author | nipkow |
Fri, 23 Nov 2012 23:07:38 +0100 | |
changeset 50180 | c6626861c31a |
parent 50009 | e48de0410307 |
child 51436 | 790310525e97 |
permissions | -rw-r--r-- |
45812 | 1 |
(* Author: Tobias Nipkow *) |
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theory Live_True |
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imports "~~/src/HOL/Library/While_Combinator" Vars Big_Step |
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begin |
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subsection "True Liveness Analysis" |
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fun L :: "com \<Rightarrow> vname set \<Rightarrow> vname set" where |
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"L SKIP X = X" | |
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"L (x ::= a) X = (if x \<in> X then X - {x} \<union> vars a else X)" | |
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"L (c\<^isub>1; c\<^isub>2) X = (L c\<^isub>1 \<circ> L c\<^isub>2) X" | |
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"L (IF b THEN c\<^isub>1 ELSE c\<^isub>2) X = vars b \<union> L c\<^isub>1 X \<union> L c\<^isub>2 X" | |
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"L (WHILE b DO c) X = lfp(\<lambda>Y. vars b \<union> X \<union> L c Y)" |
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lemma L_mono: "mono (L c)" |
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proof- |
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{ fix X Y have "X \<subseteq> Y \<Longrightarrow> L c X \<subseteq> L c Y" |
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proof(induction c arbitrary: X Y) |
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case (While b c) |
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show ?case |
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proof(simp, rule lfp_mono) |
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fix Z show "vars b \<union> X \<union> L c Z \<subseteq> vars b \<union> Y \<union> L c Z" |
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using While by auto |
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qed |
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next |
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case If thus ?case by(auto simp: subset_iff) |
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qed auto |
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} thus ?thesis by(rule monoI) |
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qed |
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lemma mono_union_L: |
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"mono (\<lambda>Y. X \<union> L c Y)" |
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by (metis (no_types) L_mono mono_def order_eq_iff set_eq_subset sup_mono) |
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lemma L_While_unfold: |
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"L (WHILE b DO c) X = vars b \<union> X \<union> L c (L (WHILE b DO c) X)" |
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by(metis lfp_unfold[OF mono_union_L] L.simps(5)) |
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subsection "Soundness" |
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theorem L_sound: |
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"(c,s) \<Rightarrow> s' \<Longrightarrow> s = t on L c X \<Longrightarrow> |
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\<exists> t'. (c,t) \<Rightarrow> t' & s' = t' on X" |
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proof (induction arbitrary: X t rule: big_step_induct) |
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case Skip then show ?case by auto |
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next |
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case Assign then show ?case |
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by (auto simp: ball_Un) |
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next |
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case (Seq c1 s1 s2 c2 s3 X t1) |
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from Seq.IH(1) Seq.prems obtain t2 where |
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t12: "(c1, t1) \<Rightarrow> t2" and s2t2: "s2 = t2 on L c2 X" |
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by simp blast |
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from Seq.IH(2)[OF s2t2] obtain t3 where |
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t23: "(c2, t2) \<Rightarrow> t3" and s3t3: "s3 = t3 on X" |
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by auto |
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show ?case using t12 t23 s3t3 by auto |
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next |
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case (IfTrue b s c1 s' c2) |
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hence "s = t on vars b" and "s = t on L c1 X" by auto |
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from bval_eq_if_eq_on_vars[OF this(1)] IfTrue(1) have "bval b t" by simp |
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from IfTrue.IH[OF `s = t on L c1 X`] obtain t' where |
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"(c1, t) \<Rightarrow> t'" "s' = t' on X" by auto |
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thus ?case using `bval b t` by auto |
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next |
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case (IfFalse b s c2 s' c1) |
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hence "s = t on vars b" "s = t on L c2 X" by auto |
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from bval_eq_if_eq_on_vars[OF this(1)] IfFalse(1) have "~bval b t" by simp |
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from IfFalse.IH[OF `s = t on L c2 X`] obtain t' where |
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"(c2, t) \<Rightarrow> t'" "s' = t' on X" by auto |
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thus ?case using `~bval b t` by auto |
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next |
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case (WhileFalse b s c) |
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hence "~ bval b t" |
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by (metis L_While_unfold UnI1 bval_eq_if_eq_on_vars) |
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thus ?case using WhileFalse.prems L_While_unfold[of b c X] by auto |
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next |
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case (WhileTrue b s1 c s2 s3 X t1) |
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let ?w = "WHILE b DO c" |
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from `bval b s1` WhileTrue.prems have "bval b t1" |
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by (metis L_While_unfold UnI1 bval_eq_if_eq_on_vars) |
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have "s1 = t1 on L c (L ?w X)" using L_While_unfold WhileTrue.prems |
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by (blast) |
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from WhileTrue.IH(1)[OF this] obtain t2 where |
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"(c, t1) \<Rightarrow> t2" "s2 = t2 on L ?w X" by auto |
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from WhileTrue.IH(2)[OF this(2)] obtain t3 where "(?w,t2) \<Rightarrow> t3" "s3 = t3 on X" |
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by auto |
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with `bval b t1` `(c, t1) \<Rightarrow> t2` show ?case by auto |
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qed |
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subsection "Executability" |
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instantiation com :: vars |
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begin |
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fun vars_com :: "com \<Rightarrow> vname set" where |
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"vars SKIP = {}" | |
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"vars (x::=e) = vars e" | |
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"vars (c\<^isub>1; c\<^isub>2) = vars c\<^isub>1 \<union> vars c\<^isub>2" | |
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"vars (IF b THEN c\<^isub>1 ELSE c\<^isub>2) = vars b \<union> vars c\<^isub>1 \<union> vars c\<^isub>2" | |
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"vars (WHILE b DO c) = vars b \<union> vars c" |
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instance .. |
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end |
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lemma L_subset_vars: "L c X \<subseteq> vars c \<union> X" |
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proof(induction c arbitrary: X) |
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case (While b c) |
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have "lfp(\<lambda>Y. vars b \<union> X \<union> L c Y) \<subseteq> vars b \<union> vars c \<union> X" |
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using While.IH[of "vars b \<union> vars c \<union> X"] |
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by (auto intro!: lfp_lowerbound) |
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thus ?case by simp |
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qed auto |
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lemma afinite[simp]: "finite(vars(a::aexp))" |
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by (induction a) auto |
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lemma bfinite[simp]: "finite(vars(b::bexp))" |
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by (induction b) auto |
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lemma cfinite[simp]: "finite(vars(c::com))" |
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by (induction c) auto |
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text{* Make @{const L} executable by replacing @{const lfp} with the @{const |
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while} combinator from theory @{theory While_Combinator}. The @{const while} |
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combinator obeys the recursion equation |
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@{thm[display] While_Combinator.while_unfold[no_vars]} |
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and is thus executable. *} |
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lemma L_While: fixes b c X |
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assumes "finite X" defines "f == \<lambda>A. vars b \<union> X \<union> L c A" |
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shows "L (WHILE b DO c) X = while (\<lambda>A. f A \<noteq> A) f {}" (is "_ = ?r") |
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proof - |
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let ?V = "vars b \<union> vars c \<union> X" |
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have "lfp f = ?r" |
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proof(rule lfp_while[where C = "?V"]) |
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show "mono f" by(simp add: f_def mono_union_L) |
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next |
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fix Y show "Y \<subseteq> ?V \<Longrightarrow> f Y \<subseteq> ?V" |
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unfolding f_def using L_subset_vars[of c] by blast |
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next |
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show "finite ?V" using `finite X` by simp |
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qed |
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thus ?thesis by (simp add: f_def) |
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qed |
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lemma L_While_set: "L (WHILE b DO c) (set xs) = |
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(let f = (\<lambda>A. vars b \<union> set xs \<union> L c A) |
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in while (\<lambda>A. f A \<noteq> A) f {})" |
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by(simp add: L_While del: L.simps(5)) |
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text{* Replace the equation for @{text "L (WHILE \<dots>)"} by the executable @{thm[source] L_While_set}: *} |
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lemmas [code] = L.simps(1-4) L_While_set |
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text{* Sorry, this syntax is odd. *} |
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text{* A test: *} |
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lemma "(let b = Less (N 0) (V ''y''); c = ''y'' ::= V ''x''; ''x'' ::= V ''z'' |
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in L (WHILE b DO c) {''y''}) = {''x'', ''y'', ''z''}" |
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by eval |
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subsection "Limiting the number of iterations" |
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text{* The final parameter is the default value: *} |
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fun iter :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
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"iter f 0 p d = d" | |
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"iter f (Suc n) p d = (if f p = p then p else iter f n (f p) d)" |
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text{* A version of @{const L} with a bounded number of iterations (here: 2) |
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in the WHILE case: *} |
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fun Lb :: "com \<Rightarrow> vname set \<Rightarrow> vname set" where |
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"Lb SKIP X = X" | |
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"Lb (x ::= a) X = (if x \<in> X then X - {x} \<union> vars a else X)" | |
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"Lb (c\<^isub>1; c\<^isub>2) X = (Lb c\<^isub>1 \<circ> Lb c\<^isub>2) X" | |
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"Lb (IF b THEN c\<^isub>1 ELSE c\<^isub>2) X = vars b \<union> Lb c\<^isub>1 X \<union> Lb c\<^isub>2 X" | |
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"Lb (WHILE b DO c) X = iter (\<lambda>A. vars b \<union> X \<union> Lb c A) 2 {} (vars b \<union> vars c \<union> X)" |
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text{* @{const Lb} (and @{const iter}) is not monotone! *} |
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lemma "let w = WHILE Bc False DO (''x'' ::= V ''y''; ''z'' ::= V ''x'') |
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in \<not> (Lb w {''z''} \<subseteq> Lb w {''y'',''z''})" |
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by eval |
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lemma lfp_subset_iter: |
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"\<lbrakk> mono f; !!X. f X \<subseteq> f' X; lfp f \<subseteq> D \<rbrakk> \<Longrightarrow> lfp f \<subseteq> iter f' n A D" |
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proof(induction n arbitrary: A) |
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case 0 thus ?case by simp |
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next |
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case Suc thus ?case by simp (metis lfp_lowerbound) |
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qed |
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lemma "L c X \<subseteq> Lb c X" |
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proof(induction c arbitrary: X) |
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case (While b c) |
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let ?f = "\<lambda>A. vars b \<union> X \<union> L c A" |
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let ?fb = "\<lambda>A. vars b \<union> X \<union> Lb c A" |
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show ?case |
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proof (simp, rule lfp_subset_iter[OF mono_union_L]) |
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show "!!X. ?f X \<subseteq> ?fb X" using While.IH by blast |
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show "lfp ?f \<subseteq> vars b \<union> vars c \<union> X" |
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by (metis (full_types) L.simps(5) L_subset_vars vars_com.simps(5)) |
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qed |
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next |
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case Seq thus ?case by simp (metis (full_types) L_mono monoD subset_trans) |
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qed auto |
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end |