distrib = thm "left_distrib"; |
14738 | 1492 |
val right_distrib = thm "right_distrib"; |
1493 |
val mult_commute = thm "mult_commute"; |
|
1494 |
val distrib = thm "distrib"; |
|
1495 |
val zero_neq_one = thm "zero_neq_one"; |
|
1496 |
val no_zero_divisors = thm "no_zero_divisors"; |
|
14331 | 1497 |
val left_inverse = thm "left_inverse"; |
14738 | 1498 |
val divide_inverse = thm "divide_inverse"; |
1499 |
val mult_zero_left = thm "mult_zero_left"; |
|
1500 |
val mult_zero_right = thm "mult_zero_right"; |
|
1501 |
val field_mult_eq_0_iff = thm "field_mult_eq_0_iff"; |
|
1502 |
val inverse_zero = thm "inverse_zero"; |
|
1503 |
val ring_distrib = thms "ring_distrib"; |
|
1504 |
val combine_common_factor = thm "combine_common_factor"; |
|
1505 |
val minus_mult_left = thm "minus_mult_left"; |
|
1506 |
val minus_mult_right = thm "minus_mult_right"; |
|
1507 |
val minus_mult_minus = thm "minus_mult_minus"; |
|
1508 |
val minus_mult_commute = thm "minus_mult_commute"; |
|
1509 |
val right_diff_distrib = thm "right_diff_distrib"; |
|
1510 |
val left_diff_distrib = thm "left_diff_distrib"; |
|
1511 |
val mult_left_mono = thm "mult_left_mono"; |
|
1512 |
val mult_right_mono = thm "mult_right_mono"; |
|
1513 |
val mult_strict_left_mono = thm "mult_strict_left_mono"; |
|
1514 |
val mult_strict_right_mono = thm "mult_strict_right_mono"; |
|
1515 |
val mult_mono = thm "mult_mono"; |
|
1516 |
val mult_strict_mono = thm "mult_strict_mono"; |
|
1517 |
val abs_if = thm "abs_if"; |
|
1518 |
val zero_less_one = thm "zero_less_one"; |
|
1519 |
val eq_add_iff1 = thm "eq_add_iff1"; |
|
1520 |
val eq_add_iff2 = thm "eq_add_iff2"; |
|
1521 |
val less_add_iff1 = thm "less_add_iff1"; |
|
1522 |
val less_add_iff2 = thm "less_add_iff2"; |
|
1523 |
val le_add_iff1 = thm "le_add_iff1"; |
|
1524 |
val le_add_iff2 = thm "le_add_iff2"; |
|
1525 |
val mult_left_le_imp_le = thm "mult_left_le_imp_le"; |
|
1526 |
val mult_right_le_imp_le = thm "mult_right_le_imp_le"; |
|
1527 |
val mult_left_less_imp_less = thm "mult_left_less_imp_less"; |
|
1528 |
val mult_right_less_imp_less = thm "mult_right_less_imp_less"; |
|
1529 |
val mult_strict_left_mono_neg = thm "mult_strict_left_mono_neg"; |
|
1530 |
val mult_left_mono_neg = thm "mult_left_mono_neg"; |
|
1531 |
val mult_strict_right_mono_neg = thm "mult_strict_right_mono_neg"; |
|
1532 |
val mult_right_mono_neg = thm "mult_right_mono_neg"; |
|
1533 |
val mult_pos = thm "mult_pos"; |
|
1534 |
val mult_pos_le = thm "mult_pos_le"; |
|
1535 |
val mult_pos_neg = thm "mult_pos_neg"; |
|
1536 |
val mult_pos_neg_le = thm "mult_pos_neg_le"; |
|
1537 |
val mult_pos_neg2 = thm "mult_pos_neg2"; |
|
1538 |
val mult_pos_neg2_le = thm "mult_pos_neg2_le"; |
|
1539 |
val mult_neg = thm "mult_neg"; |
|
1540 |
val mult_neg_le = thm "mult_neg_le"; |
|
1541 |
val zero_less_mult_pos = thm "zero_less_mult_pos"; |
|
1542 |
val zero_less_mult_pos2 = thm "zero_less_mult_pos2"; |
|
1543 |
val zero_less_mult_iff = thm "zero_less_mult_iff"; |
|
1544 |
val mult_eq_0_iff = thm "mult_eq_0_iff"; |
|
1545 |
val zero_le_mult_iff = thm "zero_le_mult_iff"; |
|
1546 |
val mult_less_0_iff = thm "mult_less_0_iff"; |
|
1547 |
val mult_le_0_iff = thm "mult_le_0_iff"; |
|
1548 |
val split_mult_pos_le = thm "split_mult_pos_le"; |
|
1549 |
val split_mult_neg_le = thm "split_mult_neg_le"; |
|
1550 |
val zero_le_square = thm "zero_le_square"; |
|
1551 |
val zero_le_one = thm "zero_le_one"; |
|
1552 |
val not_one_le_zero = thm "not_one_le_zero"; |
|
1553 |
val not_one_less_zero = thm "not_one_less_zero"; |
|
1554 |
val mult_left_mono_neg = thm "mult_left_mono_neg"; |
|
1555 |
val mult_right_mono_neg = thm "mult_right_mono_neg"; |
|
1556 |
val mult_strict_mono = thm "mult_strict_mono"; |
|
1557 |
val mult_strict_mono' = thm "mult_strict_mono'"; |
|
1558 |
val mult_mono = thm "mult_mono"; |
|
1559 |
val less_1_mult = thm "less_1_mult"; |
|
1560 |
val mult_less_cancel_right = thm "mult_less_cancel_right"; |
|
1561 |
val mult_less_cancel_left = thm "mult_less_cancel_left"; |
|
1562 |
val mult_le_cancel_right = thm "mult_le_cancel_right"; |
|
1563 |
val mult_le_cancel_left = thm "mult_le_cancel_left"; |
|
1564 |
val mult_less_imp_less_left = thm "mult_less_imp_less_left"; |
|
1565 |
val mult_less_imp_less_right = thm "mult_less_imp_less_right"; |
|
1566 |
val mult_cancel_right = thm "mult_cancel_right"; |
|
1567 |
val mult_cancel_left = thm "mult_cancel_left"; |
|
1568 |
val ring_eq_simps = thms "ring_eq_simps"; |
|
1569 |
val right_inverse = thm "right_inverse"; |
|
1570 |
val right_inverse_eq = thm "right_inverse_eq"; |
|
1571 |
val nonzero_inverse_eq_divide = thm "nonzero_inverse_eq_divide"; |
|
1572 |
val divide_self = thm "divide_self"; |
|
1573 |
val divide_zero = thm "divide_zero"; |
|
1574 |
val divide_zero_left = thm "divide_zero_left"; |
|
1575 |
val inverse_eq_divide = thm "inverse_eq_divide"; |
|
1576 |
val add_divide_distrib = thm "add_divide_distrib"; |
|
1577 |
val field_mult_eq_0_iff = thm "field_mult_eq_0_iff"; |
|
1578 |
val field_mult_cancel_right_lemma = thm "field_mult_cancel_right_lemma"; |
|
1579 |
val field_mult_cancel_right = thm "field_mult_cancel_right"; |
|
1580 |
val field_mult_cancel_left = thm "field_mult_cancel_left"; |
|
1581 |
val nonzero_imp_inverse_nonzero = thm "nonzero_imp_inverse_nonzero"; |
|
1582 |
val inverse_zero_imp_zero = thm "inverse_zero_imp_zero"; |
|
1583 |
val inverse_nonzero_imp_nonzero = thm "inverse_nonzero_imp_nonzero"; |
|
1584 |
val inverse_nonzero_iff_nonzero = thm "inverse_nonzero_iff_nonzero"; |
|
1585 |
val nonzero_inverse_minus_eq = thm "nonzero_inverse_minus_eq"; |
|
1586 |
val inverse_minus_eq = thm "inverse_minus_eq"; |
|
1587 |
val nonzero_inverse_eq_imp_eq = thm "nonzero_inverse_eq_imp_eq"; |
|
1588 |
val inverse_eq_imp_eq = thm "inverse_eq_imp_eq"; |
|
1589 |
val inverse_eq_iff_eq = thm "inverse_eq_iff_eq"; |
|
1590 |
val nonzero_inverse_inverse_eq = thm "nonzero_inverse_inverse_eq"; |
|
1591 |
val inverse_inverse_eq = thm "inverse_inverse_eq"; |
|
1592 |
val inverse_1 = thm "inverse_1"; |
|
1593 |
val nonzero_inverse_mult_distrib = thm "nonzero_inverse_mult_distrib"; |
|
1594 |
val inverse_mult_distrib = thm "inverse_mult_distrib"; |
|
1595 |
val inverse_add = thm "inverse_add"; |
|
1596 |
val inverse_divide = thm "inverse_divide"; |
|
1597 |
val nonzero_mult_divide_cancel_left = thm "nonzero_mult_divide_cancel_left"; |
|
1598 |
val mult_divide_cancel_left = thm "mult_divide_cancel_left"; |
|
1599 |
val nonzero_mult_divide_cancel_right = thm "nonzero_mult_divide_cancel_right"; |
|
1600 |
val mult_divide_cancel_right = thm "mult_divide_cancel_right"; |
|
1601 |
val mult_divide_cancel_eq_if = thm "mult_divide_cancel_eq_if"; |
|
1602 |
val divide_1 = thm "divide_1"; |
|
1603 |
val times_divide_eq_right = thm "times_divide_eq_right"; |
|
1604 |
val times_divide_eq_left = thm "times_divide_eq_left"; |
|
1605 |
val divide_divide_eq_right = thm "divide_divide_eq_right"; |
|
1606 |
val divide_divide_eq_left = thm "divide_divide_eq_left"; |
|
1607 |
val nonzero_minus_divide_left = thm "nonzero_minus_divide_left"; |
|
1608 |
val nonzero_minus_divide_right = thm "nonzero_minus_divide_right"; |
|
1609 |
val nonzero_minus_divide_divide = thm "nonzero_minus_divide_divide"; |
|
1610 |
val minus_divide_left = thm "minus_divide_left"; |
|
1611 |
val minus_divide_right = thm "minus_divide_right"; |
|
1612 |
val minus_divide_divide = thm "minus_divide_divide"; |
|
1613 |
val diff_divide_distrib = thm "diff_divide_distrib"; |
|
1614 |
val positive_imp_inverse_positive = thm "positive_imp_inverse_positive"; |
|
1615 |
val negative_imp_inverse_negative = thm "negative_imp_inverse_negative"; |
|
1616 |
val inverse_le_imp_le = thm "inverse_le_imp_le"; |
|
1617 |
val inverse_positive_imp_positive = thm "inverse_positive_imp_positive"; |
|
1618 |
val inverse_positive_iff_positive = thm "inverse_positive_iff_positive"; |
|
1619 |
val inverse_negative_imp_negative = thm "inverse_negative_imp_negative"; |
|
1620 |
val inverse_negative_iff_negative = thm "inverse_negative_iff_negative"; |
|
1621 |
val inverse_nonnegative_iff_nonnegative = thm "inverse_nonnegative_iff_nonnegative"; |
|
1622 |
val inverse_nonpositive_iff_nonpositive = thm "inverse_nonpositive_iff_nonpositive"; |
|
1623 |
val less_imp_inverse_less = thm "less_imp_inverse_less"; |
|
1624 |
val inverse_less_imp_less = thm "inverse_less_imp_less"; |
|
1625 |
val inverse_less_iff_less = thm "inverse_less_iff_less"; |
|
1626 |
val le_imp_inverse_le = thm "le_imp_inverse_le"; |
|
1627 |
val inverse_le_iff_le = thm "inverse_le_iff_le"; |
|
1628 |
val inverse_le_imp_le_neg = thm "inverse_le_imp_le_neg"; |
|
1629 |
val less_imp_inverse_less_neg = thm "less_imp_inverse_less_neg"; |
|
1630 |
val inverse_less_imp_less_neg = thm "inverse_less_imp_less_neg"; |
|
1631 |
val inverse_less_iff_less_neg = thm "inverse_less_iff_less_neg"; |
|
1632 |
val le_imp_inverse_le_neg = thm "le_imp_inverse_le_neg"; |
|
1633 |
val inverse_le_iff_le_neg = thm "inverse_le_iff_le_neg"; |
|
1634 |
val one_less_inverse_iff = thm "one_less_inverse_iff"; |
|
1635 |
val inverse_eq_1_iff = thm "inverse_eq_1_iff"; |
|
1636 |
val one_le_inverse_iff = thm "one_le_inverse_iff"; |
|
1637 |
val inverse_less_1_iff = thm "inverse_less_1_iff"; |
|
1638 |
val inverse_le_1_iff = thm "inverse_le_1_iff"; |
|
1639 |
val zero_less_divide_iff = thm "zero_less_divide_iff"; |
|
1640 |
val divide_less_0_iff = thm "divide_less_0_iff"; |
|
1641 |
val zero_le_divide_iff = thm "zero_le_divide_iff"; |
|
1642 |
val divide_le_0_iff = thm "divide_le_0_iff"; |
|
1643 |
val divide_eq_0_iff = thm "divide_eq_0_iff"; |
|
1644 |
val pos_le_divide_eq = thm "pos_le_divide_eq"; |
|
1645 |
val neg_le_divide_eq = thm "neg_le_divide_eq"; |
|
1646 |
val le_divide_eq = thm "le_divide_eq"; |
|
1647 |
val pos_divide_le_eq = thm "pos_divide_le_eq"; |
|
1648 |
val neg_divide_le_eq = thm "neg_divide_le_eq"; |
|
1649 |
val divide_le_eq = thm "divide_le_eq"; |
|
1650 |
val pos_less_divide_eq = thm "pos_less_divide_eq"; |
|
1651 |
val neg_less_divide_eq = thm "neg_less_divide_eq"; |
|
1652 |
val less_divide_eq = thm "less_divide_eq"; |
|
1653 |
val pos_divide_less_eq = thm "pos_divide_less_eq"; |
|
1654 |
val neg_divide_less_eq = thm "neg_divide_less_eq"; |
|
1655 |
val divide_less_eq = thm "divide_less_eq"; |
|
1656 |
val nonzero_eq_divide_eq = thm "nonzero_eq_divide_eq"; |
|
1657 |
val eq_divide_eq = thm "eq_divide_eq"; |
|
1658 |
val nonzero_divide_eq_eq = thm "nonzero_divide_eq_eq"; |
|
1659 |
val divide_eq_eq = thm "divide_eq_eq"; |
|
1660 |
val divide_cancel_right = thm "divide_cancel_right"; |
|
1661 |
val divide_cancel_left = thm "divide_cancel_left"; |
|
1662 |
val divide_eq_1_iff = thm "divide_eq_1_iff"; |
|
1663 |
val one_eq_divide_iff = thm "one_eq_divide_iff"; |
|
1664 |
val zero_eq_1_divide_iff = thm "zero_eq_1_divide_iff"; |
|
1665 |
val one_divide_eq_0_iff = thm "one_divide_eq_0_iff"; |
|
1666 |
val divide_strict_right_mono = thm "divide_strict_right_mono"; |
|
1667 |
val divide_right_mono = thm "divide_right_mono"; |
|
1668 |
val divide_strict_left_mono = thm "divide_strict_left_mono"; |
|
1669 |
val divide_left_mono = thm "divide_left_mono"; |
|
1670 |
val divide_strict_left_mono_neg = thm "divide_strict_left_mono_neg"; |
|
1671 |
val divide_strict_right_mono_neg = thm "divide_strict_right_mono_neg"; |
|
1672 |
val less_add_one = thm "less_add_one"; |
|
1673 |
val zero_less_two = thm "zero_less_two"; |
|
1674 |
val less_half_sum = thm "less_half_sum"; |
|
1675 |
val gt_half_sum = thm "gt_half_sum"; |
|
1676 |
val dense = thm "dense"; |
|
1677 |
val abs_one = thm "abs_one"; |
|
1678 |
val abs_le_mult = thm "abs_le_mult"; |
|
1679 |
val abs_eq_mult = thm "abs_eq_mult"; |
|
1680 |
val abs_mult = thm "abs_mult"; |
|
1681 |
val abs_mult_self = thm "abs_mult_self"; |
|
1682 |
val nonzero_abs_inverse = thm "nonzero_abs_inverse"; |
|
1683 |
val abs_inverse = thm "abs_inverse"; |
|
1684 |
val nonzero_abs_divide = thm "nonzero_abs_divide"; |
|
1685 |
val abs_divide = thm "abs_divide"; |
|
1686 |
val abs_mult_less = thm "abs_mult_less"; |
|
1687 |
val eq_minus_self_iff = thm "eq_minus_self_iff"; |
|
1688 |
val less_minus_self_iff = thm "less_minus_self_iff"; |
|
1689 |
val abs_less_iff = thm "abs_less_iff"; |
|
14331 | 1690 |
*} |
1691 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1692 |
end |