Ord.thy/.ML converted to Isar
added min_of_mono, max_of_mono, max_leastL, max_leastR to Ord.thy
moved Least_equality from Nat.ML to Ord.thy
added LeastM operator
(* Title: HOL/Ord.thy
ID: $Id$
Author: Tobias Nipkow, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Type classes for order signatures and orders.
*)
theory Ord = HOL:
axclass
ord < "term"
syntax
"op <" :: "['a::ord, 'a] => bool" ("op <")
"op <=" :: "['a::ord, 'a] => bool" ("op <=")
global
consts
"op <" :: "['a::ord, 'a] => bool" ("(_/ < _)" [50, 51] 50)
"op <=" :: "['a::ord, 'a] => bool" ("(_/ <= _)" [50, 51] 50)
local
syntax (symbols)
"op <=" :: "['a::ord, 'a] => bool" ("op \\<le>")
"op <=" :: "['a::ord, 'a] => bool" ("(_/ \\<le> _)" [50, 51] 50)
(*Tell Blast_tac about overloading of < and <= to reduce the risk of
its applying a rule for the wrong type*)
ML {*
Blast.overloaded ("op <" , domain_type);
Blast.overloaded ("op <=", domain_type);
*}
constdefs
mono :: "['a::ord => 'b::ord] => bool" (*monotonicity*)
"mono(f) == (!A B. A <= B --> f(A) <= f(B))"
lemma monoI [intro?]: "[| !!A B. A <= B ==> f(A) <= f(B) |] ==> mono(f)"
apply (unfold mono_def)
apply fast
done
lemma monoD [dest?]: "[| mono(f); A <= B |] ==> f(A) <= f(B)"
apply (unfold mono_def)
apply fast
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_leastL: "(!!x. least <= x) ==> min least x = least"
apply (simp add: min_def)
done
lemma min_of_mono:
"!x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
apply (simp add: min_def)
done
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
apply (simp add: max_def)
done
lemma max_of_mono:
"!x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
apply (simp add: max_def)
done
constdefs
LeastM :: "['a => 'b::ord, 'a => bool] => 'a"
"LeastM m P == @x. P x & (!y. P y --> m x <= m y)"
Least :: "('a::ord => bool) => 'a" (binder "LEAST " 10)
"Least == LeastM (%x. x)"
syntax
"@LeastM" :: "[pttrn, 'a=>'b::ord, bool] => 'a" ("LEAST _ WRT _. _" [0,4,10]10)
translations
"LEAST x WRT m. P" == "LeastM m (%x. P)"
lemma LeastMI2: "[| P x; !!y. P y ==> m x <= m y;
!!x. [| P x; \\<forall>y. P y \\<longrightarrow> m x \\<le> m y |] ==> Q x
|] ==> Q (LeastM m P)";
apply (unfold LeastM_def)
apply (rule someI2_ex)
apply blast
apply blast
done
section "Orders"
axclass order < ord
order_refl [iff]: "x <= x"
order_trans: "[| x <= y; y <= z |] ==> x <= z"
order_antisym: "[| x <= y; y <= x |] ==> x = y"
order_less_le: "x < y = (x <= y & x ~= y)"
(** Reflexivity **)
(*This form is useful with the classical reasoner*)
lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
apply (erule ssubst)
apply (rule order_refl)
done
lemma order_less_irrefl [simp]: "~ x < (x::'a::order)"
apply (simp (no_asm) add: order_less_le)
done
lemma order_le_less: "(x::'a::order) <= y = (x < y | x = y)"
apply (simp (no_asm) add: order_less_le)
(*NOT suitable for AddIffs, since it can cause PROOF FAILED*)
apply (blast intro!: order_refl)
done
lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
apply (simp add: order_less_le)
done
(** Asymmetry **)
lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y<x)"
apply (simp add: order_less_le order_antisym)
done
(* [| n<m; ~P ==> m<n |] ==> P *)
lemmas order_less_asym = order_less_not_sym [THEN contrapos_np, standard]
(* Transitivity *)
lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
apply (simp add: order_less_le)
apply (blast intro: order_trans order_antisym)
done
lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
apply (simp add: order_less_le)
apply (blast intro: order_trans order_antisym)
done
lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
apply (simp add: order_less_le)
apply (blast intro: order_trans order_antisym)
done
(** Useful for simplification, but too risky to include by default. **)
lemma order_less_imp_not_less: "(x::'a::order) < y ==> (~ y < x) = True"
apply (blast elim: order_less_asym)
done
lemma order_less_imp_triv: "(x::'a::order) < y ==> (y < x --> P) = True"
apply (blast elim: order_less_asym)
done
lemma order_less_imp_not_eq: "(x::'a::order) < y ==> (x = y) = False"
apply auto
done
lemma order_less_imp_not_eq2: "(x::'a::order) < y ==> (y = x) = False"
apply auto
done
(* Other operators *)
lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
apply (simp (no_asm_simp) add: min_def)
apply (blast intro: order_antisym)
done
lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
apply (simp add: max_def)
apply (blast intro: order_antisym)
done
lemma LeastM_equality:
"[| P k; !!x. P x ==> m k <= m x |] ==> m (LEAST x WRT m. P x) =
(m k::'a::order)";
apply (rule LeastMI2)
apply assumption
apply blast
apply (blast intro!: order_antisym)
done
lemma Least_equality:
"[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k";
apply (unfold Least_def)
apply (erule LeastM_equality)
apply blast
done
section "Linear/Total Orders"
axclass linorder < order
linorder_linear: "x <= y | y <= x"
lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
apply (simp (no_asm) add: order_less_le)
apply (cut_tac linorder_linear)
apply blast
done
lemma linorder_less_split:
"[| (x::'a::linorder)<y ==> P; x=y ==> P; y<x ==> P |] ==> P"
apply (cut_tac linorder_less_linear)
apply blast
done
lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
apply (simp (no_asm) add: order_less_le)
apply (cut_tac linorder_linear)
apply (blast intro: order_antisym)
done
lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
apply (simp (no_asm) add: order_less_le)
apply (cut_tac linorder_linear)
apply (blast intro: order_antisym)
done
lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
apply (cut_tac x = "x" and y = "y" in linorder_less_linear)
apply auto
done
(* eliminates ~= in premises *)
lemmas linorder_neqE = linorder_neq_iff [THEN iffD1, THEN disjE, standard]
section "min & max on (linear) orders"
lemma min_same [simp]: "min (x::'a::order) x = x"
apply (simp add: min_def)
done
lemma max_same [simp]: "max (x::'a::order) x = x"
apply (simp add: max_def)
done
lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
apply (unfold max_def)
apply (simp (no_asm))
apply (cut_tac linorder_linear)
apply (blast intro: order_trans)
done
lemma le_maxI1: "(x::'a::linorder) <= max x y"
apply (simp (no_asm) add: le_max_iff_disj)
done
lemma le_maxI2: "(y::'a::linorder) <= max x y"
apply (simp (no_asm) add: le_max_iff_disj)
done
(*CANNOT use with AddSIs because blast_tac will give PROOF FAILED.*)
lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
apply (simp (no_asm) add: max_def order_le_less)
apply (cut_tac linorder_less_linear)
apply (blast intro: order_less_trans)
done
lemma max_le_iff_conj [simp]:
"!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
apply (simp (no_asm) add: max_def)
apply (cut_tac linorder_linear)
apply (blast intro: order_trans)
done
lemma max_less_iff_conj [simp]:
"!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
apply (simp (no_asm) add: order_le_less max_def)
apply (cut_tac linorder_less_linear)
apply (blast intro: order_less_trans)
done
lemma le_min_iff_conj [simp]:
"!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
apply (simp (no_asm) add: min_def)
apply (cut_tac linorder_linear)
apply (blast intro: order_trans)
done
(* AddIffs screws up a blast_tac in MiniML *)
lemma min_less_iff_conj [simp]:
"!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
apply (simp (no_asm) add: order_le_less min_def)
apply (cut_tac linorder_less_linear)
apply (blast intro: order_less_trans)
done
lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
apply (unfold min_def)
apply (simp (no_asm))
apply (cut_tac linorder_linear)
apply (blast intro: order_trans)
done
lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
apply (unfold min_def)
apply (simp (no_asm) add: order_le_less)
apply (cut_tac linorder_less_linear)
apply (blast intro: order_less_trans)
done
lemma split_min:
"P(min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
apply (simp (no_asm) add: min_def)
done
lemma split_max:
"P(max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
apply (simp (no_asm) add: max_def)
done
section "bounded quantifiers"
syntax
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10)
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10)
"_leAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10)
"_leEx" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10)
syntax (symbols)
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3\\<forall>_<_./ _)" [0, 0, 10] 10)
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3\\<exists>_<_./ _)" [0, 0, 10] 10)
"_leAll" :: "[idt, 'a, bool] => bool" ("(3\\<forall>_\\<le>_./ _)" [0, 0, 10] 10)
"_leEx" :: "[idt, 'a, bool] => bool" ("(3\\<exists>_\\<le>_./ _)" [0, 0, 10] 10)
syntax (HOL)
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10)
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10)
"_leAll" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10)
"_leEx" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10)
translations
"ALL x<y. P" => "ALL x. x < y --> P"
"EX x<y. P" => "EX x. x < y & P"
"ALL x<=y. P" => "ALL x. x <= y --> P"
"EX x<=y. P" => "EX x. x <= y & P"
ML
{*
val mono_def = thm "mono_def";
val monoI = thm "monoI";
val monoD = thm "monoD";
val min_def = thm "min_def";
val min_of_mono = thm "min_of_mono";
val max_def = thm "max_def";
val max_of_mono = thm "max_of_mono";
val min_leastL = thm "min_leastL";
val max_leastL = thm "max_leastL";
val LeastMI2 = thm "LeastMI2";
val LeastM_equality = thm "LeastM_equality";
val Least_def = thm "Least_def";
val Least_equality = thm "Least_equality";
val min_leastR = thm "min_leastR";
val max_leastR = thm "max_leastR";
val order_eq_refl = thm "order_eq_refl";
val order_less_irrefl = thm "order_less_irrefl";
val order_le_less = thm "order_le_less";
val order_le_imp_less_or_eq = thm "order_le_imp_less_or_eq";
val order_less_imp_le = thm "order_less_imp_le";
val order_less_not_sym = thm "order_less_not_sym";
val order_less_asym = thm "order_less_asym";
val order_less_trans = thm "order_less_trans";
val order_le_less_trans = thm "order_le_less_trans";
val order_less_le_trans = thm "order_less_le_trans";
val order_less_imp_not_less = thm "order_less_imp_not_less";
val order_less_imp_triv = thm "order_less_imp_triv";
val order_less_imp_not_eq = thm "order_less_imp_not_eq";
val order_less_imp_not_eq2 = thm "order_less_imp_not_eq2";
val linorder_less_linear = thm "linorder_less_linear";
val linorder_less_split = thm "linorder_less_split";
val linorder_not_less = thm "linorder_not_less";
val linorder_not_le = thm "linorder_not_le";
val linorder_neq_iff = thm "linorder_neq_iff";
val linorder_neqE = thm "linorder_neqE";
val min_same = thm "min_same";
val max_same = thm "max_same";
val le_max_iff_disj = thm "le_max_iff_disj";
val le_maxI1 = thm "le_maxI1";
val le_maxI2 = thm "le_maxI2";
val less_max_iff_disj = thm "less_max_iff_disj";
val max_le_iff_conj = thm "max_le_iff_conj";
val max_less_iff_conj = thm "max_less_iff_conj";
val le_min_iff_conj = thm "le_min_iff_conj";
val min_less_iff_conj = thm "min_less_iff_conj";
val min_le_iff_disj = thm "min_le_iff_disj";
val min_less_iff_disj = thm "min_less_iff_disj";
val split_min = thm "split_min";
val split_max = thm "split_max";
val order_refl = thm "order_refl";
val order_trans = thm "order_trans";
val order_antisym = thm "order_antisym";
val order_less_le = thm "order_less_le";
val linorder_linear = thm "linorder_linear";
*}
end