src/HOL/IMP/AExp.thy
author huffman
Thu Aug 11 09:11:15 2011 -0700 (2011-08-11)
changeset 44165 d26a45f3c835
parent 44036 d03f9f28d01d
child 44923 b80108b346a9
permissions -rw-r--r--
remove lemma stupid_ext
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header "Arithmetic and Boolean Expressions"
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theory AExp imports Main begin
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subsection "Arithmetic Expressions"
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type_synonym name = string
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type_synonym val = int
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type_synonym state = "name \<Rightarrow> val"
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datatype aexp = N int | V name | Plus aexp aexp
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fun aval :: "aexp \<Rightarrow> state \<Rightarrow> val" where
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"aval (N n) _ = n" |
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"aval (V x) s = s x" |
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"aval (Plus a1 a2) s = aval a1 s + aval a2 s"
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value "aval (Plus (V ''x'') (N 5)) (%x. if x = ''x'' then 7 else 0)"
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text {* The same state more concisely: *}
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value "aval (Plus (V ''x'') (N 5)) ((%x. 0) (''x'':= 7))"
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text {* A little syntax magic to write larger states compactly: *}
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definition
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  "null_heap \<equiv> \<lambda>x. 0"
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syntax 
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  "_State" :: "updbinds => 'a" ("<_>")
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translations
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  "_State ms" => "_Update (CONST null_heap) ms"
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text {* 
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  We can now write a series of updates to the function @{text "\<lambda>x. 0"} compactly:
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*}
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lemma "<a := Suc 0, b := 2> = (null_heap (a := Suc 0)) (b := 2)"
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  by (rule refl)
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value "aval (Plus (V ''x'') (N 5)) <''x'' := 7>"
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text {* Variables that are not mentioned in the state are 0 by default in 
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  the @{term "<a := b::int>"} syntax: 
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*}   
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value "aval (Plus (V ''x'') (N 5)) <''y'' := 7>"
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subsection "Optimization"
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text{* Evaluate constant subsexpressions: *}
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fun asimp_const :: "aexp \<Rightarrow> aexp" where
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"asimp_const (N n) = N n" |
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"asimp_const (V x) = V x" |
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"asimp_const (Plus a1 a2) =
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  (case (asimp_const a1, asimp_const a2) of
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    (N n1, N n2) \<Rightarrow> N(n1+n2) |
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    (a1',a2') \<Rightarrow> Plus a1' a2')"
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theorem aval_asimp_const[simp]:
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  "aval (asimp_const a) s = aval a s"
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apply(induct a)
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apply (auto split: aexp.split)
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done
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text{* Now we also eliminate all occurrences 0 in additions. The standard
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method: optimized versions of the constructors: *}
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fun plus :: "aexp \<Rightarrow> aexp \<Rightarrow> aexp" where
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"plus (N i1) (N i2) = N(i1+i2)" |
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"plus (N i) a = (if i=0 then a else Plus (N i) a)" |
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"plus a (N i) = (if i=0 then a else Plus a (N i))" |
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"plus a1 a2 = Plus a1 a2"
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code_thms plus
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code_thms plus
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(* FIXME: dropping subsumed code eqns?? *)
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lemma aval_plus[simp]:
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  "aval (plus a1 a2) s = aval a1 s + aval a2 s"
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apply(induct a1 a2 rule: plus.induct)
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apply simp_all (* just for a change from auto *)
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done
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code_thms plus
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fun asimp :: "aexp \<Rightarrow> aexp" where
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"asimp (N n) = N n" |
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"asimp (V x) = V x" |
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"asimp (Plus a1 a2) = plus (asimp a1) (asimp a2)"
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text{* Note that in @{const asimp_const} the optimized constructor was
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inlined. Making it a separate function @{const plus} improves modularity of
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the code and the proofs. *}
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value "asimp (Plus (Plus (N 0) (N 0)) (Plus (V ''x'') (N 0)))"
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theorem aval_asimp[simp]:
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  "aval (asimp a) s = aval a s"
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apply(induct a)
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apply simp_all
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done
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end