(* Title: HOL/Orderings.thy
ID: $Id$
Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)
header {* Syntactic and abstract orders *}
theory Orderings
imports HOL
begin
subsection {* Order syntax *}
class ord =
fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubseteq>" 50)
and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubset>" 50)
begin
notation
less_eq ("op \<^loc><=") and
less_eq ("(_/ \<^loc><= _)" [51, 51] 50) and
less ("op \<^loc><") and
less ("(_/ \<^loc>< _)" [51, 51] 50)
notation (xsymbols)
less_eq ("op \<^loc>\<le>") and
less_eq ("(_/ \<^loc>\<le> _)" [51, 51] 50)
notation (HTML output)
less_eq ("op \<^loc>\<le>") and
less_eq ("(_/ \<^loc>\<le> _)" [51, 51] 50)
abbreviation (input)
greater (infix "\<^loc>>" 50) where
"x \<^loc>> y \<equiv> y \<^loc>< x"
abbreviation (input)
greater_eq (infix "\<^loc>>=" 50) where
"x \<^loc>>= y \<equiv> y \<^loc><= x"
notation (input)
greater_eq (infix "\<^loc>\<ge>" 50)
end
notation
less_eq ("op <=") and
less_eq ("(_/ <= _)" [51, 51] 50) and
less ("op <") and
less ("(_/ < _)" [51, 51] 50)
notation (xsymbols)
less_eq ("op \<le>") and
less_eq ("(_/ \<le> _)" [51, 51] 50)
notation (HTML output)
less_eq ("op \<le>") and
less_eq ("(_/ \<le> _)" [51, 51] 50)
abbreviation (input)
greater (infix ">" 50) where
"x > y \<equiv> y < x"
abbreviation (input)
greater_eq (infix ">=" 50) where
"x >= y \<equiv> y <= x"
notation (input)
greater_eq (infix "\<ge>" 50)
subsection {* Quasiorders (preorders) *}
class preorder = ord +
assumes less_le: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y"
and order_refl [iff]: "x \<sqsubseteq> x"
and order_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
begin
text {* Reflexivity. *}
lemma eq_refl: "x = y \<Longrightarrow> x \<sqsubseteq> y"
-- {* This form is useful with the classical reasoner. *}
by (erule ssubst) (rule order_refl)
lemma less_irrefl [iff]: "\<not> x \<sqsubset> x"
by (simp add: less_le)
lemma le_less: "x \<sqsubseteq> y \<longleftrightarrow> x \<sqsubset> y \<or> x = y"
-- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
by (simp add: less_le) blast
lemma le_imp_less_or_eq: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubset> y \<or> x = y"
unfolding less_le by blast
lemma less_imp_le: "x \<sqsubset> y \<Longrightarrow> x \<sqsubseteq> y"
unfolding less_le by blast
lemma less_imp_neq: "x \<sqsubset> y \<Longrightarrow> x \<noteq> y"
by (erule contrapos_pn, erule subst, rule less_irrefl)
text {* Useful for simplification, but too risky to include by default. *}
lemma less_imp_not_eq: "x \<sqsubset> y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
by auto
lemma less_imp_not_eq2: "x \<sqsubset> y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
by auto
text {* Transitivity rules for calculational reasoning *}
lemma neq_le_trans: "\<lbrakk> a \<noteq> b; a \<sqsubseteq> b \<rbrakk> \<Longrightarrow> a \<sqsubset> b"
by (simp add: less_le)
lemma le_neq_trans: "\<lbrakk> a \<sqsubseteq> b; a \<noteq> b \<rbrakk> \<Longrightarrow> a \<sqsubset> b"
by (simp add: less_le)
end
subsection {* Partial orderings *}
class order = preorder +
assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
begin
text {* Asymmetry. *}
lemma less_not_sym: "x \<sqsubset> y \<Longrightarrow> \<not> (y \<sqsubset> x)"
by (simp add: less_le antisym)
lemma less_asym: "x \<sqsubset> y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<sqsubset> x) \<Longrightarrow> P"
by (drule less_not_sym, erule contrapos_np) simp
lemma eq_iff: "x = y \<longleftrightarrow> x \<sqsubseteq> y \<and> y \<sqsubseteq> x"
by (blast intro: antisym)
lemma antisym_conv: "y \<sqsubseteq> x \<Longrightarrow> x \<sqsubseteq> y \<longleftrightarrow> x = y"
by (blast intro: antisym)
lemma less_imp_neq: "x \<sqsubset> y \<Longrightarrow> x \<noteq> y"
by (erule contrapos_pn, erule subst, rule less_irrefl)
text {* Transitivity. *}
lemma less_trans: "\<lbrakk> x \<sqsubset> y; y \<sqsubset> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z"
by (simp add: less_le) (blast intro: order_trans antisym)
lemma le_less_trans: "\<lbrakk> x \<sqsubseteq> y; y \<sqsubset> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z"
by (simp add: less_le) (blast intro: order_trans antisym)
lemma less_le_trans: "\<lbrakk> x \<sqsubset> y; y \<sqsubseteq> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z"
by (simp add: less_le) (blast intro: order_trans antisym)
text {* Useful for simplification, but too risky to include by default. *}
lemma less_imp_not_less: "x \<sqsubset> y \<Longrightarrow> (\<not> y \<sqsubset> x) \<longleftrightarrow> True"
by (blast elim: less_asym)
lemma less_imp_triv: "x \<sqsubset> y \<Longrightarrow> (y \<sqsubset> x \<longrightarrow> P) \<longleftrightarrow> True"
by (blast elim: less_asym)
text {* Transitivity rules for calculational reasoning *}
lemma less_asym': "\<lbrakk> a \<sqsubset> b; b \<sqsubset> a \<rbrakk> \<Longrightarrow> P"
by (rule less_asym)
end
subsection {* Linear (total) orders *}
class linorder = order +
assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
begin
lemma less_linear: "x \<sqsubset> y \<or> x = y \<or> y \<sqsubset> x"
unfolding less_le using less_le linear by blast
lemma le_less_linear: "x \<sqsubseteq> y \<or> y \<sqsubset> x"
by (simp add: le_less less_linear)
lemma le_cases [case_names le ge]:
"\<lbrakk> x \<sqsubseteq> y \<Longrightarrow> P; y \<sqsubseteq> x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
using linear by blast
lemma linorder_cases [case_names less equal greater]:
"\<lbrakk> x \<sqsubset> y \<Longrightarrow> P; x = y \<Longrightarrow> P; y \<sqsubset> x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
using less_linear by blast
lemma not_less: "\<not> x \<sqsubset> y \<longleftrightarrow> y \<sqsubseteq> x"
apply (simp add: less_le)
using linear apply (blast intro: antisym)
done
lemma not_le: "\<not> x \<sqsubseteq> y \<longleftrightarrow> y \<sqsubset> x"
apply (simp add: less_le)
using linear apply (blast intro: antisym)
done
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<sqsubset> y \<or> y \<sqsubset> x"
by (cut_tac x = x and y = y in less_linear, auto)
lemma neqE: "\<lbrakk> x \<noteq> y; x \<sqsubset> y \<Longrightarrow> R; y \<sqsubset> x \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
by (simp add: neq_iff) blast
lemma antisym_conv1: "\<not> x \<sqsubset> y \<Longrightarrow> x \<sqsubseteq> y \<longleftrightarrow> x = y"
by (blast intro: antisym dest: not_less [THEN iffD1])
lemma antisym_conv2: "x \<sqsubseteq> y \<Longrightarrow> \<not> x \<sqsubset> y \<longleftrightarrow> x = y"
by (blast intro: antisym dest: not_less [THEN iffD1])
lemma antisym_conv3: "\<not> y \<sqsubset> x \<Longrightarrow> \<not> x \<sqsubset> y \<longleftrightarrow> x = y"
by (blast intro: antisym dest: not_less [THEN iffD1])
text{*Replacing the old Nat.leI*}
lemma leI: "\<not> x \<sqsubset> y \<Longrightarrow> y \<sqsubseteq> x"
unfolding not_less .
lemma leD: "y \<sqsubseteq> x \<Longrightarrow> \<not> x \<sqsubset> y"
unfolding not_less .
(*FIXME inappropriate name (or delete altogether)*)
lemma not_leE: "\<not> y \<sqsubseteq> x \<Longrightarrow> x \<sqsubset> y"
unfolding not_le .
text {* min/max *}
definition
min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
"min a b = (if a \<sqsubseteq> b then a else b)"
definition
max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
"max a b = (if a \<sqsubseteq> b then b else a)"
end
context linorder
begin
lemma min_le_iff_disj:
"min x y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
unfolding min_def using linear by (auto intro: order_trans)
lemma le_max_iff_disj:
"z \<sqsubseteq> max x y \<longleftrightarrow> z \<sqsubseteq> x \<or> z \<sqsubseteq> y"
unfolding max_def using linear by (auto intro: order_trans)
lemma min_less_iff_disj:
"min x y \<sqsubset> z \<longleftrightarrow> x \<sqsubset> z \<or> y \<sqsubset> z"
unfolding min_def le_less using less_linear by (auto intro: less_trans)
lemma less_max_iff_disj:
"z \<sqsubset> max x y \<longleftrightarrow> z \<sqsubset> x \<or> z \<sqsubset> y"
unfolding max_def le_less using less_linear by (auto intro: less_trans)
lemma min_less_iff_conj [simp]:
"z \<sqsubset> min x y \<longleftrightarrow> z \<sqsubset> x \<and> z \<sqsubset> y"
unfolding min_def le_less using less_linear by (auto intro: less_trans)
lemma max_less_iff_conj [simp]:
"max x y \<sqsubset> z \<longleftrightarrow> x \<sqsubset> z \<and> y \<sqsubset> z"
unfolding max_def le_less using less_linear by (auto intro: less_trans)
lemma split_min:
"P (min i j) \<longleftrightarrow> (i \<sqsubseteq> j \<longrightarrow> P i) \<and> (\<not> i \<sqsubseteq> j \<longrightarrow> P j)"
by (simp add: min_def)
lemma split_max:
"P (max i j) \<longleftrightarrow> (i \<sqsubseteq> j \<longrightarrow> P j) \<and> (\<not> i \<sqsubseteq> j \<longrightarrow> P i)"
by (simp add: max_def)
end
subsection {* Name duplicates *}
(*lemmas order_refl [iff] = preorder_class.order_refl
lemmas order_trans = preorder_class.order_trans*)
lemmas order_less_le = less_le
lemmas order_eq_refl = preorder_class.eq_refl
lemmas order_less_irrefl = preorder_class.less_irrefl
lemmas order_le_less = preorder_class.le_less
lemmas order_le_imp_less_or_eq = preorder_class.le_imp_less_or_eq
lemmas order_less_imp_le = preorder_class.less_imp_le
lemmas order_less_imp_not_eq = preorder_class.less_imp_not_eq
lemmas order_less_imp_not_eq2 = preorder_class.less_imp_not_eq2
lemmas order_neq_le_trans = preorder_class.neq_le_trans
lemmas order_le_neq_trans = preorder_class.le_neq_trans
lemmas order_antisym = antisym
lemmas order_less_not_sym = order_class.less_not_sym
lemmas order_less_asym = order_class.less_asym
lemmas order_eq_iff = order_class.eq_iff
lemmas order_antisym_conv = order_class.antisym_conv
lemmas less_imp_neq = order_class.less_imp_neq
lemmas order_less_trans = order_class.less_trans
lemmas order_le_less_trans = order_class.le_less_trans
lemmas order_less_le_trans = order_class.less_le_trans
lemmas order_less_imp_not_less = order_class.less_imp_not_less
lemmas order_less_imp_triv = order_class.less_imp_triv
lemmas order_less_asym' = order_class.less_asym'
lemmas linorder_linear = 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_cases = linorder_class.linorder_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
lemmas leI = linorder_class.leI
lemmas leD = linorder_class.leD
lemmas not_leE = linorder_class.not_leE
subsection {* Reasoning tools setup *}
ML {*
local
fun decomp_gen sort thy (Trueprop $ t) =
let
fun of_sort t =
let
val T = type_of t
in
(* exclude numeric types: linear arithmetic subsumes transitivity *)
T <> HOLogic.natT andalso T <> HOLogic.intT
andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort)
end;
fun dec (Const ("Not", _) $ t) = (case dec t
of NONE => NONE
| SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
| dec (Const ("op =", _) $ t1 $ t2) =
if of_sort t1
then SOME (t1, "=", t2)
else NONE
| dec (Const ("Orderings.less_eq", _) $ t1 $ t2) =
if of_sort t1
then SOME (t1, "<=", t2)
else NONE
| dec (Const ("Orderings.less", _) $ t1 $ t2) =
if of_sort t1
then SOME (t1, "<", t2)
else NONE
| dec _ = NONE;
in dec t end;
in
(* The setting up of Quasi_Tac serves as a demo. Since there is no
class for quasi orders, the tactics Quasi_Tac.trans_tac and
Quasi_Tac.quasi_tac are not of much use. *)
structure Quasi_Tac = Quasi_Tac_Fun (
struct
val le_trans = thm "order_trans";
val le_refl = thm "order_refl";
val eqD1 = thm "order_eq_refl";
val eqD2 = thm "sym" RS thm "order_eq_refl";
val less_reflE = thm "order_less_irrefl" RS thm "notE";
val less_imp_le = thm "order_less_imp_le";
val le_neq_trans = thm "order_le_neq_trans";
val neq_le_trans = thm "order_neq_le_trans";
val less_imp_neq = thm "less_imp_neq";
val decomp_trans = decomp_gen ["Orderings.order"];
val decomp_quasi = decomp_gen ["Orderings.order"];
end);
structure Order_Tac = Order_Tac_Fun (
struct
val less_reflE = thm "order_less_irrefl" RS thm "notE";
val le_refl = thm "order_refl";
val less_imp_le = thm "order_less_imp_le";
val not_lessI = thm "linorder_not_less" RS thm "iffD2";
val not_leI = thm "linorder_not_le" RS thm "iffD2";
val not_lessD = thm "linorder_not_less" RS thm "iffD1";
val not_leD = thm "linorder_not_le" RS thm "iffD1";
val eqI = thm "order_antisym";
val eqD1 = thm "order_eq_refl";
val eqD2 = thm "sym" RS thm "order_eq_refl";
val less_trans = thm "order_less_trans";
val less_le_trans = thm "order_less_le_trans";
val le_less_trans = thm "order_le_less_trans";
val le_trans = thm "order_trans";
val le_neq_trans = thm "order_le_neq_trans";
val neq_le_trans = thm "order_neq_le_trans";
val less_imp_neq = thm "less_imp_neq";
val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
val not_sym = thm "not_sym";
val decomp_part = decomp_gen ["Orderings.order"];
val decomp_lin = decomp_gen ["Orderings.linorder"];
end);
end;
*}
setup {*
let
val order_antisym_conv = thm "order_antisym_conv"
val linorder_antisym_conv1 = thm "linorder_antisym_conv1"
val linorder_antisym_conv2 = thm "linorder_antisym_conv2"
val linorder_antisym_conv3 = thm "linorder_antisym_conv3"
fun prp t thm = (#prop (rep_thm thm) = t);
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
let val prems = prems_of_ss ss;
val less = Const("Orderings.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 linorder_antisym_conv1))
end
| SOME thm => SOME(mk_meta_eq(thm RS order_antisym_conv))
end
handle THM _ => NONE;
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
let val prems = prems_of_ss ss;
val le = Const("Orderings.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 linorder_antisym_conv3))
end
| SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv2))
end
handle THM _ => NONE;
fun add_simprocs procs thy =
(Simplifier.change_simpset_of thy (fn ss => ss
addsimprocs (map (fn (name, raw_ts, proc) =>
Simplifier.simproc thy name raw_ts proc)) procs); thy);
fun add_solver name tac thy =
(Simplifier.change_simpset_of thy (fn ss => ss addSolver
(mk_solver name (K tac))); thy);
in
add_simprocs [
("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
]
#> add_solver "Trans_linear" Order_Tac.linear_tac
#> add_solver "Trans_partial" Order_Tac.partial_tac
(* 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). *)
end
*}
subsection {* Bounded quantifiers *}
syntax
"_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 (xsymbols)
"_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 (HOL)
"_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)
syntax (HTML output)
"_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)
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 {*
let
val All_binder = Syntax.binder_name @{const_syntax "All"};
val Ex_binder = Syntax.binder_name @{const_syntax "Ex"};
val impl = @{const_syntax "op -->"};
val conj = @{const_syntax "op &"};
val less = @{const_syntax "less"};
val less_eq = @{const_syntax "less_eq"};
val trans =
[((All_binder, impl, less), ("_All_less", "_All_greater")),
((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
fun matches_bound v t =
case t of (Const ("_bound", _) $ Free (v', _)) => (v = v')
| _ => false
fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P
fun tr' q = (q,
fn [Const ("_bound", _) $ Free (v, _), 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 l u P
else if matches_bound v u andalso not (contains_var v t) then mk v g t P
else raise Match)
| _ => raise Match);
in [tr' All_binder, tr' Ex_binder] end
*}
subsection {* Transitivity reasoning *}
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)
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 (order_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 (order_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 (order_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 (order_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 (order_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 (order_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 {*
Note that this list of rules is in reverse order of priorities.
*}
lemmas order_trans_rules [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
order_neq_le_trans
order_le_neq_trans
order_less_trans
order_less_asym'
order_le_less_trans
order_less_le_trans
order_trans
order_antisym
ord_le_eq_trans
ord_eq_le_trans
ord_less_eq_trans
ord_eq_less_trans
trans
(* FIXME cleanup *)
text {* These support proving chains of decreasing inequalities
a >= b >= c ... in Isar proofs. *}
lemma xt1:
"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:
"(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: "(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: "(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: "(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: "(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: "(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: "(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: "(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 {* Order on bool *}
instance bool :: linorder
le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
by default (auto simp add: le_bool_def less_bool_def)
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
by (simp add: le_bool_def)
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
by (simp add: le_bool_def)
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
by (simp add: le_bool_def)
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
by (simp add: le_bool_def)
lemma [code func]:
"False \<le> b \<longleftrightarrow> True"
"True \<le> b \<longleftrightarrow> b"
"False < b \<longleftrightarrow> b"
"True < b \<longleftrightarrow> False"
unfolding le_bool_def less_bool_def by simp_all
subsection {* Monotonicity, syntactic least value operator and min/max *}
locale mono =
fixes f
assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
lemmas monoI [intro?] = mono.intro
and monoD [dest?] = mono.mono
constdefs
Least :: "('a::ord => bool) => 'a" (binder "LEAST " 10)
"Least P == THE x. P x & (ALL y. P y --> x <= y)"
-- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
lemma LeastI2_order:
"[| P (x::'a::order);
!!y. P y ==> x <= y;
!!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
==> Q (Least P)"
apply (unfold Least_def)
apply (rule theI2)
apply (blast intro: order_antisym)+
done
lemma Least_equality:
"[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
apply (simp add: Least_def)
apply (rule the_equality)
apply (auto intro!: order_antisym)
done
constdefs
min :: "['a::ord, 'a] => 'a"
"min a b == (if a <= b then a else b)"
max :: "['a::ord, 'a] => 'a"
"max a b == (if a <= b then b else a)"
lemma min_linorder:
"linorder.min (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = min"
by rule+ (simp add: min_def linorder_class.min_def)
lemma max_linorder:
"linorder.max (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = max"
by rule+ (simp add: max_def linorder_class.max_def)
lemmas min_le_iff_disj = linorder_class.min_le_iff_disj [unfolded min_linorder]
lemmas le_max_iff_disj = linorder_class.le_max_iff_disj [unfolded max_linorder]
lemmas min_less_iff_disj = linorder_class.min_less_iff_disj [unfolded min_linorder]
lemmas less_max_iff_disj = linorder_class.less_max_iff_disj [unfolded max_linorder]
lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj [unfolded min_linorder]
lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj [unfolded max_linorder]
lemmas split_min = linorder_class.split_min [unfolded min_linorder]
lemmas split_max = linorder_class.split_max [unfolded max_linorder]
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
by (simp add: min_def)
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
by (simp add: max_def)
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
apply (simp add: min_def)
apply (blast intro: order_antisym)
done
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
apply (simp add: max_def)
apply (blast intro: order_antisym)
done
lemma min_of_mono:
"(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
by (simp add: min_def)
lemma max_of_mono:
"(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
by (simp add: max_def)
subsection {* Basic ML bindings *}
ML {*
val leD = thm "leD";
val leI = thm "leI";
val linorder_neqE = thm "linorder_neqE";
val linorder_neq_iff = thm "linorder_neq_iff";
val linorder_not_le = thm "linorder_not_le";
val linorder_not_less = thm "linorder_not_less";
val monoD = thm "monoD";
val monoI = thm "monoI";
val order_antisym = thm "order_antisym";
val order_less_irrefl = thm "order_less_irrefl";
val order_refl = thm "order_refl";
val order_trans = thm "order_trans";
val split_max = thm "split_max";
val split_min = thm "split_min";
*}
ML {*
structure HOL =
struct
val thy = theory "HOL";
end;
*} -- "belongs to theory HOL"
end