author | chaieb |
Wed, 31 Oct 2007 12:19:41 +0100 | |
changeset 25252 | 833abbc3e733 |
parent 25158 | 271e499f2d03 |
child 26046 | 1624b3304bb9 |
permissions | -rw-r--r-- |
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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 ("Tools/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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pcpodef (open) 'a maybe = "UNIV::(one ++ 'a u) set" |
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by simp |
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constdefs |
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fail :: "'a maybe" |
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"fail \<equiv> Abs_maybe (sinl\<cdot>ONE)" |
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constdefs |
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return :: "'a \<rightarrow> 'a maybe" where |
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"return \<equiv> \<Lambda> x. Abs_maybe (sinr\<cdot>(up\<cdot>x))" |
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definition |
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maybe_when :: "'b \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a maybe \<rightarrow> 'b::pcpo" where |
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"maybe_when = (\<Lambda> f r m. sscase\<cdot>(\<Lambda> x. f)\<cdot>(fup\<cdot>r)\<cdot>(Rep_maybe m))" |
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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 (cases p, rename_tac r) |
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apply (rule_tac p=r in ssumE, simp add: Abs_maybe_strict) |
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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 add: cont_Abs_maybe) |
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done |
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lemma return_defined [simp]: "return\<cdot>x \<noteq> \<bottom>" |
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by (simp add: return_def cont_Abs_maybe Abs_maybe_defined) |
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lemma fail_defined [simp]: "fail \<noteq> \<bottom>" |
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by (simp add: fail_def Abs_maybe_defined) |
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lemma return_eq [simp]: "(return\<cdot>x = return\<cdot>y) = (x = y)" |
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by (simp add: return_def cont_Abs_maybe Abs_maybe_inject) |
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lemma return_neq_fail [simp]: |
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"return\<cdot>x \<noteq> fail" "fail \<noteq> return\<cdot>x" |
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by (simp_all add: return_def fail_def cont_Abs_maybe Abs_maybe_inject) |
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lemma maybe_when_rews [simp]: |
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"maybe_when\<cdot>f\<cdot>r\<cdot>\<bottom> = \<bottom>" |
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"maybe_when\<cdot>f\<cdot>r\<cdot>fail = f" |
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"maybe_when\<cdot>f\<cdot>r\<cdot>(return\<cdot>x) = r\<cdot>x" |
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by (simp_all add: return_def fail_def maybe_when_def cont_Rep_maybe |
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cont_Abs_maybe Abs_maybe_inverse Rep_maybe_strict) |
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translations |
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"case m of fail \<Rightarrow> t1 | return\<cdot>x \<Rightarrow> t2" == "CONST maybe_when\<cdot>t1\<cdot>(\<Lambda> x. t2)\<cdot>m" |
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subsubsection {* Monadic bind operator *} |
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definition |
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bind :: "'a maybe \<rightarrow> ('a \<rightarrow> 'b maybe) \<rightarrow> 'b maybe" where |
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"bind = (\<Lambda> m f. case m of fail \<Rightarrow> fail | return\<cdot>x \<Rightarrow> f\<cdot>x)" |
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text {* monad laws *} |
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lemma bind_strict [simp]: "bind\<cdot>\<bottom>\<cdot>f = \<bottom>" |
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by (simp add: bind_def) |
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lemma bind_fail [simp]: "bind\<cdot>fail\<cdot>f = fail" |
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by (simp add: bind_def) |
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lemma left_unit [simp]: "bind\<cdot>(return\<cdot>a)\<cdot>k = k\<cdot>a" |
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by (simp add: bind_def) |
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lemma right_unit [simp]: "bind\<cdot>m\<cdot>return = m" |
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by (rule_tac p=m in maybeE, simp_all) |
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lemma bind_assoc: |
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"bind\<cdot>(bind\<cdot>m\<cdot>k)\<cdot>h = bind\<cdot>m\<cdot>(\<Lambda> a. bind\<cdot>(k\<cdot>a)\<cdot>h)" |
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by (rule_tac p=m in maybeE, simp_all) |
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subsubsection {* Run operator *} |
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definition |
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run:: "'a maybe \<rightarrow> 'a::pcpo" where |
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"run = maybe_when\<cdot>\<bottom>\<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) |
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lemma run_return [simp]: "run\<cdot>(return\<cdot>x) = x" |
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by (simp add: run_def) |
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subsubsection {* Monad plus operator *} |
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definition |
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mplus :: "'a maybe \<rightarrow> 'a maybe \<rightarrow> 'a maybe" where |
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"mplus = (\<Lambda> m1 m2. case m1 of fail \<Rightarrow> m2 | return\<cdot>x \<Rightarrow> m1)" |
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abbreviation |
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mplus_syn :: "['a maybe, 'a maybe] \<Rightarrow> 'a maybe" (infixr "+++" 65) where |
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"m1 +++ m2 == mplus\<cdot>m1\<cdot>m2" |
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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) |
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lemma mplus_return [simp]: "return\<cdot>x +++ m = return\<cdot>x" |
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by (simp add: mplus_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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subsubsection {* Fatbar combinator *} |
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definition |
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fatbar :: "('a \<rightarrow> 'b maybe) \<rightarrow> ('a \<rightarrow> 'b maybe) \<rightarrow> ('a \<rightarrow> 'b maybe)" where |
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"fatbar = (\<Lambda> a b x. a\<cdot>x +++ b\<cdot>x)" |
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abbreviation |
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fatbar_syn :: "['a \<rightarrow> 'b maybe, 'a \<rightarrow> 'b maybe] \<Rightarrow> 'a \<rightarrow> 'b maybe" (infixr "\<parallel>" 60) where |
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"m1 \<parallel> m2 == fatbar\<cdot>m1\<cdot>m2" |
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lemma fatbar1: "m\<cdot>x = \<bottom> \<Longrightarrow> (m \<parallel> ms)\<cdot>x = \<bottom>" |
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by (simp add: fatbar_def) |
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lemma fatbar2: "m\<cdot>x = fail \<Longrightarrow> (m \<parallel> ms)\<cdot>x = ms\<cdot>x" |
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by (simp add: fatbar_def) |
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lemma fatbar3: "m\<cdot>x = return\<cdot>y \<Longrightarrow> (m \<parallel> ms)\<cdot>x = return\<cdot>y" |
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by (simp add: fatbar_def) |
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lemmas fatbar_simps = fatbar1 fatbar2 fatbar3 |
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lemma run_fatbar1: "m\<cdot>x = \<bottom> \<Longrightarrow> run\<cdot>((m \<parallel> ms)\<cdot>x) = \<bottom>" |
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by (simp add: fatbar_def) |
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lemma run_fatbar2: "m\<cdot>x = fail \<Longrightarrow> run\<cdot>((m \<parallel> ms)\<cdot>x) = run\<cdot>(ms\<cdot>x)" |
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by (simp add: fatbar_def) |
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lemma run_fatbar3: "m\<cdot>x = return\<cdot>y \<Longrightarrow> run\<cdot>((m \<parallel> ms)\<cdot>x) = y" |
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by (simp add: fatbar_def) |
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lemmas run_fatbar_simps [simp] = run_fatbar1 run_fatbar2 run_fatbar3 |
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subsection {* Case branch combinator *} |
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constdefs |
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branch :: "('a \<rightarrow> 'b maybe) \<Rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'c maybe)" |
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"branch p \<equiv> \<Lambda> r x. bind\<cdot>(p\<cdot>x)\<cdot>(\<Lambda> y. return\<cdot>(r\<cdot>y))" |
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lemma branch_rews: |
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"p\<cdot>x = \<bottom> \<Longrightarrow> branch p\<cdot>r\<cdot>x = \<bottom>" |
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"p\<cdot>x = fail \<Longrightarrow> branch p\<cdot>r\<cdot>x = fail" |
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"p\<cdot>x = return\<cdot>y \<Longrightarrow> branch p\<cdot>r\<cdot>x = return\<cdot>(r\<cdot>y)" |
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by (simp_all add: branch_def) |
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lemma branch_return [simp]: "branch return\<cdot>r\<cdot>x = return\<cdot>(r\<cdot>x)" |
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by (simp add: branch_def) |
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subsection {* Case syntax *} |
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nonterminals |
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Case_syn Cases_syn |
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"_Case_syntax":: "['a, Cases_syn] => 'b" ("(Case _ of/ _)" 10) |
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"_Case1" :: "['a, 'b] => Case_syn" ("(2_ =>/ _)" 10) |
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"" :: "Case_syn => Cases_syn" ("_") |
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"_Case2" :: "[Case_syn, Cases_syn] => Cases_syn" ("_/ | _") |
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syntax (xsymbols) |
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"_Case1" :: "['a, 'b] => Case_syn" ("(2_ \<Rightarrow>/ _)" 10) |
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translations |
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"_Case_syntax x ms" == "CONST Fixrec.run\<cdot>(ms\<cdot>x)" |
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"_Case2 m ms" == "m \<parallel> ms" |
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198 |
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text {* Parsing Case expressions *} |
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syntax |
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"_pat" :: "'a" |
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"_var" :: "'a" |
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translations |
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"_Case1 p r" => "XCONST branch (_pat p)\<cdot>(_var p r)" |
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"_var (_args x y) r" => "XCONST csplit\<cdot>(_var x (_var y r))" |
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"_var () r" => "XCONST unit_when\<cdot>r" |
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209 |
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parse_translation {* |
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(* rewrites (_pat x) => (return) *) |
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(* rewrites (_var x t) => (Abs_CFun (%x. t)) *) |
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[("_pat", K (Syntax.const "Fixrec.return")), |
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mk_binder_tr ("_var", "Abs_CFun")]; |
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*} |
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216 |
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text {* Printing Case expressions *} |
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218 |
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syntax |
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"_match" :: "'a" |
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print_translation {* |
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let |
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fun dest_LAM (Const (@{const_syntax Rep_CFun},_) $ Const (@{const_syntax unit_when},_) $ t) = |
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(Syntax.const @{const_syntax Unity}, t) |
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| dest_LAM (Const (@{const_syntax Rep_CFun},_) $ Const (@{const_syntax csplit},_) $ t) = |
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let |
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val (v1, t1) = dest_LAM t; |
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val (v2, t2) = dest_LAM t1; |
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in (Syntax.const "_args" $ v1 $ v2, t2) end |
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| dest_LAM (Const (@{const_syntax Abs_CFun},_) $ t) = |
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let |
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val abs = case t of Abs abs => abs |
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| _ => ("x", dummyT, incr_boundvars 1 t $ Bound 0); |
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val (x, t') = atomic_abs_tr' abs; |
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in (Syntax.const "_var" $ x, t') end |
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| dest_LAM _ = raise Match; (* too few vars: abort translation *) |
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fun Case1_tr' [Const(@{const_syntax branch},_) $ p, r] = |
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let val (v, t) = dest_LAM r; |
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in Syntax.const "_Case1" $ (Syntax.const "_match" $ p $ v) $ t end; |
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in [(@{const_syntax Rep_CFun}, Case1_tr')] end; |
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*} |
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246 |
translations |
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"x" <= "_match Fixrec.return (_var x)" |
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subsection {* Pattern combinators for data constructors *} |
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types ('a, 'b) pat = "'a \<rightarrow> 'b maybe" |
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definition |
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cpair_pat :: "('a, 'c) pat \<Rightarrow> ('b, 'd) pat \<Rightarrow> ('a \<times> 'b, 'c \<times> 'd) pat" where |
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"cpair_pat p1 p2 = (\<Lambda>\<langle>x, y\<rangle>. |
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bind\<cdot>(p1\<cdot>x)\<cdot>(\<Lambda> a. bind\<cdot>(p2\<cdot>y)\<cdot>(\<Lambda> b. return\<cdot>\<langle>a, b\<rangle>)))" |
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definition |
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spair_pat :: |
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"('a, 'c) pat \<Rightarrow> ('b, 'd) pat \<Rightarrow> ('a::pcpo \<otimes> 'b::pcpo, 'c \<times> 'd) pat" where |
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"spair_pat p1 p2 = (\<Lambda>(:x, y:). cpair_pat p1 p2\<cdot>\<langle>x, y\<rangle>)" |
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definition |
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sinl_pat :: "('a, 'c) pat \<Rightarrow> ('a::pcpo \<oplus> 'b::pcpo, 'c) pat" where |
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"sinl_pat p = sscase\<cdot>p\<cdot>(\<Lambda> x. fail)" |
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definition |
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sinr_pat :: "('b, 'c) pat \<Rightarrow> ('a::pcpo \<oplus> 'b::pcpo, 'c) pat" where |
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"sinr_pat p = sscase\<cdot>(\<Lambda> x. fail)\<cdot>p" |
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271 |
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definition |
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up_pat :: "('a, 'b) pat \<Rightarrow> ('a u, 'b) pat" where |
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"up_pat p = fup\<cdot>p" |
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definition |
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TT_pat :: "(tr, unit) pat" where |
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"TT_pat = (\<Lambda> b. If b then return\<cdot>() else fail fi)" |
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definition |
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FF_pat :: "(tr, unit) pat" where |
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"FF_pat = (\<Lambda> b. If b then fail else return\<cdot>() fi)" |
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definition |
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ONE_pat :: "(one, unit) pat" where |
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"ONE_pat = (\<Lambda> ONE. return\<cdot>())" |
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text {* Parse translations (patterns) *} |
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"_pat (XCONST cpair\<cdot>x\<cdot>y)" => "XCONST cpair_pat (_pat x) (_pat y)" |
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"_pat (XCONST spair\<cdot>x\<cdot>y)" => "XCONST spair_pat (_pat x) (_pat y)" |
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"_pat (XCONST sinl\<cdot>x)" => "XCONST sinl_pat (_pat x)" |
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"_pat (XCONST sinr\<cdot>x)" => "XCONST sinr_pat (_pat x)" |
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"_pat (XCONST up\<cdot>x)" => "XCONST up_pat (_pat x)" |
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"_pat (XCONST TT)" => "XCONST TT_pat" |
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"_pat (XCONST FF)" => "XCONST FF_pat" |
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"_pat (XCONST ONE)" => "XCONST ONE_pat" |
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text {* Parse translations (variables) *} |
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"_var (XCONST cpair\<cdot>x\<cdot>y) r" => "_var (_args x y) r" |
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"_var (XCONST spair\<cdot>x\<cdot>y) r" => "_var (_args x y) r" |
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"_var (XCONST sinl\<cdot>x) r" => "_var x r" |
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"_var (XCONST sinr\<cdot>x) r" => "_var x r" |
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"_var (XCONST up\<cdot>x) r" => "_var x r" |
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"_var (XCONST TT) r" => "_var () r" |
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"_var (XCONST FF) r" => "_var () r" |
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"_var (XCONST ONE) r" => "_var () r" |
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text {* Print translations *} |
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"CONST cpair\<cdot>(_match p1 v1)\<cdot>(_match p2 v2)" |
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<= "_match (CONST cpair_pat p1 p2) (_args v1 v2)" |
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"CONST spair\<cdot>(_match p1 v1)\<cdot>(_match p2 v2)" |
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<= "_match (CONST spair_pat p1 p2) (_args v1 v2)" |
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316 |
"CONST sinl\<cdot>(_match p1 v1)" <= "_match (CONST sinl_pat p1) v1" |
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317 |
"CONST sinr\<cdot>(_match p1 v1)" <= "_match (CONST sinr_pat p1) v1" |
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318 |
"CONST up\<cdot>(_match p1 v1)" <= "_match (CONST up_pat p1) v1" |
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"CONST TT" <= "_match (CONST TT_pat) ()" |
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"CONST FF" <= "_match (CONST FF_pat) ()" |
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321 |
"CONST ONE" <= "_match (CONST ONE_pat) ()" |
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322 |
|
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323 |
lemma cpair_pat1: |
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"branch p\<cdot>r\<cdot>x = \<bottom> \<Longrightarrow> branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>\<langle>x, y\<rangle> = \<bottom>" |
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apply (simp add: branch_def cpair_pat_def) |
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326 |
apply (rule_tac p="p\<cdot>x" in maybeE, simp_all) |
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327 |
done |
18097 | 328 |
|
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329 |
lemma cpair_pat2: |
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330 |
"branch p\<cdot>r\<cdot>x = fail \<Longrightarrow> branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>\<langle>x, y\<rangle> = fail" |
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331 |
apply (simp add: branch_def cpair_pat_def) |
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332 |
apply (rule_tac p="p\<cdot>x" in maybeE, simp_all) |
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333 |
done |
18097 | 334 |
|
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335 |
lemma cpair_pat3: |
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336 |
"branch p\<cdot>r\<cdot>x = return\<cdot>s \<Longrightarrow> |
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337 |
branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>\<langle>x, y\<rangle> = branch q\<cdot>s\<cdot>y" |
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338 |
apply (simp add: branch_def cpair_pat_def) |
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339 |
apply (rule_tac p="p\<cdot>x" in maybeE, simp_all) |
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340 |
apply (rule_tac p="q\<cdot>y" in maybeE, simp_all) |
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341 |
done |
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|
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343 |
lemmas cpair_pat [simp] = |
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344 |
cpair_pat1 cpair_pat2 cpair_pat3 |
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|
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346 |
lemma spair_pat [simp]: |
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347 |
"branch (spair_pat p1 p2)\<cdot>r\<cdot>\<bottom> = \<bottom>" |
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348 |
"\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> |
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349 |
\<Longrightarrow> branch (spair_pat p1 p2)\<cdot>r\<cdot>(:x, y:) = |
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350 |
branch (cpair_pat p1 p2)\<cdot>r\<cdot>\<langle>x, y\<rangle>" |
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351 |
by (simp_all add: branch_def spair_pat_def) |
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|
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353 |
lemma sinl_pat [simp]: |
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354 |
"branch (sinl_pat p)\<cdot>r\<cdot>\<bottom> = \<bottom>" |
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355 |
"x \<noteq> \<bottom> \<Longrightarrow> branch (sinl_pat p)\<cdot>r\<cdot>(sinl\<cdot>x) = branch p\<cdot>r\<cdot>x" |
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356 |
"y \<noteq> \<bottom> \<Longrightarrow> branch (sinl_pat p)\<cdot>r\<cdot>(sinr\<cdot>y) = fail" |
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357 |
by (simp_all add: branch_def sinl_pat_def) |
18097 | 358 |
|
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359 |
lemma sinr_pat [simp]: |
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360 |
"branch (sinr_pat p)\<cdot>r\<cdot>\<bottom> = \<bottom>" |
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361 |
"x \<noteq> \<bottom> \<Longrightarrow> branch (sinr_pat p)\<cdot>r\<cdot>(sinl\<cdot>x) = fail" |
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362 |
"y \<noteq> \<bottom> \<Longrightarrow> branch (sinr_pat p)\<cdot>r\<cdot>(sinr\<cdot>y) = branch p\<cdot>r\<cdot>y" |
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363 |
by (simp_all add: branch_def sinr_pat_def) |
18097 | 364 |
|
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365 |
lemma up_pat [simp]: |
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366 |
"branch (up_pat p)\<cdot>r\<cdot>\<bottom> = \<bottom>" |
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367 |
"branch (up_pat p)\<cdot>r\<cdot>(up\<cdot>x) = branch p\<cdot>r\<cdot>x" |
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368 |
by (simp_all add: branch_def up_pat_def) |
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369 |
|
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370 |
lemma TT_pat [simp]: |
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371 |
"branch TT_pat\<cdot>(unit_when\<cdot>r)\<cdot>\<bottom> = \<bottom>" |
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372 |
"branch TT_pat\<cdot>(unit_when\<cdot>r)\<cdot>TT = return\<cdot>r" |
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373 |
"branch TT_pat\<cdot>(unit_when\<cdot>r)\<cdot>FF = fail" |
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374 |
by (simp_all add: branch_def TT_pat_def) |
18097 | 375 |
|
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376 |
lemma FF_pat [simp]: |
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377 |
"branch FF_pat\<cdot>(unit_when\<cdot>r)\<cdot>\<bottom> = \<bottom>" |
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378 |
"branch FF_pat\<cdot>(unit_when\<cdot>r)\<cdot>TT = fail" |
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379 |
"branch FF_pat\<cdot>(unit_when\<cdot>r)\<cdot>FF = return\<cdot>r" |
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380 |
by (simp_all add: branch_def FF_pat_def) |
18097 | 381 |
|
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382 |
lemma ONE_pat [simp]: |
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383 |
"branch ONE_pat\<cdot>(unit_when\<cdot>r)\<cdot>\<bottom> = \<bottom>" |
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384 |
"branch ONE_pat\<cdot>(unit_when\<cdot>r)\<cdot>ONE = return\<cdot>r" |
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385 |
by (simp_all add: branch_def ONE_pat_def) |
18097 | 386 |
|
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387 |
|
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388 |
subsection {* Wildcards, as-patterns, and lazy patterns *} |
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389 |
|
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390 |
syntax |
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391 |
"_as_pat" :: "[idt, 'a] \<Rightarrow> 'a" (infixr "\<as>" 10) |
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392 |
"_lazy_pat" :: "'a \<Rightarrow> 'a" ("\<lazy> _" [1000] 1000) |
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393 |
|
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394 |
definition |
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395 |
wild_pat :: "'a \<rightarrow> unit maybe" where |
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396 |
"wild_pat = (\<Lambda> x. return\<cdot>())" |
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397 |
|
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398 |
definition |
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399 |
as_pat :: "('a \<rightarrow> 'b maybe) \<Rightarrow> 'a \<rightarrow> ('a \<times> 'b) maybe" where |
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400 |
"as_pat p = (\<Lambda> x. bind\<cdot>(p\<cdot>x)\<cdot>(\<Lambda> a. return\<cdot>\<langle>x, a\<rangle>))" |
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401 |
|
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402 |
definition |
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403 |
lazy_pat :: "('a \<rightarrow> 'b::pcpo maybe) \<Rightarrow> ('a \<rightarrow> 'b maybe)" where |
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404 |
"lazy_pat p = (\<Lambda> x. return\<cdot>(run\<cdot>(p\<cdot>x)))" |
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405 |
|
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406 |
text {* Parse translations (patterns) *} |
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407 |
translations |
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408 |
"_pat _" => "XCONST wild_pat" |
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409 |
"_pat (_as_pat x y)" => "XCONST as_pat (_pat y)" |
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410 |
"_pat (_lazy_pat x)" => "XCONST lazy_pat (_pat x)" |
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411 |
|
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412 |
text {* Parse translations (variables) *} |
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413 |
translations |
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414 |
"_var _ r" => "_var () r" |
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415 |
"_var (_as_pat x y) r" => "_var (_args x y) r" |
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416 |
"_var (_lazy_pat x) r" => "_var x r" |
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417 |
|
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418 |
text {* Print translations *} |
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419 |
translations |
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420 |
"_" <= "_match (CONST wild_pat) ()" |
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421 |
"_as_pat x (_match p v)" <= "_match (CONST as_pat p) (_args (_var x) v)" |
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422 |
"_lazy_pat (_match p v)" <= "_match (CONST lazy_pat p) v" |
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423 |
|
19327 | 424 |
text {* Lazy patterns in lambda abstractions *} |
425 |
translations |
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426 |
"_cabs (_lazy_pat p) r" == "CONST Fixrec.run oo (_Case1 (_lazy_pat p) r)" |
19327 | 427 |
|
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428 |
lemma wild_pat [simp]: "branch wild_pat\<cdot>(unit_when\<cdot>r)\<cdot>x = return\<cdot>r" |
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429 |
by (simp add: branch_def wild_pat_def) |
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430 |
|
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431 |
lemma as_pat [simp]: |
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432 |
"branch (as_pat p)\<cdot>(csplit\<cdot>r)\<cdot>x = branch p\<cdot>(r\<cdot>x)\<cdot>x" |
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433 |
apply (simp add: branch_def as_pat_def) |
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434 |
apply (rule_tac p="p\<cdot>x" in maybeE, simp_all) |
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435 |
done |
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|
436 |
|
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437 |
lemma lazy_pat [simp]: |
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438 |
"branch p\<cdot>r\<cdot>x = \<bottom> \<Longrightarrow> branch (lazy_pat p)\<cdot>r\<cdot>x = return\<cdot>(r\<cdot>\<bottom>)" |
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439 |
"branch p\<cdot>r\<cdot>x = fail \<Longrightarrow> branch (lazy_pat p)\<cdot>r\<cdot>x = return\<cdot>(r\<cdot>\<bottom>)" |
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|
440 |
"branch p\<cdot>r\<cdot>x = return\<cdot>s \<Longrightarrow> branch (lazy_pat p)\<cdot>r\<cdot>x = return\<cdot>s" |
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441 |
apply (simp_all add: branch_def lazy_pat_def) |
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442 |
apply (rule_tac [!] p="p\<cdot>x" in maybeE, simp_all) |
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|
443 |
done |
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|
444 |
|
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445 |
|
16221 | 446 |
subsection {* Match functions for built-in types *} |
447 |
||
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|
448 |
defaultsort pcpo |
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449 |
|
25131
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450 |
definition |
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|
451 |
match_UU :: "'a \<rightarrow> unit maybe" where |
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|
452 |
"match_UU = (\<Lambda> x. fail)" |
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|
453 |
|
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|
454 |
definition |
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|
455 |
match_cpair :: "'a::cpo \<times> 'b::cpo \<rightarrow> ('a \<times> 'b) maybe" where |
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|
456 |
"match_cpair = csplit\<cdot>(\<Lambda> x y. return\<cdot><x,y>)" |
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|
457 |
|
25131
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|
458 |
definition |
2c8caac48ade
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|
459 |
match_spair :: "'a \<otimes> 'b \<rightarrow> ('a \<times> 'b) maybe" where |
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|
460 |
"match_spair = ssplit\<cdot>(\<Lambda> x y. return\<cdot><x,y>)" |
16221 | 461 |
|
25131
2c8caac48ade
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|
462 |
definition |
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|
463 |
match_sinl :: "'a \<oplus> 'b \<rightarrow> 'a maybe" where |
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|
464 |
"match_sinl = sscase\<cdot>return\<cdot>(\<Lambda> y. fail)" |
16551 | 465 |
|
25131
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|
466 |
definition |
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|
467 |
match_sinr :: "'a \<oplus> 'b \<rightarrow> 'b maybe" where |
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|
468 |
"match_sinr = sscase\<cdot>(\<Lambda> x. fail)\<cdot>return" |
16551 | 469 |
|
25131
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|
470 |
definition |
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|
471 |
match_up :: "'a::cpo u \<rightarrow> 'a maybe" where |
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|
472 |
"match_up = fup\<cdot>return" |
16221 | 473 |
|
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|
474 |
definition |
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|
475 |
match_ONE :: "one \<rightarrow> unit maybe" where |
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|
476 |
"match_ONE = (\<Lambda> ONE. return\<cdot>())" |
18094 | 477 |
|
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|
478 |
definition |
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|
479 |
match_TT :: "tr \<rightarrow> unit maybe" where |
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|
480 |
"match_TT = (\<Lambda> b. If b then return\<cdot>() else fail fi)" |
18094 | 481 |
|
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|
482 |
definition |
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|
483 |
match_FF :: "tr \<rightarrow> unit maybe" where |
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|
484 |
"match_FF = (\<Lambda> b. If b then fail else return\<cdot>() fi)" |
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|
485 |
|
16776
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generalized types of monadic operators to class cpo; added match function for UU
huffman
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|
486 |
lemma match_UU_simps [simp]: |
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|
487 |
"match_UU\<cdot>x = fail" |
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|
488 |
by (simp add: match_UU_def) |
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parents:
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|
489 |
|
16221 | 490 |
lemma match_cpair_simps [simp]: |
491 |
"match_cpair\<cdot><x,y> = return\<cdot><x,y>" |
|
492 |
by (simp add: match_cpair_def) |
|
493 |
||
16551 | 494 |
lemma match_spair_simps [simp]: |
495 |
"\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> match_spair\<cdot>(:x,y:) = return\<cdot><x,y>" |
|
496 |
"match_spair\<cdot>\<bottom> = \<bottom>" |
|
497 |
by (simp_all add: match_spair_def) |
|
498 |
||
499 |
lemma match_sinl_simps [simp]: |
|
500 |
"x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinl\<cdot>x) = return\<cdot>x" |
|
501 |
"x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinr\<cdot>x) = fail" |
|
502 |
"match_sinl\<cdot>\<bottom> = \<bottom>" |
|
503 |
by (simp_all add: match_sinl_def) |
|
504 |
||
505 |
lemma match_sinr_simps [simp]: |
|
506 |
"x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinr\<cdot>x) = return\<cdot>x" |
|
507 |
"x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinl\<cdot>x) = fail" |
|
508 |
"match_sinr\<cdot>\<bottom> = \<bottom>" |
|
509 |
by (simp_all add: match_sinr_def) |
|
510 |
||
16221 | 511 |
lemma match_up_simps [simp]: |
512 |
"match_up\<cdot>(up\<cdot>x) = return\<cdot>x" |
|
513 |
"match_up\<cdot>\<bottom> = \<bottom>" |
|
514 |
by (simp_all add: match_up_def) |
|
515 |
||
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huffman
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|
516 |
lemma match_ONE_simps [simp]: |
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huffman
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|
517 |
"match_ONE\<cdot>ONE = return\<cdot>()" |
72a08d509d62
added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
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changeset
|
518 |
"match_ONE\<cdot>\<bottom> = \<bottom>" |
18094 | 519 |
by (simp_all add: match_ONE_def) |
16460
72a08d509d62
added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
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|
520 |
|
72a08d509d62
added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
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changeset
|
521 |
lemma match_TT_simps [simp]: |
72a08d509d62
added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents:
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changeset
|
522 |
"match_TT\<cdot>TT = return\<cdot>()" |
72a08d509d62
added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents:
16417
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changeset
|
523 |
"match_TT\<cdot>FF = fail" |
72a08d509d62
added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents:
16417
diff
changeset
|
524 |
"match_TT\<cdot>\<bottom> = \<bottom>" |
18094 | 525 |
by (simp_all add: match_TT_def) |
16460
72a08d509d62
added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents:
16417
diff
changeset
|
526 |
|
72a08d509d62
added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents:
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diff
changeset
|
527 |
lemma match_FF_simps [simp]: |
72a08d509d62
added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents:
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diff
changeset
|
528 |
"match_FF\<cdot>FF = return\<cdot>()" |
72a08d509d62
added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents:
16417
diff
changeset
|
529 |
"match_FF\<cdot>TT = fail" |
72a08d509d62
added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents:
16417
diff
changeset
|
530 |
"match_FF\<cdot>\<bottom> = \<bottom>" |
18094 | 531 |
by (simp_all add: match_FF_def) |
16460
72a08d509d62
added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents:
16417
diff
changeset
|
532 |
|
16401
57c35ede00b9
fixrec package now handles mutually-recursive definitions
huffman
parents:
16229
diff
changeset
|
533 |
subsection {* Mutual recursion *} |
57c35ede00b9
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huffman
parents:
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changeset
|
534 |
|
57c35ede00b9
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huffman
parents:
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diff
changeset
|
535 |
text {* |
57c35ede00b9
fixrec package now handles mutually-recursive definitions
huffman
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diff
changeset
|
536 |
The following rules are used to prove unfolding theorems from |
57c35ede00b9
fixrec package now handles mutually-recursive definitions
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changeset
|
537 |
fixed-point definitions of mutually recursive functions. |
57c35ede00b9
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huffman
parents:
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diff
changeset
|
538 |
*} |
57c35ede00b9
fixrec package now handles mutually-recursive definitions
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parents:
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changeset
|
539 |
|
57c35ede00b9
fixrec package now handles mutually-recursive definitions
huffman
parents:
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changeset
|
540 |
lemma cpair_equalI: "\<lbrakk>x \<equiv> cfst\<cdot>p; y \<equiv> csnd\<cdot>p\<rbrakk> \<Longrightarrow> <x,y> \<equiv> p" |
57c35ede00b9
fixrec package now handles mutually-recursive definitions
huffman
parents:
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diff
changeset
|
541 |
by (simp add: surjective_pairing_Cprod2) |
57c35ede00b9
fixrec package now handles mutually-recursive definitions
huffman
parents:
16229
diff
changeset
|
542 |
|
57c35ede00b9
fixrec package now handles mutually-recursive definitions
huffman
parents:
16229
diff
changeset
|
543 |
lemma cpair_eqD1: "<x,y> = <x',y'> \<Longrightarrow> x = x'" |
57c35ede00b9
fixrec package now handles mutually-recursive definitions
huffman
parents:
16229
diff
changeset
|
544 |
by simp |
57c35ede00b9
fixrec package now handles mutually-recursive definitions
huffman
parents:
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diff
changeset
|
545 |
|
57c35ede00b9
fixrec package now handles mutually-recursive definitions
huffman
parents:
16229
diff
changeset
|
546 |
lemma cpair_eqD2: "<x,y> = <x',y'> \<Longrightarrow> y = y'" |
57c35ede00b9
fixrec package now handles mutually-recursive definitions
huffman
parents:
16229
diff
changeset
|
547 |
by simp |
57c35ede00b9
fixrec package now handles mutually-recursive definitions
huffman
parents:
16229
diff
changeset
|
548 |
|
16463
342d74ca8815
fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents:
16460
diff
changeset
|
549 |
text {* lemma for proving rewrite rules *} |
342d74ca8815
fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents:
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diff
changeset
|
550 |
|
342d74ca8815
fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents:
16460
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changeset
|
551 |
lemma ssubst_lhs: "\<lbrakk>t = s; P s = Q\<rbrakk> \<Longrightarrow> P t = Q" |
342d74ca8815
fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents:
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diff
changeset
|
552 |
by simp |
342d74ca8815
fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents:
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diff
changeset
|
553 |
|
16221 | 554 |
|
16758 | 555 |
subsection {* Initializing the fixrec package *} |
16221 | 556 |
|
23152 | 557 |
use "Tools/fixrec_package.ML" |
16221 | 558 |
|
19439 | 559 |
hide (open) const return bind fail run |
19104 | 560 |
|
16221 | 561 |
end |