(* Author: Tobias Nipkow *) section "Live Variable Analysis" theory Live imports Vars Big_Step begin subsection "Liveness Analysis" fun L :: "com ⇒ vname set ⇒ vname set" where "L SKIP X = X" | "L (x ::= a) X = vars a ∪ (X - {x})" | "L (c⇩_{1};; c⇩_{2}) X = L c⇩_{1}(L c⇩_{2}X)" | "L (IF b THEN c⇩_{1}ELSE c⇩_{2}) X = vars b ∪ L c⇩_{1}X ∪ L c⇩_{2}X" | "L (WHILE b DO c) X = vars b ∪ X ∪ L c X" value "show (L (''y'' ::= V ''z'';; ''x'' ::= Plus (V ''y'') (V ''z'')) {''x''})" value "show (L (WHILE Less (V ''x'') (V ''x'') DO ''y'' ::= V ''z'') {''x''})" fun "kill" :: "com ⇒ vname set" where "kill SKIP = {}" | "kill (x ::= a) = {x}" | "kill (c⇩_{1};; c⇩_{2}) = kill c⇩_{1}∪ kill c⇩_{2}" | "kill (IF b THEN c⇩_{1}ELSE c⇩_{2}) = kill c⇩_{1}∩ kill c⇩_{2}" | "kill (WHILE b DO c) = {}" fun gen :: "com ⇒ vname set" where "gen SKIP = {}" | "gen (x ::= a) = vars a" | "gen (c⇩_{1};; c⇩_{2}) = gen c⇩_{1}∪ (gen c⇩_{2}- kill c⇩_{1})" | "gen (IF b THEN c⇩_{1}ELSE c⇩_{2}) = vars b ∪ gen c⇩_{1}∪ gen c⇩_{2}" | "gen (WHILE b DO c) = vars b ∪ gen c" lemma L_gen_kill: "L c X = gen c ∪ (X - kill c)" by(induct c arbitrary:X) auto lemma L_While_pfp: "L c (L (WHILE b DO c) X) ⊆ L (WHILE b DO c) X" by(auto simp add:L_gen_kill) lemma L_While_lpfp: "vars b ∪ X ∪ L c P ⊆ P ⟹ L (WHILE b DO c) X ⊆ P" by(simp add: L_gen_kill) lemma L_While_vars: "vars b ⊆ L (WHILE b DO c) X" by auto lemma L_While_X: "X ⊆ L (WHILE b DO c) X" by auto text‹Disable L WHILE equation and reason only with L WHILE constraints› declare L.simps(5)[simp del] subsection "Correctness" theorem L_correct: "(c,s) ⇒ s' ⟹ s = t on L c X ⟹ ∃ t'. (c,t) ⇒ t' & s' = t' on X" proof (induction arbitrary: X t rule: big_step_induct) case Skip then show ?case by auto next case Assign then show ?case by (auto simp: ball_Un) next case (Seq c1 s1 s2 c2 s3 X t1) from Seq.IH(1) Seq.prems obtain t2 where t12: "(c1, t1) ⇒ t2" and s2t2: "s2 = t2 on L c2 X" by simp blast from Seq.IH(2)[OF s2t2] obtain t3 where t23: "(c2, t2) ⇒ t3" and s3t3: "s3 = t3 on X" by auto show ?case using t12 t23 s3t3 by auto next case (IfTrue b s c1 s' c2) hence "s = t on vars b" "s = t on L c1 X" by auto from bval_eq_if_eq_on_vars[OF this(1)] IfTrue(1) have "bval b t" by simp from IfTrue.IH[OF ‹s = t on L c1 X›] obtain t' where "(c1, t) ⇒ t'" "s' = t' on X" by auto thus ?case using ‹bval b t› by auto next case (IfFalse b s c2 s' c1) hence "s = t on vars b" "s = t on L c2 X" by auto from bval_eq_if_eq_on_vars[OF this(1)] IfFalse(1) have "~bval b t" by simp from IfFalse.IH[OF ‹s = t on L c2 X›] obtain t' where "(c2, t) ⇒ t'" "s' = t' on X" by auto thus ?case using ‹~bval b t› by auto next case (WhileFalse b s c) hence "~ bval b t" by (metis L_While_vars bval_eq_if_eq_on_vars set_mp) thus ?case by(metis WhileFalse.prems L_While_X big_step.WhileFalse set_mp) next case (WhileTrue b s1 c s2 s3 X t1) let ?w = "WHILE b DO c" from ‹bval b s1› WhileTrue.prems have "bval b t1" by (metis L_While_vars bval_eq_if_eq_on_vars set_mp) have "s1 = t1 on L c (L ?w X)" using L_While_pfp WhileTrue.prems by (blast) from WhileTrue.IH(1)[OF this] obtain t2 where "(c, t1) ⇒ t2" "s2 = t2 on L ?w X" by auto from WhileTrue.IH(2)[OF this(2)] obtain t3 where "(?w,t2) ⇒ t3" "s3 = t3 on X" by auto with ‹bval b t1› ‹(c, t1) ⇒ t2› show ?case by auto qed subsection "Program Optimization" text‹Burying assignments to dead variables:› fun bury :: "com ⇒ vname set ⇒ com" where "bury SKIP X = SKIP" | "bury (x ::= a) X = (if x ∈ X then x ::= a else SKIP)" | "bury (c⇩_{1};; c⇩_{2}) X = (bury c⇩_{1}(L c⇩_{2}X);; bury c⇩_{2}X)" | "bury (IF b THEN c⇩_{1}ELSE c⇩_{2}) X = IF b THEN bury c⇩_{1}X ELSE bury c⇩_{2}X" | "bury (WHILE b DO c) X = WHILE b DO bury c (L (WHILE b DO c) X)" text‹We could prove the analogous lemma to @{thm[source]L_correct}, and the proof would be very similar. However, we phrase it as a semantics preservation property:› theorem bury_correct: "(c,s) ⇒ s' ⟹ s = t on L c X ⟹ ∃ t'. (bury c X,t) ⇒ t' & s' = t' on X" proof (induction arbitrary: X t rule: big_step_induct) case Skip then show ?case by auto next case Assign then show ?case by (auto simp: ball_Un) next case (Seq c1 s1 s2 c2 s3 X t1) from Seq.IH(1) Seq.prems obtain t2 where t12: "(bury c1 (L c2 X), t1) ⇒ t2" and s2t2: "s2 = t2 on L c2 X" by simp blast from Seq.IH(2)[OF s2t2] obtain t3 where t23: "(bury c2 X, t2) ⇒ t3" and s3t3: "s3 = t3 on X" by auto show ?case using t12 t23 s3t3 by auto next case (IfTrue b s c1 s' c2) hence "s = t on vars b" "s = t on L c1 X" by auto from bval_eq_if_eq_on_vars[OF this(1)] IfTrue(1) have "bval b t" by simp from IfTrue.IH[OF ‹s = t on L c1 X›] obtain t' where "(bury c1 X, t) ⇒ t'" "s' =t' on X" by auto thus ?case using ‹bval b t› by auto next case (IfFalse b s c2 s' c1) hence "s = t on vars b" "s = t on L c2 X" by auto from bval_eq_if_eq_on_vars[OF this(1)] IfFalse(1) have "~bval b t" by simp from IfFalse.IH[OF ‹s = t on L c2 X›] obtain t' where "(bury c2 X, t) ⇒ t'" "s' = t' on X" by auto thus ?case using ‹~bval b t› by auto next case (WhileFalse b s c) hence "~ bval b t" by (metis L_While_vars bval_eq_if_eq_on_vars set_mp) thus ?case by simp (metis L_While_X WhileFalse.prems big_step.WhileFalse set_mp) next case (WhileTrue b s1 c s2 s3 X t1) let ?w = "WHILE b DO c" from ‹bval b s1› WhileTrue.prems have "bval b t1" by (metis L_While_vars bval_eq_if_eq_on_vars set_mp) have "s1 = t1 on L c (L ?w X)" using L_While_pfp WhileTrue.prems by blast from WhileTrue.IH(1)[OF this] obtain t2 where "(bury c (L ?w X), t1) ⇒ t2" "s2 = t2 on L ?w X" by auto from WhileTrue.IH(2)[OF this(2)] obtain t3 where "(bury ?w X,t2) ⇒ t3" "s3 = t3 on X" by auto with ‹bval b t1› ‹(bury c (L ?w X), t1) ⇒ t2› show ?case by auto qed corollary final_bury_correct: "(c,s) ⇒ s' ⟹ (bury c UNIV,s) ⇒ s'" using bury_correct[of c s s' UNIV] by (auto simp: fun_eq_iff[symmetric]) text‹Now the opposite direction.› lemma SKIP_bury[simp]: "SKIP = bury c X ⟷ c = SKIP | (∃x a. c = x::=a & x ∉ X)" by (cases c) auto lemma Assign_bury[simp]: "x::=a = bury c X ⟷ c = x::=a ∧ x ∈ X" by (cases c) auto lemma Seq_bury[simp]: "bc⇩_{1};;bc⇩_{2}= bury c X ⟷ (∃c⇩_{1}c⇩_{2}. c = c⇩_{1};;c⇩_{2}& bc⇩_{2}= bury c⇩_{2}X & bc⇩_{1}= bury c⇩_{1}(L c⇩_{2}X))" by (cases c) auto lemma If_bury[simp]: "IF b THEN bc1 ELSE bc2 = bury c X ⟷ (∃c1 c2. c = IF b THEN c1 ELSE c2 & bc1 = bury c1 X & bc2 = bury c2 X)" by (cases c) auto lemma While_bury[simp]: "WHILE b DO bc' = bury c X ⟷ (∃c'. c = WHILE b DO c' & bc' = bury c' (L (WHILE b DO c') X))" by (cases c) auto theorem bury_correct2: "(bury c X,s) ⇒ s' ⟹ s = t on L c X ⟹ ∃ t'. (c,t) ⇒ t' & s' = t' on X" proof (induction "bury c X" s s' arbitrary: c X t rule: big_step_induct) case Skip then show ?case by auto next case Assign then show ?case by (auto simp: ball_Un) next case (Seq bc1 s1 s2 bc2 s3 c X t1) then obtain c1 c2 where c: "c = c1;;c2" and bc2: "bc2 = bury c2 X" and bc1: "bc1 = bury c1 (L c2 X)" by auto note IH = Seq.hyps(2,4) from IH(1)[OF bc1, of t1] Seq.prems c obtain t2 where t12: "(c1, t1) ⇒ t2" and s2t2: "s2 = t2 on L c2 X" by auto from IH(2)[OF bc2 s2t2] obtain t3 where t23: "(c2, t2) ⇒ t3" and s3t3: "s3 = t3 on X" by auto show ?case using c t12 t23 s3t3 by auto next case (IfTrue b s bc1 s' bc2) then obtain c1 c2 where c: "c = IF b THEN c1 ELSE c2" and bc1: "bc1 = bury c1 X" and bc2: "bc2 = bury c2 X" by auto have "s = t on vars b" "s = t on L c1 X" using IfTrue.prems c by auto from bval_eq_if_eq_on_vars[OF this(1)] IfTrue(1) have "bval b t" by simp note IH = IfTrue.hyps(3) from IH[OF bc1 ‹s = t on L c1 X›] obtain t' where "(c1, t) ⇒ t'" "s' =t' on X" by auto thus ?case using c ‹bval b t› by auto next case (IfFalse b s bc2 s' bc1) then obtain c1 c2 where c: "c = IF b THEN c1 ELSE c2" and bc1: "bc1 = bury c1 X" and bc2: "bc2 = bury c2 X" by auto have "s = t on vars b" "s = t on L c2 X" using IfFalse.prems c by auto from bval_eq_if_eq_on_vars[OF this(1)] IfFalse(1) have "~bval b t" by simp note IH = IfFalse.hyps(3) from IH[OF bc2 ‹s = t on L c2 X›] obtain t' where "(c2, t) ⇒ t'" "s' =t' on X" by auto thus ?case using c ‹~bval b t› by auto next case (WhileFalse b s c) hence "~ bval b t" by auto (metis L_While_vars bval_eq_if_eq_on_vars set_rev_mp) thus ?case using WhileFalse by auto (metis L_While_X big_step.WhileFalse set_mp) next case (WhileTrue b s1 bc' s2 s3 w X t1) then obtain c' where w: "w = WHILE b DO c'" and bc': "bc' = bury c' (L (WHILE b DO c') X)" by auto from ‹bval b s1› WhileTrue.prems w have "bval b t1" by auto (metis L_While_vars bval_eq_if_eq_on_vars set_mp) note IH = WhileTrue.hyps(3,5) have "s1 = t1 on L c' (L w X)" using L_While_pfp WhileTrue.prems w by blast with IH(1)[OF bc', of t1] w obtain t2 where "(c', t1) ⇒ t2" "s2 = t2 on L w X" by auto from IH(2)[OF WhileTrue.hyps(6), of t2] w this(2) obtain t3 where "(w,t2) ⇒ t3" "s3 = t3 on X" by auto with ‹bval b t1› ‹(c', t1) ⇒ t2› w show ?case by auto qed corollary final_bury_correct2: "(bury c UNIV,s) ⇒ s' ⟹ (c,s) ⇒ s'" using bury_correct2[of c UNIV] by (auto simp: fun_eq_iff[symmetric]) corollary bury_sim: "bury c UNIV ∼ c" by(metis final_bury_correct final_bury_correct2) end