# HG changeset patch
# User oheimb
# Date 982249282 -3600
# Node ID a46eaedbeb2d9da4d480272a9b299019c22c38ff
# Parent  b092ad5cd510ced2e803b7508299c00b30ec6cda
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

diff -r b092ad5cd510 -r a46eaedbeb2d src/HOL/Ord.thy
--- a/src/HOL/Ord.thy	Thu Feb 15 16:01:07 2001 +0100
+++ b/src/HOL/Ord.thy	Thu Feb 15 16:01:22 2001 +0100
@@ -6,69 +6,344 @@
 Type classes for order signatures and orders.
 *)
 
-Ord = HOL +
+theory Ord = HOL:
 
 
 axclass
-  ord < term
+  ord < "term"
 
 syntax
-  "op <"        :: ['a::ord, 'a] => bool             ("op <")
-  "op <="       :: ['a::ord, 'a] => bool             ("op <=")
+  "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)
+  "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)
+  "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)"
 
-consts
-  mono          :: ['a::ord => 'b::ord] => bool      (*monotonicity*)
-  min, max      :: ['a::ord, 'a] => 'a
-  Least         :: ('a::ord=>bool) => 'a             (binder "LEAST " 10)
+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
 
-defs
-  mono_def      "mono(f) == (!A B. A <= B --> f(A) <= f(B))"
-  min_def       "min a b == (if a <= b then a else b)"
-  max_def       "max a b == (if a <= b then b else a)"
-  Least_def     "Least P == @x. P(x) & (ALL y. P(y) --> x <= y)"
+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    "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)"
+  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"
+  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
 
 
-(* bounded quantifiers *)
+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)
+  "_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)
+  "_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)
+  "_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"
@@ -77,4 +352,62 @@
  "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