--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MicroJava/BV/Product.thy Mon Nov 20 16:37:42 2000 +0100
@@ -0,0 +1,144 @@
+(* Title: HOL/BCV/Product.thy
+ ID: $Id$
+ Author: Tobias Nipkow
+ Copyright 2000 TUM
+
+Products as semilattices
+*)
+
+header "Products as Semilattices"
+
+theory Product = Err:
+
+constdefs
+ le :: "'a ord => 'b ord => ('a * 'b) ord"
+"le rA rB == %(a,b) (a',b'). a <=_rA a' & b <=_rB b'"
+
+ sup :: "'a ebinop => 'b ebinop => ('a * 'b)ebinop"
+"sup f g == %(a1,b1)(a2,b2). Err.sup Pair (a1 +_f a2) (b1 +_g b2)"
+
+ esl :: "'a esl => 'b esl => ('a * 'b ) esl"
+"esl == %(A,rA,fA) (B,rB,fB). (A <*> B, le rA rB, sup fA fB)"
+
+syntax "@lesubprod" :: "'a*'b => 'a ord => 'b ord => 'b => bool"
+ ("(_ /<='(_,_') _)" [50, 0, 0, 51] 50)
+translations "p <=(rA,rB) q" == "p <=_(Product.le rA rB) q"
+
+lemma unfold_lesub_prod:
+ "p <=(rA,rB) q == le rA rB p q"
+ by (simp add: lesub_def)
+
+lemma le_prod_Pair_conv [iff]:
+ "((a1,b1) <=(rA,rB) (a2,b2)) = (a1 <=_rA a2 & b1 <=_rB b2)"
+ by (simp add: lesub_def le_def)
+
+lemma less_prod_Pair_conv:
+ "((a1,b1) <_(Product.le rA rB) (a2,b2)) =
+ (a1 <_rA a2 & b1 <=_rB b2 | a1 <=_rA a2 & b1 <_rB b2)"
+apply (unfold lesssub_def)
+apply simp
+apply blast
+done
+
+lemma order_le_prod [iff]:
+ "order(Product.le rA rB) = (order rA & order rB)"
+apply (unfold order_def)
+apply simp
+apply blast
+done
+
+
+lemma acc_le_prodI [intro!]:
+ "[| acc rA; acc rB |] ==> acc(Product.le rA rB)"
+apply (unfold acc_def)
+apply (rule wf_subset)
+ apply (erule wf_lex_prod)
+ apply assumption
+apply (auto simp add: lesssub_def less_prod_Pair_conv lex_prod_def)
+done
+
+
+lemma closed_lift2_sup:
+ "[| closed (err A) (lift2 f); closed (err B) (lift2 g) |] ==>
+ closed (err(A<*>B)) (lift2(sup f g))";
+apply (unfold closed_def plussub_def lift2_def err_def sup_def)
+apply (simp split: err.split)
+apply blast
+done
+
+lemma unfold_plussub_lift2:
+ "e1 +_(lift2 f) e2 == lift2 f e1 e2"
+ by (simp add: plussub_def)
+
+
+lemma plus_eq_Err_conv [simp]:
+ "[| x:A; y:A; semilat(err A, Err.le r, lift2 f) |]
+ ==> (x +_f y = Err) = (~(? z:A. x <=_r z & y <=_r z))"
+proof -
+ have plus_le_conv2:
+ "!!r f z. [| z : err A; semilat (err A, r, f); OK x : err A; OK y : err A;
+ OK x +_f OK y <=_r z|] ==> OK x <=_r z \<and> OK y <=_r z"
+ by (rule plus_le_conv [THEN iffD1])
+ case antecedent
+ thus ?thesis
+ apply (rule_tac iffI)
+ apply clarify
+ apply (drule OK_le_err_OK [THEN iffD2])
+ apply (drule OK_le_err_OK [THEN iffD2])
+ apply (drule semilat_lub)
+ apply assumption
+ apply assumption
+ apply simp
+ apply simp
+ apply simp
+ apply simp
+ apply (case_tac "x +_f y")
+ apply assumption
+ apply (rename_tac "z")
+ apply (subgoal_tac "OK z: err A")
+ apply (frule plus_le_conv2)
+ apply assumption
+ apply simp
+ apply blast
+ apply simp
+ apply (blast dest: semilatDorderI order_refl)
+ apply blast
+ apply (erule subst)
+ apply (unfold semilat_def err_def closed_def)
+ apply simp
+ done
+qed
+
+lemma err_semilat_Product_esl:
+ "!!L1 L2. [| err_semilat L1; err_semilat L2 |] ==> err_semilat(Product.esl L1 L2)"
+apply (unfold esl_def Err.sl_def)
+apply (simp (no_asm_simp) only: split_tupled_all)
+apply simp
+apply (simp (no_asm) only: semilat_Def)
+apply (simp (no_asm_simp) only: semilatDclosedI closed_lift2_sup)
+apply (simp (no_asm) only: unfold_lesub_err Err.le_def unfold_plussub_lift2 sup_def)
+apply (auto elim: semilat_le_err_OK1 semilat_le_err_OK2
+ simp add: lift2_def split: err.split)
+apply (blast dest: semilatDorderI)
+apply (blast dest: semilatDorderI)
+
+apply (rule OK_le_err_OK [THEN iffD1])
+apply (erule subst, subst OK_lift2_OK [symmetric], rule semilat_lub)
+apply simp
+apply simp
+apply simp
+apply simp
+apply simp
+apply simp
+
+apply (rule OK_le_err_OK [THEN iffD1])
+apply (erule subst, subst OK_lift2_OK [symmetric], rule semilat_lub)
+apply simp
+apply simp
+apply simp
+apply simp
+apply simp
+apply simp
+done
+
+end