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A Auxiliary Definitions A Auxiliary Definitions This appendix contains auxiliary definitions omitted from the main text. Variables fun lvars :: com ⇒ vname set where lvars SKIP = {} lvars (x ::= e)={x} lvars (c1;; c2)=lvars c1 ∪ lvars c2 lvars (IF b THEN c1 ELSE c2)=lvars c1 ∪ lvars c2 lvars (WHILE b DO c)=lvars c fun rvars :: com ⇒ vname set where rvars SKIP = {} rvars (x ::= e)=vars e rvars (c1;; c2)=rvars c1 ∪ rvars c2 rvars (IF b THEN c1 ELSE c2)=vars b ∪ rvars c1 ∪ rvars c2 rvars (WHILE b DO c)=vars b ∪ rvars c definition vars :: com ⇒ vname set where vars c = lvars c ∪ rvars c Abstract Interpretation fun strip :: aacom⇒ com where strip (SKIP {P})=SKIP strip (x ::= e {P})=x ::= e © Springer International Publishing Switzerland 2014 281 T. Nipkow and G. Klein, Concrete Semantics, DOI 10.1007/978-3-319-10542-0 282 A Auxiliary Definitions strip (C 1;;C 2)=strip C 1;; strip C 2 strip (IF b THEN {P 1} C 1 ELSE {P 2} C 2 {P})= IF b THEN strip C 1 ELSE strip C 2 strip ({I } WHILE b DO {P} C {Q})=WHILE b DO strip C fun annos :: aacom⇒ a list where annos (SKIP {P})=[P] annos (x ::= e {P})=[P] annos (C 1;;C 2)=annos C 1 @ annos C 2 annos (IF b THEN {P 1} C 1 ELSE {P 2} C 2 {Q})= P 1 # annos C 1 @ P 2 # annos C 2 @ [Q] annos ({I } WHILE b DO {P} C {Q})=I # P # annos C @ [Q] fun asize :: com ⇒ nat where asize SKIP = 1 asize (x ::= e)=1 asize (C 1;;C 2)=asize C 1 + asize C 2 asize (IF b THEN C 1 ELSE C 2)=asize C 1 + asize C 2 + 3 asize (WHILE b DO C )=asize C + 3 definition shift :: (nat ⇒ a) ⇒ nat ⇒ nat ⇒ a where shift f n =(λp. f (p+n)) fun annotate :: (nat ⇒ a) ⇒ com ⇒ aacomwhere annotate f SKIP = SKIP {f 0} annotate f (x ::= e)=x ::= e {f 0} annotate f (c1;;c2)=annotate f c1;; annotate (shift f (asize c1)) c2 annotate f (IF b THEN c1 ELSE c2)= IF b THEN {f 0} annotate (shift f 1) c1 ELSE {f (asize c1 + 1)} annotate (shift f (asize c1 + 2)) c2 {f (asize c1 + asize c2 + 2)} annotate f (WHILE b DO c)= {f 0} WHILE b DO {f 1} annotate (shift f 2) c {f (asize c + 2)} fun map_acom :: ( a ⇒ b) ⇒ aacom⇒ bacomwhere map_acom f (SKIP {P})=SKIP {fP} map_acom f (x ::= e {P})=x ::= e {fP} map_acom f (C 1;;C 2)=map_acom f C 1;; map_acom f C 2 map_acom f (IF b THEN {P 1} C 1 ELSE {P 2} C 2 {Q})= IF b THEN {fP1} map_acom f C 1 ELSE {fP2} map_acom f C 2 {fQ} map_acom f ({I } WHILE b DO {P} C {Q})= {fI} WHILE b DO {fP} map_acom f C {fQ} B Symbols [[ [| \<lbrakk> ]] |] \<rbrakk> =⇒ ==> \<Longrightarrow> !! \<And> ≡ == \<equiv> λ % \<lambda> ⇒ => \<Rightarrow> ∧ & \<and> ∨ | \<or> −→ --> \<longrightarrow> → -> \<rightarrow> ¬ ~ \<not> = ~= \<noteq> ∀ ALL \<forall> ∃ EX \<exists> <= \<le> × * \<times> ∈ : \<in> ∈/ ~: \<notin> ⊆ <= \<subseteq> ⊂ < \<subset> ∪ Un \<union> ∩ Int \<inter> UN, Union \<Union> INT, Inter \<Inter> sup \<squnion> inf \<sqinter> SUP, Sup \<Squnion> INF, Inf \<Sqinter> \<top> ⊥ \<bottom> Table B.1. Mathematical symbols, their ascii equivalents and internal names © Springer International Publishing Switzerland 2014 283 T. Nipkow and G. Klein, Concrete Semantics, DOI 10.1007/978-3-319-10542-0 C Theories The following table shows which sections are based on which theories in the directory src/HOL/IMP of the Isabelle distribution. 3.1 AExp 12.1 Hoare_Examples 3.2 BExp 12.2.2 Hoare 3.3 ASM 12.2.3 Hoare_Examples 7.1 Com 12.3 Hoare_Sound_Complete 7.2 Big_Step 12.4 VCG 7.3 Small_Step 12.5 Hoare_Total 8.1 Compiler 13.2 ACom 8.2 Compiler 13.3 Collecting 8.3 Compiler 13.3.3 Complete_Lattice 8.4 Compiler2 13.4 Abs_Int1_parity 9.1 Types 13.4.2 Abs_Int0 9.2.1 Sec_Type_Expr 13.5 Abs_Int0 9.2.2 Sec_Typing 13.5.1 Collecting 9.2.6 Sec_TypingT 13.6 Abs_Int1 10.1.1 Def_Init 13.6.1 Abs_Int1_parity 10.1.2 Def_Init_Exp 13.6.2 Abs_Int1_const 10.1.3 Def_Init_Small 13.6.3 Abs_State 10.1.4 Def_Init_Big 13.7 Abs_Int2 10.2 Fold 13.8 Abs_Int2_ivl 10.3 Live 13.9 Abs_Int3 10.4 Live_True 11.0 Denotational © Springer International Publishing Switzerland 2014 285 T. Nipkow and G. Klein, Concrete Semantics, DOI 10.1007/978-3-319-10542-0 References 1. 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