(* 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 "~~/src/Provers/order.ML"
ML_file "~~/src/Provers/quasi.ML" (* FIXME unused? *)
subsection \<open>Abstract ordering\<close>
locale ordering =
fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<^bold>\<le>" 50)
and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<^bold><" 50)
assumes strict_iff_order: "a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b"
assumes refl: "a \<^bold>\<le> a" \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close>
and antisym: "a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> a \<Longrightarrow> a = b"
and trans: "a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>\<le> c"
begin
lemma strict_implies_order:
"a \<^bold>< b \<Longrightarrow> a \<^bold>\<le> b"
by (simp add: strict_iff_order)
lemma strict_implies_not_eq:
"a \<^bold>< b \<Longrightarrow> a \<noteq> b"
by (simp add: strict_iff_order)
lemma not_eq_order_implies_strict:
"a \<noteq> b \<Longrightarrow> a \<^bold>\<le> b \<Longrightarrow> a \<^bold>< b"
by (simp add: strict_iff_order)
lemma order_iff_strict:
"a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> 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>
"\<not> a \<^bold>< a"
by (simp add: strict_iff_order)
lemma asym:
"a \<^bold>< b \<Longrightarrow> b \<^bold>< a \<Longrightarrow> False"
by (auto simp add: strict_iff_order intro: antisym)
lemma strict_trans1:
"a \<^bold>\<le> b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c"
by (auto simp add: strict_iff_order intro: trans antisym)
lemma strict_trans2:
"a \<^bold>< b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>< c"
by (auto simp add: strict_iff_order intro: trans antisym)
lemma strict_trans:
"a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c"
by (auto intro: strict_trans1 strict_implies_order)
end
text \<open>Alternative introduction rule with bias towards strict order\<close>
lemma ordering_strictI:
fixes less_eq (infix "\<^bold>\<le>" 50)
and less (infix "\<^bold><" 50)
assumes less_eq_less: "\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b"
assumes asym: "\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a"
assumes irrefl: "\<And>a. \<not> a \<^bold>< a"
assumes trans: "\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c"
shows "ordering less_eq less"
proof
fix a b
show "a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b"
by (auto simp add: less_eq_less asym irrefl)
next
fix a
show "a \<^bold>\<le> a"
by (auto simp add: less_eq_less)
next
fix a b c
assume "a \<^bold>\<le> b" and "b \<^bold>\<le> c" then show "a \<^bold>\<le> c"
by (auto simp add: less_eq_less intro: trans)
next
fix a b
assume "a \<^bold>\<le> b" and "b \<^bold>\<le> a" then show "a = b"
by (auto simp add: less_eq_less asym)
qed
lemma ordering_dualI:
fixes less_eq (infix "\<^bold>\<le>" 50)
and less (infix "\<^bold><" 50)
assumes "ordering (\<lambda>a b. b \<^bold>\<le> a) (\<lambda>a b. b \<^bold>< a)"
shows "ordering less_eq less"
proof -
from assms interpret ordering "\<lambda>a b. b \<^bold>\<le> a" "\<lambda>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>\<top>")
assumes extremum [simp]: "a \<^bold>\<le> \<^bold>\<top>"
begin
lemma extremum_uniqueI:
"\<^bold>\<top> \<^bold>\<le> a \<Longrightarrow> a = \<^bold>\<top>"
by (rule antisym) auto
lemma extremum_unique:
"\<^bold>\<top> \<^bold>\<le> a \<longleftrightarrow> a = \<^bold>\<top>"
by (auto intro: antisym)
lemma extremum_strict [simp]:
"\<not> (\<^bold>\<top> \<^bold>< a)"
using extremum [of a] by (auto simp add: order_iff_strict intro: asym irrefl)
lemma not_eq_extremum:
"a \<noteq> \<^bold>\<top> \<longleftrightarrow> a \<^bold>< \<^bold>\<top>"
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 \<Rightarrow> 'a \<Rightarrow> bool"
and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
begin
notation
less_eq ("op \<le>") and
less_eq ("(_/ \<le> _)" [51, 51] 50) and
less ("op <") and
less ("(_/ < _)" [51, 51] 50)
abbreviation (input)
greater_eq (infix "\<ge>" 50)
where "x \<ge> y \<equiv> y \<le> x"
abbreviation (input)
greater (infix ">" 50)
where "x > y \<equiv> y < x"
notation (ASCII)
less_eq ("op <=") 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 \<longleftrightarrow> x \<le> y \<and> \<not> (y \<le> x)"
and order_refl [iff]: "x \<le> x"
and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
begin
text \<open>Reflexivity.\<close>
lemma eq_refl: "x = y \<Longrightarrow> x \<le> y"
\<comment> \<open>This form is useful with the classical reasoner.\<close>
by (erule ssubst) (rule order_refl)
lemma less_irrefl [iff]: "\<not> x < x"
by (simp add: less_le_not_le)
lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y"
by (simp add: less_le_not_le)
text \<open>Asymmetry.\<close>
lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)"
by (simp add: less_le_not_le)
lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P"
by (drule less_not_sym, erule contrapos_np) simp
text \<open>Transitivity.\<close>
lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
by (auto simp add: less_le_not_le intro: order_trans)
lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
by (auto simp add: less_le_not_le intro: order_trans)
lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> 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 \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True"
by (blast elim: less_asym)
lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True"
by (blast elim: less_asym)
text \<open>Transitivity rules for calculational reasoning\<close>
lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P"
by (rule less_asym)
text \<open>Dual order\<close>
lemma dual_preorder:
"class.preorder (op \<ge>) (op >)"
by standard (auto simp add: less_le_not_le intro: order_trans)
end
subsection \<open>Partial orders\<close>
class order = preorder +
assumes antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
begin
lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> 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 \<le> y \<longleftrightarrow> x < y \<or> 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 \<le> y \<Longrightarrow> x < y \<or> 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 \<Longrightarrow> (x = y) \<longleftrightarrow> False"
by auto
lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
by auto
text \<open>Transitivity rules for calculational reasoning\<close>
lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"
by (fact order.not_eq_order_implies_strict)
lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b"
by (rule order.not_eq_order_implies_strict)
text \<open>Asymmetry.\<close>
lemma eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x"
by (blast intro: antisym)
lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
by (blast intro: antisym)
lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y"
by (fact order.strict_implies_not_eq)
text \<open>Least value operator\<close>
definition (in ord)
Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where
"Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))"
lemma Least_equality:
assumes "P x"
and "\<And>y. P y \<Longrightarrow> x \<le> y"
shows "Least P = x"
unfolding Least_def by (rule the_equality)
(blast intro: assms antisym)+
lemma LeastI2_order:
assumes "P x"
and "\<And>y. P y \<Longrightarrow> x \<le> y"
and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x"
shows "Q (Least P)"
unfolding Least_def by (rule theI2)
(blast intro: assms antisym)+
end
lemma ordering_orderI:
fixes less_eq (infix "\<^bold>\<le>" 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 "\<sqsubset>" 50)
and less_eq (infix "\<sqsubseteq>" 50)
assumes "\<And>a b. a \<sqsubseteq> b \<longleftrightarrow> a \<sqsubset> b \<or> a = b"
assumes "\<And>a b. a \<sqsubset> b \<Longrightarrow> \<not> b \<sqsubset> a"
assumes "\<And>a. \<not> a \<sqsubset> a"
assumes "\<And>a b c. a \<sqsubset> b \<Longrightarrow> b \<sqsubset> c \<Longrightarrow> a \<sqsubset> 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 (op \<ge>) (op >)"
using dual_order.ordering_axioms by (rule ordering_orderI)
end
subsection \<open>Linear (total) orders\<close>
class linorder = order +
assumes linear: "x \<le> y \<or> y \<le> x"
begin
lemma less_linear: "x < y \<or> x = y \<or> y < x"
unfolding less_le using less_le linear by blast
lemma le_less_linear: "x \<le> y \<or> y < x"
by (simp add: le_less less_linear)
lemma le_cases [case_names le ge]:
"(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P"
using linear by blast
lemma (in linorder) le_cases3:
"\<lbrakk>\<lbrakk>x \<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"
by (blast intro: le_cases)
lemma linorder_cases [case_names less equal greater]:
"(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P"
using less_linear by blast
lemma linorder_wlog[case_names le sym]:
"(\<And>a b. a \<le> b \<Longrightarrow> P a b) \<Longrightarrow> (\<And>a b. P b a \<Longrightarrow> P a b) \<Longrightarrow> P a b"
by (cases rule: le_cases[of a b]) blast+
lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x"
apply (simp add: less_le)
using linear apply (blast intro: antisym)
done
lemma not_less_iff_gr_or_eq:
"\<not>(x < y) \<longleftrightarrow> (x > y | x = y)"
apply(simp add:not_less le_less)
apply blast
done
lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x"
apply (simp add: less_le)
using linear apply (blast intro: antisym)
done
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x"
by (cut_tac x = x and y = y in less_linear, auto)
lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R"
by (simp add: neq_iff) blast
lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
by (blast intro: antisym dest: not_less [THEN iffD1])
lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
by (blast intro: antisym dest: not_less [THEN iffD1])
lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
by (blast intro: antisym dest: not_less [THEN iffD1])
lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"
unfolding not_less .
lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"
unfolding not_less .
lemma not_le_imp_less: "\<not> y \<le> x \<Longrightarrow> x < y"
unfolding not_le .
text \<open>Dual order\<close>
lemma dual_linorder:
"class.linorder (op \<ge>) (op >)"
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>\<le>" 50)
and less (infix "\<^bold><" 50)
assumes "class.order less_eq less"
assumes trichotomy: "\<And>a b. a \<^bold>< b \<or> a = b \<or> 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>\<le> b \<or> b \<^bold>\<le> 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 print_orders}
"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 Trueprop} $ 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 Not}, _) $ 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 op =} below is necessary since the operation
is not a parameter of the locale. *)
declare less_irrefl [THEN notE, order add less_reflE: order "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "op <=" "op <"]
declare order_refl [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare less_trans [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare less_le_trans [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare le_less_trans [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare order_trans [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
declare not_sym [order add not_sym: order "op = :: 'a => 'a => bool" "op <=" "op <"]
end
context linorder
begin
declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]]
declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare less_imp_le [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare not_less [THEN iffD2, order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare not_le [THEN iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare less_trans [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare less_le_trans [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare le_less_trans [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare order_trans [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare neq_le_trans [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare less_imp_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
declare not_sym [order add not_sym: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
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 less}, 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 linorder_class.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 less_eq}, 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 linorder_class.antisym_conv2})))
end handle THM _ => NONE)
| _ => NONE);
end;
\<close>
simproc_setup antisym_le ("(x::'a::order) \<le> y") = "K antisym_le_simproc"
simproc_setup antisym_less ("\<not> (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)
syntax
"_All_less" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3\<forall>_>_./ _)" [0, 0, 10] 10)
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3\<exists>_>_./ _)" [0, 0, 10] 10)
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [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)
translations
"ALL x<y. P" => "ALL x. x < y \<longrightarrow> P"
"EX x<y. P" => "EX x. x < y \<and> P"
"ALL x<=y. P" => "ALL x. x <= y \<longrightarrow> P"
"EX x<=y. P" => "EX x. x <= y \<and> P"
"ALL x>y. P" => "ALL x. x > y \<longrightarrow> P"
"EX x>y. P" => "EX x. x > y \<and> P"
"ALL x>=y. P" => "ALL x. x >= y \<longrightarrow> P"
"EX x>=y. P" => "EX x. x >= y \<and> P"
print_translation \<open>
let
val All_binder = Mixfix.binder_name @{const_syntax All};
val Ex_binder = Mixfix.binder_name @{const_syntax Ex};
val impl = @{const_syntax HOL.implies};
val conj = @{const_syntax HOL.conj};
val less = @{const_syntax less};
val less_eq = @{const_syntax less_eq};
val trans =
[((All_binder, impl, less),
(@{syntax_const "_All_less"}, @{syntax_const "_All_greater"})),
((All_binder, impl, less_eq),
(@{syntax_const "_All_less_eq"}, @{syntax_const "_All_greater_eq"})),
((Ex_binder, conj, less),
(@{syntax_const "_Ex_less"}, @{syntax_const "_Ex_greater"})),
((Ex_binder, conj, less_eq),
(@{syntax_const "_Ex_less_eq"}, @{syntax_const "_Ex_greater_eq"}))];
fun matches_bound v t =
(case t of
Const (@{syntax_const "_bound"}, _) $ 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 "_bound"}, _) $ Free (v, T),
Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
(case AList.lookup (op =) 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 \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c"
by (rule subst)
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
by (rule ssubst)
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c"
by (rule subst)
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> 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 \<Rightarrow> 'b::order) \<Rightarrow> bool" where
"mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
lemma monoI [intro?]:
fixes f :: "'a \<Rightarrow> 'b::order"
shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f"
unfolding mono_def by iprover
lemma monoD [dest?]:
fixes f :: "'a \<Rightarrow> 'b::order"
shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
unfolding mono_def by iprover
lemma monoE:
fixes f :: "'a \<Rightarrow> 'b::order"
assumes "mono f"
assumes "x \<le> y"
obtains "f x \<le> f y"
proof
from assms show "f x \<le> f y" by (simp add: mono_def)
qed
definition antimono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
"antimono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<ge> f y)"
lemma antimonoI [intro?]:
fixes f :: "'a \<Rightarrow> 'b::order"
shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> antimono f"
unfolding antimono_def by iprover
lemma antimonoD [dest?]:
fixes f :: "'a \<Rightarrow> 'b::order"
shows "antimono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<ge> f y"
unfolding antimono_def by iprover
lemma antimonoE:
fixes f :: "'a \<Rightarrow> 'b::order"
assumes "antimono f"
assumes "x \<le> y"
obtains "f x \<ge> f y"
proof
from assms show "f x \<ge> f y" by (simp add: antimono_def)
qed
definition strict_mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
"strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)"
lemma strict_monoI [intro?]:
assumes "\<And>x y. x < y \<Longrightarrow> f x < f y"
shows "strict_mono f"
using assms unfolding strict_mono_def by auto
lemma strict_monoD [dest?]:
"strict_mono f \<Longrightarrow> x < y \<Longrightarrow> 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 \<le> y"
show "f x \<le> 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 \<Rightarrow> 'b::order"
assumes "mono f"
assumes "f x < f y"
obtains "x \<le> y"
proof
show "x \<le> y"
proof (rule ccontr)
assume "\<not> x \<le> y"
then have "y \<le> 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 \<longleftrightarrow> 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 \<le> f y \<longleftrightarrow> x \<le> y"
proof
assume "x \<le> y"
with assms strict_mono_mono monoD show "f x \<le> f y" by auto
next
assume "f x \<le> f y"
show "x \<le> y" proof (rule ccontr)
assume "\<not> x \<le> 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 \<longleftrightarrow> 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 \<Rightarrow> 'a \<Rightarrow> 'a" where
"min a b = (if a \<le> b then a else b)"
definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
"max a b = (if a \<le> b then b else a)"
lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x"
by (simp add: min_def)
lemma max_absorb2: "x \<le> y \<Longrightarrow> max x y = y"
by (simp add: max_def)
lemma min_absorb2: "(y::'a::order) \<le> x \<Longrightarrow> min x y = y"
by (simp add:min_def)
lemma max_absorb1: "(y::'a::order) \<le> x \<Longrightarrow> 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 ("\<bottom>")
class order_bot = order + bot +
assumes bot_least: "\<bottom> \<le> a"
begin
sublocale bot: ordering_top greater_eq greater bot
by standard (fact bot_least)
lemma le_bot:
"a \<le> \<bottom> \<Longrightarrow> a = \<bottom>"
by (fact bot.extremum_uniqueI)
lemma bot_unique:
"a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>"
by (fact bot.extremum_unique)
lemma not_less_bot:
"\<not> a < \<bottom>"
by (fact bot.extremum_strict)
lemma bot_less:
"a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a"
by (fact bot.not_eq_extremum)
end
class top =
fixes top :: 'a ("\<top>")
class order_top = order + top +
assumes top_greatest: "a \<le> \<top>"
begin
sublocale top: ordering_top less_eq less top
by standard (fact top_greatest)
lemma top_le:
"\<top> \<le> a \<Longrightarrow> a = \<top>"
by (fact top.extremum_uniqueI)
lemma top_unique:
"\<top> \<le> a \<longleftrightarrow> a = \<top>"
by (fact top.extremum_unique)
lemma not_top_less:
"\<not> \<top> < a"
by (fact top.extremum_strict)
lemma less_top:
"a \<noteq> \<top> \<longleftrightarrow> a < \<top>"
by (fact top.not_eq_extremum)
end
subsection \<open>Dense orders\<close>
class dense_order = order +
assumes dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"
class dense_linorder = linorder + dense_order
begin
lemma dense_le:
fixes y z :: 'a
assumes "\<And>x. x < y \<Longrightarrow> x \<le> z"
shows "y \<le> z"
proof (rule ccontr)
assume "\<not> ?thesis"
hence "z < y" by simp
from dense[OF this]
obtain x where "x < y" and "z < x" by safe
moreover have "x \<le> z" using assms[OF \<open>x < y\<close>] .
ultimately show False by auto
qed
lemma dense_le_bounded:
fixes x y z :: 'a
assumes "x < y"
assumes *: "\<And>w. \<lbrakk> x < w ; w < y \<rbrakk> \<Longrightarrow> w \<le> z"
shows "y \<le> 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 \<le> z"
proof (rule disjE)
assume "u \<le> w"
from less_le_trans[OF \<open>x < u\<close> \<open>u \<le> w\<close>] \<open>w < y\<close>
show "w \<le> z" by (rule *)
next
assume "w \<le> u"
from \<open>w \<le> u\<close> *[OF \<open>x < u\<close> \<open>u < y\<close>]
show "w \<le> z" by (rule order_trans)
qed
qed
lemma dense_ge:
fixes y z :: 'a
assumes "\<And>x. z < x \<Longrightarrow> y \<le> x"
shows "y \<le> z"
proof (rule ccontr)
assume "\<not> ?thesis"
hence "z < y" by simp
from dense[OF this]
obtain x where "x < y" and "z < x" by safe
moreover have "y \<le> x" using assms[OF \<open>z < x\<close>] .
ultimately show False by auto
qed
lemma dense_ge_bounded:
fixes x y z :: 'a
assumes "z < x"
assumes *: "\<And>w. \<lbrakk> z < w ; w < x \<rbrakk> \<Longrightarrow> y \<le> w"
shows "y \<le> 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 \<le> w"
proof (rule disjE)
assume "w \<le> u"
from \<open>z < w\<close> le_less_trans[OF \<open>w \<le> u\<close> \<open>u < x\<close>]
show "y \<le> w" by (rule *)
next
assume "u \<le> w"
from *[OF \<open>z < u\<close> \<open>u < x\<close>] \<open>u \<le> w\<close>
show "y \<le> w" by (rule order_trans)
qed
qed
end
class no_top = order +
assumes gt_ex: "\<exists>y. x < y"
class no_bot = order +
assumes lt_ex: "\<exists>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]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> 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) \<le> k"
proof -
have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> 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: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)"
have "\<And>y. P y \<Longrightarrow> x \<le> y"
proof (rule classical)
fix y
assume "P y" and "\<not> x \<le> y"
with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
by (auto simp add: not_le)
with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
by auto
then show "x \<le> 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) \<le> k" by auto
qed
\<comment> "The following 3 lemmas are due to Brian Huffman"
lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)"
by (erule exE) (erule LeastI)
lemma LeastI2:
"P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
by (blast intro: LeastI)
lemma LeastI2_ex:
"\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
by (blast intro: LeastI_ex)
lemma LeastI2_wellorder:
assumes "P a"
and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> 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 \<le> y" by (rule Least_le)
next
fix x assume "P x" "\<forall>y. P y \<longrightarrow> x \<le> y" thus "Q x" by (rule assms(2))
qed
lemma LeastI2_wellorder_ex:
assumes "\<exists>x. P x"
and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a"
shows "Q (Least P)"
using assms by clarify (blast intro!: LeastI2_wellorder)
lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k"
apply (simp add: not_le [symmetric])
apply (erule contrapos_nn)
apply (erule Least_le)
done
lemma exists_least_iff: "(\<exists>n. P n) \<longleftrightarrow> (\<exists>n. P n \<and> (\<forall>m < n. \<not> P m))" (is "?lhs \<longleftrightarrow> ?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 "\<not> P m" . }
with LeastI_ex[OF H] show ?rhs by blast
qed
end
subsection \<open>Order on @{typ bool}\<close>
instantiation bool :: "{order_bot, order_top, linorder}"
begin
definition
le_bool_def [simp]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q"
definition
[simp]: "(P::bool) < Q \<longleftrightarrow> \<not> P \<and> Q"
definition
[simp]: "\<bottom> \<longleftrightarrow> False"
definition
[simp]: "\<top> \<longleftrightarrow> True"
instance proof
qed auto
end
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
by simp
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
by simp
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
by simp
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
by simp
lemma bot_boolE: "\<bottom> \<Longrightarrow> P"
by simp
lemma top_boolI: \<top>
by simp
lemma [code]:
"False \<le> b \<longleftrightarrow> True"
"True \<le> b \<longleftrightarrow> b"
"False < b \<longleftrightarrow> b"
"True < b \<longleftrightarrow> False"
by simp_all
subsection \<open>Order on @{typ "_ \<Rightarrow> _"}\<close>
instantiation "fun" :: (type, ord) ord
begin
definition
le_fun_def: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)"
definition
"(f::'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> \<not> (g \<le> 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
"\<bottom> = (\<lambda>x. \<bottom>)"
instance ..
end
instantiation "fun" :: (type, order_bot) order_bot
begin
lemma bot_apply [simp, code]:
"\<bottom> x = \<bottom>"
by (simp add: bot_fun_def)
instance proof
qed (simp add: le_fun_def)
end
instantiation "fun" :: (type, top) top
begin
definition
[no_atp]: "\<top> = (\<lambda>x. \<top>)"
instance ..
end
instantiation "fun" :: (type, order_top) order_top
begin
lemma top_apply [simp, code]:
"\<top> x = \<top>"
by (simp add: top_fun_def)
instance proof
qed (simp add: le_fun_def)
end
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
unfolding le_fun_def by simp
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
unfolding le_fun_def by simp
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
by (rule le_funE)
lemma mono_compose: "mono Q \<Longrightarrow> mono (\<lambda>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: "\<And>x. P x \<Longrightarrow> Q x"
shows "P \<le> Q"
apply (rule le_funI)
apply (rule le_boolI)
apply (rule PQ)
apply assumption
done
lemma predicate1D:
"P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
apply (erule le_funE)
apply (erule le_boolE)
apply assumption+
done
lemma rev_predicate1D:
"P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x"
by (rule predicate1D)
lemma predicate2I:
assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
shows "P \<le> Q"
apply (rule le_funI)+
apply (rule le_boolI)
apply (rule PQ)
apply assumption
done
lemma predicate2D:
"P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
apply (erule le_funE)+
apply (erule le_boolE)
apply assumption+
done
lemma rev_predicate2D:
"P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y"
by (rule predicate2D)
lemma bot1E [no_atp]: "\<bottom> x \<Longrightarrow> P"
by (simp add: bot_fun_def)
lemma bot2E: "\<bottom> x y \<Longrightarrow> P"
by (simp add: bot_fun_def)
lemma top1I: "\<top> x"
by (simp add: top_fun_def)
lemma top2I: "\<top> 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
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
end