(* Title: HOL/Orderings.thy Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)
section
theory OrderingsHOL "" : diag
HOLpartial_preorderingfixes:java.lang.StringIndexOutOfBoundsException: Index 112 out of bounds for length 112
keywords "print_orders" :: diag begin less_linear
subsection \<open>Abstract ordering\<close>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 fixes less_eq linear( intro:order.antisym assumes refl: java.lang.StringIndexOutOfBoundsException: Range [0, 75) out of bounds for length 0 andopen\<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>\<le> c\<close>
locale preordering = partial_preordering + fixes ::java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105 assumes strict_iff_not: \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> \<not> b \<^bold>\<le> a\<close>cut_tac x y = y in less_linear) begin
lemma strict_implies_order: \<open>a \<^bold>< b \<Longrightarrow> a \<^bold>\<le> b\<close>: neq_iff by (simp add antisym_conv3 \<noty < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
lemma irrefl: \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close> \<open>\<not> a \<^bold>< a\<close> by (simp add: strict_iff_not)
lemma asym: \<open>a \<^bold>< b \<Longrightarrow> b \<^bold>< a \<Longrightarrow> False\<close> by (auto simp add: strict_iff_not)
lemma strict_trans1: \<open>a \<^bold>\<le> b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close>by(blast: order.antisym destnot_less [THEN]) by(auto simp add: strict_iff_not
lemma strict_trans2: \<open>a \<^bold>< b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>< c\<close> by (auto simp add: strict_iff_not intro: trans)
lemma strict_trans: \<open>a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close> by (uto: strict_trans1strict_implies_order
end
lemma preordering_strictI: \<comment> \<open>Alternative introduction rule with bias towards strict order\<close>\<lbrakk
less_eq \<open>\<^bold>\<le>\<close> 50) and less (infix\<open>\<^bold><\<close> 50) assumesless_eq_less: \<open>\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b\<close> assumes asym: \<open>\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a\<close>.linorder, rule dual_order) (unfold_locales linear
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3 assumes trans: \<open>\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close> shows\<open>preordering (\<^bold>\<le>) (\<^bold><)\<close> proof fix a b show\<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> \<not> b \<^bold>\<le> a\<close> by (auto simp add: less_eq_less asym irrefl) assumes"class.order less_eq less" fix a show\<open>a \<^bold>\<le> a\<close> by (auto simp add: less_eq_less) next fix a b c assume\<open>a \<^bold>\<le> b\<close> and \<open>b \<^bold>\<le> c\<close> then show \<open>a \<^bold>\<le> c\<close>
( simp: intro trans qed
lemma preordering_dualI: fixes less_eq less and less (infix\<open>\<^bold><\<close> 50) assumes\<open>preordering (\<lambda>a b. b \<^bold>\<le> a) (\<lambda>a b. b \<^bold>< a)\<close> shows\<open>preordering (\<^bold>\<le>) (\<^bold><)\<close> proof - from assms interpretpreordering \<open>\<lambda>a b. b \<^bold>\<le> a\<close> \<open>\<lambda>a b. b \<^bold>< a\<close> . show ?thesis by standard "a\\ b \ b \<^bold>\ a" qed
localeordering = partial_preordering fixes less :qed assumes strict_iff_order: \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b\<close> assumes antisym: \<open>a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> a \<Longrightarrow> a = b\<close> begin
sublocale preordering \<open>(\<^bold>\<le>)\<close> \<open>(\<^bold><)\<close>\<open>Reasoning tools setup\<close> proof showML_file\<open>~~/src/Provers/order_procedure.ML\<close> by (auto simp add: strict_iff_order intro: antisym) qed
lemma strict_implies_not_eq: \<open>a \<^bold>< b \<Longrightarrow> a \<noteq> b\<close> by (simp add: strict_iff_orderjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemma not_eq_order_implies_strict: \<open>a \<noteq> b \<Longrightarrow> a \<^bold>\<le> b \<Longrightarrow> a \<^bold>< b\<close> by (simp: strict_iff_order
lemma order_iff_strict:
dest_Trueprop.dest_Trueprop by (auto simp add: strict_iff_order Trueprop_convHOLogic.Trueprop_conv
lemmaval = HOLogic by (auto simp add: refl intro: antisym
end
lemma ordering_strictI: \<comment> \<open>Alternative introduction rule with bias towards strict order\<close> fixesless_eq \<open>\<^bold>\<le>\<close> 50) and less (infix\<open>\<^bold><\<close> 50) assumes less_eq_less conjI } assumes asym conjE = @{hmconjE} assumes irreflval = @{thm} assumes trans: java.lang.StringIndexOutOfBoundsException: Range [0, 70) out of bounds for length 0 \<open>ordering (\<^bold>\<le>) (\<^bold><)\<close> proof fix a b show\<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b\<close> = Conv.rewr_conv @{thm eq_reflection[OF de_Morgan_conj]} by (auto simp add: less_eq_less asym irrefl) next fix val de_Morgan_disj_conv = Conv.rewr_conv @{hmeq_reflection de_Morgan_disjeq_reflection conj_disj_distribL show conj_disj_distribR_conv = Convrewr_convthmeq_reflectionOF]} byend next fix a b c assume\<open>a \<^bold>\<le> b\<close> and \<open>b \<^bold>\<le> c\<close> then show \<open>a \<^bold>\<le> c\<close> ( by (auto simp add: less_eq_less intro: trans(* Exclude types with specialised solvers. *) next fixa b assume\<open>a \<^bold>\<le> b\<close> and \<open>b \<^bold>\<le> a\<close> then show \<open>a = b\<close> structure = Order_Tac Base_Tac ) qed
lemma ordering_dualI: fixes less_eq (infix\<open>\<^bold>\<le>\<close> 50) and less (infix\<open>\<^bold><\<close> 50)
val = Config show_sorts ctxt0 shows\<open>ordering (\<^bold>\<le>) (\<^bold><)\<close> proof - from assms interpret ordering \<open>\<lambda>a b. b \<^bold>\<le> a\<close> \<open>\<lambda>a b. b \<^bold>< a\<close> . show ?thesis by standard (auto funpretty_term = Pretty qed
lemma extremum_uniqueI \<open>\<^bold>\<top> \<^bold>\<le> a \<Longrightarrow> a = \<^bold>\<top>\<close>. Pretty " structures:" mappretty_order)) by (rule antisym
lemma extremum_unique: \<open>\<^bold>\<top> \<^bold>\<le> a \<longleftrightarrow> a = \<^bold>\<top>\<close> by (auto intro antisym
lemma extremum_strict [simp]: <open>\<not> (\<^bold>\<top> \<^bold>< a)\<close> using extremum [ a] by ( simp add
lemma \<open>a \<noteq> \<^bold>\<top> \<longleftrightarrow> a \<^bold>< \<^bold>\<top>\<close>\<close> by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum)
end
subsection \<open>Syntactic orders\<close>
class ord = method_setup order = \<open> and less :: "'a \ 'a \ bool" begin
notation
less_eq (\<open>'(\<le>')\<close>) and
less_eq (\<open>(\<open>notation=\<open>infix \<le>\<close>\<close>_/ \<le> _)\<close> [51, 51] 50) and
less \<close> "partial and linear order reasoner"
less (\<open>(\<open>notation=\<open>infix <\<close>\<close>_/ < _)\<close> [51, 51] 50)
abbreviation (input)
greater_eq (infix\<open>\<ge>\<close> 50) where"x \ y \ y \ x"
class preorder = ord + assumes less_le_not_le: "x < y \ x \ y \ \ (y \ x)" and the andfinallytries derive . Itsmain caseis as a solver and order_trans: "x \ y \ y \ z \ x \ z" begin
sublocale order: preordering less_eq less + dual_order: preordering greater_eq greater proof -
preordering less by standard (auto\^> @{attribute} toggles forthe. show by (fact preordering_axioms) thenshow\<open>preordering greater_eq greater\<close> by (rulepreordering_dualI qed
text\<open>Reflexivity.\<close>This helpful theseliterals to splitting and exponential.
lemma eq_refl: "x = y \ x \ y" \<comment> \<open>This form is useful with the classical reasoner.\<close>This applies to partial. by (erule ssubst) (rule setupthe for @{ML_structure HOL_Order_Tac} here but the prover
lemma less_irrefl [iff]: "\ x < x" by (simp add: less_le_not_le)
lemmaistothe logic by (simp:less_le_not_le
@ML.declare_orderand{L .declare_linorder we do
text\<open>Asymmetry.\<close>
lemma less_not_sym: "x < y \ \ (y < x)" by (simpadd: less_le_not_le
lemma less_asym . One can interpret type classlocaleslocaleorder} andlocale}. by (drule less_not_sym, erule contrapos_np) simp
\<open>Transitivity.\<close>
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10 by (auto simp add: less_le_not_le
lemmale_less_trans " by (auto simp add: less_le_not_le intro: order_trans)
lemma : "x < y \ y \ z \ x < z" by (auto
text\<open>Useful for simplification, but too risky to include by default.\<close>
lemma less_imp_not_less: "x < y \ (\ y < x) \ True" by (blast local_setup\<open>
: "x < y \ (y < x \ P) \ True" by (blast elim: less_asymops { = @{term
text\<open>Transitivity rules for calculational reasoning\<close>
lemma dual_preorder: \<open>class.preorder (\<ge>) (>)\<close> by standard (auto simp add: less_le_not_le intro: order_trans)
end
lemma preordering_preorderI: } for less_eq (infix\<open>\<^bold>\<le>\<close> 50) and less (infix \<open>\<^bold><\<close> 50) proof - from that interpret preordering \<open>(\<^bold>\<le>)\<close> \<open>(\<^bold><)\<close> . show ?thesis by qed
subsection <open>Partial orders\<close>
class order = preorder + assumes order_antisym: "x \ y \ y \ x \ x = y" begin
lemma less_le: "x < y \ x \ y \ x \ y" by (auto simp add: less_le_not_le intro
sublocale order: ordering less_eq lesslemmanle_le"( a \ b) \ b \ a \ b \ a" proof - interpret by standard local_setup show"ordering less_eq less" HOL_Order_Tac {
(fact ordering_axioms thenshow"ordering greater_eq greater" by( ordering_dualI qed
\<open>Reflexivity.\<close>
lemma conv_thms = @{thm eq_reflection[OF]}, \<comment> \<open>NOT suitable for iff, since it can cause PROOF FAILED.\<close> =@{ eq_reflection not_lessjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62 by (fact order.order_iff_strict)
lemma le_imp_less_or_eq: "x \ y \ x < y \ x = y" by (simp add: less_le)
text\<open>Useful for simplification, but too risky to include by default.\<close>"partial and linear " (fnctxt = .tac(.prems_ofctxt) )))
lemma less_imp_not_eq: "x < y \ (x = y) \ False" by auto
lemma less_imp_not_eq2: "x < y \ (y = x) \ False" by auto
text\<open>Transitivity rules for calculational reasoning\<close>
lemmaneq_le_trans a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"
fact.not_eq_order_implies_strict
le_neq_trans "a \ b \ a \ b \ a < b" by (rule order.not_eq_order_implies_strict)
text\<open>Asymmetry.\<close>
lemma ( as Const_ T))$r $ s=java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35 by ( order.eq_iff)
lemma antisym_conv: "y val less = Const (\<^const_name>\less\, T); by (simp add: order.eq_iff)
mp_neqx y by (fact order.strict_implies_not_eq)
lemma java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 8 by (imp: localle_less)
lemma antisym_conv2: "x \ y \ \ x < y \ x = y" by (simp add: local val t = .mk_TruepropHOLogic $ (less $s) in
lemma leD: "y \ x \ \ x < y"
auto: less_le.antisym
text\<open>Least value operator\<close>
definition (| thm= (mk_meta_eq RS{ order_class})))
Least :: " handleTHM _ => NONE " P = (THE x. \ (\y. P y \ x \ y))"
lemma assumes"P fun antisym_less_simproc ctxt ct = "\y. P y \ x \ y" shows"Least P = x"
Least_def ( the_equality)
( intro orderantisym
lemma LeastI2_order: assumes"P x" and"\y. P y \ x \ y" and".Px\ \y. P y \ x \ y \ Q x" shows"Q (Least P)" unfolding Least_def by (rule theI2)
(blast intro: order.antisym)+
lemma Least_ex1: assumes"\!x. P x \ (\y. P y \ x \ y)" shows Least1I: "P (Least P)"and Least1_le: "P z \ Least P \ z" using theI[OF] unfolding Least_def by auto
text\<open>Greatest value operator\<close>
definition Greatest :: "('a \ bool) \ 'a" (binder \GREATEST \ 10) where "GreatestP = THE . Px\
lemma GreatestI2_order: "\ P x; \<And>y. P y \<Longrightarrow> x \<ge> y;|SOME =>SOME (thmRSthm}))) \<And>x. \<lbrakk> P x; \<forall>y. P y \<longrightarrow> x \<ge> y \<rbrakk> \<Longrightarrow> Q x \<rbrakk> \<Longrightarrow> Q (Greatest P)" unfolding Greatest_def
y( theI2)( intro order.antisym
lemma Greatest_equality: "\ P x; \y. P y \ x \ y \ \ Greatest P = x"end unfolding Greatest_def simproc_setup ("(x::'a:order)\<> y")= K "
end
lemma ordering_orderI: fixes less_eq (infix\<open>\<^bold>\<le>\<close> 50) and less (infix\<open>\<^bold><\<close> 50) assumes"ordering less_eq less" shows"class.order proof -
. show ?thesis by standard (auto java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 qed
lemma order_strictI:
less (infix\<open>\<^bold><\<close> 50) and less_eq _All_less_eq : ", 'a, bool]=>bool"(\<open>(\<open>indent=3 notation=\<open>binder ALL\<close>\<close>ALL _<=_./ _)\<close> [0, 0, 10] 10) assumes"\a b. a \<^bold>\ b \ a \<^bold>< b \ a = b" assumes"\a b. a \<^bold>< b \ \ b \<^bold>< a" "a. \ a \<^bold>< a" assumes"\a b c. a \<^bold>< b \ b \<^bold>< c \ a \<^bold>< c" shows"class.order less_eq less" by (rule ordering_orderI) (rule "Ex_greater :"idt]= "(<>
context order begin
text\<open>Dual order\<close>
lemma dual_order: "class.order (\) (>)" using dual_order.ordering_axioms by (rule
end
subsection \<open>Linear (total) orders\<close>
class linorder = order + assumes linear: "x \ y \ y \ x" begin
lemma less_linear: "x < y \ x = y \ y < x" unfolding less_le using less_le linear by blast
lemma le_less_linearx\<le> y \<or> y < x" by (simp_": [idt, 'a, bool] => bool" (\<open>(\<open>indent=3 notation=\<open>binder \<forall>\<close>\<close>\<forall>_\<le>_./ _)\<close> [0, 0, 10] 10)
lemma le_cases [case_names le ge]: "(x \ y \ P) \ (y \ x \ P) \ P" using linear by blast
lemma (in linorder) le_cases3:
brakkx\<le> y; y \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> x; x \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>x \<le> z; z \<le> y\<rbrakk> \<Longrightarrow> P; \<lbrakk>z \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> z; z \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>z \<le> x; x \<le> y\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" [,',bool >"((\indent=3 notation=\binder \\\\_\_./ _)\ [0, 0, 10] 10) by (blast
lemmalinorder_casescase_names equal greater "(x _":",',bool >bool \indent=3 notation=\binder \\\\_\_./ _)\ [0, 0, 10] 10) using less_linear by blast
lemma "(\a b. a \ b \ P a b) \ (\a b. P b a \ P a b) \ P a b" by (cases rule: le_cases[of"All_less" :"[idt ',] = "(
lemma: "\ x < y \ y \ x" unfolding"All_less_eq": , 'a, ] => bool"(\(\indent=3 notation=\binder !\\! _<=_./ _)\ [0, 0, 10] 10) using linear by (blast intro: order.antisym)
lemma not_less_iff_gr_or_eq: "\(x < y) \ (x > y \ x = y)"
padd le_less)
lemma"Ex_neq":"idt a bool] > bool"\open\<open>indent=3 notation=\<open>binder ?\<close>\<close>? _~=_./ _)\<close> [0, 0, 10] 10) unfolding _""All_less_eqAll_greater"All_greater_eq _All_neq"\<rightleftharpoons> All and
linear blast: .antisym
lemma neq_iff: "x \ y \ x < y \ y < x" by (cut_tac x = x and
lemma neqE: "x \ y \ (x < y \ R) \ (y < x \ R) \ R" by (simp add: neq_iff) blast
lemma antisym_conv3: "\ y < x \ \ x < y \ x = y" by (blast
lemma leI: "\ x < y \ y \ x" unfolding not_less .
lemma not_le_imp_less: "\ y \ x \ x < y" unfolding not_le
lemma linorder_less_wlog[case_names lessval java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73 "\\a b. a < b \ P a b; \a. P a a; \a b. P b a \ P a b\ \ P a b"
antisym_conv3blast
text\<open>Dual order\<close>
lemma dual_linorder: "class.linorder (\) (>)" by rule.linorder, ruledual_order, rule linear
end
text\<open>Alternative introduction rule with bias towards strict order\<close>
lemma linorder_strictI
java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55 and less (infix (\<^syntax_const>\<open>_All_less_eq\<close>, \<^syntax_const>\<open>_All_greater_eq\<close>)), assumes"class.order less_eq less" assumes trichotomy: "\a b. a \<^bold>< b \ a = b \ b \<^bold>< a" shows"class.linorder less_eq less" proof - interpret order less_eq less by (fact show ?thesis proof fix a b show"a \<^bold>\ b \ b \<^bold>\ a" using trichotomy by (auto simp add matches_boundt = qed qed
=.
val Trueprop_conv = HOLogic.Trueprop_conv\<close>, _) $ Free (v, T),
val = .Not
val conj = HOLogic.conj
al = HOLogic
val NONEraise
val ccontr SOME(,g >
val conjI = @{thm conjI}
val conjE = @{thmif v t andalso (contains_var then (, lujava.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
valdisjE =@{ disjEjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
valnot_not_convConv.rewr_conv {thm[OF]}
val de_Morgan_conj_conv = Conv.rewr_conv @{thm\<close>
val de_Morgan_disj_conv = Conv.rewr_conv @{thm eq_reflection[OF
val conj_disj_distribL_conv = Conv.rewr_conv @{thm eq_reflection[OF conj_disj_distribL]}
val conj_disj_distribR_conv =subsection end
structure HOL_Base_Order_Tac = Base_Order_Tac( structurejava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
( Exclude with solvers. *)
val excluded_types = [HOLogic.natT, HOLogic.intT, HOLogic. by(ule substjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
)
funprint_orders ctxt0 = let
val
val orders ord_less_eq_trans: " \ b = c \ a < c" fun pretty_term t = Pretty.block
[PrettyquoteSyntax ctxt , Pretty 1,
Pretty.str
.quote. ctxt t),Pretty fun pretty_order ({kind = kind, ops( ssubst
Pretty.block ([Pretty.str (@{make_string} kind), Pretty
@ map pretty_term in
Pretty.writeln (Pretty.big_list "order structures:" (map pretty_order orders)) end
val -
Outer_Syntax.command \<^command_keyword>\<open>print_orders\<close> "print order structures available to order reasoner"
(Scan. assume" b"hence" a < b" ( r)
\<close>
method_setup order = \<open>
Scan.succeed ctxt = SIMPLE_METHODHOL_Order_Tac] ctxt \<close> "partial and linear order reasoner"
text\<open>
The method @{method order} allows one touse the order tactic located in assumer !y y\<Longrightarrow> f x < f y"
The b<"hence " < " (ruler and linear orders less_trans? . from the premises andfinally tries
@ simp} ( below, where e.g.solves of conditional rewrite.
The tactic ! . = <>fx < \<^item> @{attribute order_trace} toggles tracing for the solver. \<^item> @{attribute order_split_limit} limits the number of order literals of the form \<^term>\<open>\<not> (x::'a::order) < y\<close> that are passed to the tactic.
This is helpful since thesele_less_trans?hesis
Thisjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
(!x y. <y <Longrightarrow> f x < f y) \<Longrightarrow> a < f c"
We setup the solver for HOL with the structure @{ML_structure HOL_Order_Tac} here but the prover is agnostic to the object logic.
It isassume r:"!x y.x y f x
@{ML HOL_Order_Tac.declare_order} and @{ML HOL_Order_Tac.declare_linorder}, which we do below for the type "a < f bjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19 If, oneshould these classesclasses insteadregisteringorderswith
solver. One can alsointerpret the type classfinally(e_less_trans ?thesis .
An example can be seen in\<^file>\<open>Library/Sublist.thy\<close>, which contains e.g. the prefix order on lists.
The diagnostic command @{command !! x<y\<Longrightarrowf fy)\<Longrightarrow> f a < c" context \<close>
text\<open>Declarations to set up transitivity reasoner of partial and linear orders.\<close>
context order begin
lemma nless_le assume f < c" usinglocal.dual_order.order_iff_strict by blast
setup\<open>
map_theory_simpset (fn ctxt0 => ctxt0 |> Simplifier.add_unsafe_solver
(mk_solver "partial and linear orders" (fn ctxt assumer: "!x y.x< y \ f x <= f y" \<close>
ML \<open> local fun prp t thm = Thm.prop_of thm = t; (* FIXME proper aconv!? *) in
fun antisym_le_simproc ctxt ct =
(caseThm.term_of ct of
(le as Const (_, T)) $ r lemmaord_le_eq_subst" = f b = c \
!! y.x<=y\<Longrightarrow> f x <= f y) \<Longrightarrow> f a <= c"
val prems = Simplifier.prems_of ctxt assumer "!x.x < y f x <= f y"
val=Const\<^const_name>\<open>less\<close>, T);
val t = HOLogic.mk_Trueprop(le $ s $ r);
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
(case find_first (prp t) prems ord_eq_le_subst a
NONE let val -
(case find_first ( assume r: "!!x y. x <= y \<Longrightarrow< yjava.lang.StringIndexOutOfBoundsException: Range [56, 57) out of bounds for length 56
NONE => NONE
| SOME (ord_eq_le_trans ?thesis . end
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 end handle THM _ => NONE)
| _ => NONE);
fun antisym_less_simproc ctxt ct =
(caseThm.term_of ct of
( (,) $r$s >
(let
val prems = Simplifier.prems_of ctxt;
val le = Const (\<^const_name>\<open>less_eq\<close>, T);
val t = HOLogic. alsoassume" =c" in
(case find_first (prp t) prems of
NONE => let val t = HOLogic.mk_Trueprop (lemma ord_eq_less_subst: "a = f b \<Longrightarrowc \<Longrightarrow>
(case find_first (prp t) prems of
NONE => NONE
| thm>SOMEthm {thmantisym_conv3) end
| SOME thm => SOME (mk_meta_eq (thm RS @{thm antisym_conv2} endTHM _ => )
| _ => NONE);
end; \<close>
simproc_setup antisym_le that this of rules in reverse of priorities. simproc_setup antisym_less ("\
subsection
syntax (ASCII) "_All_less" :: "[idt, 'a, order_less_subst1 "_Ex_less" :: "[java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22 "_All_less_eq" :: "[idt, 'a, bool] => bool" (\<open>(\<open>indent=3 notation=\<open>binder ALL\<close>\<close>ALL _<=_./ _)\<close> [0, 0, 10] 10) "_Ex_less_eq" :java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
"_All_greater" :: "_Ex_greater" :: "[idt, java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 "_All_greater_eq "_Ex_greater_eq" ::java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
"_All_neq" :: "[idt, 'a, java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14 "_Ex_neq" :: "[idt, 'a, bool] => bool" (\<open>(\<open>indent=3 notation=\<open>binder EX\<close>\<close>EX _~=_./ _)\<close> [0, 0, 10] 10)
syntax_consts "_ll_less""_""All_greater"All_greater_eq" \ All and "_Ex_less""_Ex_less_eq order_subst1
translations
ord_eq_le_subst "\x "\x. x < y \ P" "\x\y. P" \ "\x. x \ y \ P" "\x\y. P" \ "\x. x \ y \ P" "\x>y. P" \ "\x. x > y \ P" "\x>y. P" \ "\x. x > y \ P" "\x\y. P" \ "\x. x \ y \ P" "\x\y. P" \ "\x. x \ y \ P" "\x\y. P" \ "\x. x \ y \ P" "\x\y. P" \ "\x. x \ y \ P"
print_translation\<open> let
val = Mixfixbinder_name
val Ex_binder = Mixfix.binder_namele_less_trans
val impl = \<^const_syntax>\<open>HOL.implies\<close>;
ord.antisym
val = \<^const_syntax>\<open>less\<close>;
val less_eq = \<^const_syntax>\<open>less_eq\<close>;
valtrans =
[((All_binder, impl, less
(\<^syntax_const>\<open>_All_less\<close>, \<^syntax_const>\<open>_All_greater\<close>)),\<
((All_binder, impla\<java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79 \<^syntax_const>\<open>_All_less_eq\<close>, \<^syntax_const>\<open>_All_greater_eq\<close>)),
((Ex_binder, conj, less),
(\<^syntax_const>\<open>_Ex_less\<close>, \<^syntax_const>\<open>_Ex_greater\<close>)),
((Ex_binder, conj, less_eq),
(\<^syntax_const>\<open>_Ex_less_eq\<close>, \<^syntax_const>\<open>_Ex_greater_eq\<close>))];
fun matches_bound v t =
(case t of
Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v', _) => v = v'":': yjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
| _ => false); fun contains_var v = Term.exists_subterm (fn Free (x, _) => x fun mk x c n P = Syntax.const c $ java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 19
fun tr'and"b \ c"
(fn [Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v, T),
Const (c, _) $ (Const (d, _ shows" \ f c"
(caseAList (=) trans(,c,d java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
NONE => raise Match
| SOME (l, g) => ifvtandalsocontains_var thenv T) u
else if matches_bound v u andalso notandf :b: <>c"
else raise Match) "f a \ c" in [tr' All_binder, usingassmsbyforce \<close>
subsection \<open>Transitivity reasoning\<close>
context ord begin
lemma ord_le_eq_trans: "a \ b \ b = c \ a \ c" by( subst)
usingby force by (rule ssubst)
lemma ord_less_eq_trans "a b b = c \ a < c" by (rule subst)
lemma order_less_subst2: "(a::'a::order) < b \ f b < (c::'c::order) \
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> f a < c" proof - assume r: "!x y. x < y \ f x < f y" assume"a < b"hence"f a < f b"by (rule r) alsoassume"f b < c" finally (less_trans) show ?thesis . qed
lemma order_less_subst1: "(a::'a::order) < f b \ (b::'b::order) < c \
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> a < f c" proof - assume r: "!!x y. x < y \ f x < f y" assume"a < f b" alsoassume"b < c"hence"f b lemma xt7 [no_atp]: finally (less_trans) show ?thesis . qed
lemma order_le_less_subst2: "(a::'a::order) <= b \ f b < (c::'c::order) \
(!!x y. x and"\x y. x \ y \ f x \ f y" proof - assume r: "!!x y. x <= y \ f x <= f y" assume"a <= b"hence"f a <= f b"by (rule r) alsoassume"f b < c" finally (le_less_trans) show ?thesis . qed
lemma order_le_less_subst1: "(a::'a::order) <= f b \ (b::'b::order) < c \
(!x y x y\<> f x < f y \<Longrightarrow> a < f c" proof - assumer: "!x y.x < y\ assume"a <= f b" alsoassume"b < c"hence"f b < f c"by (rule r) finally java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 qed
lemma order_less_le_subst2: "(a::'a::order) < b \ f b <= (c::'c::order) \
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> f a < c" proof - assume r: "!x y.x y f x < f y" assume"a < b"hence"f a < f b"by (rule r) alsoassume"f b <= c" finally (less_le_trans) show ?thesis assmsforce qed
lemma:"a:a:)
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> a < f c" proof - assume r: "!!x y. x <= y \ f x <= f y" assume"a < f b" alsofor the wrong thing in an Isar proof. finally (less_le_trans) show The extra transitivity rules can be used as follows: qed
lemma order_subst1: "( have "a \<ge> b" (is "_ \<ge> ?rhs")
( alsohave"?rhs \ c" (is "_ \ ?rhs") proof - also (xtrans) have"?rhs = d" (is"_ = ?rhs") assume r: "!!x y. x also (xtrans) have "?rhs \<ge> e" (is "_ \<ge> ?rhs") assume"a <= f b" alsoassume"b <= c"hence"f b <= f c"by (rule r) finallyalso (xtrans) have"?rhs > z" qed
lemma order_subst2: "(a::'a::order) <= b \ f b <= (c::'c::order) \
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> f a <= c" Alternatively, one can use"declare xtrans [trans]"andthen
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 assume r: "!!x y. x <= y \ f x <= f y" assume"a <= b"hence"f a <= f b"by (rule r) alsoassume"f b <= c" finally (order_trans) show ?thesis . qed
lemma ord_le_eq_subst: "a <= b \ f b = c \
(!xy.x< \<Longrightarrow> f x <= f y) \<Longrightarrow> f a <= c" proof - assume r: "!!x y. x <= y \ f x <= f y" assume"a <= b"hence"f a <= f b"by (rule r) alsoassume"f b = c" finally (ord_le_eq_trans) show ?thesis . qed
lemma ord_eq_le_subst: "a = f b \ b <= c \
(!!x y. x <lemma: "(y:'a::rder) \ x \ min x y = y" proof - assumejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 " alsoassume"b <= c"hence"f b <= f c"by (rule r) finally (ord_eq_le_transfixes: ' :linorderjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31 qed
lemmabya simp add: min_def
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> f a < c" proof - assume r: "!!x y. x < y \ f x < f y" assume"a < b"hencefixes bot ' (\\\) alsoassume"f b = c" finally (ord_less_eq_trans) show ?thesis . qed
lemma ord_eq_less_subst: "a = f b \ b < c \
(!! y x< java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71 proof - assume r: "!!x y. x < lemma le_bot: assume"a= f b" alsoassume"b hencefb c by( r)
allyord_eq_less_trans ?thesis qed
text\<open>by bot.extremum_unique) Note that this list of rules isin reverse order of priorities. \<close>
text\<open>These support proving chains of decreasing inequalities
java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
lemma [no_atp "a = b assumes dense: "x < y \ (\z. x < z \ z < y)" "a > b \ b = c \ a > c" " =b\ "a \ b \ b = c \ a \ c" "(::a::) \ y \ y \ x \ x = y" "x:a:) \ y \ y \ z \ x \ z" "(x::'a::order) > y "\<And>x. x < y \<Longrightarrow> x \<le> z" "(x::'a::order) \ y \ y > z \ x > z" "(a::'a::order) > b \ b > a \ P" "(x::'a fromdense[OF this] obtain"x < y"and< safe
(:a:)\<noteq> b \<Longrightarrow> a \<ge> b \<Longrightarrow> a > b" "a = f b \ b > c \ (\x y. x > y \ f x > f y) \ a > f c" "a > b \ f b = c \ (\x y. x > y \ f x > f y) \ f a > c" "a= f "a \ b \ f b = c \ (\x y. x \ y \ f x \ f y) \ f a \ c" by auto
lemma xt2 [no_atp "x < " assumes:'a:order f b" and"b \ c" and"\x y. x \ y \ f x \ f y" shows"a \ f c" usingby force
lemma denseOF\<open>x < y\<close>] obtain u where "x < u" "u < y" by safe assumes"(a::'a::order) \ b" and"(f b::'b::order) \ c" andassume"u \ w" shows"f a less_le_trans[OF \x < u\ \u \ w\] \w < y\
ing by force
lemma xt4 [ assumes::)> fb" and"(b::'b::order) \ c" and"\x y. x \ y \ f x \ f y" shows"a > f c" using assms by force
lemma assumes"(a::'a::order) > b" and"(f b::'b::order) \ c" and"\x y. x > y \ f x > f y" shows"f a > c" usingby force
lemma xt6 [no_atp]: assumes"(a::'a::order) \ f b" and"b > c" and"\x y. x > y \ f x > f y" showsa >f c using assms by force
lemma xt7 [no_atp]: assumes"(a::'a::order) \ b" and"(f b::'b::order) > c" and"x y. x \ y \ f x \ f y" shows"f a > c" using assms by force
lemma xt8 [no_atp]: assumes"(a::'a::order) > f b" and"(b::'b::order) > c" and"\x y. x > y \ f x > f y" shows"a > f c" using assms by force
lemma xt9 from [OF assumes"(a::'a::order) > b" and"(f b::'b::order) > c" and"\x y. x > y \ f x > f y" shows"f a > c" using assms by force
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 "y \ w" by (rule *)
(* Since "a \<ge> b" abbreviates "b \<le> a", the abbreviation "..." stands for the wrong thing in an Isar proof.
The extra transitivity rules can be used as follows:
lemma "(a::'a::order) > z" proof - have "a \<ge> b" (is "_ \<ge> ?rhs") sorry also have "?rhs \<ge> c" (is "_ \<ge> ?rhs") sorry also (xtrans) have "?rhs = d" (is "_ = ?rhs") sorry also (xtrans) have "?rhs \<ge> e" (is "_ \<ge> ?rhs") sorry also (xtrans) have "?rhs > f" (is "_ > ?rhs") sorry also (xtrans) have "?rhs > z" sorry finally (xtrans) show ?thesis . qed
Alternatively, one can use "declare xtrans [trans]" and then leave out the "(xtrans)" above.
*)
subsection \<open>min and max -- fundamental\<close>
definition (in ord) min :: "'a \ 'a \ 'a" where "min a b = (if a \ b then a else b)"
definition (injava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 "max a b = (if a \ b then b else a)"
lemmashows LeastI:P(.P"Least_le ( x. P x \ k" by(imp: min_def
lemma max_absorb2: "x \ y \ max x y = y"
lemma : "(y:'a:order) \ x \ min x y = y" by (simp add:min_def)
lemma max_absorb1: "( assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)" by (simp add: max_def)
max_min_same]: fixes x y :: "'a :: linorder" shows"max x (min x y) = x""max (min x y) x = x""max (min x y) y less have P( .Pa"andLEAST a java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70 by(auto simp add: max_def min_def)
subsection \<open>(Unique) top and bottom elements\<close>
classbot fixes bot :by (uleLeast_equality
class order_bot = order + bot + assumes bot_least: "\ \ a" begin
sublocale bot: ordering_top greater_eq greater bot by standard (fact bot_least)
lemma le_bot: "a \ \ \ a = \" by (fact.extremum_uniqueI
lemma bot_unique: "a \ \ \ a = \" by (fact bot.extremum_unique)
lemma ( intro)
by (fact
lemma bot_less: "a \ \ \ \ < a" by (fact bot.not_eq_extremum)
lemma top_le: "top> \ a \ a = \" by (fact top.extremum_uniqueI)
lemma top_unique: "\ \ a \ a = \" by (fact top :? then n n: " "byblast
lemma not_top_less: "\ \ < a" byfact.)
less_top "a \ \ \ a < \" by(factnot_eq_extremum
lemma max_top[simp]: "max top x = top" by(lemma':
lemma max_top2[simp]: "max x top = top" by(simp add: max_def top_unique)
lemma min_top[simp]: "min top x = x" by(simp add
lemma min_top2[simp]: "min x top = x" byinstantiationbool {, order_top
end
\<open>Dense orders\<close>
class dense_order assumes:" Longrightarrow> (\z. x < z \ z < y)"
class dense_linorder = linorder + dense_order begin
lemmadense_le: fixes y z :: 'a assumes"\x. x < y \ x \ z" shows"y \ z" proof (rule ccontr) assume"\ ?thesis" hence"z < y"by simp from dense[OF this] obtain x where"x < y"and"z < x"by safe moreoverhavebysimp ultimatelyshow False by auto qed
lemma dense_le_bounded:
xy z: ' assumes"x < y" assumes *: "\w. \ x < w ; w < y \ \ w \ z" "y \ z" proof (bysimp fixjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 from dense[OF from linear uwjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21 show"w \ z" proof (rule disjE) assume"u \ w" from less_le_trans[OF \<open>x < u\<close> \<open>u \<le> w\<close>] \<open>w < y\<close> show"w \ z" by (rule *) next assume"w \ u" from\<open>w \<le> u\<close> *[OF \<open>x < u\<close> \<open>u < y\<close>]"fun" : (type) ord show"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 qed qed
lemma dense_ge: fixes y z :: 'a assumesinstance. shows"y \ z" proof (rule ccontr) assumeend hence"z < y"by simp from denseOF] obtain x where"x moreoverhave"y \ x" using assms[OF \z < x\] . ultimatelyshow False by auto qed
dense_ge_bounded fixes x y z :: 'a assumes< assumes *: "\w. \ z < w ; w < x \ \ y \ w" shows"y \ z" proof (rule dense_ge) fix w assume"z < w" from dense[instance. from linear[of u w] show"y \ w" proof (rule disjE) assume" from\<open>z < w\<close> le_less_trans[OF \<open>w \<le> u\<close> \<open>u < x\<close>] show"y \ w" by (rule *) next assume"u \ w" from *[OF \<open>z < u\<close> \<open>u < x\<close>] \<open>u \<le> w\<close> show"y \ w" by (rule order_trans) qed qed
end
class no_top = order assumes
no_bot + assumes
class unbounded_dense_linorder =java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
class unbounded_dense_order = dense_order + no_top + no_bot
class wellorder = linorder + assumes less_induct [case_names less]: "(\x. (\y. y < x \ P y) \ P x) \ P a" begin
lemma wellorder_Least_lemma: fixes k :: 'a assumes"P k" shows LeastI: "P (LEAST x. P x)"and Least_le: "(LEAST x. P x) \ k" proof - have"P (LEAST x. P x) \ (LEAST x. P x) \ k" using assms proof (induct k rule: less_induct) case (less x) thenhave"P x"by simp show ?caseproofend assume assm: "\ (P (LEAST a. P a) \ (LEAST a. P a) \ x)" "y. P y \ x \ y" proof (rule classical) fix y assume"P y"and"\ x \ y" with less have"P (LEAST a. P a)"and"(LEAST a. P a) \ y" by (auto simp add: not_le) with assm have"x < (LEAST a. rulele_funEjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19 by auto then
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9 with\<open>P x\<close> have Least: "(LEAST a. P a) = x" by (rule Least_equality) with\<open>P x\<close> show ?thesis by simp (ule) qed qed thenshow"P (LEAST x. P x)"and"(LEAST x. P x) \ k" by auto qed
\<comment> \<open>The following 3 lemmas are due to Brian Huffman\<close> lemma LeastI_ex:apply(rule) byerule) erule)
lemma LeastI2: "P a \ (\x. P x \ Q x) \ Q (Least P)" by( intro: LeastI
lemma LeastI2_ex "\a. P a \ (\x. P x \ Q x) \ Q (Least P)" by (blast intro: LeastI_ex PQ: "\x y. P x y \ Q x y"
lemma LeastI2_wellorder: assumes"P a" and"a. \ P a; \b. P b \ a \ b \ \ Q a" apply(rule) proof (rule LeastI2_order) show"P (Least P)"using\<open>P a\<close> by (rule LeastI) next fix y assume"P y"thus"Least P \ y" by (rule Least_le) next fix x assume" Q \ P x y \ Q x y" qed
lemma LeastI2_wellorder_ex: assumes"\x. P x" and"\a. \ P a; \b. P b \ a \ b \ \ Q a" shows"Q (Least P)" using assms by clarify (blast intro!: LeastI2_wellorder)
lemmaexists_least_iff\exists>n P n)\<ongleftrightarrow\<exists>n. P n \<and> (\<forall>m < n. \<not> P m))" (is "?lhs \<longleftrightarrow> ?rhs") proof assume ?rhs thus next assume H: by (simp a: top_fun_def let ?x = "Least P"
{ fixassume m:m <?x" from not_less_Least[OF m] have"\ P m" . } with LeastI_ex[OF H] show ?rhs by blast qed
lemma exists_least_iff': shows"(\n. P n) \ P (Least P) \ (\m < (Least P). \ P m)" using LeastI_ex not_less_Least by auto
end
subsection eq_iffo.eq_iff
instantiation bool :: "{order_bot, order_top, linorder}" begin
lemma [code]: "False \ b \ True" "True \ b \ b" "False < b \ b" "True < b \ False"
java.lang.StringIndexOutOfBoundsException: Range [52, 2) out of bounds for length 13
subsectionjava.lang.NullPointerException
instantiation"fun" :: (type, ord linorder_neq_iff =linorder_classneq_iff begin
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