author  huffman 
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parent 16758  c32334d98fcd 
child 16779  ac1dc3d4746a 
permissions  rwrr 
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(* Title: HOLCF/Fixrec.thy 
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ID: $Id$ 

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Author: Amber Telfer and Brian Huffman 

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*) 

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header "Package for defining recursive functions in HOLCF" 

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theory Fixrec 

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imports Sprod Ssum Up One Tr Fix 
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uses ("fixrec_package.ML") 
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begin 
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subsection {* Maybe monad type *} 

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defaultsort cpo 
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types 'a maybe = "one ++ 'a u" 
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constdefs 

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fail :: "'a maybe" 

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"fail \<equiv> sinl\<cdot>ONE" 

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return :: "'a \<rightarrow> 'a maybe" 

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"return \<equiv> sinr oo up" 

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lemma maybeE: 

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"\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = fail \<Longrightarrow> Q; \<And>x. p = return\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" 

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apply (unfold fail_def return_def) 

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apply (rule_tac p=p in ssumE, simp) 

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apply (rule_tac p=x in oneE, simp, simp) 

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apply (rule_tac p=y in upE, simp, simp) 
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done 
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subsection {* Monadic bind operator *} 

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constdefs 

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bind :: "'a maybe \<rightarrow> ('a \<rightarrow> 'b maybe) \<rightarrow> 'b maybe" 

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"bind \<equiv> \<Lambda> m f. sscase\<cdot>sinl\<cdot>(fup\<cdot>f)\<cdot>m" 

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syntax 

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"_bind" :: "'a maybe \<Rightarrow> ('a \<rightarrow> 'b maybe) \<Rightarrow> 'b maybe" 

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("(_ >>= _)" [50, 51] 50) 

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translations "m >>= k" == "bind\<cdot>m\<cdot>k" 

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nonterminals 

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maybebind maybebinds 

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syntax 

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"_MBIND" :: "pttrn \<Rightarrow> 'a maybe \<Rightarrow> maybebind" ("(2_ </ _)" 10) 

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"" :: "maybebind \<Rightarrow> maybebinds" ("_") 

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"_MBINDS" :: "[maybebind, maybebinds] \<Rightarrow> maybebinds" ("_;/ _") 

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"_MDO" :: "[maybebinds, 'a maybe] \<Rightarrow> 'a maybe" ("(do _;/ (_))" 10) 

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translations 

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"_MDO (_MBINDS b bs) e" == "_MDO b (_MDO bs e)" 

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"do (x,y) < m; e" == "m >>= (LAM <x,y>. e)" 

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"do x < m; e" == "m >>= (LAM x. e)" 

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text {* monad laws *} 

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lemma bind_strict [simp]: "UU >>= f = UU" 

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by (simp add: bind_def) 

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lemma bind_fail [simp]: "fail >>= f = fail" 

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by (simp add: bind_def fail_def) 

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lemma left_unit [simp]: "(return\<cdot>a) >>= k = k\<cdot>a" 

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by (simp add: bind_def return_def) 

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lemma right_unit [simp]: "m >>= return = m" 

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by (rule_tac p=m in maybeE, simp_all) 

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lemma bind_assoc [simp]: 

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"(do a < m; b < k\<cdot>a; h\<cdot>b) = (do b < (do a < m; k\<cdot>a); h\<cdot>b)" 

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by (rule_tac p=m in maybeE, simp_all) 

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subsection {* Run operator *} 

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constdefs 

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run:: "'a::pcpo maybe \<rightarrow> 'a" 
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"run \<equiv> sscase\<cdot>\<bottom>\<cdot>(fup\<cdot>ID)" 
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text {* rewrite rules for run *} 

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lemma run_strict [simp]: "run\<cdot>\<bottom> = \<bottom>" 

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by (simp add: run_def) 

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lemma run_fail [simp]: "run\<cdot>fail = \<bottom>" 

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by (simp add: run_def fail_def) 

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lemma run_return [simp]: "run\<cdot>(return\<cdot>x) = x" 

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by (simp add: run_def return_def) 

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subsection {* Monad plus operator *} 

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constdefs 

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mplus :: "'a maybe \<rightarrow> 'a maybe \<rightarrow> 'a maybe" 

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"mplus \<equiv> \<Lambda> m1 m2. sscase\<cdot>(\<Lambda> x. m2)\<cdot>(fup\<cdot>return)\<cdot>m1" 

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syntax "+++" :: "'a maybe \<Rightarrow> 'a maybe \<Rightarrow> 'a maybe" (infixr 65) 

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translations "x +++ y" == "mplus\<cdot>x\<cdot>y" 

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text {* rewrite rules for mplus *} 

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lemma mplus_strict [simp]: "\<bottom> +++ m = \<bottom>" 

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by (simp add: mplus_def) 

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lemma mplus_fail [simp]: "fail +++ m = m" 

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by (simp add: mplus_def fail_def) 

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lemma mplus_return [simp]: "return\<cdot>x +++ m = return\<cdot>x" 

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by (simp add: mplus_def return_def) 

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lemma mplus_fail2 [simp]: "m +++ fail = m" 
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by (rule_tac p=m in maybeE, simp_all) 
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lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)" 
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by (rule_tac p=x in maybeE, simp_all) 

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subsection {* Match functions for builtin types *} 

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defaultsort pcpo 
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constdefs 
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match_UU :: "'a \<rightarrow> unit maybe" 
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"match_UU \<equiv> \<Lambda> x. fail" 
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match_cpair :: "'a::cpo \<times> 'b::cpo \<rightarrow> ('a \<times> 'b) maybe" 
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"match_cpair \<equiv> csplit\<cdot>(\<Lambda> x y. return\<cdot><x,y>)" 
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match_spair :: "'a \<otimes> 'b \<rightarrow> ('a \<times> 'b) maybe" 
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"match_spair \<equiv> ssplit\<cdot>(\<Lambda> x y. return\<cdot><x,y>)" 

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match_sinl :: "'a \<oplus> 'b \<rightarrow> 'a maybe" 

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"match_sinl \<equiv> sscase\<cdot>return\<cdot>(\<Lambda> y. fail)" 

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match_sinr :: "'a \<oplus> 'b \<rightarrow> 'b maybe" 

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"match_sinr \<equiv> sscase\<cdot>(\<Lambda> x. fail)\<cdot>return" 

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match_up :: "'a::cpo u \<rightarrow> 'a maybe" 
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"match_up \<equiv> fup\<cdot>return" 
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match_ONE :: "one \<rightarrow> unit maybe" 
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"match_ONE \<equiv> flift1 (\<lambda>u. return\<cdot>())" 
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match_TT :: "tr \<rightarrow> unit maybe" 
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"match_TT \<equiv> flift1 (\<lambda>b. if b then return\<cdot>() else fail)" 
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match_FF :: "tr \<rightarrow> unit maybe" 
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"match_FF \<equiv> flift1 (\<lambda>b. if b then fail else return\<cdot>())" 
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lemma match_UU_simps [simp]: 
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"match_UU\<cdot>x = fail" 
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by (simp add: match_UU_def) 
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lemma match_cpair_simps [simp]: 
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"match_cpair\<cdot><x,y> = return\<cdot><x,y>" 

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by (simp add: match_cpair_def) 

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lemma match_spair_simps [simp]: 
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"\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> match_spair\<cdot>(:x,y:) = return\<cdot><x,y>" 

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"match_spair\<cdot>\<bottom> = \<bottom>" 

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by (simp_all add: match_spair_def) 

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lemma match_sinl_simps [simp]: 

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"x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinl\<cdot>x) = return\<cdot>x" 

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"x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinr\<cdot>x) = fail" 

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"match_sinl\<cdot>\<bottom> = \<bottom>" 

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by (simp_all add: match_sinl_def) 

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lemma match_sinr_simps [simp]: 

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"x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinr\<cdot>x) = return\<cdot>x" 

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"x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinl\<cdot>x) = fail" 

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"match_sinr\<cdot>\<bottom> = \<bottom>" 

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by (simp_all add: match_sinr_def) 

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lemma match_up_simps [simp]: 
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"match_up\<cdot>(up\<cdot>x) = return\<cdot>x" 

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"match_up\<cdot>\<bottom> = \<bottom>" 

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by (simp_all add: match_up_def) 

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lemma match_ONE_simps [simp]: 
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"match_ONE\<cdot>ONE = return\<cdot>()" 
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"match_ONE\<cdot>\<bottom> = \<bottom>" 
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by (simp_all add: ONE_def match_ONE_def) 
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lemma match_TT_simps [simp]: 
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"match_TT\<cdot>TT = return\<cdot>()" 
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"match_TT\<cdot>FF = fail" 
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"match_TT\<cdot>\<bottom> = \<bottom>" 
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by (simp_all add: TT_def FF_def match_TT_def) 
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lemma match_FF_simps [simp]: 
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"match_FF\<cdot>FF = return\<cdot>()" 
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"match_FF\<cdot>TT = fail" 
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"match_FF\<cdot>\<bottom> = \<bottom>" 
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by (simp_all add: TT_def FF_def match_FF_def) 
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subsection {* Mutual recursion *} 
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text {* 
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The following rules are used to prove unfolding theorems from 
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fixedpoint definitions of mutually recursive functions. 
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*} 
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lemma cpair_equalI: "\<lbrakk>x \<equiv> cfst\<cdot>p; y \<equiv> csnd\<cdot>p\<rbrakk> \<Longrightarrow> <x,y> \<equiv> p" 
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by (simp add: surjective_pairing_Cprod2) 
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lemma cpair_eqD1: "<x,y> = <x',y'> \<Longrightarrow> x = x'" 
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by simp 
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lemma cpair_eqD2: "<x,y> = <x',y'> \<Longrightarrow> y = y'" 
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by simp 
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text {* lemma for proving rewrite rules *} 
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lemma ssubst_lhs: "\<lbrakk>t = s; P s = Q\<rbrakk> \<Longrightarrow> P t = Q" 
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by simp 
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ML {* 
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val cpair_equalI = thm "cpair_equalI"; 
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val cpair_eqD1 = thm "cpair_eqD1"; 
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val cpair_eqD2 = thm "cpair_eqD2"; 
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val ssubst_lhs = thm "ssubst_lhs"; 
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*} 
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subsection {* Initializing the fixrec package *} 
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use "fixrec_package.ML" 
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end 