(* Title: HOL/Orderings.thy
Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)
section \<open>Abstract orderings\<close>
theory Orderings
imports HOL
keywords "print_orders" :: diag
begin
ML_file \<open>~~/src/Provers/order.ML\<close>
subsection \<open>Abstract ordering\<close>
locale ordering =
fixes less_eq :: "'a \ 'a \ bool" (infix "\<^bold>\" 50)
and less :: "'a \ 'a \ bool" (infix "\<^bold><" 50)
assumes strict_iff_order: "a \<^bold>< b \ a \<^bold>\ b \ a \ b"
assumes refl: "a \<^bold>\ a" \ \not \iff\: makes problems due to multiple (dual) interpretations\
and antisym: "a \<^bold>\ b \ b \<^bold>\ a \ a = b"
and trans: "a \<^bold>\ b \ b \<^bold>\ c \ a \<^bold>\ c"
begin
lemma strict_implies_order:
"a \<^bold>< b \ a \<^bold>\ b"
by (simp add: strict_iff_order)
lemma strict_implies_not_eq:
"a \<^bold>< b \ a \ b"
by (simp add: strict_iff_order)
lemma not_eq_order_implies_strict:
"a \ b \ a \<^bold>\ b \ a \<^bold>< b"
by (simp add: strict_iff_order)
lemma order_iff_strict:
"a \<^bold>\ b \ a \<^bold>< b \ a = b"
by (auto simp add: strict_iff_order refl)
lemma irrefl: \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close>
"\ a \<^bold>< a"
by (simp add: strict_iff_order)
lemma asym:
"a \<^bold>< b \ b \<^bold>< a \ False"
by (auto simp add: strict_iff_order intro: antisym)
lemma strict_trans1:
"a \<^bold>\ b \ b \<^bold>< c \ a \<^bold>< c"
by (auto simp add: strict_iff_order intro: trans antisym)
lemma strict_trans2:
"a \<^bold>< b \ b \<^bold>\ c \ a \<^bold>< c"
by (auto simp add: strict_iff_order intro: trans antisym)
lemma strict_trans:
"a \<^bold>< b \ b \<^bold>< c \ a \<^bold>< c"
by (auto intro: strict_trans1 strict_implies_order)
lemma eq_iff: "a = b \ a \<^bold>\ b \ b \<^bold>\ a"
by (auto simp add: refl intro: antisym)
end
text \<open>Alternative introduction rule with bias towards strict order\<close>
lemma ordering_strictI:
fixes less_eq (infix "\<^bold>\" 50)
and less (infix "\<^bold><" 50)
assumes less_eq_less: "\a b. a \<^bold>\ b \ a \<^bold>< b \ a = b"
assumes asym: "\a b. a \<^bold>< b \ \ b \<^bold>< a"
assumes irrefl: "\a. \ a \<^bold>< a"
assumes trans: "\a b c. a \<^bold>< b \ b \<^bold>< c \ a \<^bold>< c"
shows "ordering less_eq less"
proof
fix a b
show "a \<^bold>< b \ a \<^bold>\ b \ a \ b"
by (auto simp add: less_eq_less asym irrefl)
next
fix a
show "a \<^bold>\ a"
by (auto simp add: less_eq_less)
next
fix a b c
assume "a \<^bold>\ b" and "b \<^bold>\ c" then show "a \<^bold>\ c"
by (auto simp add: less_eq_less intro: trans)
next
fix a b
assume "a \<^bold>\ b" and "b \<^bold>\ a" then show "a = b"
by (auto simp add: less_eq_less asym)
qed
lemma ordering_dualI:
fixes less_eq (infix "\<^bold>\" 50)
and less (infix "\<^bold><" 50)
assumes "ordering (\a b. b \<^bold>\ a) (\a b. b \<^bold>< a)"
shows "ordering less_eq less"
proof -
from assms interpret ordering "\a b. b \<^bold>\ a" "\a b. b \<^bold>< a" .
show ?thesis
by standard (auto simp: strict_iff_order refl intro: antisym trans)
qed
locale ordering_top = ordering +
fixes top :: "'a" ("\<^bold>\")
assumes extremum [simp]: "a \<^bold>\ \<^bold>\"
begin
lemma extremum_uniqueI:
"\<^bold>\ \<^bold>\ a \ a = \<^bold>\"
by (rule antisym) auto
lemma extremum_unique:
"\<^bold>\ \<^bold>\ a \ a = \<^bold>\"
by (auto intro: antisym)
lemma extremum_strict [simp]:
"\ (\<^bold>\ \<^bold>< a)"
using extremum [of a] by (auto simp add: order_iff_strict intro: asym irrefl)
lemma not_eq_extremum:
"a \ \<^bold>\ \ a \<^bold>< \<^bold>\"
by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum)
end
subsection \<open>Syntactic orders\<close>
class ord =
fixes less_eq :: "'a \ 'a \ bool"
and less :: "'a \ 'a \ bool"
begin
notation
less_eq ("'(\')") and
less_eq ("(_/ \ _)" [51, 51] 50) and
less ("'(<')") and
less ("(_/ < _)" [51, 51] 50)
abbreviation (input)
greater_eq (infix "\" 50)
where "x \ y \ y \ x"
abbreviation (input)
greater (infix ">" 50)
where "x > y \ y < x"
notation (ASCII)
less_eq ("'(<=')") and
less_eq ("(_/ <= _)" [51, 51] 50)
notation (input)
greater_eq (infix ">=" 50)
end
subsection \<open>Quasi orders\<close>
class preorder = ord +
assumes less_le_not_le: "x < y \ x \ y \ \ (y \ x)"
and order_refl [iff]: "x \ x"
and order_trans: "x \ y \ y \ z \ x \ z"
begin
text \<open>Reflexivity.\<close>
lemma eq_refl: "x = y \ x \ y"
\<comment> \<open>This form is useful with the classical reasoner.\<close>
by (erule ssubst) (rule order_refl)
lemma less_irrefl [iff]: "\ x < x"
by (simp add: less_le_not_le)
lemma less_imp_le: "x < y \ x \ y"
by (simp add: less_le_not_le)
text \<open>Asymmetry.\<close>
lemma less_not_sym: "x < y \ \ (y < x)"
by (simp add: less_le_not_le)
lemma less_asym: "x < y \ (\ P \ y < x) \ P"
by (drule less_not_sym, erule contrapos_np) simp
text \<open>Transitivity.\<close>
lemma less_trans: "x < y \ y < z \ x < z"
by (auto simp add: less_le_not_le intro: order_trans)
lemma le_less_trans: "x \ y \ y < z \ x < z"
by (auto simp add: less_le_not_le intro: order_trans)
lemma less_le_trans: "x < y \ y \ z \ x < z"
by (auto simp add: less_le_not_le intro: order_trans)
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 elim: less_asym)
lemma less_imp_triv: "x < y \ (y < x \ P) \ True"
by (blast elim: less_asym)
text \<open>Transitivity rules for calculational reasoning\<close>
lemma less_asym': "a < b \ b < a \ P"
by (rule less_asym)
text \<open>Dual order\<close>
lemma dual_preorder:
"class.preorder (\) (>)"
by standard (auto simp add: less_le_not_le intro: order_trans)
end
subsection \<open>Partial orders\<close>
class order = preorder +
assumes 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: antisym)
sublocale order: ordering less_eq less + dual_order: ordering greater_eq greater
proof -
interpret ordering less_eq less
by standard (auto intro: antisym order_trans simp add: less_le)
show "ordering less_eq less"
by (fact ordering_axioms)
then show "ordering greater_eq greater"
by (rule ordering_dualI)
qed
text \<open>Reflexivity.\<close>
lemma le_less: "x \ y \ x < y \ x = y"
\<comment> \<open>NOT suitable for iff, since it can cause PROOF FAILED.\<close>
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>
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>
lemma neq_le_trans: "a \ b \ a \ b \ a < b"
by (fact order.not_eq_order_implies_strict)
lemma le_neq_trans: "a \ b \ a \ b \ a < b"
by (rule order.not_eq_order_implies_strict)
text \<open>Asymmetry.\<close>
lemma eq_iff: "x = y \ x \ y \ y \ x"
by (fact order.eq_iff)
lemma antisym_conv: "y \ x \ x \ y \ x = y"
by (simp add: eq_iff)
lemma less_imp_neq: "x < y \ x \ y"
by (fact order.strict_implies_not_eq)
lemma antisym_conv1: "\ x < y \ x \ y \ x = y"
by (simp add: local.le_less)
lemma antisym_conv2: "x \ y \ \ x < y \ x = y"
by (simp add: local.less_le)
lemma leD: "y \ x \ \ x < y"
by (auto simp: less_le antisym)
text \<open>Least value operator\<close>
definition (in ord)
Least :: "('a \ bool) \ 'a" (binder "LEAST " 10) where
"Least P = (THE x. P x \ (\y. P y \ x \ y))"
lemma Least_equality:
assumes "P x"
and "\y. P y \ x \ y"
shows "Least P = x"
unfolding Least_def by (rule the_equality)
(blast intro: assms antisym)+
lemma LeastI2_order:
assumes "P x"
and "\y. P y \ x \ y"
and "\x. P x \ \y. P y \ x \ y \ Q x"
shows "Q (Least P)"
unfolding Least_def by (rule theI2)
(blast intro: assms 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 assms]
unfolding Least_def
by auto
text \<open>Greatest value operator\<close>
definition Greatest :: "('a \ bool) \ 'a" (binder "GREATEST " 10) where
"Greatest P = (THE x. P x \ (\y. P y \ x \ y))"
lemma GreatestI2_order:
"\ P x;
\<And>y. P y \<Longrightarrow> x \<ge> y;
\<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
by (rule theI2) (blast intro: antisym)+
lemma Greatest_equality:
"\ P x; \y. P y \ x \ y \ \ Greatest P = x"
unfolding Greatest_def
by (rule the_equality) (blast intro: antisym)+
end
lemma ordering_orderI:
fixes less_eq (infix "\<^bold>\" 50)
and less (infix "\<^bold><" 50)
assumes "ordering less_eq less"
shows "class.order less_eq less"
proof -
from assms interpret ordering less_eq less .
show ?thesis
by standard (auto intro: antisym trans simp add: refl strict_iff_order)
qed
lemma order_strictI:
fixes less (infix "\" 50)
and less_eq (infix "\" 50)
assumes "\a b. a \ b \ a \ b \ a = b"
assumes "\a b. a \ b \ \ b \ a"
assumes "\a. \ a \ a"
assumes "\a b c. a \ b \ b \ c \ a \ c"
shows "class.order less_eq less"
by (rule ordering_orderI) (rule ordering_strictI, (fact assms)+)
context order
begin
text \<open>Dual order\<close>
lemma dual_order:
"class.order (\) (>)"
using dual_order.ordering_axioms by (rule ordering_orderI)
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_linear: "x \ y \ y < x"
by (simp add: le_less less_linear)
lemma le_cases [case_names le ge]:
"(x \ y \ P) \ (y \ x \ P) \ P"
using linear by blast
lemma (in linorder) le_cases3:
"\\x \ y; y \ z\ \ P; \y \ x; x \ z\ \ P; \x \ z; z \ y\ \ 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"
by (blast intro: le_cases)
lemma linorder_cases [case_names less equal greater]:
"(x < y \ P) \ (x = y \ P) \ (y < x \ P) \ P"
using less_linear by blast
lemma linorder_wlog[case_names le sym]:
"(\a b. a \ b \ P a b) \ (\a b. P b a \ P a b) \ P a b"
by (cases rule: le_cases[of a b]) blast+
lemma not_less: "\ x < y \ y \ x"
unfolding less_le
using linear by (blast intro: antisym)
lemma not_less_iff_gr_or_eq: "\(x < y) \ (x > y \ x = y)"
by (auto simp add:not_less le_less)
lemma not_le: "\ x \ y \ y < x"
unfolding less_le
using linear by (blast intro: antisym)
lemma neq_iff: "x \ y \ x < y \ y < x"
by (cut_tac x = x and y = y in less_linear, auto)
lemma neqE: "x \ y \ (x < y \ R) \ (y < x \ R) \ R"
by (simp add: neq_iff) blast
lemma antisym_conv3: "\ y < x \ \ x < y \ x = y"
by (blast intro: antisym dest: not_less [THEN iffD1])
lemma leI: "\ x < y \ y \ x"
unfolding not_less .
lemma not_le_imp_less: "\ y \ x \ x < y"
unfolding not_le .
lemma linorder_less_wlog[case_names less refl sym]:
"\\a b. a < b \ P a b; \a. P a a; \a b. P b a \ P a b\ \ P a b"
using antisym_conv3 by blast
text \<open>Dual order\<close>
lemma dual_linorder:
"class.linorder (\) (>)"
by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear)
end
text \<open>Alternative introduction rule with bias towards strict order\<close>
lemma linorder_strictI:
fixes less_eq (infix "\<^bold>\" 50)
and less (infix "\<^bold><" 50)
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 \<open>class.order less_eq less\<close>)
show ?thesis
proof
fix a b
show "a \<^bold>\ b \ b \<^bold>\ a"
using trichotomy by (auto simp add: le_less)
qed
qed
subsection \<open>Reasoning tools setup\<close>
ML \<open>
signature ORDERS =
sig
val print_structures: Proof.context -> unit
val order_tac: Proof.context -> thm list -> int -> tactic
val add_struct: string * term list -> string -> attribute
val del_struct: string * term list -> attribute
end;
structure Orders: ORDERS =
struct
(* context data *)
fun struct_eq ((s1: string, ts1), (s2, ts2)) =
s1 = s2 andalso eq_list (op aconv) (ts1, ts2);
structure Data = Generic_Data
(
type T = ((string * term list) * Order_Tac.less_arith) list;
(* Order structures:
identifier of the structure, list of operations and record of theorems
needed to set up the transitivity reasoner,
identifier and operations identify the structure uniquely. *)
val empty = [];
val extend = I;
fun merge data = AList.join struct_eq (K fst) data;
);
fun print_structures ctxt =
let
val structs = Data.get (Context.Proof ctxt);
fun pretty_term t = Pretty.block
[Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
Pretty.str "::", Pretty.brk 1,
Pretty.quote (Syntax.pretty_typ ctxt (type_of t))];
fun pretty_struct ((s, ts), _) = Pretty.block
[Pretty.str s, Pretty.str ":", Pretty.brk 1,
Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
in
Pretty.writeln (Pretty.big_list "order structures:" (map pretty_struct structs))
end;
val _ =
Outer_Syntax.command \<^command_keyword>\<open>print_orders\<close>
"print order structures available to transitivity reasoner"
(Scan.succeed (Toplevel.keep (print_structures o Toplevel.context_of)));
(* tactics *)
fun struct_tac ((s, ops), thms) ctxt facts =
let
val [eq, le, less] = ops;
fun decomp thy (\<^const>\<open>Trueprop\<close> $ t) =
let
fun excluded t =
(* exclude numeric types: linear arithmetic subsumes transitivity *)
let val T = type_of t
in
T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
end;
fun rel (bin_op $ t1 $ t2) =
if excluded t1 then NONE
else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
else NONE
| rel _ = NONE;
fun dec (Const (\<^const_name>\<open>Not\<close>, _) $ t) =
(case rel t of NONE =>
NONE
| SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
| dec x = rel x;
in dec t end
| decomp _ _ = NONE;
in
(case s of
"order" => Order_Tac.partial_tac decomp thms ctxt facts
| "linorder" => Order_Tac.linear_tac decomp thms ctxt facts
| _ => error ("Unknown order kind " ^ quote s ^ " encountered in transitivity reasoner"))
end
fun order_tac ctxt facts =
FIRST' (map (fn s => CHANGED o struct_tac s ctxt facts) (Data.get (Context.Proof ctxt)));
(* attributes *)
fun add_struct s tag =
Thm.declaration_attribute
(fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm)));
fun del_struct s =
Thm.declaration_attribute
(fn _ => Data.map (AList.delete struct_eq s));
end;
\<close>
attribute_setup order = \<open>
Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --|
Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name --
Scan.repeat Args.term
>> (fn ((SOME tag, n), ts) => Orders.add_struct (n, ts) tag
| ((NONE, n), ts) => Orders.del_struct (n, ts))
\<close> "theorems controlling transitivity reasoner"
method_setup order = \<open>
Scan.succeed (fn ctxt => SIMPLE_METHOD' (Orders.order_tac ctxt []))
\<close> "transitivity reasoner"
text \<open>Declarations to set up transitivity reasoner of partial and linear orders.\<close>
context order
begin
(* The type constraint on @{term (=}) below is necessary since the operation
is not a parameter of the locale. *)
declare less_irrefl [THEN notE, order add less_reflE: order "(=) :: 'a \ 'a \ bool" "(<=)" "(<)"]
declare order_refl [order add le_refl: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare less_imp_le [order add less_imp_le: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare antisym [order add eqI: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare eq_refl [order add eqD1: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare sym [THEN eq_refl, order add eqD2: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare less_trans [order add less_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare less_le_trans [order add less_le_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare le_less_trans [order add le_less_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare order_trans [order add le_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare le_neq_trans [order add le_neq_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare neq_le_trans [order add neq_le_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare less_imp_neq [order add less_imp_neq: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare not_sym [order add not_sym: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
end
context linorder
begin
declare [[order del: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]]
declare less_irrefl [THEN notE, order add less_reflE: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare order_refl [order add le_refl: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare less_imp_le [order add less_imp_le: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare not_less [THEN iffD2, order add not_lessI: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare not_le [THEN iffD2, order add not_leI: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare not_less [THEN iffD1, order add not_lessD: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare not_le [THEN iffD1, order add not_leD: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare antisym [order add eqI: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare eq_refl [order add eqD1: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare sym [THEN eq_refl, order add eqD2: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare less_trans [order add less_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare less_le_trans [order add less_le_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare le_less_trans [order add le_less_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare order_trans [order add le_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare le_neq_trans [order add le_neq_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare neq_le_trans [order add neq_le_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare less_imp_neq [order add less_imp_neq: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
declare not_sym [order add not_sym: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
end
setup \<open>
map_theory_simpset (fn ctxt0 => ctxt0 addSolver
mk_solver "Transitivity" (fn ctxt => Orders.order_tac ctxt (Simplifier.prems_of ctxt)))
(*Adding the transitivity reasoners also as safe solvers showed a slight
speed up, but the reasoning strength appears to be not higher (at least
no breaking of additional proofs in the entire HOL distribution, as
of 5 March 2004, was observed).*)
\<close>
ML \<open>
local
fun prp t thm = Thm.prop_of thm = t; (* FIXME proper aconv!? *)
in
fun antisym_le_simproc ctxt ct =
(case Thm.term_of ct of
(le as Const (_, T)) $ r $ s =>
(let
val prems = Simplifier.prems_of ctxt;
val less = Const (\<^const_name>\<open>less\<close>, T);
val t = HOLogic.mk_Trueprop(le $ s $ r);
in
(case find_first (prp t) prems of
NONE =>
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) in
(case find_first (prp t) prems of
NONE => NONE
| SOME thm => SOME(mk_meta_eq(thm RS @{thm antisym_conv1})))
end
| SOME thm => SOME (mk_meta_eq (thm RS @{thm order_class.antisym_conv})))
end handle THM _ => NONE)
| _ => NONE);
fun antisym_less_simproc ctxt ct =
(case Thm.term_of ct of
NotC $ ((less as Const(_,T)) $ r $ s) =>
(let
val prems = Simplifier.prems_of ctxt;
val le = Const (\<^const_name>\<open>less_eq\<close>, T);
val t = HOLogic.mk_Trueprop(le $ r $ s);
in
(case find_first (prp t) prems of
NONE =>
let val t = HOLogic.mk_Trueprop (NotC $ (less $ s $ r)) in
(case find_first (prp t) prems of
NONE => NONE
| SOME thm => SOME (mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})))
end
| SOME thm => SOME (mk_meta_eq (thm RS @{thm antisym_conv2})))
end handle THM _ => NONE)
| _ => NONE);
end;
\<close>
simproc_setup antisym_le ("(x::'a::order) \ y") = "K antisym_le_simproc"
simproc_setup antisym_less ("\ (x::'a::linorder) < y") = "K antisym_less_simproc"
subsection \<open>Bounded quantifiers\<close>
syntax (ASCII)
"_All_less" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10)
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3ALL _>_./ _)" [0, 0, 10] 10)
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3EX _>_./ _)" [0, 0, 10] 10)
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _>=_./ _)" [0, 0, 10] 10)
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10)
"_All_neq" :: "[idt, 'a, bool] => bool" ("(3ALL _~=_./ _)" [0, 0, 10] 10)
"_Ex_neq" :: "[idt, 'a, bool] => bool" ("(3EX _~=_./ _)" [0, 0, 10] 10)
syntax
"_All_less" :: "[idt, 'a, bool] => bool" ("(3\_<_./ _)" [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3\_<_./ _)" [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10)
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3\_>_./ _)" [0, 0, 10] 10)
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3\_>_./ _)" [0, 0, 10] 10)
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10)
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10)
"_All_neq" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10)
"_Ex_neq" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10)
syntax (input)
"_All_less" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10)
"_All_neq" :: "[idt, 'a, bool] => bool" ("(3! _~=_./ _)" [0, 0, 10] 10)
"_Ex_neq" :: "[idt, 'a, bool] => bool" ("(3? _~=_./ _)" [0, 0, 10] 10)
translations
"\x "\x. x < y \ P"
"\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 All_binder = Mixfix.binder_name \<^const_syntax>\<open>All\<close>;
val Ex_binder = Mixfix.binder_name \<^const_syntax>\<open>Ex\<close>;
val impl = \<^const_syntax>\<open>HOL.implies\<close>;
val conj = \<^const_syntax>\<open>HOL.conj\<close>;
val less = \<^const_syntax>\<open>less\<close>;
val less_eq = \<^const_syntax>\<open>less_eq\<close>;
val trans =
[((All_binder, impl, less),
(\<^syntax_const>\<open>_All_less\<close>, \<^syntax_const>\<open>_All_greater\<close>)),
((All_binder, impl, less_eq),
(\<^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'
| _ => false);
fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false);
fun mk x c n P = Syntax.const c $ Syntax_Trans.mark_bound_body x $ n $ P;
fun tr' q = (q, fn _ =>
(fn [Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v, T),
Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
(case AList.lookup (=) trans (q, c, d) of
NONE => raise Match
| SOME (l, g) =>
if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P
else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P
else raise Match)
| _ => raise Match));
in [tr' All_binder, tr' Ex_binder] end
\<close>
subsection \<open>Transitivity reasoning\<close>
context ord
begin
lemma ord_le_eq_trans: "a \ b \ b = c \ a \ c"
by (rule subst)
lemma ord_eq_le_trans: "a = b \ b \ c \ a \ c"
by (rule ssubst)
lemma ord_less_eq_trans: "a < b \ b = c \ a < c"
by (rule subst)
lemma ord_eq_less_trans: "a = b \ b < c \ a < c"
by (rule ssubst)
end
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
(!!x y. x < y ==> f x < f y) ==> 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)
also assume "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 ==> f x < f y) ==> a < f c"
proof -
assume r: "!!x y. x < y ==> f x < f y"
assume "a < f b"
also assume "b < c" hence "f b < f c" by (rule r)
finally (less_trans) show ?thesis .
qed
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
(!!x y. x <= y ==> f x <= f y) ==> 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)
also assume "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) ==> a < f c"
proof -
assume r: "!!x y. x < y ==> f x < f y"
assume "a <= f b"
also assume "b < c" hence "f b < f c" by (rule r)
finally (le_less_trans) show ?thesis .
qed
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
(!!x y. x < y ==> f x < f y) ==> 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)
also assume "f b <= c"
finally (less_le_trans) show ?thesis .
qed
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
(!!x y. x <= y ==> f x <= f y) ==> a < f c"
proof -
assume r: "!!x y. x <= y ==> f x <= f y"
assume "a < f b"
also assume "b <= c" hence "f b <= f c" by (rule r)
finally (less_le_trans) show ?thesis .
qed
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
(!!x y. x <= y ==> f x <= f y) ==> a <= f c"
proof -
assume r: "!!x y. x <= y ==> f x <= f y"
assume "a <= f b"
also assume "b <= c" hence "f b <= f c" by (rule r)
finally (order_trans) show ?thesis .
qed
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
(!!x y. x <= y ==> f x <= f y) ==> 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)
also assume "f b <= c"
finally (order_trans) show ?thesis .
qed
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
(!!x y. x <= y ==> f x <= f y) ==> 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)
also assume "f b = c"
finally (ord_le_eq_trans) show ?thesis .
qed
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
(!!x y. x <= y ==> f x <= f y) ==> a <= f c"
proof -
assume r: "!!x y. x <= y ==> f x <= f y"
assume "a = f b"
also assume "b <= c" hence "f b <= f c" by (rule r)
finally (ord_eq_le_trans) show ?thesis .
qed
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
(!!x y. x < y ==> f x < f y) ==> 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)
also assume "f b = c"
finally (ord_less_eq_trans) show ?thesis .
qed
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
(!!x y. x < y ==> f x < f y) ==> a < f c"
proof -
assume r: "!!x y. x < y ==> f x < f y"
assume "a = f b"
also assume "b < c" hence "f b < f c" by (rule r)
finally (ord_eq_less_trans) show ?thesis .
qed
text \<open>
Note that this list of rules is in reverse order of priorities.
\<close>
lemmas [trans] =
order_less_subst2
order_less_subst1
order_le_less_subst2
order_le_less_subst1
order_less_le_subst2
order_less_le_subst1
order_subst2
order_subst1
ord_le_eq_subst
ord_eq_le_subst
ord_less_eq_subst
ord_eq_less_subst
forw_subst
back_subst
rev_mp
mp
lemmas (in order) [trans] =
neq_le_trans
le_neq_trans
lemmas (in preorder) [trans] =
less_trans
less_asym'
le_less_trans
less_le_trans
order_trans
lemmas (in order) [trans] =
antisym
lemmas (in ord) [trans] =
ord_le_eq_trans
ord_eq_le_trans
ord_less_eq_trans
ord_eq_less_trans
lemmas [trans] =
trans
lemmas order_trans_rules =
order_less_subst2
order_less_subst1
order_le_less_subst2
order_le_less_subst1
order_less_le_subst2
order_less_le_subst1
order_subst2
order_subst1
ord_le_eq_subst
ord_eq_le_subst
ord_less_eq_subst
ord_eq_less_subst
forw_subst
back_subst
rev_mp
mp
neq_le_trans
le_neq_trans
less_trans
less_asym'
le_less_trans
less_le_trans
order_trans
antisym
ord_le_eq_trans
ord_eq_le_trans
ord_less_eq_trans
ord_eq_less_trans
trans
text \<open>These support proving chains of decreasing inequalities
a >= b >= c ... in Isar proofs.\<close>
lemma xt1 [no_atp]:
"a = b \ b > c \ a > c"
"a > b \ b = c \ a > c"
"a = b \ b \ c \ a \ c"
"a \ b \ b = c \ a \ c"
"(x::'a::order) \ y \ y \ x \ x = y"
"(x::'a::order) \ y \ y \ z \ x \ z"
"(x::'a::order) > y \ y \ z \ x > z"
"(x::'a::order) \ y \ y > z \ x > z"
"(a::'a::order) > b \ b > a \ P"
"(x::'a::order) > y \ y > z \ x > z"
"(a::'a::order) \ b \ a \ b \ a > b"
"(a::'a::order) \ b \ a \ b \ 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 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"
by auto
lemma xt2 [no_atp]:
"(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
by (subgoal_tac "f b >= f c", force, force)
lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>
(!!x y. x >= y ==> f x >= f y) ==> f a >= c"
by (subgoal_tac "f a >= f b", force, force)
lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
(!!x y. x >= y ==> f x >= f y) ==> a > f c"
by (subgoal_tac "f b >= f c", force, force)
lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
(!!x y. x > y ==> f x > f y) ==> f a > c"
by (subgoal_tac "f a > f b", force, force)
lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==>
(!!x y. x > y ==> f x > f y) ==> a > f c"
by (subgoal_tac "f b > f c", force, force)
lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
(!!x y. x >= y ==> f x >= f y) ==> f a > c"
by (subgoal_tac "f a >= f b", force, force)
lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
(!!x y. x > y ==> f x > f y) ==> a > f c"
by (subgoal_tac "f b > f c", force, force)
lemma xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
(!!x y. x > y ==> f x > f y) ==> f a > c"
by (subgoal_tac "f a > f b", force, force)
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
(*
Since "a >= b" abbreviates "b <= 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 >= b" (is "_ >= ?rhs")
sorry
also have "?rhs >= c" (is "_ >= ?rhs")
sorry
also (xtrans) have "?rhs = d" (is "_ = ?rhs")
sorry
also (xtrans) have "?rhs >= e" (is "_ >= ?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>Monotonicity\<close>
context order
begin
definition mono :: "('a \ 'b::order) \ bool" where
"mono f \ (\x y. x \ y \ f x \ f y)"
lemma monoI [intro?]:
fixes f :: "'a \ 'b::order"
shows "(\x y. x \ y \ f x \ f y) \ mono f"
unfolding mono_def by iprover
lemma monoD [dest?]:
fixes f :: "'a \ 'b::order"
shows "mono f \ x \ y \ f x \ f y"
unfolding mono_def by iprover
lemma monoE:
fixes f :: "'a \ 'b::order"
assumes "mono f"
assumes "x \ y"
obtains "f x \ f y"
proof
from assms show "f x \ f y" by (simp add: mono_def)
qed
definition antimono :: "('a \ 'b::order) \ bool" where
"antimono f \ (\x y. x \ y \ f x \ f y)"
lemma antimonoI [intro?]:
fixes f :: "'a \ 'b::order"
shows "(\x y. x \ y \ f x \ f y) \ antimono f"
unfolding antimono_def by iprover
lemma antimonoD [dest?]:
fixes f :: "'a \ 'b::order"
shows "antimono f \ x \ y \ f x \ f y"
unfolding antimono_def by iprover
lemma antimonoE:
fixes f :: "'a \ 'b::order"
assumes "antimono f"
assumes "x \ y"
obtains "f x \ f y"
proof
from assms show "f x \ f y" by (simp add: antimono_def)
qed
definition strict_mono :: "('a \ 'b::order) \ bool" where
"strict_mono f \ (\x y. x < y \ f x < f y)"
lemma strict_monoI [intro?]:
assumes "\x y. x < y \ f x < f y"
shows "strict_mono f"
using assms unfolding strict_mono_def by auto
lemma strict_monoD [dest?]:
"strict_mono f \ x < y \ f x < f y"
unfolding strict_mono_def by auto
lemma strict_mono_mono [dest?]:
assumes "strict_mono f"
shows "mono f"
proof (rule monoI)
fix x y
assume "x \ y"
show "f x \ f y"
proof (cases "x = y")
case True then show ?thesis by simp
next
case False with \<open>x \<le> y\<close> have "x < y" by simp
with assms strict_monoD have "f x < f y" by auto
then show ?thesis by simp
qed
qed
end
context linorder
begin
lemma mono_invE:
fixes f :: "'a \ 'b::order"
assumes "mono f"
assumes "f x < f y"
obtains "x \ y"
proof
show "x \ y"
proof (rule ccontr)
assume "\ x \ y"
then have "y \ x" by simp
with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE)
with \<open>f x < f y\<close> show False by simp
qed
qed
lemma mono_strict_invE:
fixes f :: "'a \ 'b::order"
assumes "mono f"
assumes "f x < f y"
obtains "x < y"
proof
show "x < y"
proof (rule ccontr)
assume "\ x < y"
then have "y \ x" by simp
with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE)
with \<open>f x < f y\<close> show False by simp
qed
qed
lemma strict_mono_eq:
assumes "strict_mono f"
shows "f x = f y \ x = y"
proof
assume "f x = f y"
show "x = y" proof (cases x y rule: linorder_cases)
case less with assms strict_monoD have "f x < f y" by auto
with \<open>f x = f y\<close> show ?thesis by simp
next
case equal then show ?thesis .
next
case greater with assms strict_monoD have "f y < f x" by auto
with \<open>f x = f y\<close> show ?thesis by simp
qed
qed simp
lemma strict_mono_less_eq:
assumes "strict_mono f"
shows "f x \ f y \ x \ y"
proof
assume "x \ y"
with assms strict_mono_mono monoD show "f x \ f y" by auto
next
assume "f x \ f y"
show "x \ y" proof (rule ccontr)
assume "\ x \ y" then have "y < x" by simp
with assms strict_monoD have "f y < f x" by auto
with \<open>f x \<le> f y\<close> show False by simp
qed
qed
lemma strict_mono_less:
assumes "strict_mono f"
shows "f x < f y \ x < y"
using assms
by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq)
end
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 (in ord) max :: "'a \ 'a \ 'a" where
"max a b = (if a \ b then b else a)"
lemma min_absorb1: "x \ y \ min x y = x"
by (simp add: min_def)
lemma max_absorb2: "x \ y \ max x y = y"
by (simp add: max_def)
lemma min_absorb2: "(y::'a::order) \ x \ min x y = y"
by (simp add:min_def)
lemma max_absorb1: "(y::'a::order) \ x \ max x y = x"
by (simp add: max_def)
lemma max_min_same [simp]:
fixes x y :: "'a :: linorder"
shows "max x (min x y) = x" "max (min x y) x = x" "max (min x y) y = y" "max y (min x y) = y"
by(auto simp add: max_def min_def)
subsection \<open>(Unique) top and bottom elements\<close>
class bot =
fixes bot :: 'a ("\")
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 bot.extremum_uniqueI)
lemma bot_unique:
"a \ \ \ a = \"
by (fact bot.extremum_unique)
lemma not_less_bot:
"\ a < \"
by (fact bot.extremum_strict)
lemma bot_less:
"a \ \ \ \ < a"
by (fact bot.not_eq_extremum)
lemma max_bot[simp]: "max bot x = x"
by(simp add: max_def bot_unique)
lemma max_bot2[simp]: "max x bot = x"
by(simp add: max_def bot_unique)
lemma min_bot[simp]: "min bot x = bot"
by(simp add: min_def bot_unique)
lemma min_bot2[simp]: "min x bot = bot"
by(simp add: min_def bot_unique)
end
class top =
fixes top :: 'a ("\")
class order_top = order + top +
assumes top_greatest: "a \ \"
begin
sublocale top: ordering_top less_eq less top
by standard (fact top_greatest)
lemma top_le:
"\ \ a \ a = \"
by (fact top.extremum_uniqueI)
lemma top_unique:
"\ \ a \ a = \"
by (fact top.extremum_unique)
lemma not_top_less:
"\ \ < a"
by (fact top.extremum_strict)
lemma less_top:
"a \ \ \ a < \"
by (fact top.not_eq_extremum)
lemma max_top[simp]: "max top x = top"
by(simp add: max_def top_unique)
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: min_def top_unique)
lemma min_top2[simp]: "min x top = x"
by(simp add: min_def top_unique)
end
subsection \<open>Dense orders\<close>
class dense_order = order +
assumes dense: "x < y \ (\z. x < z \ z < y)"
class dense_linorder = linorder + dense_order
begin
lemma dense_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
moreover have "x \ z" using assms[OF \x < y\] .
ultimately show False by auto
qed
lemma dense_le_bounded:
fixes x y z :: 'a
assumes "x < y"
assumes *: "\w. \ x < w ; w < y \ \ w \ z"
shows "y \ z"
proof (rule dense_le)
fix w assume "w < y"
from dense[OF \<open>x < y\<close>] obtain u where "x < u" "u < y" by safe
from linear[of u w]
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>]
show "w \ z" by (rule order_trans)
qed
qed
lemma dense_ge:
fixes y z :: 'a
assumes "\x. z < x \ y \ x"
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
moreover have "y \ x" using assms[OF \z < x\] .
ultimately show False by auto
qed
lemma dense_ge_bounded:
fixes x y z :: 'a
assumes "z < x"
assumes *: "\w. \ z < w ; w < x \ \ y \ w"
shows "y \ z"
proof (rule dense_ge)
fix w assume "z < w"
from dense[OF \<open>z < x\<close>] obtain u where "z < u" "u < x" by safe
from linear[of u w]
show "y \ w"
proof (rule disjE)
assume "w \ u"
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 gt_ex: "\y. x < y"
class no_bot = order +
assumes lt_ex: "\y. y < x"
class unbounded_dense_linorder = dense_linorder + no_top + no_bot
subsection \<open>Wellorders\<close>
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) then have "P x" by simp
show ?case proof (rule classical)
assume assm: "\ (P (LEAST a. P a) \ (LEAST a. P a) \ x)"
have "\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. P a)" and "(LEAST a. P a) \ y"
by auto
then show "x \ y" by auto
qed
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
qed
qed
then show "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: "\x. P x \ P (Least P)"
by (erule exE) (erule LeastI)
lemma LeastI2:
"P a \ (\x. P x \ Q x) \ Q (Least P)"
by (blast intro: LeastI)
lemma LeastI2_ex:
"\a. P a \ (\x. P x \ Q x) \ Q (Least P)"
by (blast intro: LeastI_ex)
lemma LeastI2_wellorder:
assumes "P a"
and "\a. \ P a; \b. P b \ a \ b \ \ Q a"
shows "Q (Least P)"
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 "P x" "\y. P y \ x \ y" thus "Q x" by (rule assms(2))
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)
lemma not_less_Least: "k < (LEAST x. P x) \ \ P k"
apply (simp add: not_le [symmetric])
apply (erule contrapos_nn)
apply (erule Least_le)
done
lemma exists_least_iff: "(\n. P n) \ (\n. P n \ (\m < n. \ P m))" (is "?lhs \ ?rhs")
proof
assume ?rhs thus ?lhs by blast
next
assume H: ?lhs then obtain n where n: "P n" by blast
let ?x = "Least P"
{ fix m assume m: "m < ?x"
from not_less_Least[OF m] have "\ P m" . }
with LeastI_ex[OF H] show ?rhs by blast
qed
end
subsection \<open>Order on \<^typ>\<open>bool\<close>\<close>
instantiation bool :: "{order_bot, order_top, linorder}"
begin
definition
le_bool_def [simp]: "P \ Q \ P \ Q"
definition
[simp]: "(P::bool) < Q \ \ P \ Q"
definition
[simp]: "\ \ False"
definition
[simp]: "\ \ True"
instance proof
qed auto
end
lemma le_boolI: "(P \ Q) \ P \ Q"
by simp
lemma le_boolI': "P \ Q \ P \ Q"
by simp
lemma le_boolE: "P \ Q \ P \ (Q \ R) \ R"
by simp
lemma le_boolD: "P \ Q \ P \ Q"
by simp
lemma bot_boolE: "\ \ P"
by simp
lemma top_boolI: \<top>
by simp
lemma [code]:
"False \ b \ True"
"True \ b \ b"
"False < b \ b"
"True < b \ False"
by simp_all
subsection \<open>Order on \<^typ>\<open>_ \<Rightarrow> _\<close>\<close>
instantiation "fun" :: (type, ord) ord
begin
definition
le_fun_def: "f \ g \ (\x. f x \ g x)"
definition
"(f::'a \ 'b) < g \ f \ g \ \ (g \ f)"
instance ..
end
instance "fun" :: (type, preorder) preorder proof
qed (auto simp add: le_fun_def less_fun_def
intro: order_trans antisym)
instance "fun" :: (type, order) order proof
qed (auto simp add: le_fun_def intro: antisym)
instantiation "fun" :: (type, bot) bot
begin
definition
"\ = (\x. \)"
instance ..
end
instantiation "fun" :: (type, order_bot) order_bot
begin
lemma bot_apply [simp, code]:
"\ x = \"
by (simp add: bot_fun_def)
instance proof
qed (simp add: le_fun_def)
end
instantiation "fun" :: (type, top) top
begin
definition
[no_atp]: "\ = (\x. \)"
instance ..
end
instantiation "fun" :: (type, order_top) order_top
begin
lemma top_apply [simp, code]:
"\ x = \"
by (simp add: top_fun_def)
instance proof
qed (simp add: le_fun_def)
end
lemma le_funI: "(\x. f x \ g x) \ f \ g"
unfolding le_fun_def by simp
lemma le_funE: "f \ g \ (f x \ g x \ P) \ P"
unfolding le_fun_def by simp
lemma le_funD: "f \ g \ f x \ g x"
by (rule le_funE)
lemma mono_compose: "mono Q \ mono (\i x. Q i (f x))"
unfolding mono_def le_fun_def by auto
subsection \<open>Order on unary and binary predicates\<close>
lemma predicate1I:
assumes PQ: "\x. P x \ Q x"
shows "P \ Q"
apply (rule le_funI)
apply (rule le_boolI)
apply (rule PQ)
apply assumption
done
lemma predicate1D:
"P \ Q \ P x \ Q x"
apply (erule le_funE)
apply (erule le_boolE)
apply assumption+
done
lemma rev_predicate1D:
"P x \ P \ Q \ Q x"
by (rule predicate1D)
lemma predicate2I:
assumes PQ: "\x y. P x y \ Q x y"
shows "P \ Q"
apply (rule le_funI)+
apply (rule le_boolI)
apply (rule PQ)
apply assumption
done
lemma predicate2D:
"P \ Q \ P x y \ Q x y"
apply (erule le_funE)+
apply (erule le_boolE)
apply assumption+
done
lemma rev_predicate2D:
"P x y \ P \ Q \ Q x y"
by (rule predicate2D)
lemma bot1E [no_atp]: "\ x \ P"
by (simp add: bot_fun_def)
lemma bot2E: "\ x y \ P"
by (simp add: bot_fun_def)
lemma top1I: "\ x"
by (simp add: top_fun_def)
lemma top2I: "\ x y"
by (simp add: top_fun_def)
subsection \<open>Name duplicates\<close>
lemmas order_eq_refl = preorder_class.eq_refl
lemmas order_less_irrefl = preorder_class.less_irrefl
lemmas order_less_imp_le = preorder_class.less_imp_le
lemmas order_less_not_sym = preorder_class.less_not_sym
lemmas order_less_asym = preorder_class.less_asym
lemmas order_less_trans = preorder_class.less_trans
lemmas order_le_less_trans = preorder_class.le_less_trans
lemmas order_less_le_trans = preorder_class.less_le_trans
lemmas order_less_imp_not_less = preorder_class.less_imp_not_less
lemmas order_less_imp_triv = preorder_class.less_imp_triv
lemmas order_less_asym' = preorder_class.less_asym'
lemmas order_less_le = order_class.less_le
lemmas order_le_less = order_class.le_less
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
lemmas order_neq_le_trans = order_class.neq_le_trans
lemmas order_le_neq_trans = order_class.le_neq_trans
lemmas order_antisym = order_class.antisym
lemmas order_eq_iff = order_class.eq_iff
lemmas order_antisym_conv = order_class.antisym_conv
lemmas linorder_linear = linorder_class.linear
lemmas linorder_less_linear = linorder_class.less_linear
lemmas linorder_le_less_linear = linorder_class.le_less_linear
lemmas linorder_le_cases = linorder_class.le_cases
lemmas linorder_not_less = linorder_class.not_less
lemmas linorder_not_le = linorder_class.not_le
lemmas linorder_neq_iff = linorder_class.neq_iff
lemmas linorder_neqE = linorder_class.neqE
end
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