# HG changeset patch # User paulson # Date 1401373669 -3600 # Node ID f00a299fa522b39812d3ccc0e533132140b3f24d # Parent 7e95523302e68b810220480ed56609d71ff5c512# Parent 70395c65c0e36e8505fa2704caaa9dc63f1e8409 Merge diff -r 7e95523302e6 -r f00a299fa522 NEWS --- a/NEWS Thu May 29 14:39:19 2014 +0100 +++ b/NEWS Thu May 29 15:27:49 2014 +0100 @@ -755,6 +755,8 @@ * Renamed abbreviation integral\<^sup>P to integral\<^sup>N. +* Library/Kleene-Algebra was removed because AFP/Kleene_Algebra subsumes it. + *** Scala *** * The signature and semantics of Document.Snapshot.cumulate_markup / diff -r 7e95523302e6 -r f00a299fa522 src/HOL/Library/Kleene_Algebra.thy --- a/src/HOL/Library/Kleene_Algebra.thy Thu May 29 14:39:19 2014 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,485 +0,0 @@ -(* Title: HOL/Library/Kleene_Algebra.thy - Author: Alexander Krauss, TU Muenchen - Author: Tjark Weber, University of Cambridge -*) - -header {* Kleene Algebras *} - -theory Kleene_Algebra -imports Main -begin - -text {* WARNING: This is work in progress. Expect changes in the future. *} - -text {* Various lemmas correspond to entries in a database of theorems - about Kleene algebras and related structures maintained by Peter - H\"ofner: see - @{url "http://www.informatik.uni-augsburg.de/~hoefnepe/kleene_db/lemmas/index.html"}. *} - -subsection {* Preliminaries *} - -text {* A class where addition is idempotent. *} - -class idem_add = plus + - assumes add_idem [simp]: "x + x = x" - -text {* A class of idempotent abelian semigroups (written additively). *} - -class idem_ab_semigroup_add = ab_semigroup_add + idem_add -begin - -lemma add_idem2 [simp]: "x + (x + y) = x + y" -unfolding add_assoc[symmetric] by simp - -lemma add_idem3 [simp]: "x + (y + x) = x + y" -by (simp add: add_commute) - -end - -text {* A class where order is defined in terms of addition. *} - -class order_by_add = plus + ord + - assumes order_def: "x \ y \ x + y = y" - assumes strict_order_def: "x < y \ x \ y \ \ y \ x" -begin - -lemma ord_simp [simp]: "x \ y \ x + y = y" - unfolding order_def . - -lemma ord_intro: "x + y = y \ x \ y" - unfolding order_def . - -end - -text {* A class of idempotent abelian semigroups (written additively) - where order is defined in terms of addition. *} - -class ordered_idem_ab_semigroup_add = idem_ab_semigroup_add + order_by_add -begin - -lemma ord_simp2 [simp]: "x \ y \ y + x = y" - unfolding order_def add_commute . - -subclass order proof - fix x y z :: 'a - show "x \ x" - unfolding order_def by simp - show "x \ y \ y \ z \ x \ z" - unfolding order_def by (metis add_assoc) - show "x \ y \ y \ x \ x = y" - unfolding order_def by (simp add: add_commute) - show "x < y \ x \ y \ \ y \ x" - by (fact strict_order_def) -qed - -subclass ordered_ab_semigroup_add proof - fix a b c :: 'a - assume "a \ b" show "c + a \ c + b" - proof (rule ord_intro) - have "c + a + (c + b) = a + b + c" by (simp add: add_ac) - also have "\ = c + b" by (simp add: `a \ b` add_ac) - finally show "c + a + (c + b) = c + b" . - qed -qed - -lemma plus_leI [simp]: - "x \ z \ y \ z \ x + y \ z" - unfolding order_def by (simp add: add_assoc) - -lemma less_add [simp]: "x \ x + y" "y \ x + y" -unfolding order_def by (auto simp: add_ac) - -lemma add_est1 [elim]: "x + y \ z \ x \ z" -using less_add(1) by (rule order_trans) - -lemma add_est2 [elim]: "x + y \ z \ y \ z" -using less_add(2) by (rule order_trans) - -lemma add_supremum: "(x + y \ z) = (x \ z \ y \ z)" -by auto - -end - -text {* A class of commutative monoids (written additively) where - order is defined in terms of addition. *} - -class ordered_comm_monoid_add = comm_monoid_add + order_by_add -begin - -lemma zero_minimum [simp]: "0 \ x" -unfolding order_def by simp - -end - -text {* A class of idempotent commutative monoids (written additively) - where order is defined in terms of addition. *} - -class ordered_idem_comm_monoid_add = ordered_comm_monoid_add + idem_add -begin - -subclass ordered_idem_ab_semigroup_add .. - -lemma sum_is_zero: "(x + y = 0) = (x = 0 \ y = 0)" -by (simp add: add_supremum eq_iff) - -end - -subsection {* A class of Kleene algebras *} - -text {* Class @{text pre_kleene} provides all operations of Kleene - algebras except for the Kleene star. *} - -class pre_kleene = semiring_1 + idem_add + order_by_add -begin - -subclass ordered_idem_comm_monoid_add .. - -subclass ordered_semiring proof - fix a b c :: 'a - assume "a \ b" - - show "c * a \ c * b" - proof (rule ord_intro) - from `a \ b` have "c * (a + b) = c * b" by simp - thus "c * a + c * b = c * b" by (simp add: distrib_left) - qed - - show "a * c \ b * c" - proof (rule ord_intro) - from `a \ b` have "(a + b) * c = b * c" by simp - thus "a * c + b * c = b * c" by (simp add: distrib_right) - qed -qed - -end - -text {* A class that provides a star operator. *} - -class star = - fixes star :: "'a \ 'a" - -text {* Finally, a class of Kleene algebras. *} - -class kleene = pre_kleene + star + - assumes star1: "1 + a * star a \ star a" - and star2: "1 + star a * a \ star a" - and star3: "a * x \ x \ star a * x \ x" - and star4: "x * a \ x \ x * star a \ x" -begin - -lemma star3' [simp]: - assumes a: "b + a * x \ x" - shows "star a * b \ x" -by (metis assms less_add mult_left_mono order_trans star3 zero_minimum) - -lemma star4' [simp]: - assumes a: "b + x * a \ x" - shows "b * star a \ x" -by (metis assms less_add mult_right_mono order_trans star4 zero_minimum) - -lemma star_unfold_left: "1 + a * star a = star a" -proof (rule antisym, rule star1) - have "1 + a * (1 + a * star a) \ 1 + a * star a" - by (metis add_left_mono mult_left_mono star1 zero_minimum) - with star3' have "star a * 1 \ 1 + a * star a" . - thus "star a \ 1 + a * star a" by simp -qed - -lemma star_unfold_right: "1 + star a * a = star a" -proof (rule antisym, rule star2) - have "1 + (1 + star a * a) * a \ 1 + star a * a" - by (metis add_left_mono mult_right_mono star2 zero_minimum) - with star4' have "1 * star a \ 1 + star a * a" . - thus "star a \ 1 + star a * a" by simp -qed - -lemma star_zero [simp]: "star 0 = 1" -by (fact star_unfold_left[of 0, simplified, symmetric]) - -lemma star_one [simp]: "star 1 = 1" -by (metis add_idem2 eq_iff mult_1_right ord_simp2 star3 star_unfold_left) - -lemma one_less_star [simp]: "1 \ star x" -by (metis less_add(1) star_unfold_left) - -lemma ka1 [simp]: "x * star x \ star x" -by (metis less_add(2) star_unfold_left) - -lemma star_mult_idem [simp]: "star x * star x = star x" -by (metis add_commute add_est1 eq_iff mult_1_right distrib_left star3 star_unfold_left) - -lemma less_star [simp]: "x \ star x" -by (metis less_add(2) mult_1_right mult_left_mono one_less_star order_trans star_unfold_left zero_minimum) - -lemma star_simulation_leq_1: - assumes a: "a * x \ x * b" - shows "star a * x \ x * star b" -proof (rule star3', rule order_trans) - from a have "a * x * star b \ x * b * star b" - by (rule mult_right_mono) simp - thus "x + a * (x * star b) \ x + x * b * star b" - using add_left_mono by (auto simp: mult_assoc) - show "\ \ x * star b" - by (metis add_supremum ka1 mult.right_neutral mult_assoc mult_left_mono one_less_star zero_minimum) -qed - -lemma star_simulation_leq_2: - assumes a: "x * a \ b * x" - shows "x * star a \ star b * x" -proof (rule star4', rule order_trans) - from a have "star b * x * a \ star b * b * x" - by (metis mult_assoc mult_left_mono zero_minimum) - thus "x + star b * x * a \ x + star b * b * x" - using add_mono by auto - show "\ \ star b * x" - by (metis add_supremum distrib_right less_add mult.left_neutral mult_assoc mult_right_mono star_unfold_right zero_minimum) -qed - -lemma star_simulation [simp]: - assumes a: "a * x = x * b" - shows "star a * x = x * star b" -by (metis antisym assms order_refl star_simulation_leq_1 star_simulation_leq_2) - -lemma star_slide2 [simp]: "star x * x = x * star x" -by (metis star_simulation) - -lemma star_idemp [simp]: "star (star x) = star x" -by (metis add_idem2 eq_iff less_star mult_1_right star3' star_mult_idem star_unfold_left) - -lemma star_slide [simp]: "star (x * y) * x = x * star (y * x)" -by (metis mult_assoc star_simulation) - -lemma star_one': - assumes "p * p' = 1" "p' * p = 1" - shows "p' * star a * p = star (p' * a * p)" -proof - - from assms - have "p' * star a * p = p' * star (p * p' * a) * p" - by simp - also have "\ = p' * p * star (p' * a * p)" - by (simp add: mult_assoc) - also have "\ = star (p' * a * p)" - by (simp add: assms) - finally show ?thesis . -qed - -lemma x_less_star [simp]: "x \ x * star a" -by (metis mult.right_neutral mult_left_mono one_less_star zero_minimum) - -lemma star_mono [simp]: "x \ y \ star x \ star y" -by (metis add_commute eq_iff less_star ord_simp2 order_trans star3 star4' star_idemp star_mult_idem x_less_star) - -lemma star_sub: "x \ 1 \ star x = 1" -by (metis add_commute ord_simp star_idemp star_mono star_mult_idem star_one star_unfold_left) - -lemma star_unfold2: "star x * y = y + x * star x * y" -by (subst star_unfold_right[symmetric]) (simp add: mult_assoc distrib_right) - -lemma star_absorb_one [simp]: "star (x + 1) = star x" -by (metis add_commute eq_iff distrib_right less_add mult_1_left mult_assoc star3 star_mono star_mult_idem star_unfold2 x_less_star) - -lemma star_absorb_one' [simp]: "star (1 + x) = star x" -by (subst add_commute) (fact star_absorb_one) - -lemma ka16: "(y * star x) * star (y * star x) \ star x * star (y * star x)" -by (metis ka1 less_add(1) mult_assoc order_trans star_unfold2) - -lemma ka16': "(star x * y) * star (star x * y) \ star (star x * y) * star x" -by (metis ka1 mult_assoc order_trans star_slide x_less_star) - -lemma ka17: "(x * star x) * star (y * star x) \ star x * star (y * star x)" -by (metis ka1 mult_assoc mult_right_mono zero_minimum) - -lemma ka18: "(x * star x) * star (y * star x) + (y * star x) * star (y * star x) - \ star x * star (y * star x)" -by (metis ka16 ka17 distrib_right mult_assoc plus_leI) - -lemma star_decomp: "star (x + y) = star x * star (y * star x)" -proof (rule antisym) - have "1 + (x + y) * star x * star (y * star x) \ - 1 + x * star x * star (y * star x) + y * star x * star (y * star x)" - by (metis add_commute add_left_commute eq_iff distrib_right mult_assoc) - also have "\ \ star x * star (y * star x)" - by (metis add_commute add_est1 add_left_commute ka18 plus_leI star_unfold_left x_less_star) - finally show "star (x + y) \ star x * star (y * star x)" - by (metis mult_1_right mult_assoc star3') -next - show "star x * star (y * star x) \ star (x + y)" - by (metis add_assoc add_est1 add_est2 add_left_commute less_star mult_mono' - star_absorb_one star_absorb_one' star_idemp star_mono star_mult_idem zero_minimum) -qed - -lemma ka22: "y * star x \ star x * star y \ star y * star x \ star x * star y" -by (metis mult_assoc mult_right_mono plus_leI star3' star_mult_idem x_less_star zero_minimum) - -lemma ka23: "star y * star x \ star x * star y \ y * star x \ star x * star y" -by (metis less_star mult_right_mono order_trans zero_minimum) - -lemma ka24: "star (x + y) \ star (star x * star y)" -by (metis add_est1 add_est2 less_add(1) mult_assoc order_def plus_leI star_absorb_one star_mono star_slide2 star_unfold2 star_unfold_left x_less_star) - -lemma ka25: "star y * star x \ star x * star y \ star (star y * star x) \ star x * star y" -proof - - assume "star y * star x \ star x * star y" - hence "\x\<^sub>1. star y * (star x * x\<^sub>1) \ star x * (star y * x\<^sub>1)" by (metis mult_assoc mult_right_mono zero_minimum) - hence "star y * (star x * star y) \ star x * star y" by (metis star_mult_idem) - hence "\x\<^sub>1. star (star y * star x) * star x\<^sub>1 \ star x * star y" by (metis star_decomp star_idemp star_simulation_leq_2 star_slide) - hence "\x\<^sub>1\star (star y * star x). x\<^sub>1 \ star x * star y" by (metis x_less_star) - thus "star (star y * star x) \ star x * star y" by (metis order_trans) -qed - -lemma church_rosser: - "star y * star x \ star x * star y \ star (x + y) \ star x * star y" -by (metis add_commute ka24 ka25 order_trans) - -lemma kleene_bubblesort: "y * x \ x * y \ star (x + y) \ star x * star y" -by (metis church_rosser star_simulation_leq_1 star_simulation_leq_2) - -lemma ka27: "star (x + star y) = star (x + y)" -by (metis add_commute star_decomp star_idemp) - -lemma ka28: "star (star x + star y) = star (x + y)" -by (metis add_commute ka27) - -lemma ka29: "(y * (1 + x) \ (1 + x) * star y) = (y * x \ (1 + x) * star y)" -by (metis add_supremum distrib_right less_add(1) less_star mult.left_neutral mult.right_neutral order_trans distrib_left) - -lemma ka30: "star x * star y \ star (x + y)" -by (metis mult_left_mono star_decomp star_mono x_less_star zero_minimum) - -lemma simple_simulation: "x * y = 0 \ star x * y = y" -by (metis mult.right_neutral mult_zero_right star_simulation star_zero) - -lemma ka32: "star (x * y) = 1 + x * star (y * x) * y" -by (metis mult_assoc star_slide star_unfold_left) - -lemma ka33: "x * y + 1 \ y \ star x \ y" -by (metis add_commute mult.right_neutral star3') - -end - -subsection {* Complete lattices are Kleene algebras *} - -lemma (in complete_lattice) SUP_upper': - assumes "l \ M i" - shows "l \ (SUP i. M i)" - using assms by (rule order_trans) (rule SUP_upper [OF UNIV_I]) - -class kleene_by_complete_lattice = pre_kleene - + complete_lattice + power + star + - assumes star_cont: "a * star b * c = SUPREMUM UNIV (\n. a * b ^ n * c)" -begin - -subclass kleene -proof - fix a x :: 'a - - have [simp]: "1 \ star a" - unfolding star_cont[of 1 a 1, simplified] - by (subst power_0[symmetric]) (rule SUP_upper [OF UNIV_I]) - - have "a * star a \ star a" - using star_cont[of a a 1] star_cont[of 1 a 1] - by (auto simp add: power_Suc[symmetric] simp del: power_Suc - intro: SUP_least SUP_upper) - - then show "1 + a * star a \ star a" - by simp - - then show "1 + star a * a \ star a" - using star_cont[of a a 1] star_cont[of 1 a a] - by (simp add: power_commutes) - - show "a * x \ x \ star a * x \ x" - proof - - assume a: "a * x \ x" - - { - fix n - have "a ^ (Suc n) * x \ a ^ n * x" - proof (induct n) - case 0 thus ?case by (simp add: a) - next - case (Suc n) - hence "a * (a ^ Suc n * x) \ a * (a ^ n * x)" - by (auto intro: mult_mono) - thus ?case - by (simp add: mult_assoc) - qed - } - note a = this - - { - fix n have "a ^ n * x \ x" - proof (induct n) - case 0 show ?case by simp - next - case (Suc n) with a[of n] - show ?case by simp - qed - } - note b = this - - show "star a * x \ x" - unfolding star_cont[of 1 a x, simplified] - by (rule SUP_least) (rule b) - qed - - show "x * a \ x \ x * star a \ x" (* symmetric *) - proof - - assume a: "x * a \ x" - - { - fix n - have "x * a ^ (Suc n) \ x * a ^ n" - proof (induct n) - case 0 thus ?case by (simp add: a) - next - case (Suc n) - hence "(x * a ^ Suc n) * a \ (x * a ^ n) * a" - by (auto intro: mult_mono) - thus ?case - by (simp add: power_commutes mult_assoc) - qed - } - note a = this - - { - fix n have "x * a ^ n \ x" - proof (induct n) - case 0 show ?case by simp - next - case (Suc n) with a[of n] - show ?case by simp - qed - } - note b = this - - show "x * star a \ x" - unfolding star_cont[of x a 1, simplified] - by (rule SUP_least) (rule b) - qed -qed - -end - -subsection {* Transitive closure *} - -context kleene -begin - -definition - tcl_def: "tcl x = star x * x" - -lemma tcl_zero: "tcl 0 = 0" -unfolding tcl_def by simp - -lemma tcl_unfold_right: "tcl a = a + tcl a * a" -by (metis star_slide2 star_unfold2 tcl_def) - -lemma less_tcl: "a \ tcl a" -by (metis star_slide2 tcl_def x_less_star) - -end - -end diff -r 7e95523302e6 -r f00a299fa522 src/HOL/Library/Library.thy --- a/src/HOL/Library/Library.thy Thu May 29 14:39:19 2014 +0100 +++ b/src/HOL/Library/Library.thy Thu May 29 15:27:49 2014 +0100 @@ -34,7 +34,6 @@ Lattice_Syntax ListVector Lubs_Glbs - Kleene_Algebra Mapping Monad_Syntax Multiset