author | nipkow |
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permissions | -rw-r--r-- |
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(* Title: HOL/Ring_and_Field.thy |
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ID: $Id$ |
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Author: Gertrud Bauer, Steven Obua, Lawrence C Paulson and Markus Wenzel |
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*) |
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header {* (Ordered) Rings and Fields *} |
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theory Ring_and_Field |
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imports OrderedGroup |
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begin |
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text {* |
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The theory of partially ordered rings is taken from the books: |
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\begin{itemize} |
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\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 |
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\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 |
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\end{itemize} |
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Most of the used notions can also be looked up in |
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\begin{itemize} |
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\item \url{http://www.mathworld.com} by Eric Weisstein et. al. |
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\item \emph{Algebra I} by van der Waerden, Springer. |
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\end{itemize} |
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*} |
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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 semiring_0_cancel \<subseteq> semiring_0, cancel_ab_semigroup_add |
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||
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axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult |
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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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||
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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 comm_semiring_0_cancel \<subseteq> comm_semiring_0, cancel_ab_semigroup_add |
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||
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instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel .. |
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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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||
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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 semiring_1_cancel \<subseteq> semiring_1, cancel_ab_semigroup_add |
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instance semiring_1_cancel \<subseteq> semiring_0_cancel .. |
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axclass comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *) |
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instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel .. |
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instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel .. |
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axclass ring \<subseteq> semiring, ab_group_add |
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instance ring \<subseteq> semiring_0_cancel .. |
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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_cancel .. |
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axclass ring_1 \<subseteq> ring, semiring_1 |
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instance ring_1 \<subseteq> semiring_1_cancel .. |
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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)" |
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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)" |
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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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instance pordered_cancel_semiring \<subseteq> semiring_0_cancel .. |
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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> semiring_0_cancel .. |
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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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instance lordered_ring \<subseteq> lordered_ab_group_meet .. |
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instance lordered_ring \<subseteq> lordered_ab_group_join .. |
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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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|
14738 | 246 |
axclass ordered_field \<subseteq> field, ordered_idom |
14272
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Tidying of the integer development; towards removing the
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diff
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|
247 |
|
15923 | 248 |
lemmas linorder_neqE_ordered_idom = |
249 |
linorder_neqE[where 'a = "?'b::ordered_idom"] |
|
250 |
||
14272
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parents:
14270
diff
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|
251 |
lemma eq_add_iff1: |
5efbb548107d
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diff
changeset
|
252 |
"(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))" |
14738 | 253 |
apply (simp add: diff_minus left_distrib) |
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) |
14738 | 255 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
14272
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 eq_add_iff2: |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
259 |
"(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
|
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 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
264 |
lemma less_add_iff1: |
14738 | 265 |
"(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
|
266 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
267 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
268 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
269 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
270 |
lemma less_add_iff2: |
14738 | 271 |
"(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
|
272 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
273 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
274 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
275 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
276 |
lemma le_add_iff1: |
14738 | 277 |
"(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
|
278 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
279 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
280 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
281 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
282 |
lemma le_add_iff2: |
14738 | 283 |
"(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
|
284 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
285 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
286 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
287 |
|
14270 | 288 |
subsection {* Ordering Rules for Multiplication *} |
289 |
||
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
290 |
lemma mult_left_le_imp_le: |
14738 | 291 |
"[|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
|
292 |
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
|
293 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
294 |
lemma mult_right_le_imp_le: |
14738 | 295 |
"[|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
|
296 |
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
|
297 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
298 |
lemma mult_left_less_imp_less: |
14738 | 299 |
"[|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
|
300 |
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
|
301 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
302 |
lemma mult_right_less_imp_less: |
14738 | 303 |
"[|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
|
304 |
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
|
305 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
306 |
lemma mult_strict_left_mono_neg: |
14738 | 307 |
"[|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
|
308 |
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
|
309 |
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
|
310 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
311 |
|
14738 | 312 |
lemma mult_left_mono_neg: |
313 |
"[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::pordered_ring)" |
|
314 |
apply (drule mult_left_mono [of _ _ "-c"]) |
|
315 |
apply (simp_all add: minus_mult_left [symmetric]) |
|
316 |
done |
|
317 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
318 |
lemma mult_strict_right_mono_neg: |
14738 | 319 |
"[|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
|
320 |
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
|
321 |
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
|
322 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
323 |
|
14738 | 324 |
lemma mult_right_mono_neg: |
325 |
"[|b \<le> a; c \<le> 0|] ==> a * c \<le> (b::'a::pordered_ring) * c" |
|
326 |
apply (drule mult_right_mono [of _ _ "-c"]) |
|
327 |
apply (simp) |
|
328 |
apply (simp_all add: minus_mult_right [symmetric]) |
|
329 |
done |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
330 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
331 |
subsection{* Products of Signs *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
332 |
|
14738 | 333 |
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
|
334 |
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
|
335 |
|
14738 | 336 |
lemma mult_pos_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b" |
337 |
by (drule mult_left_mono [of 0 b], auto) |
|
338 |
||
339 |
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
|
340 |
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
|
341 |
|
14738 | 342 |
lemma mult_pos_neg_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0" |
343 |
by (drule mult_left_mono [of b 0], auto) |
|
344 |
||
345 |
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" |
|
346 |
by (drule mult_strict_right_mono[of b 0], auto) |
|
347 |
||
348 |
lemma mult_pos_neg2_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" |
|
349 |
by (drule mult_right_mono[of b 0], auto) |
|
350 |
||
351 |
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
|
352 |
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
|
353 |
|
14738 | 354 |
lemma mult_neg_le: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b" |
355 |
by (drule mult_right_mono_neg[of a 0 b ], auto) |
|
356 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
357 |
lemma zero_less_mult_pos: |
14738 | 358 |
"[| 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
|
359 |
apply (case_tac "b\<le>0") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
360 |
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
|
361 |
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
|
362 |
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
|
363 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
364 |
|
14738 | 365 |
lemma zero_less_mult_pos2: |
366 |
"[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)" |
|
367 |
apply (case_tac "b\<le>0") |
|
368 |
apply (auto simp add: order_le_less linorder_not_less) |
|
369 |
apply (drule_tac mult_pos_neg2 [of a b]) |
|
370 |
apply (auto dest: order_less_not_sym) |
|
371 |
done |
|
372 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
373 |
lemma zero_less_mult_iff: |
14738 | 374 |
"((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
|
375 |
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
|
376 |
apply (blast dest: zero_less_mult_pos) |
14738 | 377 |
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
|
378 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
379 |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
380 |
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
|
381 |
assumption of an ordering. See @{text field_mult_eq_0_iff} below.*} |
14738 | 382 |
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
|
383 |
apply (case_tac "a < 0") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
384 |
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
|
385 |
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
|
386 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
387 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
388 |
lemma zero_le_mult_iff: |
14738 | 389 |
"((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
|
390 |
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
|
391 |
zero_less_mult_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
392 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
393 |
lemma mult_less_0_iff: |
14738 | 394 |
"(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
|
395 |
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
|
396 |
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
|
397 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
398 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
399 |
lemma mult_le_0_iff: |
14738 | 400 |
"(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
|
401 |
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
|
402 |
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
|
403 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
404 |
|
14738 | 405 |
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)" |
406 |
by (auto simp add: mult_pos_le mult_neg_le) |
|
407 |
||
408 |
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)" |
|
409 |
by (auto simp add: mult_pos_neg_le mult_pos_neg2_le) |
|
410 |
||
411 |
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
|
412 |
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
|
413 |
|
14738 | 414 |
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom} |
415 |
theorems available to members of @{term ordered_idom} *} |
|
416 |
||
417 |
instance ordered_idom \<subseteq> ordered_semidom |
|
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
418 |
proof |
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
419 |
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
|
420 |
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
|
421 |
qed |
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset
|
422 |
|
14738 | 423 |
instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors |
424 |
by (intro_classes, simp) |
|
425 |
||
426 |
instance ordered_idom \<subseteq> idom .. |
|
427 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
428 |
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
|
429 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
430 |
declare zero_neq_one [THEN not_sym, simp] |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
431 |
|
14738 | 432 |
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
|
433 |
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
|
434 |
|
14738 | 435 |
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0" |
436 |
by (simp add: linorder_not_le) |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
437 |
|
14738 | 438 |
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0" |
439 |
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
|
440 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
441 |
subsection{*More Monotonicity*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
442 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
443 |
text{*Strict monotonicity in both arguments*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
444 |
lemma mult_strict_mono: |
14738 | 445 |
"[|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
|
446 |
apply (case_tac "c=0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
447 |
apply (simp add: mult_pos) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
448 |
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
|
449 |
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
|
450 |
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
|
451 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
452 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
453 |
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
|
454 |
lemma mult_strict_mono': |
14738 | 455 |
"[| 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
|
456 |
apply (rule mult_strict_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
457 |
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
|
458 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
459 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
460 |
lemma mult_mono: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
461 |
"[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] |
14738 | 462 |
==> 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
|
463 |
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
|
464 |
apply (erule mult_left_mono, assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
465 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
466 |
|
14738 | 467 |
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
|
468 |
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
|
469 |
apply (simp add: order_less_trans [OF zero_less_one]) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
470 |
done |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
471 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
472 |
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
|
473 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
474 |
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
|
475 |
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
|
476 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
477 |
text{*These ``disjunction'' versions produce two cases when the comparison is |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
478 |
an assumption, but effectively four when the comparison is a goal.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
479 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
480 |
lemma mult_less_cancel_right_disj: |
14738 | 481 |
"(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
|
482 |
apply (case_tac "c = 0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
483 |
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
|
484 |
mult_strict_right_mono_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
485 |
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
|
486 |
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
|
487 |
linorder_not_le [symmetric, of a]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
488 |
apply (erule_tac [!] notE) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
489 |
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
|
490 |
mult_right_mono_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
491 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
492 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
493 |
lemma mult_less_cancel_left_disj: |
14738 | 494 |
"(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))" |
495 |
apply (case_tac "c = 0") |
|
496 |
apply (auto simp add: linorder_neq_iff mult_strict_left_mono |
|
497 |
mult_strict_left_mono_neg) |
|
498 |
apply (auto simp add: linorder_not_less |
|
499 |
linorder_not_le [symmetric, of "c*a"] |
|
500 |
linorder_not_le [symmetric, of a]) |
|
501 |
apply (erule_tac [!] notE) |
|
502 |
apply (auto simp add: order_less_imp_le mult_left_mono |
|
503 |
mult_left_mono_neg) |
|
504 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
505 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
506 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
507 |
text{*The ``conjunction of implication'' lemmas produce two cases when the |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
508 |
comparison is a goal, but give four when the comparison is an assumption.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
509 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
510 |
lemma mult_less_cancel_right: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
511 |
fixes c :: "'a :: ordered_ring_strict" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
512 |
shows "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
513 |
by (insert mult_less_cancel_right_disj [of a c b], auto) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
514 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
515 |
lemma mult_less_cancel_left: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
516 |
fixes c :: "'a :: ordered_ring_strict" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
517 |
shows "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
518 |
by (insert mult_less_cancel_left_disj [of c a b], auto) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
519 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
520 |
lemma mult_le_cancel_right: |
14738 | 521 |
"(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
522 |
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
523 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
524 |
lemma mult_le_cancel_left: |
14738 | 525 |
"(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
526 |
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
527 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
528 |
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
|
529 |
assumes less: "c*a < c*b" and nonneg: "0 \<le> c" |
14738 | 530 |
shows "a < (b::'a::ordered_semiring_strict)" |
14377 | 531 |
proof (rule ccontr) |
532 |
assume "~ a < b" |
|
533 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
534 |
hence "c*b \<le> c*a" by (rule mult_left_mono) |
|
535 |
with this and less show False |
|
536 |
by (simp add: linorder_not_less [symmetric]) |
|
537 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
538 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
539 |
lemma mult_less_imp_less_right: |
14738 | 540 |
assumes less: "a*c < b*c" and nonneg: "0 <= c" |
541 |
shows "a < (b::'a::ordered_semiring_strict)" |
|
542 |
proof (rule ccontr) |
|
543 |
assume "~ a < b" |
|
544 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
545 |
hence "b*c \<le> a*c" by (rule mult_right_mono) |
|
546 |
with this and less show False |
|
547 |
by (simp add: linorder_not_less [symmetric]) |
|
548 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
549 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
550 |
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
|
551 |
lemma mult_cancel_right [simp]: |
14738 | 552 |
"(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
|
553 |
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
|
554 |
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
|
555 |
simp add: linorder_neq_iff) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
556 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
557 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
558 |
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
|
559 |
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
|
560 |
lemma mult_cancel_left [simp]: |
14738 | 561 |
"(c*a = c*b) = (c = (0::'a::ordered_ring_strict) | a=b)" |
562 |
apply (cut_tac linorder_less_linear [of 0 c]) |
|
563 |
apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono |
|
564 |
simp add: linorder_neq_iff) |
|
565 |
done |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
566 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
567 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
568 |
subsubsection{*Special Cancellation Simprules for Multiplication*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
569 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
570 |
text{*These also produce two cases when the comparison is a goal.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
571 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
572 |
lemma mult_le_cancel_right1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
573 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
574 |
shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
575 |
by (insert mult_le_cancel_right [of 1 c b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
576 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
577 |
lemma mult_le_cancel_right2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
578 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
579 |
shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
580 |
by (insert mult_le_cancel_right [of a c 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
581 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
582 |
lemma mult_le_cancel_left1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
583 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
584 |
shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
585 |
by (insert mult_le_cancel_left [of c 1 b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
586 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
587 |
lemma mult_le_cancel_left2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
588 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
589 |
shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
590 |
by (insert mult_le_cancel_left [of c a 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
591 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
592 |
lemma mult_less_cancel_right1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
593 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
594 |
shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
595 |
by (insert mult_less_cancel_right [of 1 c b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
596 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
597 |
lemma mult_less_cancel_right2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
598 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
599 |
shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
600 |
by (insert mult_less_cancel_right [of a c 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
601 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
602 |
lemma mult_less_cancel_left1: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
603 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
604 |
shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
605 |
by (insert mult_less_cancel_left [of c 1 b], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
606 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
607 |
lemma mult_less_cancel_left2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
608 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
609 |
shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
610 |
by (insert mult_less_cancel_left [of c a 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
611 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
612 |
lemma mult_cancel_right1 [simp]: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
613 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
614 |
shows "(c = b*c) = (c = 0 | b=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
615 |
by (insert mult_cancel_right [of 1 c b], force) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
616 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
617 |
lemma mult_cancel_right2 [simp]: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
618 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
619 |
shows "(a*c = c) = (c = 0 | a=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
620 |
by (insert mult_cancel_right [of a c 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
621 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
622 |
lemma mult_cancel_left1 [simp]: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
623 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
624 |
shows "(c = c*b) = (c = 0 | b=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
625 |
by (insert mult_cancel_left [of c 1 b], force) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
626 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
627 |
lemma mult_cancel_left2 [simp]: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
628 |
fixes c :: "'a :: ordered_idom" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
629 |
shows "(c*a = c) = (c = 0 | a=1)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
630 |
by (insert mult_cancel_left [of c a 1], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
631 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
632 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
633 |
text{*Simprules for comparisons where common factors can be cancelled.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
634 |
lemmas mult_compare_simps = |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
635 |
mult_le_cancel_right mult_le_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
636 |
mult_le_cancel_right1 mult_le_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
637 |
mult_le_cancel_left1 mult_le_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
638 |
mult_less_cancel_right mult_less_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
639 |
mult_less_cancel_right1 mult_less_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
640 |
mult_less_cancel_left1 mult_less_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
641 |
mult_cancel_right mult_cancel_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
642 |
mult_cancel_right1 mult_cancel_right2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
643 |
mult_cancel_left1 mult_cancel_left2 |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
644 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
645 |
|
14738 | 646 |
text{*This list of rewrites decides ring equalities by ordered rewriting.*} |
15178 | 647 |
lemmas ring_eq_simps = |
648 |
(* mult_ac*) |
|
14738 | 649 |
left_distrib right_distrib left_diff_distrib right_diff_distrib |
15178 | 650 |
group_eq_simps |
651 |
(* add_ac |
|
14738 | 652 |
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 |
15178 | 653 |
diff_eq_eq eq_diff_eq *) |
14738 | 654 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
655 |
subsection {* Fields *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
656 |
|
14288 | 657 |
lemma right_inverse [simp]: |
658 |
assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1" |
|
659 |
proof - |
|
660 |
have "a * inverse a = inverse a * a" by (simp add: mult_ac) |
|
661 |
also have "... = 1" using not0 by simp |
|
662 |
finally show ?thesis . |
|
663 |
qed |
|
664 |
||
665 |
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" |
|
666 |
proof |
|
667 |
assume neq: "b \<noteq> 0" |
|
668 |
{ |
|
669 |
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) |
|
670 |
also assume "a / b = 1" |
|
671 |
finally show "a = b" by simp |
|
672 |
next |
|
673 |
assume "a = b" |
|
674 |
with neq show "a / b = 1" by (simp add: divide_inverse) |
|
675 |
} |
|
676 |
qed |
|
677 |
||
678 |
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" |
|
679 |
by (simp add: divide_inverse) |
|
680 |
||
15228 | 681 |
lemma divide_self: "a \<noteq> 0 ==> a / (a::'a::field) = 1" |
14288 | 682 |
by (simp add: divide_inverse) |
683 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
684 |
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
|
685 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
686 |
|
15228 | 687 |
lemma divide_self_if [simp]: |
688 |
"a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)" |
|
689 |
by (simp add: divide_self) |
|
690 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
691 |
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
|
692 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
693 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
694 |
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
|
695 |
by (simp add: divide_inverse) |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
696 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
697 |
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c" |
14293 | 698 |
by (simp add: divide_inverse left_distrib) |
699 |
||
700 |
||
14270 | 701 |
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement |
702 |
of an ordering.*} |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
703 |
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)" |
14377 | 704 |
proof cases |
705 |
assume "a=0" thus ?thesis by simp |
|
706 |
next |
|
707 |
assume anz [simp]: "a\<noteq>0" |
|
708 |
{ assume "a * b = 0" |
|
709 |
hence "inverse a * (a * b) = 0" by simp |
|
710 |
hence "b = 0" by (simp (no_asm_use) add: mult_assoc [symmetric])} |
|
711 |
thus ?thesis by force |
|
712 |
qed |
|
14270 | 713 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
714 |
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
|
715 |
lemma field_mult_cancel_right_lemma: |
14269 | 716 |
assumes cnz: "c \<noteq> (0::'a::field)" |
717 |
and eq: "a*c = b*c" |
|
718 |
shows "a=b" |
|
14377 | 719 |
proof - |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
720 |
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
|
721 |
by (simp add: eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
722 |
thus "a=b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
723 |
by (simp add: mult_assoc cnz) |
14377 | 724 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
725 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
726 |
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
|
727 |
"(a*c = b*c) = (c = (0::'a::field) | a=b)" |
14377 | 728 |
proof cases |
729 |
assume "c=0" thus ?thesis by simp |
|
730 |
next |
|
731 |
assume "c\<noteq>0" |
|
732 |
thus ?thesis by (force dest: field_mult_cancel_right_lemma) |
|
733 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
734 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
735 |
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
|
736 |
"(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
|
737 |
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
|
738 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
739 |
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)" |
14377 | 740 |
proof |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
741 |
assume ianz: "inverse a = 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
742 |
assume "a \<noteq> 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
743 |
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
|
744 |
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
|
745 |
finally have "1 = (0::'a::field)" . |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
746 |
thus False by (simp add: eq_commute) |
14377 | 747 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
748 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
749 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
750 |
subsection{*Basic Properties of @{term inverse}*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
751 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
752 |
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
|
753 |
apply (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
754 |
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
|
755 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
756 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
757 |
lemma inverse_nonzero_imp_nonzero: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
758 |
"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
|
759 |
apply (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
760 |
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
|
761 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
762 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
763 |
lemma inverse_nonzero_iff_nonzero [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
764 |
"(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
|
765 |
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
|
766 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
767 |
lemma nonzero_inverse_minus_eq: |
14269 | 768 |
assumes [simp]: "a\<noteq>0" shows "inverse(-a) = -inverse(a::'a::field)" |
14377 | 769 |
proof - |
770 |
have "-a * inverse (- a) = -a * - inverse a" |
|
771 |
by simp |
|
772 |
thus ?thesis |
|
773 |
by (simp only: field_mult_cancel_left, simp) |
|
774 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
775 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
776 |
lemma inverse_minus_eq [simp]: |
14377 | 777 |
"inverse(-a) = -inverse(a::'a::{field,division_by_zero})"; |
778 |
proof cases |
|
779 |
assume "a=0" thus ?thesis by (simp add: inverse_zero) |
|
780 |
next |
|
781 |
assume "a\<noteq>0" |
|
782 |
thus ?thesis by (simp add: nonzero_inverse_minus_eq) |
|
783 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
784 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
785 |
lemma nonzero_inverse_eq_imp_eq: |
14269 | 786 |
assumes inveq: "inverse a = inverse b" |
787 |
and anz: "a \<noteq> 0" |
|
788 |
and bnz: "b \<noteq> 0" |
|
789 |
shows "a = (b::'a::field)" |
|
14377 | 790 |
proof - |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
791 |
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
|
792 |
by (simp add: inveq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
793 |
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
|
794 |
by simp |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
795 |
thus "a = b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
796 |
by (simp add: mult_assoc anz bnz) |
14377 | 797 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
798 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
799 |
lemma inverse_eq_imp_eq: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
800 |
"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
|
801 |
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
|
802 |
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
|
803 |
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
|
804 |
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
|
805 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
806 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
807 |
lemma inverse_eq_iff_eq [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
808 |
"(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
|
809 |
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
|
810 |
|
14270 | 811 |
lemma nonzero_inverse_inverse_eq: |
812 |
assumes [simp]: "a \<noteq> 0" shows "inverse(inverse (a::'a::field)) = a" |
|
813 |
proof - |
|
814 |
have "(inverse (inverse a) * inverse a) * a = a" |
|
815 |
by (simp add: nonzero_imp_inverse_nonzero) |
|
816 |
thus ?thesis |
|
817 |
by (simp add: mult_assoc) |
|
818 |
qed |
|
819 |
||
820 |
lemma inverse_inverse_eq [simp]: |
|
821 |
"inverse(inverse (a::'a::{field,division_by_zero})) = a" |
|
822 |
proof cases |
|
823 |
assume "a=0" thus ?thesis by simp |
|
824 |
next |
|
825 |
assume "a\<noteq>0" |
|
826 |
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) |
|
827 |
qed |
|
828 |
||
829 |
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)" |
|
830 |
proof - |
|
831 |
have "inverse 1 * 1 = (1::'a::field)" |
|
832 |
by (rule left_inverse [OF zero_neq_one [symmetric]]) |
|
833 |
thus ?thesis by simp |
|
834 |
qed |
|
835 |
||
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
836 |
lemma inverse_unique: |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
837 |
assumes ab: "a*b = 1" |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
838 |
shows "inverse a = (b::'a::field)" |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
839 |
proof - |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
840 |
have "a \<noteq> 0" using ab by auto |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
841 |
moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
842 |
ultimately show ?thesis by (simp add: mult_assoc [symmetric]) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
843 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset
|
844 |
|
14270 | 845 |
lemma nonzero_inverse_mult_distrib: |
846 |
assumes anz: "a \<noteq> 0" |
|
847 |
and bnz: "b \<noteq> 0" |
|
848 |
shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)" |
|
849 |
proof - |
|
850 |
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" |
|
851 |
by (simp add: field_mult_eq_0_iff anz bnz) |
|
852 |
hence "inverse(a*b) * a = inverse(b)" |
|
853 |
by (simp add: mult_assoc bnz) |
|
854 |
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" |
|
855 |
by simp |
|
856 |
thus ?thesis |
|
857 |
by (simp add: mult_assoc anz) |
|
858 |
qed |
|
859 |
||
860 |
text{*This version builds in division by zero while also re-orienting |
|
861 |
the right-hand side.*} |
|
862 |
lemma inverse_mult_distrib [simp]: |
|
863 |
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" |
|
864 |
proof cases |
|
865 |
assume "a \<noteq> 0 & b \<noteq> 0" |
|
866 |
thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute) |
|
867 |
next |
|
868 |
assume "~ (a \<noteq> 0 & b \<noteq> 0)" |
|
869 |
thus ?thesis by force |
|
870 |
qed |
|
871 |
||
872 |
text{*There is no slick version using division by zero.*} |
|
873 |
lemma inverse_add: |
|
874 |
"[|a \<noteq> 0; b \<noteq> 0|] |
|
875 |
==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)" |
|
876 |
apply (simp add: left_distrib mult_assoc) |
|
877 |
apply (simp add: mult_commute [of "inverse a"]) |
|
878 |
apply (simp add: mult_assoc [symmetric] add_commute) |
|
879 |
done |
|
880 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
881 |
lemma inverse_divide [simp]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
882 |
"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
|
883 |
by (simp add: divide_inverse mult_commute) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
884 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
885 |
lemma nonzero_mult_divide_cancel_left: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
886 |
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
|
887 |
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
|
888 |
proof - |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
889 |
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
|
890 |
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
|
891 |
nonzero_inverse_mult_distrib) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
892 |
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
|
893 |
by (simp only: mult_ac) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
894 |
also have "... = a * inverse b" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
895 |
by simp |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
896 |
finally show ?thesis |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
897 |
by (simp add: divide_inverse) |
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 mult_divide_cancel_left: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
901 |
"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
|
902 |
apply (case_tac "b = 0") |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
903 |
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
|
904 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
905 |
|
14321 | 906 |
lemma nonzero_mult_divide_cancel_right: |
907 |
"[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)" |
|
908 |
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) |
|
909 |
||
910 |
lemma mult_divide_cancel_right: |
|
911 |
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" |
|
912 |
apply (case_tac "b = 0") |
|
913 |
apply (simp_all add: nonzero_mult_divide_cancel_right) |
|
914 |
done |
|
915 |
||
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
916 |
(*For ExtractCommonTerm*) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
917 |
lemma mult_divide_cancel_eq_if: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
918 |
"(c*a) / (c*b) = |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
919 |
(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
|
920 |
by (simp add: mult_divide_cancel_left) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
921 |
|
14284
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
922 |
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
|
923 |
by (simp add: divide_inverse) |
14284
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
924 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
925 |
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
926 |
by (simp add: divide_inverse mult_assoc) |
14288 | 927 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
928 |
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
|
929 |
by (simp add: divide_inverse mult_ac) |
14288 | 930 |
|
931 |
lemma divide_divide_eq_right [simp]: |
|
932 |
"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
|
933 |
by (simp add: divide_inverse mult_ac) |
14288 | 934 |
|
935 |
lemma divide_divide_eq_left [simp]: |
|
936 |
"(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
|
937 |
by (simp add: divide_inverse mult_assoc) |
14288 | 938 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
939 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
940 |
subsubsection{*Special Cancellation Simprules for Division*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
941 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
942 |
lemma mult_divide_cancel_left_if [simp]: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
943 |
fixes c :: "'a :: {field,division_by_zero}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
944 |
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
945 |
by (simp add: mult_divide_cancel_left) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
946 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
947 |
lemma mult_divide_cancel_right_if [simp]: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
948 |
fixes c :: "'a :: {field,division_by_zero}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
949 |
shows "(a*c) / (b*c) = (if c=0 then 0 else a/b)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
950 |
by (simp add: mult_divide_cancel_right) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
951 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
952 |
lemma mult_divide_cancel_left_if1 [simp]: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
953 |
fixes c :: "'a :: {field,division_by_zero}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
954 |
shows "c / (c*b) = (if c=0 then 0 else 1/b)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
955 |
apply (insert mult_divide_cancel_left_if [of c 1 b]) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
956 |
apply (simp del: mult_divide_cancel_left_if) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
957 |
done |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
958 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
959 |
lemma mult_divide_cancel_left_if2 [simp]: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
960 |
fixes c :: "'a :: {field,division_by_zero}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
961 |
shows "(c*a) / c = (if c=0 then 0 else a)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
962 |
apply (insert mult_divide_cancel_left_if [of c a 1]) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
963 |
apply (simp del: mult_divide_cancel_left_if) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
964 |
done |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
965 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
966 |
lemma mult_divide_cancel_right_if1 [simp]: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
967 |
fixes c :: "'a :: {field,division_by_zero}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
968 |
shows "c / (b*c) = (if c=0 then 0 else 1/b)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
969 |
apply (insert mult_divide_cancel_right_if [of 1 c b]) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
970 |
apply (simp del: mult_divide_cancel_right_if) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
971 |
done |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
972 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
973 |
lemma mult_divide_cancel_right_if2 [simp]: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
974 |
fixes c :: "'a :: {field,division_by_zero}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
975 |
shows "(a*c) / c = (if c=0 then 0 else a)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
976 |
apply (insert mult_divide_cancel_right_if [of a c 1]) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
977 |
apply (simp del: mult_divide_cancel_right_if) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
978 |
done |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
979 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
980 |
text{*Two lemmas for cancelling the denominator*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
981 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
982 |
lemma times_divide_self_right [simp]: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
983 |
fixes a :: "'a :: {field,division_by_zero}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
984 |
shows "a * (b/a) = (if a=0 then 0 else b)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
985 |
by (simp add: times_divide_eq_right) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
986 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
987 |
lemma times_divide_self_left [simp]: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
988 |
fixes a :: "'a :: {field,division_by_zero}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
989 |
shows "(b/a) * a = (if a=0 then 0 else b)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
990 |
by (simp add: times_divide_eq_left) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
991 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
992 |
|
14293 | 993 |
subsection {* Division and Unary Minus *} |
994 |
||
995 |
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)" |
|
996 |
by (simp add: divide_inverse minus_mult_left) |
|
997 |
||
998 |
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)" |
|
999 |
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) |
|
1000 |
||
1001 |
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)" |
|
1002 |
by (simp add: divide_inverse nonzero_inverse_minus_eq) |
|
1003 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1004 |
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
|
1005 |
by (simp add: divide_inverse minus_mult_left [symmetric]) |
14293 | 1006 |
|
1007 |
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
|
1008 |
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
|
1009 |
|
14293 | 1010 |
|
1011 |
text{*The effect is to extract signs from divisions*} |
|
1012 |
declare minus_divide_left [symmetric, simp] |
|
1013 |
declare minus_divide_right [symmetric, simp] |
|
1014 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1015 |
text{*Also, extract signs from products*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1016 |
declare minus_mult_left [symmetric, simp] |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1017 |
declare minus_mult_right [symmetric, simp] |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1018 |
|
14293 | 1019 |
lemma minus_divide_divide [simp]: |
1020 |
"(-a)/(-b) = a / (b::'a::{field,division_by_zero})" |
|
1021 |
apply (case_tac "b=0", simp) |
|
1022 |
apply (simp add: nonzero_minus_divide_divide) |
|
1023 |
done |
|
1024 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1025 |
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
|
1026 |
by (simp add: diff_minus add_divide_distrib) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1027 |
|
14293 | 1028 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1029 |
subsection {* Ordered Fields *} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1030 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1031 |
lemma positive_imp_inverse_positive: |
14269 | 1032 |
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
|
1033 |
proof - |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1034 |
have "0 < a * inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1035 |
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
|
1036 |
thus "0 < inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1037 |
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
|
1038 |
qed |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1039 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1040 |
lemma negative_imp_inverse_negative: |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1041 |
"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
|
1042 |
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
|
1043 |
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
|
1044 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1045 |
lemma inverse_le_imp_le: |
14269 | 1046 |
assumes invle: "inverse a \<le> inverse b" |
1047 |
and apos: "0 < a" |
|
1048 |
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
|
1049 |
proof (rule classical) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1050 |
assume "~ b \<le> a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1051 |
hence "a < b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1052 |
by (simp add: linorder_not_le) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1053 |
hence bpos: "0 < b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1054 |
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
|
1055 |
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
|
1056 |
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
|
1057 |
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
|
1058 |
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
|
1059 |
thus "b \<le> a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1060 |
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
|
1061 |
qed |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1062 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1063 |
lemma inverse_positive_imp_positive: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1064 |
assumes inv_gt_0: "0 < inverse a" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1065 |
and [simp]: "a \<noteq> 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1066 |
shows "0 < (a::'a::ordered_field)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1067 |
proof - |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1068 |
have "0 < inverse (inverse a)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1069 |
by (rule positive_imp_inverse_positive) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1070 |
thus "0 < a" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1071 |
by (simp add: nonzero_inverse_inverse_eq) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1072 |
qed |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1073 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1074 |
lemma inverse_positive_iff_positive [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1075 |
"(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
|
1076 |
apply (case_tac "a = 0", simp) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1077 |
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
|
1078 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1079 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1080 |
lemma inverse_negative_imp_negative: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1081 |
assumes inv_less_0: "inverse a < 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1082 |
and [simp]: "a \<noteq> 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1083 |
shows "a < (0::'a::ordered_field)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1084 |
proof - |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1085 |
have "inverse (inverse a) < 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1086 |
by (rule negative_imp_inverse_negative) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1087 |
thus "a < 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1088 |
by (simp add: nonzero_inverse_inverse_eq) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1089 |
qed |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1090 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1091 |
lemma inverse_negative_iff_negative [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1092 |
"(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
|
1093 |
apply (case_tac "a = 0", simp) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1094 |
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
|
1095 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1096 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1097 |
lemma inverse_nonnegative_iff_nonnegative [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1098 |
"(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
|
1099 |
by (simp add: linorder_not_less [symmetric]) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1100 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1101 |
lemma inverse_nonpositive_iff_nonpositive [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1102 |
"(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
|
1103 |
by (simp add: linorder_not_less [symmetric]) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1104 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1105 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1106 |
subsection{*Anti-Monotonicity of @{term inverse}*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1107 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1108 |
lemma less_imp_inverse_less: |
14269 | 1109 |
assumes less: "a < b" |
1110 |
and apos: "0 < a" |
|
1111 |
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
|
1112 |
proof (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1113 |
assume "~ inverse b < inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1114 |
hence "inverse a \<le> inverse b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1115 |
by (simp add: linorder_not_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1116 |
hence "~ (a < b)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1117 |
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
|
1118 |
thus False |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1119 |
by (rule notE [OF _ less]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1120 |
qed |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1121 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1122 |
lemma inverse_less_imp_less: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1123 |
"[|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
|
1124 |
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
|
1125 |
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
|
1126 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1127 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1128 |
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
|
1129 |
lemma inverse_less_iff_less [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1130 |
"[|0 < a; 0 < b|] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1131 |
==> (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
|
1132 |
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
|
1133 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1134 |
lemma le_imp_inverse_le: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1135 |
"[|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
|
1136 |
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
|
1137 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1138 |
lemma inverse_le_iff_le [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1139 |
"[|0 < a; 0 < b|] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1140 |
==> (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
|
1141 |
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
|
1142 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1143 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1144 |
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
|
1145 |
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
|
1146 |
lemma inverse_le_imp_le_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1147 |
"[|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
|
1148 |
apply (rule classical) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1149 |
apply (subgoal_tac "a < 0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1150 |
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
|
1151 |
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
|
1152 |
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
|
1153 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1154 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1155 |
lemma less_imp_inverse_less_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1156 |
"[|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
|
1157 |
apply (subgoal_tac "a < 0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1158 |
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
|
1159 |
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
|
1160 |
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
|
1161 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1162 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1163 |
lemma inverse_less_imp_less_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1164 |
"[|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
|
1165 |
apply (rule classical) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1166 |
apply (subgoal_tac "a < 0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1167 |
prefer 2 |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1168 |
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
|
1169 |
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
|
1170 |
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
|
1171 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1172 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1173 |
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
|
1174 |
"[|a < 0; b < 0|] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1175 |
==> (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
|
1176 |
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
|
1177 |
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
|
1178 |
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
|
1179 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1180 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1181 |
lemma le_imp_inverse_le_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1182 |
"[|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
|
1183 |
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
|
1184 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1185 |
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
|
1186 |
"[|a < 0; b < 0|] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1187 |
==> (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
|
1188 |
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
|
1189 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1190 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1191 |
subsection{*Inverses and the Number One*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1192 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1193 |
lemma one_less_inverse_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1194 |
"(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
|
1195 |
assume "0 < x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1196 |
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
|
1197 |
show ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1198 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1199 |
assume notless: "~ (0 < x)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1200 |
have "~ (1 < inverse x)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1201 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1202 |
assume "1 < inverse x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1203 |
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
|
1204 |
also have "... < 1" by (rule zero_less_one) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1205 |
finally show False by auto |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1206 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1207 |
with notless show ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1208 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1209 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1210 |
lemma inverse_eq_1_iff [simp]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1211 |
"(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
|
1212 |
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
|
1213 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1214 |
lemma one_le_inverse_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1215 |
"(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
|
1216 |
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
|
1217 |
eq_commute [of 1]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1218 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1219 |
lemma inverse_less_1_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1220 |
"(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
|
1221 |
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
|
1222 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1223 |
lemma inverse_le_1_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1224 |
"(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
|
1225 |
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
|
1226 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1227 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1228 |
subsection{*Division and Signs*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1229 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1230 |
lemma zero_less_divide_iff: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1231 |
"((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
|
1232 |
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
|
1233 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1234 |
lemma divide_less_0_iff: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1235 |
"(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
|
1236 |
(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
|
1237 |
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
|
1238 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1239 |
lemma zero_le_divide_iff: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1240 |
"((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
|
1241 |
(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
|
1242 |
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
|
1243 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1244 |
lemma divide_le_0_iff: |
14288 | 1245 |
"(a/b \<le> (0::'a::{ordered_field,division_by_zero})) = |
1246 |
(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
|
1247 |
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
|
1248 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1249 |
lemma divide_eq_0_iff [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1250 |
"(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
|
1251 |
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
|
1252 |
|
14288 | 1253 |
|
1254 |
subsection{*Simplification of Inequalities Involving Literal Divisors*} |
|
1255 |
||
1256 |
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)" |
|
1257 |
proof - |
|
1258 |
assume less: "0<c" |
|
1259 |
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" |
|
1260 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1261 |
also have "... = (a*c \<le> b)" |
|
1262 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1263 |
finally show ?thesis . |
|
1264 |
qed |
|
1265 |
||
1266 |
||
1267 |
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)" |
|
1268 |
proof - |
|
1269 |
assume less: "c<0" |
|
1270 |
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" |
|
1271 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1272 |
also have "... = (b \<le> a*c)" |
|
1273 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1274 |
finally show ?thesis . |
|
1275 |
qed |
|
1276 |
||
1277 |
lemma le_divide_eq: |
|
1278 |
"(a \<le> b/c) = |
|
1279 |
(if 0 < c then a*c \<le> b |
|
1280 |
else if c < 0 then b \<le> a*c |
|
1281 |
else a \<le> (0::'a::{ordered_field,division_by_zero}))" |
|
1282 |
apply (case_tac "c=0", simp) |
|
1283 |
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) |
|
1284 |
done |
|
1285 |
||
1286 |
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)" |
|
1287 |
proof - |
|
1288 |
assume less: "0<c" |
|
1289 |
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" |
|
1290 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1291 |
also have "... = (b \<le> a*c)" |
|
1292 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1293 |
finally show ?thesis . |
|
1294 |
qed |
|
1295 |
||
1296 |
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)" |
|
1297 |
proof - |
|
1298 |
assume less: "c<0" |
|
1299 |
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" |
|
1300 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1301 |
also have "... = (a*c \<le> b)" |
|
1302 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1303 |
finally show ?thesis . |
|
1304 |
qed |
|
1305 |
||
1306 |
lemma divide_le_eq: |
|
1307 |
"(b/c \<le> a) = |
|
1308 |
(if 0 < c then b \<le> a*c |
|
1309 |
else if c < 0 then a*c \<le> b |
|
1310 |
else 0 \<le> (a::'a::{ordered_field,division_by_zero}))" |
|
1311 |
apply (case_tac "c=0", simp) |
|
1312 |
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) |
|
1313 |
done |
|
1314 |
||
1315 |
||
1316 |
lemma pos_less_divide_eq: |
|
1317 |
"0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)" |
|
1318 |
proof - |
|
1319 |
assume less: "0<c" |
|
1320 |
hence "(a < b/c) = (a*c < (b/c)*c)" |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1321 |
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) |
14288 | 1322 |
also have "... = (a*c < b)" |
1323 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1324 |
finally show ?thesis . |
|
1325 |
qed |
|
1326 |
||
1327 |
lemma neg_less_divide_eq: |
|
1328 |
"c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)" |
|
1329 |
proof - |
|
1330 |
assume less: "c<0" |
|
1331 |
hence "(a < b/c) = ((b/c)*c < a*c)" |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1332 |
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) |
14288 | 1333 |
also have "... = (b < a*c)" |
1334 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1335 |
finally show ?thesis . |
|
1336 |
qed |
|
1337 |
||
1338 |
lemma less_divide_eq: |
|
1339 |
"(a < b/c) = |
|
1340 |
(if 0 < c then a*c < b |
|
1341 |
else if c < 0 then b < a*c |
|
1342 |
else a < (0::'a::{ordered_field,division_by_zero}))" |
|
1343 |
apply (case_tac "c=0", simp) |
|
1344 |
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) |
|
1345 |
done |
|
1346 |
||
1347 |
lemma pos_divide_less_eq: |
|
1348 |
"0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)" |
|
1349 |
proof - |
|
1350 |
assume less: "0<c" |
|
1351 |
hence "(b/c < a) = ((b/c)*c < a*c)" |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1352 |
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) |
14288 | 1353 |
also have "... = (b < a*c)" |
1354 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1355 |
finally show ?thesis . |
|
1356 |
qed |
|
1357 |
||
1358 |
lemma neg_divide_less_eq: |
|
1359 |
"c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)" |
|
1360 |
proof - |
|
1361 |
assume less: "c<0" |
|
1362 |
hence "(b/c < a) = (a*c < (b/c)*c)" |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1363 |
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) |
14288 | 1364 |
also have "... = (a*c < b)" |
1365 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1366 |
finally show ?thesis . |
|
1367 |
qed |
|
1368 |
||
1369 |
lemma divide_less_eq: |
|
1370 |
"(b/c < a) = |
|
1371 |
(if 0 < c then b < a*c |
|
1372 |
else if c < 0 then a*c < b |
|
1373 |
else 0 < (a::'a::{ordered_field,division_by_zero}))" |
|
1374 |
apply (case_tac "c=0", simp) |
|
1375 |
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) |
|
1376 |
done |
|
1377 |
||
1378 |
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)" |
|
1379 |
proof - |
|
1380 |
assume [simp]: "c\<noteq>0" |
|
1381 |
have "(a = b/c) = (a*c = (b/c)*c)" |
|
1382 |
by (simp add: field_mult_cancel_right) |
|
1383 |
also have "... = (a*c = b)" |
|
1384 |
by (simp add: divide_inverse mult_assoc) |
|
1385 |
finally show ?thesis . |
|
1386 |
qed |
|
1387 |
||
1388 |
lemma eq_divide_eq: |
|
1389 |
"((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)" |
|
1390 |
by (simp add: nonzero_eq_divide_eq) |
|
1391 |
||
1392 |
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)" |
|
1393 |
proof - |
|
1394 |
assume [simp]: "c\<noteq>0" |
|
1395 |
have "(b/c = a) = ((b/c)*c = a*c)" |
|
1396 |
by (simp add: field_mult_cancel_right) |
|
1397 |
also have "... = (b = a*c)" |
|
1398 |
by (simp add: divide_inverse mult_assoc) |
|
1399 |
finally show ?thesis . |
|
1400 |
qed |
|
1401 |
||
1402 |
lemma divide_eq_eq: |
|
1403 |
"(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)" |
|
1404 |
by (force simp add: nonzero_divide_eq_eq) |
|
1405 |
||
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1406 |
|
14288 | 1407 |
subsection{*Cancellation Laws for Division*} |
1408 |
||
1409 |
lemma divide_cancel_right [simp]: |
|
1410 |
"(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))" |
|
1411 |
apply (case_tac "c=0", simp) |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1412 |
apply (simp add: divide_inverse field_mult_cancel_right) |
14288 | 1413 |
done |
1414 |
||
1415 |
lemma divide_cancel_left [simp]: |
|
1416 |
"(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" |
|
1417 |
apply (case_tac "c=0", simp) |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
1418 |
apply (simp add: divide_inverse field_mult_cancel_left) |
14288 | 1419 |
done |
1420 |
||
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1421 |
subsection {* Division and the Number One *} |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1422 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1423 |
text{*Simplify expressions equated with 1*} |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1424 |
lemma divide_eq_1_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1425 |
"(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
|
1426 |
apply (case_tac "b=0", simp) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1427 |
apply (simp add: right_inverse_eq) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1428 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1429 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1430 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1431 |
lemma one_eq_divide_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1432 |
"(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
|
1433 |
by (simp add: eq_commute [of 1]) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1434 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1435 |
lemma zero_eq_1_divide_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1436 |
"((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
|
1437 |
apply (case_tac "a=0", simp) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1438 |
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
|
1439 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1440 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1441 |
lemma one_divide_eq_0_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1442 |
"(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
|
1443 |
apply (case_tac "a=0", simp) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1444 |
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
|
1445 |
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
|
1446 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1447 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1448 |
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
|
1449 |
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
|
1450 |
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
|
1451 |
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
|
1452 |
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
|
1453 |
|
14288 | 1454 |
|
14293 | 1455 |
subsection {* Ordering Rules for Division *} |
1456 |
||
1457 |
lemma divide_strict_right_mono: |
|
1458 |
"[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)" |
|
1459 |
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono |
|
1460 |
positive_imp_inverse_positive) |
|
1461 |
||
1462 |
lemma divide_right_mono: |
|
1463 |
"[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})" |
|
1464 |
by (force simp add: divide_strict_right_mono order_le_less) |
|
1465 |
||
1466 |
||
1467 |
text{*The last premise ensures that @{term a} and @{term b} |
|
1468 |
have the same sign*} |
|
1469 |
lemma divide_strict_left_mono: |
|
1470 |
"[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)" |
|
1471 |
by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono |
|
1472 |
order_less_imp_not_eq order_less_imp_not_eq2 |
|
1473 |
less_imp_inverse_less less_imp_inverse_less_neg) |
|
1474 |
||
1475 |
lemma divide_left_mono: |
|
1476 |
"[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)" |
|
1477 |
apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") |
|
1478 |
prefer 2 |
|
1479 |
apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) |
|
1480 |
apply (case_tac "c=0", simp add: divide_inverse) |
|
1481 |
apply (force simp add: divide_strict_left_mono order_le_less) |
|
1482 |
done |
|
1483 |
||
1484 |
lemma divide_strict_left_mono_neg: |
|
1485 |
"[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)" |
|
1486 |
apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") |
|
1487 |
prefer 2 |
|
1488 |
apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) |
|
1489 |
apply (drule divide_strict_left_mono [of _ _ "-c"]) |
|
1490 |
apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric]) |
|
1491 |
done |
|
1492 |
||
1493 |
lemma divide_strict_right_mono_neg: |
|
1494 |
"[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)" |
|
1495 |
apply (drule divide_strict_right_mono [of _ _ "-c"], simp) |
|
1496 |
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) |
|
1497 |
done |
|
1498 |
||
1499 |
||
1500 |
subsection {* Ordered Fields are Dense *} |
|
1501 |
||
14738 | 1502 |
lemma less_add_one: "a < (a+1::'a::ordered_semidom)" |
14293 | 1503 |
proof - |
14738 | 1504 |
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
|
1505 |
by (blast intro: zero_less_one add_strict_left_mono) |
14293 | 1506 |
thus ?thesis by simp |
1507 |
qed |
|
1508 |
||
14738 | 1509 |
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
|
1510 |
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
|
1511 |
|
14293 | 1512 |
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)" |
1513 |
by (simp add: zero_less_two pos_less_divide_eq right_distrib) |
|
1514 |
||
1515 |
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b" |
|
1516 |
by (simp add: zero_less_two pos_divide_less_eq right_distrib) |
|
1517 |
||
1518 |
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b" |
|
1519 |
by (blast intro!: less_half_sum gt_half_sum) |
|
1520 |
||
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1521 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1522 |
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1523 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1524 |
text{*It's not obvious whether these should be simprules or not. |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1525 |
Their effect is to gather terms into one big fraction, like |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1526 |
a*b*c / x*y*z. The rationale for that is unclear, but many proofs |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1527 |
seem to need them.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1528 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1529 |
declare times_divide_eq [simp] |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1530 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1531 |
|
14293 | 1532 |
subsection {* Absolute Value *} |
1533 |
||
14738 | 1534 |
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
|
1535 |
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
|
1536 |
|
14738 | 1537 |
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" |
1538 |
proof - |
|
1539 |
let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b" |
|
1540 |
let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b" |
|
1541 |
have a: "(abs a) * (abs b) = ?x" |
|
1542 |
by (simp only: abs_prts[of a] abs_prts[of b] ring_eq_simps) |
|
1543 |
{ |
|
1544 |
fix u v :: 'a |
|
15481 | 1545 |
have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> |
1546 |
u * v = pprt a * pprt b + pprt a * nprt b + |
|
1547 |
nprt a * pprt b + nprt a * nprt b" |
|
14738 | 1548 |
apply (subst prts[of u], subst prts[of v]) |
1549 |
apply (simp add: left_distrib right_distrib add_ac) |
|
1550 |
done |
|
1551 |
} |
|
1552 |
note b = this[OF refl[of a] refl[of b]] |
|
1553 |
note addm = add_mono[of "0::'a" _ "0::'a", simplified] |
|
1554 |
note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified] |
|
1555 |
have xy: "- ?x <= ?y" |
|
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1556 |
apply (simp) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1557 |
apply (rule_tac y="0::'a" in order_trans) |
16568 | 1558 |
apply (rule addm2) |
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1559 |
apply (simp_all add: mult_pos_le mult_neg_le) |
16568 | 1560 |
apply (rule addm) |
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1561 |
apply (simp_all add: mult_pos_le mult_neg_le) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1562 |
done |
14738 | 1563 |
have yx: "?y <= ?x" |
16568 | 1564 |
apply (simp add:diff_def) |
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1565 |
apply (rule_tac y=0 in order_trans) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1566 |
apply (rule addm2, (simp add: mult_pos_neg_le mult_pos_neg2_le)+) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1567 |
apply (rule addm, (simp add: mult_pos_neg_le mult_pos_neg2_le)+) |
14738 | 1568 |
done |
1569 |
have i1: "a*b <= abs a * abs b" by (simp only: a b yx) |
|
1570 |
have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy) |
|
1571 |
show ?thesis |
|
1572 |
apply (rule abs_leI) |
|
1573 |
apply (simp add: i1) |
|
1574 |
apply (simp add: i2[simplified minus_le_iff]) |
|
1575 |
done |
|
1576 |
qed |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1577 |
|
14738 | 1578 |
lemma abs_eq_mult: |
1579 |
assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)" |
|
1580 |
shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)" |
|
1581 |
proof - |
|
1582 |
have s: "(0 <= a*b) | (a*b <= 0)" |
|
1583 |
apply (auto) |
|
1584 |
apply (rule_tac split_mult_pos_le) |
|
1585 |
apply (rule_tac contrapos_np[of "a*b <= 0"]) |
|
1586 |
apply (simp) |
|
1587 |
apply (rule_tac split_mult_neg_le) |
|
1588 |
apply (insert prems) |
|
1589 |
apply (blast) |
|
1590 |
done |
|
1591 |
have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)" |
|
1592 |
by (simp add: prts[symmetric]) |
|
1593 |
show ?thesis |
|
1594 |
proof cases |
|
1595 |
assume "0 <= a * b" |
|
1596 |
then show ?thesis |
|
1597 |
apply (simp_all add: mulprts abs_prts) |
|
1598 |
apply (insert prems) |
|
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1599 |
apply (auto simp add: |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1600 |
ring_eq_simps |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1601 |
iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt] |
15197 | 1602 |
iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id]) |
1603 |
apply(drule (1) mult_pos_neg_le[of a b], simp) |
|
1604 |
apply(drule (1) mult_pos_neg2_le[of b a], simp) |
|
14738 | 1605 |
done |
1606 |
next |
|
1607 |
assume "~(0 <= a*b)" |
|
1608 |
with s have "a*b <= 0" by simp |
|
1609 |
then show ?thesis |
|
1610 |
apply (simp_all add: mulprts abs_prts) |
|
1611 |
apply (insert prems) |
|
15580 | 1612 |
apply (auto simp add: ring_eq_simps) |
15197 | 1613 |
apply(drule (1) mult_pos_le[of a b],simp) |
1614 |
apply(drule (1) mult_neg_le[of a b],simp) |
|
14738 | 1615 |
done |
1616 |
qed |
|
1617 |
qed |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1618 |
|
14738 | 1619 |
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" |
1620 |
by (simp add: abs_eq_mult linorder_linear) |
|
14293 | 1621 |
|
14738 | 1622 |
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)" |
1623 |
by (simp add: abs_if) |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1624 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1625 |
lemma nonzero_abs_inverse: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1626 |
"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
|
1627 |
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
|
1628 |
negative_imp_inverse_negative) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1629 |
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
|
1630 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1631 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1632 |
lemma abs_inverse [simp]: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1633 |
"abs (inverse (a::'a::{ordered_field,division_by_zero})) = |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1634 |
inverse (abs a)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1635 |
apply (case_tac "a=0", simp) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1636 |
apply (simp add: nonzero_abs_inverse) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1637 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1638 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1639 |
lemma nonzero_abs_divide: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1640 |
"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
|
1641 |
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
|
1642 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1643 |
lemma abs_divide [simp]: |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1644 |
"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
|
1645 |
apply (case_tac "b=0", simp) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1646 |
apply (simp add: nonzero_abs_divide) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1647 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1648 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1649 |
lemma abs_mult_less: |
14738 | 1650 |
"[| 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
|
1651 |
proof - |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1652 |
assume ac: "abs a < c" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1653 |
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
|
1654 |
assume "abs b < d" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1655 |
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
|
1656 |
qed |
14293 | 1657 |
|
14738 | 1658 |
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))" |
1659 |
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff) |
|
1660 |
||
1661 |
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))" |
|
1662 |
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff) |
|
1663 |
||
1664 |
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" |
|
1665 |
apply (simp add: order_less_le abs_le_iff) |
|
1666 |
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff) |
|
1667 |
apply (simp add: le_minus_self_iff linorder_neq_iff) |
|
1668 |
done |
|
1669 |
||
15178 | 1670 |
lemma linprog_dual_estimate: |
1671 |
assumes |
|
1672 |
"A * x \<le> (b::'a::lordered_ring)" |
|
1673 |
"0 \<le> y" |
|
1674 |
"abs (A - A') \<le> \<delta>A" |
|
1675 |
"b \<le> b'" |
|
1676 |
"abs (c - c') \<le> \<delta>c" |
|
1677 |
"abs x \<le> r" |
|
1678 |
shows |
|
1679 |
"c * x \<le> y * b' + (y * \<delta>A + abs (y * A' - c') + \<delta>c) * r" |
|
1680 |
proof - |
|
1681 |
from prems have 1: "y * b <= y * b'" by (simp add: mult_left_mono) |
|
1682 |
from prems have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono) |
|
1683 |
have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: ring_eq_simps) |
|
1684 |
from 1 2 3 have 4: "c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x <= y * b'" by simp |
|
1685 |
have 5: "c * x <= y * b' + abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)" |
|
1686 |
by (simp only: 4 estimate_by_abs) |
|
1687 |
have 6: "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <= abs (y * (A - A') + (y * A' - c') + (c'-c)) * abs x" |
|
1688 |
by (simp add: abs_le_mult) |
|
1689 |
have 7: "(abs (y * (A - A') + (y * A' - c') + (c'-c))) * abs x <= (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x" |
|
16568 | 1690 |
by(rule abs_triangle_ineq [THEN mult_right_mono]) simp |
15178 | 1691 |
have 8: " (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x <= (abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x" |
1692 |
by (simp add: abs_triangle_ineq mult_right_mono) |
|
1693 |
have 9: "(abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x" |
|
1694 |
by (simp add: abs_le_mult mult_right_mono) |
|
1695 |
have 10: "c'-c = -(c-c')" by (simp add: ring_eq_simps) |
|
1696 |
have 11: "abs (c'-c) = abs (c-c')" |
|
1697 |
by (subst 10, subst abs_minus_cancel, simp) |
|
1698 |
have 12: "(abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x" |
|
1699 |
by (simp add: 11 prems mult_right_mono) |
|
1700 |
have 13: "(abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x" |
|
1701 |
by (simp add: prems mult_right_mono mult_left_mono) |
|
1702 |
have r: "(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r" |
|
1703 |
apply (rule mult_left_mono) |
|
1704 |
apply (simp add: prems) |
|
1705 |
apply (rule_tac add_mono[of "0::'a" _ "0", simplified])+ |
|
1706 |
apply (rule mult_left_mono[of "0" "\<delta>A", simplified]) |
|
1707 |
apply (simp_all) |
|
1708 |
apply (rule order_trans[where y="abs (A-A')"], simp_all add: prems) |
|
1709 |
apply (rule order_trans[where y="abs (c-c')"], simp_all add: prems) |
|
1710 |
done |
|
1711 |
from 6 7 8 9 12 13 r have 14:" abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <=(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r" |
|
1712 |
by (simp) |
|
1713 |
show ?thesis |
|
1714 |
apply (rule_tac le_add_right_mono[of _ _ "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"]) |
|
16568 | 1715 |
apply (simp_all only: 5 14[simplified abs_of_ge_0[of y, simplified prems]]) |
15178 | 1716 |
done |
1717 |
qed |
|
1718 |
||
1719 |
lemma le_ge_imp_abs_diff_1: |
|
1720 |
assumes |
|
1721 |
"A1 <= (A::'a::lordered_ring)" |
|
1722 |
"A <= A2" |
|
1723 |
shows "abs (A-A1) <= A2-A1" |
|
1724 |
proof - |
|
1725 |
have "0 <= A - A1" |
|
1726 |
proof - |
|
1727 |
have 1: "A - A1 = A + (- A1)" by simp |
|
1728 |
show ?thesis by (simp only: 1 add_right_mono[of A1 A "-A1", simplified, simplified prems]) |
|
1729 |
qed |
|
1730 |
then have "abs (A-A1) = A-A1" by (rule abs_of_ge_0) |
|
1731 |
with prems show "abs (A-A1) <= (A2-A1)" by simp |
|
1732 |
qed |
|
1733 |
||
15580 | 1734 |
lemma mult_le_prts: |
1735 |
assumes |
|
1736 |
"a1 <= (a::'a::lordered_ring)" |
|
1737 |
"a <= a2" |
|
1738 |
"b1 <= b" |
|
1739 |
"b <= b2" |
|
1740 |
shows |
|
1741 |
"a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1" |
|
1742 |
proof - |
|
1743 |
have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" |
|
1744 |
apply (subst prts[symmetric])+ |
|
1745 |
apply simp |
|
1746 |
done |
|
1747 |
then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b" |
|
1748 |
by (simp add: ring_eq_simps) |
|
1749 |
moreover have "pprt a * pprt b <= pprt a2 * pprt b2" |
|
1750 |
by (simp_all add: prems mult_mono) |
|
1751 |
moreover have "pprt a * nprt b <= pprt a1 * nprt b2" |
|
1752 |
proof - |
|
1753 |
have "pprt a * nprt b <= pprt a * nprt b2" |
|
1754 |
by (simp add: mult_left_mono prems) |
|
1755 |
moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2" |
|
1756 |
by (simp add: mult_right_mono_neg prems) |
|
1757 |
ultimately show ?thesis |
|
1758 |
by simp |
|
1759 |
qed |
|
1760 |
moreover have "nprt a * pprt b <= nprt a2 * pprt b1" |
|
1761 |
proof - |
|
1762 |
have "nprt a * pprt b <= nprt a2 * pprt b" |
|
1763 |
by (simp add: mult_right_mono prems) |
|
1764 |
moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1" |
|
1765 |
by (simp add: mult_left_mono_neg prems) |
|
1766 |
ultimately show ?thesis |
|
1767 |
by simp |
|
1768 |
qed |
|
1769 |
moreover have "nprt a * nprt b <= nprt a1 * nprt b1" |
|
1770 |
proof - |
|
1771 |
have "nprt a * nprt b <= nprt a * nprt b1" |
|
1772 |
by (simp add: mult_left_mono_neg prems) |
|
1773 |
moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1" |
|
1774 |
by (simp add: mult_right_mono_neg prems) |
|
1775 |
ultimately show ?thesis |
|
1776 |
by simp |
|
1777 |
qed |
|
1778 |
ultimately show ?thesis |
|
1779 |
by - (rule add_mono | simp)+ |
|
1780 |
qed |
|
1781 |
||
1782 |
lemma mult_le_dual_prts: |
|
15178 | 1783 |
assumes |
1784 |
"A * x \<le> (b::'a::lordered_ring)" |
|
1785 |
"0 \<le> y" |
|
15580 | 1786 |
"A1 \<le> A" |
1787 |
"A \<le> A2" |
|
1788 |
"c1 \<le> c" |
|
1789 |
"c \<le> c2" |
|
1790 |
"r1 \<le> x" |
|
1791 |
"x \<le> r2" |
|
15178 | 1792 |
shows |
15580 | 1793 |
"c * x \<le> y * b + (let s1 = c1 - y * A2; s2 = c2 - y * A1 in pprt s2 * pprt r2 + pprt s1 * nprt r2 + nprt s2 * pprt r1 + nprt s1 * nprt r1)" |
1794 |
(is "_ <= _ + ?C") |
|
15178 | 1795 |
proof - |
15580 | 1796 |
from prems have "y * (A * x) <= y * b" by (simp add: mult_left_mono) |
1797 |
moreover have "y * (A * x) = c * x + (y * A - c) * x" by (simp add: ring_eq_simps) |
|
1798 |
ultimately have "c * x + (y * A - c) * x <= y * b" by simp |
|
1799 |
then have "c * x <= y * b - (y * A - c) * x" by (simp add: le_diff_eq) |
|
1800 |
then have cx: "c * x <= y * b + (c - y * A) * x" by (simp add: ring_eq_simps) |
|
1801 |
have s2: "c - y * A <= c2 - y * A1" |
|
1802 |
by (simp add: diff_def prems add_mono mult_left_mono) |
|
1803 |
have s1: "c1 - y * A2 <= c - y * A" |
|
1804 |
by (simp add: diff_def prems add_mono mult_left_mono) |
|
1805 |
have prts: "(c - y * A) * x <= ?C" |
|
1806 |
apply (simp add: Let_def) |
|
1807 |
apply (rule mult_le_prts) |
|
1808 |
apply (simp_all add: prems s1 s2) |
|
15178 | 1809 |
done |
15580 | 1810 |
then have "y * b + (c - y * A) * x <= y * b + ?C" |
1811 |
by simp |
|
1812 |
with cx show ?thesis |
|
1813 |
by(simp only:) |
|
15178 | 1814 |
qed |
1815 |
||
14738 | 1816 |
ML {* |
14334 | 1817 |
val left_distrib = thm "left_distrib"; |
14738 | 1818 |
val right_distrib = thm "right_distrib"; |
1819 |
val mult_commute = thm "mult_commute"; |
|
1820 |
val distrib = thm "distrib"; |
|
1821 |
val zero_neq_one = thm "zero_neq_one"; |
|
1822 |
val no_zero_divisors = thm "no_zero_divisors"; |
|
14331 | 1823 |
val left_inverse = thm "left_inverse"; |
14738 | 1824 |
val divide_inverse = thm "divide_inverse"; |
1825 |
val mult_zero_left = thm "mult_zero_left"; |
|
1826 |
val mult_zero_right = thm "mult_zero_right"; |
|
1827 |
val field_mult_eq_0_iff = thm "field_mult_eq_0_iff"; |
|
1828 |
val inverse_zero = thm "inverse_zero"; |
|
1829 |
val ring_distrib = thms "ring_distrib"; |
|
1830 |
val combine_common_factor = thm "combine_common_factor"; |
|
1831 |
val minus_mult_left = thm "minus_mult_left"; |
|
1832 |
val minus_mult_right = thm "minus_mult_right"; |
|
1833 |
val minus_mult_minus = thm "minus_mult_minus"; |
|
1834 |
val minus_mult_commute = thm "minus_mult_commute"; |
|
1835 |
val right_diff_distrib = thm "right_diff_distrib"; |
|
1836 |
val left_diff_distrib = thm "left_diff_distrib"; |
|
1837 |
val mult_left_mono = thm "mult_left_mono"; |
|
1838 |
val mult_right_mono = thm "mult_right_mono"; |
|
1839 |
val mult_strict_left_mono = thm "mult_strict_left_mono"; |
|
1840 |
val mult_strict_right_mono = thm "mult_strict_right_mono"; |
|
1841 |
val mult_mono = thm "mult_mono"; |
|
1842 |
val mult_strict_mono = thm "mult_strict_mono"; |
|
1843 |
val abs_if = thm "abs_if"; |
|
1844 |
val zero_less_one = thm "zero_less_one"; |
|
1845 |
val eq_add_iff1 = thm "eq_add_iff1"; |
|
1846 |
val eq_add_iff2 = thm "eq_add_iff2"; |
|
1847 |
val less_add_iff1 = thm "less_add_iff1"; |
|
1848 |
val less_add_iff2 = thm "less_add_iff2"; |
|
1849 |
val le_add_iff1 = thm "le_add_iff1"; |
|
1850 |
val le_add_iff2 = thm "le_add_iff2"; |
|
1851 |
val mult_left_le_imp_le = thm "mult_left_le_imp_le"; |
|
1852 |
val mult_right_le_imp_le = thm "mult_right_le_imp_le"; |
|
1853 |
val mult_left_less_imp_less = thm "mult_left_less_imp_less"; |
|
1854 |
val mult_right_less_imp_less = thm "mult_right_less_imp_less"; |
|
1855 |
val mult_strict_left_mono_neg = thm "mult_strict_left_mono_neg"; |
|
1856 |
val mult_left_mono_neg = thm "mult_left_mono_neg"; |
|
1857 |
val mult_strict_right_mono_neg = thm "mult_strict_right_mono_neg"; |
|
1858 |
val mult_right_mono_neg = thm "mult_right_mono_neg"; |
|
1859 |
val mult_pos = thm "mult_pos"; |
|
1860 |
val mult_pos_le = thm "mult_pos_le"; |
|
1861 |
val mult_pos_neg = thm "mult_pos_neg"; |
|
1862 |
val mult_pos_neg_le = thm "mult_pos_neg_le"; |
|
1863 |
val mult_pos_neg2 = thm "mult_pos_neg2"; |
|
1864 |
val mult_pos_neg2_le = thm "mult_pos_neg2_le"; |
|
1865 |
val mult_neg = thm "mult_neg"; |
|
1866 |
val mult_neg_le = thm "mult_neg_le"; |
|
1867 |
val zero_less_mult_pos = thm "zero_less_mult_pos"; |
|
1868 |
val zero_less_mult_pos2 = thm "zero_less_mult_pos2"; |
|
1869 |
val zero_less_mult_iff = thm "zero_less_mult_iff"; |
|
1870 |
val mult_eq_0_iff = thm "mult_eq_0_iff"; |
|
1871 |
val zero_le_mult_iff = thm "zero_le_mult_iff"; |
|
1872 |
val mult_less_0_iff = thm "mult_less_0_iff"; |
|
1873 |
val mult_le_0_iff = thm "mult_le_0_iff"; |
|
1874 |
val split_mult_pos_le = thm "split_mult_pos_le"; |
|
1875 |
val split_mult_neg_le = thm "split_mult_neg_le"; |
|
1876 |
val zero_le_square = thm "zero_le_square"; |
|
1877 |
val zero_le_one = thm "zero_le_one"; |
|
1878 |
val not_one_le_zero = thm "not_one_le_zero"; |
|
1879 |
val not_one_less_zero = thm "not_one_less_zero"; |
|
1880 |
val mult_left_mono_neg = thm "mult_left_mono_neg"; |
|
1881 |
val mult_right_mono_neg = thm "mult_right_mono_neg"; |
|
1882 |
val mult_strict_mono = thm "mult_strict_mono"; |
|
1883 |
val mult_strict_mono' = thm "mult_strict_mono'"; |
|
1884 |
val mult_mono = thm "mult_mono"; |
|
1885 |
val less_1_mult = thm "less_1_mult"; |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1886 |
val mult_less_cancel_right_disj = thm "mult_less_cancel_right_disj"; |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
1887 |
val mult_less_cancel_left_disj = thm "mult_less_cancel_left_disj"; |
14738 | 1888 |
val mult_less_cancel_right = thm "mult_less_cancel_right"; |
1889 |
val mult_less_cancel_left = thm "mult_less_cancel_left"; |
|
1890 |
val mult_le_cancel_right = thm "mult_le_cancel_right"; |
|
1891 |
val mult_le_cancel_left = thm "mult_le_cancel_left"; |
|
1892 |
val mult_less_imp_less_left = thm "mult_less_imp_less_left"; |
|
1893 |
val mult_less_imp_less_right = thm "mult_less_imp_less_right"; |
|
1894 |
val mult_cancel_right = thm "mult_cancel_right"; |
|
1895 |
val mult_cancel_left = thm "mult_cancel_left"; |
|
1896 |
val ring_eq_simps = thms "ring_eq_simps"; |
|
1897 |
val right_inverse = thm "right_inverse"; |
|
1898 |
val right_inverse_eq = thm "right_inverse_eq"; |
|
1899 |
val nonzero_inverse_eq_divide = thm "nonzero_inverse_eq_divide"; |
|
1900 |
val divide_self = thm "divide_self"; |
|
1901 |
val divide_zero = thm "divide_zero"; |
|
1902 |
val divide_zero_left = thm "divide_zero_left"; |
|
1903 |
val inverse_eq_divide = thm "inverse_eq_divide"; |
|
1904 |
val add_divide_distrib = thm "add_divide_distrib"; |
|
1905 |
val field_mult_eq_0_iff = thm "field_mult_eq_0_iff"; |
|
1906 |
val field_mult_cancel_right_lemma = thm "field_mult_cancel_right_lemma"; |
|
1907 |
val field_mult_cancel_right = thm "field_mult_cancel_right"; |
|
1908 |
val field_mult_cancel_left = thm "field_mult_cancel_left"; |
|
1909 |
val nonzero_imp_inverse_nonzero = thm "nonzero_imp_inverse_nonzero"; |
|
1910 |
val inverse_zero_imp_zero = thm "inverse_zero_imp_zero"; |
|
1911 |
val inverse_nonzero_imp_nonzero = thm "inverse_nonzero_imp_nonzero"; |
|
1912 |
val inverse_nonzero_iff_nonzero = thm "inverse_nonzero_iff_nonzero"; |
|
1913 |
val nonzero_inverse_minus_eq = thm "nonzero_inverse_minus_eq"; |
|
1914 |
val inverse_minus_eq = thm "inverse_minus_eq"; |
|
1915 |
val nonzero_inverse_eq_imp_eq = thm "nonzero_inverse_eq_imp_eq"; |
|
1916 |
val inverse_eq_imp_eq = thm "inverse_eq_imp_eq"; |
|
1917 |
val inverse_eq_iff_eq = thm "inverse_eq_iff_eq"; |
|
1918 |
val nonzero_inverse_inverse_eq = thm "nonzero_inverse_inverse_eq"; |
|
1919 |
val inverse_inverse_eq = thm "inverse_inverse_eq"; |
|
1920 |
val inverse_1 = thm "inverse_1"; |
|
1921 |
val nonzero_inverse_mult_distrib = thm "nonzero_inverse_mult_distrib"; |
|
1922 |
val inverse_mult_distrib = thm "inverse_mult_distrib"; |
|
1923 |
val inverse_add = thm "inverse_add"; |
|
1924 |
val inverse_divide = thm "inverse_divide"; |
|
1925 |
val nonzero_mult_divide_cancel_left = thm "nonzero_mult_divide_cancel_left"; |
|
1926 |
val mult_divide_cancel_left = thm "mult_divide_cancel_left"; |
|
1927 |
val nonzero_mult_divide_cancel_right = thm "nonzero_mult_divide_cancel_right"; |
|
1928 |
val mult_divide_cancel_right = thm "mult_divide_cancel_right"; |
|
1929 |
val mult_divide_cancel_eq_if = thm "mult_divide_cancel_eq_if"; |
|
1930 |
val divide_1 = thm "divide_1"; |
|
1931 |
val times_divide_eq_right = thm "times_divide_eq_right"; |
|
1932 |
val times_divide_eq_left = thm "times_divide_eq_left"; |
|
1933 |
val divide_divide_eq_right = thm "divide_divide_eq_right"; |
|
1934 |
val divide_divide_eq_left = thm "divide_divide_eq_left"; |
|
1935 |
val nonzero_minus_divide_left = thm "nonzero_minus_divide_left"; |
|
1936 |
val nonzero_minus_divide_right = thm "nonzero_minus_divide_right"; |
|
1937 |
val nonzero_minus_divide_divide = thm "nonzero_minus_divide_divide"; |
|
1938 |
val minus_divide_left = thm "minus_divide_left"; |
|
1939 |
val minus_divide_right = thm "minus_divide_right"; |
|
1940 |
val minus_divide_divide = thm "minus_divide_divide"; |
|
1941 |
val diff_divide_distrib = thm "diff_divide_distrib"; |
|
1942 |
val positive_imp_inverse_positive = thm "positive_imp_inverse_positive"; |
|
1943 |
val negative_imp_inverse_negative = thm "negative_imp_inverse_negative"; |
|
1944 |
val inverse_le_imp_le = thm "inverse_le_imp_le"; |
|
1945 |
val inverse_positive_imp_positive = thm "inverse_positive_imp_positive"; |
|
1946 |
val inverse_positive_iff_positive = thm "inverse_positive_iff_positive"; |
|
1947 |
val inverse_negative_imp_negative = thm "inverse_negative_imp_negative"; |
|
1948 |
val inverse_negative_iff_negative = thm "inverse_negative_iff_negative"; |
|
1949 |
val inverse_nonnegative_iff_nonnegative = thm "inverse_nonnegative_iff_nonnegative"; |
|
1950 |
val inverse_nonpositive_iff_nonpositive = thm "inverse_nonpositive_iff_nonpositive"; |
|
1951 |
val less_imp_inverse_less = thm "less_imp_inverse_less"; |
|
1952 |
val inverse_less_imp_less = thm "inverse_less_imp_less"; |
|
1953 |
val inverse_less_iff_less = thm "inverse_less_iff_less"; |
|
1954 |
val le_imp_inverse_le = thm "le_imp_inverse_le"; |
|
1955 |
val inverse_le_iff_le = thm "inverse_le_iff_le"; |
|
1956 |
val inverse_le_imp_le_neg = thm "inverse_le_imp_le_neg"; |
|
1957 |
val less_imp_inverse_less_neg = thm "less_imp_inverse_less_neg"; |
|
1958 |
val inverse_less_imp_less_neg = thm "inverse_less_imp_less_neg"; |
|
1959 |
val inverse_less_iff_less_neg = thm "inverse_less_iff_less_neg"; |
|
1960 |
val le_imp_inverse_le_neg = thm "le_imp_inverse_le_neg"; |
|
1961 |
val inverse_le_iff_le_neg = thm "inverse_le_iff_le_neg"; |
|
1962 |
val one_less_inverse_iff = thm "one_less_inverse_iff"; |
|
1963 |
val inverse_eq_1_iff = thm "inverse_eq_1_iff"; |
|
1964 |
val one_le_inverse_iff = thm "one_le_inverse_iff"; |
|
1965 |
val inverse_less_1_iff = thm "inverse_less_1_iff"; |
|
1966 |
val inverse_le_1_iff = thm "inverse_le_1_iff"; |
|
1967 |
val zero_less_divide_iff = thm "zero_less_divide_iff"; |
|
1968 |
val divide_less_0_iff = thm "divide_less_0_iff"; |
|
1969 |
val zero_le_divide_iff = thm "zero_le_divide_iff"; |
|
1970 |
val divide_le_0_iff = thm "divide_le_0_iff"; |
|
1971 |
val divide_eq_0_iff = thm "divide_eq_0_iff"; |
|
1972 |
val pos_le_divide_eq = thm "pos_le_divide_eq"; |
|
1973 |
val neg_le_divide_eq = thm "neg_le_divide_eq"; |
|
1974 |
val le_divide_eq = thm "le_divide_eq"; |
|
1975 |
val pos_divide_le_eq = thm "pos_divide_le_eq"; |
|
1976 |
val neg_divide_le_eq = thm "neg_divide_le_eq"; |
|
1977 |
val divide_le_eq = thm "divide_le_eq"; |
|
1978 |
val pos_less_divide_eq = thm "pos_less_divide_eq"; |
|
1979 |
val neg_less_divide_eq = thm "neg_less_divide_eq"; |
|
1980 |
val less_divide_eq = thm "less_divide_eq"; |
|
1981 |
val pos_divide_less_eq = thm "pos_divide_less_eq"; |
|
1982 |
val neg_divide_less_eq = thm "neg_divide_less_eq"; |
|
1983 |
val divide_less_eq = thm "divide_less_eq"; |
|
1984 |
val nonzero_eq_divide_eq = thm "nonzero_eq_divide_eq"; |
|
1985 |
val eq_divide_eq = thm "eq_divide_eq"; |
|
1986 |
val nonzero_divide_eq_eq = thm "nonzero_divide_eq_eq"; |
|
1987 |
val divide_eq_eq = thm "divide_eq_eq"; |
|
1988 |
val divide_cancel_right = thm "divide_cancel_right"; |
|
1989 |
val divide_cancel_left = thm "divide_cancel_left"; |
|
1990 |
val divide_eq_1_iff = thm "divide_eq_1_iff"; |
|
1991 |
val one_eq_divide_iff = thm "one_eq_divide_iff"; |
|
1992 |
val zero_eq_1_divide_iff = thm "zero_eq_1_divide_iff"; |
|
1993 |
val one_divide_eq_0_iff = thm "one_divide_eq_0_iff"; |
|
1994 |
val divide_strict_right_mono = thm "divide_strict_right_mono"; |
|
1995 |
val divide_right_mono = thm "divide_right_mono"; |
|
1996 |
val divide_strict_left_mono = thm "divide_strict_left_mono"; |
|
1997 |
val divide_left_mono = thm "divide_left_mono"; |
|
1998 |
val divide_strict_left_mono_neg = thm "divide_strict_left_mono_neg"; |
|
1999 |
val divide_strict_right_mono_neg = thm "divide_strict_right_mono_neg"; |
|
2000 |
val less_add_one = thm "less_add_one"; |
|
2001 |
val zero_less_two = thm "zero_less_two"; |
|
2002 |
val less_half_sum = thm "less_half_sum"; |
|
2003 |
val gt_half_sum = thm "gt_half_sum"; |
|
2004 |
val dense = thm "dense"; |
|
2005 |
val abs_one = thm "abs_one"; |
|
2006 |
val abs_le_mult = thm "abs_le_mult"; |
|
2007 |
val abs_eq_mult = thm "abs_eq_mult"; |
|
2008 |
val abs_mult = thm "abs_mult"; |
|
2009 |
val abs_mult_self = thm "abs_mult_self"; |
|
2010 |
val nonzero_abs_inverse = thm "nonzero_abs_inverse"; |
|
2011 |
val abs_inverse = thm "abs_inverse"; |
|
2012 |
val nonzero_abs_divide = thm "nonzero_abs_divide"; |
|
2013 |
val abs_divide = thm "abs_divide"; |
|
2014 |
val abs_mult_less = thm "abs_mult_less"; |
|
2015 |
val eq_minus_self_iff = thm "eq_minus_self_iff"; |
|
2016 |
val less_minus_self_iff = thm "less_minus_self_iff"; |
|
2017 |
val abs_less_iff = thm "abs_less_iff"; |
|
14331 | 2018 |
*} |
2019 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
2020 |
end |