src/HOL/Library/Extended.thy
 changeset 51338 054d1653950f child 51357 ac4c1cf1b001
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Extended.thy	Tue Mar 05 15:26:57 2013 +0100
@@ -0,0 +1,201 @@
+(*  Author:     Tobias Nipkow, TU München
+
+A theory of types extended with a greatest and a least element.
+Oriented towards numeric types, hence "\<infinity>" and "-\<infinity>".
+*)
+
+theory Extended
+imports Main
+begin
+
+datatype 'a extended = Fin 'a | Pinf ("\<infinity>") | Minf ("-\<infinity>")
+
+lemmas extended_cases2 = extended.exhaust[case_product extended.exhaust]
+lemmas extended_cases3 = extended.exhaust[case_product extended_cases2]
+
+instantiation extended :: (order)order
+begin
+
+fun less_eq_extended :: "'a extended \<Rightarrow> 'a extended \<Rightarrow> bool" where
+"Fin x \<le> Fin y = (x \<le> y)" |
+"_     \<le> Pinf  = True" |
+"Minf  \<le> _     = True" |
+"(_::'a extended) \<le> _     = False"
+
+lemma less_eq_extended_cases:
+  "x \<le> y = (case x of Fin x \<Rightarrow> (case y of Fin y \<Rightarrow> x \<le> y | Pinf \<Rightarrow> True | Minf \<Rightarrow> False)
+            | Pinf \<Rightarrow> y=Pinf | Minf \<Rightarrow> True)"
+by(induct x y rule: less_eq_extended.induct)(auto split: extended.split)
+
+definition less_extended :: "'a extended \<Rightarrow> 'a extended \<Rightarrow> bool" where
+"((x::'a extended) < y) = (x \<le> y & \<not> x \<ge> y)"
+
+instance
+proof
+  case goal1 show ?case by(rule less_extended_def)
+next
+  case goal2 show ?case by(cases x) auto
+next
+  case goal3 thus ?case by(auto simp: less_eq_extended_cases split:extended.splits)
+next
+  case goal4 thus ?case by(auto simp: less_eq_extended_cases split:extended.splits)
+qed
+
+end
+
+instance extended :: (linorder)linorder
+proof
+  case goal1 thus ?case by(auto simp: less_eq_extended_cases split:extended.splits)
+qed
+
+lemma Minf_le[simp]: "Minf \<le> y"
+by(cases y) auto
+lemma le_Pinf[simp]: "x \<le> Pinf"
+by(cases x) auto
+lemma le_Minf[simp]: "x \<le> Minf \<longleftrightarrow> x = Minf"
+by(cases x) auto
+lemma Pinf_le[simp]: "Pinf \<le> x \<longleftrightarrow> x = Pinf"
+by(cases x) auto
+
+lemma less_extended_simps[simp]:
+  "Fin x < Fin y = (x < y)"
+  "Fin x < Pinf  = True"
+  "Fin x < Minf  = False"
+  "Pinf < h      = False"
+  "Minf < Fin x  = True"
+  "Minf < Pinf   = True"
+  "l    < Minf   = False"
+
+lemma min_extended_simps[simp]:
+  "min (Fin x) (Fin y) = Fin(min x y)"
+  "min xx      Pinf    = xx"
+  "min xx      Minf    = Minf"
+  "min Pinf    yy      = yy"
+  "min Minf    yy      = Minf"
+
+lemma max_extended_simps[simp]:
+  "max (Fin x) (Fin y) = Fin(max x y)"
+  "max xx      Pinf    = Pinf"
+  "max xx      Minf    = xx"
+  "max Pinf    yy      = Pinf"
+  "max Minf    yy      = yy"
+
+
+instantiation extended :: (plus)plus
+begin
+
+text {* The following definition of of addition is totalized
+to make it asociative and commutative. Normally the sum of plus and minus infinity is undefined. *}
+
+fun plus_extended where
+"Fin x + Fin y = Fin(x+y)" |
+"Fin x + Pinf  = Pinf" |
+"Pinf  + Fin x = Pinf" |
+"Pinf  + Pinf  = Pinf" |
+"Minf  + Fin y = Minf" |
+"Fin x + Minf  = Minf" |
+"Minf  + Minf  = Minf" |
+"Minf  + Pinf  = Pinf" |
+"Pinf  + Minf  = Pinf"
+
+instance ..
+
+end
+
+
+proof
+  fix a b c :: "'a extended"
+  show "a + b = b + a"
+    by (induct a b rule: plus_extended.induct) (simp_all add: ac_simps)
+  show "a + b + c = a + (b + c)"
+    by (cases rule: extended_cases3[of a b c]) (simp_all add: ac_simps)
+qed
+
+proof
+  fix a b c :: "'a extended"
+  assume "a \<le> b" then show "c + a \<le> c + b"
+    by (cases rule: extended_cases3[of a b c]) (auto simp: add_left_mono)
+qed
+
+begin
+
+definition "0 = Fin 0"
+
+instance
+proof
+  fix x :: "'a extended" show "0 + x = x" unfolding zero_extended_def by(cases x)auto
+qed
+
+end
+
+instantiation extended :: (uminus)uminus
+begin
+
+fun uminus_extended where
+"- (Fin x) = Fin (- x)" |
+"- Pinf    = Minf" |
+"- Minf    = Pinf"
+
+instance ..
+
+end
+
+
+begin
+definition "x - y = x + -(y::'a extended)"
+instance ..
+end
+
+lemma minus_extended_simps[simp]:
+  "Fin x - Fin y = Fin(x - y)"
+  "Fin x - Pinf  = Minf"
+  "Fin x - Minf  = Pinf"
+  "Pinf  - Fin y = Pinf"
+  "Pinf  - Minf  = Pinf"
+  "Minf  - Fin y = Minf"
+  "Minf  - Pinf  = Minf"
+  "Minf  - Minf  = Pinf"
+  "Pinf  - Pinf  = Pinf"
+
+instantiation extended :: (lattice)bounded_lattice
+begin
+
+definition "bot = Minf"
+definition "top = Pinf"
+
+fun inf_extended :: "'a extended \<Rightarrow> 'a extended \<Rightarrow> 'a extended" where
+"inf_extended (Fin i) (Fin j) = Fin (inf i j)" |
+"inf_extended a Minf = Minf" |
+"inf_extended Minf a = Minf" |
+"inf_extended Pinf a = a" |
+"inf_extended a Pinf = a"
+
+fun sup_extended :: "'a extended \<Rightarrow> 'a extended \<Rightarrow> 'a extended" where
+"sup_extended (Fin i) (Fin j) = Fin (sup i j)" |
+"sup_extended a Pinf = Pinf" |
+"sup_extended Pinf a = Pinf" |
+"sup_extended Minf a = a" |
+"sup_extended a Minf = a"
+
+instance
+proof
+  fix x y z ::"'a extended"
+  show "inf x y \<le> x" "inf x y \<le> y" "\<lbrakk>x \<le> y; x \<le> z\<rbrakk> \<Longrightarrow> x \<le> inf y z"
+    "x \<le> sup x y" "y \<le> sup x y" "\<lbrakk>y \<le> x; z \<le> x\<rbrakk> \<Longrightarrow> sup y z \<le> x" "bot \<le> x" "x \<le> top"
+    apply (atomize (full))
+    apply (cases rule: extended_cases3[of x y z])
+    apply (auto simp: bot_extended_def top_extended_def)
+    done
+qed
+end
+
+end
+```